Bose Gases In and Out of Equilibrium within
the Stochastic Gross-Pitaevskii Equation
Stuart Paul Cockburn
Thesis submitted for the degree of
Doctor of Philosophy
School of Mathematics and Statistics
University of Newcastle upon Tyne
Newcastle upon Tyne
United Kingdom
March 2010
To my parents, for their constant support.
Acknowledgements
I would like to start by thanking my supervisor, Nick Proukakis, for his enthusiastic
supervision and for always being available for questions. I would also like to thank
David Toms and Gareth Roberts, as without them I would not have started this project,
for which I would also like to acknowledge funding from EPSRC. I am grateful to
Carsten Henkel for his hospitality during my research visit to Potsdam and for his efforts
towards collaborative work which was undertaken along with Antonio Negretti, whom
I would also like to thank, and my supervisor. In addition, I would like to acknowledge
contributions by Dimitri Frantzeskakis, Panayotis Kevrekidis, Hector Nistazakis and
Theodoros Horikis to work included within this thesis.
Within the School, I would like to thank the other postgraduate students for entertaining coffee/lunch times and nights out. In particular, for general chats, computing
advice and other things, I would like to thank Andrew Baggaley, Pete Milner, Anthony
Youd and events co-ordinator, Sam James.
Outside of university, I owe a lot to my family who have supported me throughout
my undergraduate and postgraduate studies, so I would like to say a big thanks to
them. Finally, I want to say a huge thank you to Becky King for her constant love and
support, and for not moving out while I was writing up.
Abstract
In this thesis, we apply the stochastic Gross-Pitaevskii equation, using the formulation of Stoof, to the study of weakly interacting, finite temperature atomic Bose gases,
both in and out of equilibrium. Explicitly maintaining the effects of phase and density
fluctuations, this state-of-the-art method is well suited to the study of low dimensional
Bose systems.
As such, we consider the non-equilibrium dynamics of dark matter wave solitons, oscillating within harmonically trapped, one-dimensional phase-fluctuating condensates,
through numerical simulations of the stochastic Gross-Pitaevskii equation. In treating
the damping due to higher lying thermal modes of the system, we include the HartreeFock contribution of the equilibrium atomic density, which is a novel feature of our
work. A stochastic approach is found to be essential in capturing the shot-to-shot
variation in soliton trajectories observed experimentally.
Neglecting stochastic effects, we find the well-known dissipative Gross-Pitaevskii
equation to recover the average dynamics of the stochastic model, on retaining the
microscopically justified form for the damping of this approach. Such a coupling is
found to be necessary in modelling the dissipative nature of finite temperature soliton
dynamics.
Furthermore, we undertake a comparison between the finite temperature equilibrium state of the stochastic Gross-Pitaevskii equation, and that predicted within a
number conserving Bogoliubov approach. Through a series of numerical experiments,
we compare the equilibrium properties of a harmonically trapped Bose gas, considering
our results in light of those of independent theoretical treatments, and additionally
assessing the sensitivity of the numerical implementation of the former method to the
numerical grid spacing. In the regime in which thermal effects dominate, the stochastic Gross-Pitaevskii approach proves to be a more consistent means of generating such
equilibrium states.
Contents
1 Introduction
1.1
1.2
Physical systems exhibiting Bose-Einstein condensation . . . . . . . . .
2
1.1.1
Atomic Bose-Einstein condensates . . . . . . . . . . . . . . . . .
3
Quasi-condensation and low dimensional systems . . . . . . . . . . . . .
6
1.2.1
2
1
Experimental realisations and applications of low-dimensional
systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.3
Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
1.4
Collaborations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
Theoretical Background
2.1
2.2
2.3
2.4
14
The Gross-Pitaevskii equation and elementary excitations . . . . . . . .
15
2.1.1
The zero temperature Bogoliubov equations . . . . . . . . . . . .
17
Coupling the Gross-Pitaevskii and Boltzmann equations . . . . . . . . .
19
2.2.1
The Boltzmann equation
19
2.2.2
Incorporating collisional particle exchange into the condensate
. . . . . . . . . . . . . . . . . . . . . .
dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
Stochastic effects in physical systems . . . . . . . . . . . . . . . . . . . .
24
2.3.1
Langevin equation of Brownian motion
. . . . . . . . . . . . . .
24
2.3.2
Fokker-Planck equation of Brownian motion . . . . . . . . . . . .
28
Stoof’s approach to a stochastic generalisation of the Gross-Pitaevskii
equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1
30
The stochastic (projected) Gross-Pitaevskii equation of C. W.
Gardiner and co-workers . . . . . . . . . . . . . . . . . . . . . . .
36
Related finite temperature approaches . . . . . . . . . . . . . . . . . . .
37
2.5.1
Dissipative Gross-Pitaevskii equation as a limit of the SGPE . .
37
2.5.2
Thermal classical field simulations . . . . . . . . . . . . . . . . .
39
2.6
Finite temperature schemes related to the SGPE applied in this thesis .
39
2.7
Interpretation of stochastic simulations: single run vs. averaged results .
41
2.8
Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
2.5
i
Contents
3 The stochastic Gross-Pitaevskii equation in one-dimension
44
3.1 Growth to equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2
Influence of the classical approximation . . . . . . . . . . . . . . . . . .
47
3.3
Equilibrium properties: an ideal thermal Bose gas . . . . . . . . . . . .
49
3.4
Equilibrium properties: an interacting, partially condensed Bose gas . .
52
3.4.1
Density profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
3.4.2
Relation between grid spacing and energy cutoff . . . . . . . . .
56
Some properties of the low-lying modes . . . . . . . . . . . . . . . . . .
57
3.5.1
Penrose-Onsager definition of the condensate mode . . . . . . . .
57
3.5.2
Extracting a quasi-condensate fraction . . . . . . . . . . . . . . .
60
3.5.3
Condensate number statistics . . . . . . . . . . . . . . . . . . . .
62
3.5.4
Modes above the condensate . . . . . . . . . . . . . . . . . . . .
63
3.5
3.6
3.7
4
3.6.1
Lerch transcendent expression of C. W. Gardiner et al. . . . . . .
72
3.6.2
Comparison between the integral and Lerch expressions for γ(z)
73
Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Finite temperature matter waves
4.1
4.2
4.3
4.4
4.5
5
Calculation of noise and damping terms within the ergodic approximation 66
74
76
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
4.1.1
Solitons in BECs . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
4.1.2
Dark soliton experiments . . . . . . . . . . . . . . . . . . . . . .
80
4.1.3
Choice of system parameters . . . . . . . . . . . . . . . . . . . .
83
Numerical simulations of dissipative dark soliton dynamics . . . . . . . .
84
4.2.1
4.2.2
Choice of initial state . . . . . . . . . . . . . . . . . . . . . . . .
Tracking a soliton . . . . . . . . . . . . . . . . . . . . . . . . . .
84
86
4.2.3
Calculating an appropriate damping term . . . . . . . . . . . . .
87
DGPE simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
4.3.1
Dynamics with damping due to γ(z) . . . . . . . . . . . . . . . .
89
4.3.2
Comparison between γ(z) and spatially independent damping . .
91
Three-dimensional analysis of dissipative soliton dynamics . . . . . . . .
95
4.4.1
Initial state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
4.4.2
Transverse stability . . . . . . . . . . . . . . . . . . . . . . . . . .
97
4.4.3
3d vs. 1d DGPE results . . . . . . . . . . . . . . . . . . . . . . .
98
4.4.4
Comparison to ZNG . . . . . . . . . . . . . . . . . . . . . . . . .
99
Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Stochastic effects in finite temperature matter waves
102
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.2
Numerical simulations of stochastic dark soliton dynamics . . . . . . . . 104
ii
Contents
5.2.1
5.3
5.4
Initial condition and damping term γ(z) . . . . . . . . . . . . . . 104
SGPE simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.3.1
Dependence of soliton speed on thermal fluctuations . . . . . . . 106
5.3.2
Extracting soliton data: averaged vs. single run densities . . . . 108
5.3.3
Tracking a soliton in a fluctuating background . . . . . . . . . . 111
5.3.4
Statistical spread of soliton trajectories . . . . . . . . . . . . . . 113
5.3.5
Comparison to the DGPE . . . . . . . . . . . . . . . . . . . . . . 115
5.3.6
5.3.7
Analysis of the soliton decay time distribution . . . . . . . . . . 119
Comparison to different forms of γ(z) . . . . . . . . . . . . . . . 123
5.3.8
Role of the heat bath in soliton dynamics . . . . . . . . . . . . . 125
Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6 Stochastic Gross-Pitaevskii and number conserving Bogoliubov methods
133
6.1
Motivating a stochastic approach . . . . . . . . . . . . . . . . . . . . . . 133
6.2
Summary of methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.3
6.4
6.2.1
Truncated Wigner plus number conserving Bogoliubov . . . . . . 135
6.2.2
Stochastic Gross-Pitaevskii plus classical approximation . . . . . 139
Equilibrium state comparison . . . . . . . . . . . . . . . . . . . . . . . . 140
6.3.1
Density profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.3.2
Spatial correlation functions . . . . . . . . . . . . . . . . . . . . . 144
Equilibration and thermometry following ergodic thermalization
. . . . 146
6.4.1
Condensate statistics . . . . . . . . . . . . . . . . . . . . . . . . . 146
6.4.2
Oscillation of the Penrose-Onsager mode . . . . . . . . . . . . . . 149
6.5
Condensate statistics and number fluctuations . . . . . . . . . . . . . . . 151
6.6
Summary of equilibrium results . . . . . . . . . . . . . . . . . . . . . . . 153
6.7
Non-equilibrium Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.7.1
Assessment of grid effects: response to trap opening . . . . . . . 157
6.8
Dynamical test between methods: response to a trap perturbation . . . 158
6.9
Centre of mass oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.10 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
7 Conclusions and future work
7.1
164
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
7.1.1
Dissipative dark solitons . . . . . . . . . . . . . . . . . . . . . . . 165
7.1.2
Stochastic dark solitons . . . . . . . . . . . . . . . . . . . . . . . 165
7.1.3
Comparison between the stochastic Gross-Pitaevskii and number
conserving Bogoliubov methods . . . . . . . . . . . . . . . . . . . 166
7.2
Future prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
iii
Contents
7.2.1
Addressing grid issues in higher dimensions . . . . . . . . . . . . 167
7.2.2
Modelling applications of ultracold gases . . . . . . . . . . . . . . 170
A Numerical methods
173
A.1 Time-stepping and spatial discretisation . . . . . . . . . . . . . . . . . . 173
A.1.1 Calculation of the mode energy in the Numerov method . . . . . 176
A.2 Condor as a tool for statistical physics . . . . . . . . . . . . . . . . . . . 177
B Supplementary results
179
B.1 Condensate statistics for different grid spacings . . . . . . . . . . . . . . 179
B.2 Tests of the effect of varying γ(z) . . . . . . . . . . . . . . . . . . . . . . 180
iv
Chapter 1
Introduction
Quantum statistics arises naturally from the notion that particles of the same species
are indistinguishable from one another. This stems from the limit to which we can know
both a particle’s position and momentum imposed by the Heisenberg uncertainty principle; in contrast to a classical picture, a quantum description is inherently probabilistic
in nature. In the quantum approach, particles are no longer viewed as discrete objects
with well defined positions, but instead as objects which have wave-like properties. The
effects of this, however, only become apparent in certain physical regimes.
The particles of a many-body system whose total wavefunction is symmetric under
the exchange of two such particles are called bosons, whereas when this operation is
anti-symmetric they are termed fermions. These particles have integer spin or half
integer spin, respectively. For fermions, the Pauli exclusion principle forbids two particles to occupy the same quantum state, however for bosons no such restriction holds,
and an arbitrary number may occupy the same quantum state. In fact, once a state
becomes occupied within a bosonic system, the probability of scattering into that state
increases.
For an atomic Bose gas at low temperatures and sufficiently high densities, quantum
features become apparent. Considering each particle to be represented by a wavefunction, which describes its properties in the required probabilistic sense, associated to
this is the de Broglie wavelength which, for a particle of mass m and temperature T ,
is [1]
λdB =
s
h2
,
2πmkB T
(1.1)
where h is Planck’s constant and kB is Boltzmann’s constant. The spatial extent of
the wavefunction is quantified by λdB , which illustrates why the quantum nature of objects is not obvious in everyday experiences. For masses and temperatures commonly
encountered, this wavelength is very small and the classical, particle-like behaviour
1
Chapter 1. Introduction
dominates. As λdB is inversely proportional to
√
T , cooling a many-body bosonic sys-
tem leads to the wavelength of individual particles increasing, and as the wavepackets of
constituent particles begin to overlap, the system enters a regime in which its behaviour
may be described in a collective way. This is the regime of quantum degeneracy, and
it is at this point that the indistinguishability of particles becomes important.
Bose-Einstein condensation is a second order phase transition which occurs at some
critical temperature, below which the lowest available energy state attains a macroscopic occupation [1]. This phenomenon is named after two physicists: Satyendra Nath
Bose and Albert Einstein. In 1924, by considering photons as a collection of identical
particles, Bose produced a paper in which he re-derived Planck’s law for black-body
radiation [2]. He sent this to Einstein, who, having recommended Bose’s work for publication [1], went on to generalise the treatment to massive particles, and so produced
a theory to describing a non-interacting gas of atoms [3]. The result of their work was
the prediction that, at low enough temperatures, a macroscopic fraction of the atoms
would be forced into the lowest single particle state - this quantum phase transition
leads to the formation of a Bose-Einstein condensate (BEC).
The onset of Bose-Einstein condensation in a three-dimensional (3d) uniform system
occurs when the inter-particle spacing, n−1/3 , where n is the number density, becomes
comparable to λdB . For a free gas of non-interacting particles with mass m, the critical
temperature is given by
Tc =
2π~2
mkB
n
ζ(3/2)
2/3
,
(1.2)
where ζ is the Riemann zeta-function, which can be translated to the requirement for
the onset of condensation in terms of the phase space density,
nλ3dB ≥ ζ(3/2).
(1.3)
The critical temperature Tc is the highest temperature at which a macroscopic occupation of the lowest energy state can be expected.
1.1
Physical systems exhibiting Bose-Einstein condensation
Following its initial prediction, the concept of Bose-Einstein condensation regained
prominence due to a remarkable insight of Fritz London [4]. London suggested this as
the mechanism behind the superfluidity of Helium II, a phenomenon discovered earlier
in experiments by Kapitza [5], and Allen and Misener in 1938 [6]. This interpretation
was not widely accepted at first, and instead Landau’s two-fluid model [7, 8], similar
to the less complete two-fluid model of Tisza [9], became the accepted theory for de2
Chapter 1. Introduction
scribing these superfluid systems. Landau’s approach was a generalisation of normal
hydrodynamics to the scenario of a superfluid component of zero viscosity, co-existing
with a normal, viscous fluid component. Landau suggested that superfluidity could be
explained by considering phonons as the fundamental excitations within a superfluid,
and that objects could therefore move without friction at speeds less than the speed
of sound [7]. Superfluidity leads to many interesting properties, such as the ability
of a liquid to flow without friction through extremely narrow capillaries. A further
striking example was predicted by Onsager [10] and Feynman [11], who showed that
rotation should lead to the appearance of quantised vortices, as subsequently observed
in an experiment of Vinen in the 1960s [12]. The existence of quantised vortices follows
from the irrotational nature of the superfluid velocity field and the requirement that
the superfluid order parameter should be single valued. These are exciting features
as they are directly observable, macroscopic consequences of the quantum nature of
superfluids.
As a liquid, Helium II is a strongly interacting system when compared to a gas.
While at low temperatures the superfluid component approaches the total system density, the BEC fraction represents only around 10% of the total density, due to interaction driven depletion of the lowest energy state [1]. This motivated the search for more
weakly interacting systems which would undergo Bose-Einstein condensation. Initially,
attention focussed upon spin-polarised hydrogen which, unlike most substances, was
expected not to solidify at low temperatures. This was proposed by Hecht [13] and
confirmed by Stwalley and Nosanow [14]. While work on spin polarised hydrogen led,
for example, to the process of evaporative cooling widely used in producing BECs, the
massive advancement in the field of laser cooling during the 1980s and 1990s, shifted
focus to the trapping and cooling of alkali atoms.
1.1.1
Atomic Bose-Einstein condensates
Experimentally, the first BEC was first achieved in an atomic sample of
87 Rb
atoms in
1995 by Eric Cornell and Carl Wiemann at JILA [15]. The coherence properties of a
23 Na
BEC [16], were subsequently verified by the interference fringes observed in the
experiment of Wolfgang Ketterle at MIT [17], earning each a share in the award of the
Nobel prize in Physics in 2001. BECs have since been formed in 7 Li [18], 41 K [19], 52 Cr
[20], 84 Sr [21, 22], 85 Rb [23], 133 Cs [24], 1 H [25] and 174 Yb [26].
In a typical experiment, the route to Bose-Einstein condensation is outlined as follows [1]: Atoms leave an oven as a thermal gas and are slowed using a Zeeman slower,
until reaching a temperature of around 1K. At this point they can be loaded into
a magneto-optical trap (MOT) in which the forces due to laser light and magnetic
fields are sufficient to trap the atoms. Doppler cooling can be employed using counter
3
Chapter 1. Introduction
propagating lasers, with frequencies chosen such that atoms moving with certain velocities interact resonantly with the photons of the laser. If the atoms absorb light with
momentum along one axis, directed consistently towards the trap centre, but emit it
spontaneously in random directions, this leads to a net decrease in momentum along
the axis of the lasers. If we imagine a cube with lasers arranged such that they point
inwards at the centres of the six faces of this cube, and a cloud of atoms held at the
centre, then the result is a cooling effect as the atoms are slowed in all directions.
There is a limit to the temperature which can be reached by this method, called
the Doppler limit. This occurs due to the fact that the momentum gain due to the
spontaneous emission process does not perfectly average out over time, and so there is a
small increase in the atoms’ momentum due to this. Therefore the lowest temperature
which can be reached by this means is set by that at which the loss of momentum due
to absorption, matches the gain in momentum due to emission.
Beyond this point, methods such as Sisyphus cooling can be used, which takes
advantage of the potential energy landscape experienced by atoms due to counterpropagating laser beams with orthogonal polarisation. The cooling process is analogous
to the energy that would be lost by a walker ascending a hill, before taking an elevator
to the bottom and beginning the process again. The atoms move up a potential which
causes them to lose kinetic energy, however on reaching the peak, they are optically
pumped to a state such that they are again at the bottom of the potential hill, and the
process continues. This allows for cooling down close to the so-called recoil limit, which
is the temperature corresponding to the energy associated with the laser frequency. A
reasonable estimate for the recoil limit of atoms is equivalent to temperatures on the
order of 1µK [1].
For typical densities, which are necessarily low to avoid losses due to three-body collisions, this is not sufficiently cold to achieve Bose-Einstein condensation. Beyond the
methods outlined, evaporative cooling often forms the final step. Cooling is achieved
in this case by removal of the hottest system atoms, thereby truncating the thermal
distribution, which on re-equilibrating acquires a lower average temperature. Experimentally, the physical process is that the spins of atoms above a certain energy are
flipped so that they enter high-field seeking states and are expelled from the trap. This
can be done selectively, so only atoms above a certain energy are affected, due to the
spatial inhomogeneity of the trapping potential. The key property is phase space density, which must reach a critical value in order for BEC to occur. This requires that
a sufficient number of atoms remain in the trap during this inherently lossy process.
Ultimately, Bose-Einstein condensation occurs when a large enough sample of atoms
are cooled so that the phase space density reaches a value on the order of unity.
Due to the weakly interacting nature of atomic BECs, the ideal gas result for the
4
Chapter 1. Introduction
critical temperature gives a reasonable estimate for that of the interacting system. For
an ideal, trapped Bose gas of N particles in 3d, the critical temperature is given by
~ω̄
Tc =
kB
N
ζ(3)
1/3
,
(1.4)
where ω̄ = (ωx ωy ωz )1/3 is the geometrical mean of the trapping frequencies.
The systems of interest in this thesis are weakly interacting, ultracold, dilute Bose
gases. To put these terms in context, dilute refers to the smallness of the atomic density
in typical experiments which is around 1013 − 1015 cm−3 , which can be compared, for
example, to air at room temperature which has a density of the order of 1022 cm−3 [1].
Though the present study is primarily centred upon thermal fluctuations in such systems, the gases we consider may nevertheless be categorised as ultracold, with quantum
phenomena becoming apparent below around 10−5 K. For liquid helium on the other
hand, such effects occur at temperatures around five orders of magnitude greater than
this. Finally, the strength of interactions is parameterised by the quantity n1/3 a in
3d, which compares the inter-particle spacing to the s-wave scattering length, a. If
n1/3 a ≪ 1, then the system is weakly interacting. It is this characteristic which leads
to the fact that depletion due to interactions is typically only around 1% in experiments, and also allows for a simple treatment of interactions, based on considering the
lowest order partial wave contribution only.
Since the achievement of BEC in the laboratory, many experiments have since been
performed in the field of degenerate quantum gases. The trapped nature of the system
allows for a diverse range of experimental possibilities. Explorations on a fundamental
level in Bose systems include the overlapping of two condensates and, subsequently, the
observation of interference fringes [17], of experiments to measure the spatial coherence
[27, 28] associated with the onset of condensation, and the introduction of rotation and
the appearance of vortices [29, 30, 31], which signify the characteristic superfluidity
intimately linked to BECs.
Due to formation of pairs of fermionic atoms which are, collectively, bosons, Fermi
gases have also proved amenable to cooling to quantum degeneracy [32]. A connection
with superconductivity and atomic BECs was highlighted in subsequent experiments
on the BEC to BCS crossover [33, 34, 35], where BCS is an abbreviation for BardeenCooper-Schrieffer. The BCS model of superconductivity proposed that electrons form
Cooper pairs, which are also composite bosons, linking this phenomenon to BoseEinstein condensation. BEC has been observed in other condensed matter settings,
with the formation of exciton-polariton condensates [36], which are the condensation
of quasi-particles formed due to the strong coupling between photons and electronic
excitations [37]. Another example is that of magnons [38], which are quasi-particle
5
Chapter 1. Introduction
excitations associated with the spin of the electron.
Exotic trapping potentials are possible, see for example [39], allowing for the creation of ultracold Bose systems in a wealth of geometries. Possible realisations include
ring traps [40, 41], which have applications to Sagnac interferometry with BECs [42],
optical lattices [43, 44, 45, 46], which are potential candidates as the basis for quantum
computation [47], and double well geometries [48], with relevance to Josephson physics
and also atom interferometry. In addition, optical lattices provide a means to study a
connection with condensed matter systems, including the crossover from a superfluid
to Mott insulator [45].
This high level of control also allows for the study of the effects of dimensionality
on these systems [49]. For example, increasing the trapping frequency in one direction
leads to an extremely flattened geometry. The system can be made to be effectively 2d,
once the energy of the trap in one direction becomes large relative to the temperature
and chemical potential, which freezes out dynamics in that direction. Similarly, by
increasing the trapping frequency in a further direction, motion is then frozen into the
ground state along all but one axis, and the system is then effectively 1d. Related to
the system dimensionality are nonlinear macroscopic excitations in the form of solitons,
which are supported in BECs due to the dispersive and nonlinear characteristics of these
systems. In particular, solitons are stable only in sufficiently elongated geometries, and
more stable excitations in the form of quantised vortices have been observed to form
due to the decay of dark solitons in a more isotropic setting [50]. Quasi-1d systems are
of relevance also in designs of so-called atom chips [51], which are BECs created above
micro-fabricated surfaces, and atom interferometers, which typically coherently split,
allow to evolve and then recombine condensates, to study their interference patterns
[52, 53]. As the work presented in this thesis focuses on 1d systems, we discuss their
characteristics in more detail in the following section.
1.2
Quasi-condensation and low dimensional systems
The onset of off-diagonal long range order (ODLRO) is a hallmark of a transition
to a new thermodynamic phase, the simplest example of which is Bose-Einstein condensation [54]. Off-diagonal order refers to the coherence between spatially separated
elements of a system, which characterises the condensed phase. On introducing the
concept of long-range order [54], Yang extended the criterion for Bose-Einstein condensation of Penrose and Onsager [55]. In turn, the criterion of Penrose and Onsager
generalises the concept of macroscopic occupation of a single quantum state, which signifies BEC in an ideal system, to interacting systems. In terms of Bose field operators
Ψ̂† (x) and Ψ̂(x), which create and annihilate, respectively, a particle at position x, the
6
Chapter 1. Introduction
criterion can be written as
hΨ̂† (x)Ψ̂(x′ )i → hΨ̂† (x)ihΨ̂(x′ )i
as |x − x′ | → ∞.
(1.5)
This represents mathematically the development of ODLRO as viewing the left hand
side of Eq. (1.5) as a matrix, called the one-body density matrix [56], it defines the
onset of coherence between the off-diagonal components. This can be illustrated for a
uniform system for which the one-body density matrix is
hΨ̂† (x)Ψ̂(x′ )i =
1 X
′
Np eip·(x−x )/~ ,
V p
(1.6)
where Np is the number of particles with momentum p. As |x − x′ | → ∞, then
only the zero-momentum state does not yield a zero contribution to the sum, hence
hΨ̂† (x)Ψ̂(x′ )i → N0 /V , where V is the system volume and N0 is the occupation of the
ground state [1]. For interacting systems, N0 should be replaced with the expectation
value hN0 i. However, in either case it is clear that the Penrose-Onsager criterion for
Bose-Einstein condensation generalises the notion of macroscopic occupation of a single
quantum state, to that of an eigenvalue of the one-body density matrix taking on a
macroscopic value.
The Mermin-Wagner-Hohenberg (MWH) theorem [57, 58] states that in low dimensional systems, long wavelength fluctuations tend to destroy coherence and so restrict
the formation of long-range order. The result is that Bose-Einstein condensation is
forbidden to occur within a 1d homogeneous system at all temperatures, and occurs
only at zero temperature in 2d homogeneous systems. The MWH theorem is applicable
only in the thermodynamic limit however. For harmonically trapped Bosons with sufficiently strong confinement, this effect can be suppressed allowing for the formation of
coherent structures [59, 60, 61]. The introduction of a harmonic trapping potential introduces a lower energy limit which restricts the formation of long-wavelength thermal
excitations, which otherwise disrupt the condensate.
Due to the finite size of trapped Bose systems, long-range order is defined when Eq.
(1.5) holds over the system size. An intermediate scenario is that in which fragmented
regions of coherence exist, which however extend over regions less than the total system
size. This is referred to as a quasi-condensate [62, 63], which can be thought of as a
phase-fluctuating condensate in which density fluctuations are largely suppressed [64].
In 2d systems, it was pointed out by Kane and Kadanoff [65], and later shown by
Berezinskii [66, 67], that a phase transition exists at sufficiently low temperatures. This
was shown to be associated with the formation of pairs of bound vortices by Kosterlitz
and Thouless [68]. Above the critical temperature for this phase transition, superfluidity is destroyed by the proliferation of free vortices although a quasi-condensate is
7
Chapter 1. Introduction
expected to persist, as observed recently experimentally [69].
One dimensional systems display a similarly rich phase diagram when cooled to
the quantum degenerate regime. There are two characteristic regimes within onedimensional trapped atomic Bose systems, dependent upon the importance of interactions between particles. This is parameterised by [64]
γ int =
mg
(~2 n)
(1.7)
where g is the effective interaction coupling constant1 . For a quasi-condensate confined
transversely within a cylindrical trap at a temperature and chemical potential such
that kB T, µ . ~ω⊥ , the interaction parameter is given by
g=
2~2 a
2 .
ml⊥
(1.8)
Here l⊥ is the transverse oscillator length, defined as
l⊥ =
s
~
,
mω⊥
(1.9)
which characterises the spatial extent of the transverse ground state. For γint ≫ 1, the
system is a Tonks-Girardeau [70, 71] strongly interacting gas, which rather counterintuitively, occurs for small atom densities. The other regime, in which we are interested
in the present work, is that of a weakly-interacting Bose gas for which γint ≪ 1.
For the weakly-interacting system, there are three main temperatures of note. Start-
ing with the 1d ideal gas case, it was found by Bagnato and Kleppner, using a semiclassical description, that Bose-Einstein condensation could occur in traps with stronger
than harmonic confinement for 1d systems [72]. Retaining the discrete summations over
energy levels, Ketterle and van Druten [73] instead considered condensation of a finite
number of particles within a harmonic potential, and found a macroscopic ground state
occupation to occur at a temperature
T1d =
~ωz N
,
kB ln(2N )
(1.10)
which is valid for large N .
In the harmonically confined, interacting case, there are two temperatures which
characterise the 1d quantum degenerate Bose gas [64]. The degeneracy temperature in
1
Note that in [64], Petrov et al. call the interaction parameter simply γ, however we introduce
the subscript here to differentiate this from γ which appears in the notation used for the stochastic
Gross-Pitaevskii equation in later chapters.
8
Chapter 1. Introduction
1d is defined as
Td =
N ~ωz
,
kB
(1.11)
below which density fluctuations become relatively suppressed. Equally, there is a
temperature at which fluctuations in the phase are reduced, and this is given by
Tφ =
N (~ωz )2
.
µkB
(1.12)
Hence, Td = (µ/~ωz ) Tφ , where µ is the chemical potential, meaning we always have
Td > Tφ for a non-zero condensate fraction. The analogous expression for highly
elongated 3d traps is [74]
Tφ,3d =
15 N (~ωz )2
,
32 µkB
(1.13)
however as we consider a purely 1d description, it is Eq. (1.11) and Eq. (1.12) which
are of most relevance here.
1.2.1
Experimental realisations and applications of low-dimensional
systems
Low dimensional weakly-interacting atomic Bose-Einstein condensates were realised in
2001. In an experiment carried out at MIT [49], Görlitz et al. prepared sodium (23 Na)
atoms in traps which were made to transition from the 3d Thomas-Fermi regime (see
Chapter 2) into the 2d and 1d regimes by reducing the number of trapped particles
though interaction with a thermal beam. Schreck et al. created a BEC of lithium
(7 Li) which was 1d in nature and additionally in thermal contact with a Fermi sea of
6 Li
atoms [75]. In a separate experiment, a lattice of 1d
87 Rb
gases was created with
tunable tunnelling between various elements of the lattice of 1d rubidium gases [28].
Releasing the atoms allowed for a study of the coherence properties of this coupled
quantum system. Phase fluctuations in a 1d system have also been observed directly in
experiments carried out at Hannover [76, 77], where they were found to decrease with
decreasing temperature and increasing numbers. Density fluctuations have also been
observed [78], and were found to decrease on entering the quasi-condensate regime.
The spatial correlations of a phase fluctuating condensate have been measured experimentally by interfering two copies of a single realisation [79], momentum Bragg
spectroscopy [80, 81] and condensate focusing [82]. The fundamental features of highly
elongated quasi-condensates are also of great practical practical interest, due to their
relevance to potential applications.
In the past interferometers may have been synonymous with experiments in optics,
however exploration based on the concept of matter waves has led to interferometers
using electrons, neutrons and atoms [53]. Due to the variation in physical properties
9
Chapter 1. Introduction
between atomic species, atom interferometers offer the potential to be tailored to the
study of a range of phenomena. A key appeal of atom interferometers is that the typical
thermal de Broglie wavelength of atoms is around 104 times smaller than that of visible
light [53]. This corresponds to a high sensitivity to external potentials, which can be
displayed as a change in the phase and contrast of interference fringes between these
matter waves.
Interferometry with atomic matter waves offers a high level of control over important properties such as the phase and amplitude, owing to the high degree of tunability within Bose-Einstein condensate experiments. Interference fringes have been
observed clearly within such systems [17, 52, 83]. Experiments using BECs may be
carried out using waveguides, within which the condensate is tightly confined in twodimensions. Confinement within tight transverse potentials leads to an increased interaction strength, and to reduced contrast in fringes after a relatively short time [84].
This may be improved, for example, by reduction of the transverse trapping frequency
[85], or by tuning of the atomic interactions via a Feshbach resonance [86]. Double well
potentials may also be used to split and recombine a single BEC. This was performed by
Shin et al. [52], who observed very clear interference fringes following the independent
phase evolution of the two samples. They additionally found that the relative phases
between the split condensates was consistent in different experimental realisations.
First physically realised in 2001 [87, 88], atom chips are ultracold bosonic systems
realised in micro-traps [89, 90, 91] near to a micro-engineered surface [92]. They offer
the opportunity for portable and robust integrated atom optics [90]. For example, atom
interferometers can be constructed on atom chips [51, 93, 94, 95, 96], with coherent
splitting and recombination on an atom chip already performed at Heidelberg [97].
As a source of coherent atoms, BECs represent the atomic equivalent to optical
lasers, and much effort has been put into experimental work towards a continuous
atom laser [98, 99, 100, 101, 102, 103]. To date, pulsed or quasi-continuous atom lasers
have been realised experimentally. Two example schemes for a continuous atom laser
are to either evaporatively cool an atom beam within a magnetic guide [104, 105], or to
continually replenish a reservoir of ultracold atoms, which in turn feed a continuously
condensed sample [101, 106].
In each of the applications highlighted, the interplay between dynamic manipulations of an interacting many-body system and fluctuations due to the effects of reduced
dimensionality, mean that an understanding of the behaviour of these systems is essential in developing such new technologies, based upon ultracold atoms.
10
Chapter 1. Introduction
1.3
Thesis overview
In this thesis we present a numerical study of the equilibrium and non-equilibrium
behaviour of weakly interacting, one-dimensional atomic Bose gases under harmonic
confinement. To include phase and density fluctuations, which are important at a broad
range of temperatures in low-dimensions, we model these systems numerically via the
stochastic Gross-Pitaevskii equation [107, 108, 109, 110]. This is a Langevin equation
which takes the form of a nonlinear Schrödinger equation with additional noise and
damping terms. These represent coherent and incoherent collisions, respectively, which
occur between condensate and noncondensate atoms.
In Chapter 2, we discuss this method in relation to approaches which retain the
concept of a condensate mean field, and motivate the use of a stochastic approach. We
initially apply the SGPE to the equilibrium properties of a trapped Bose gas in Chapter
3, where we discuss the growth to a dynamical equilibrium and also some features of
the equilibrium state which results. Comparing the sensitivity to the grid spacing used
in numerically solving the SGPE [108], we find much of the equilibrium properties to
be fairly robust to the variation of this parameter in 1d.
We then consider the dynamics of dark solitons at finite temperatures. Dark solitons
are macroscopic nonlinear excitations, which exist due to a delicate balance between the
effects of dispersion and nonlinearity within BECs. These one-dimensional excitations
embody many of the interesting features of trapped low dimensional Bose-Einstein
(quasi-)condensates at finite temperatures: For example, dark solitons are inherently
nonlinear, display interesting dynamics within harmonically trapped systems, undergo
interactions with finite temperature excitations which leads to decay, and are also
sensitive to phase and density fluctuations. We consider the non-equilibrium dynamics
of dark solitons within a phase fluctuating one-dimensional condensate in Chapters 4
and 5.
Initially, we study the decay of solitons due to interactions with thermal atoms,
modelling their evolution though a dissipative Gross-Pitaevskii equation [111, 112, 113].
We find the decay to be consistent with that expected of an excitation within a quantum
dissipative system [114]. However, recent experimental findings suggest that solitons
exist in single realisations within the laboratory for longer times than a reproducible
average can be obtained [115, 116]. This implies that there are features of interest
beyond the average dynamics, which are therefore not captured within a mean field
description. To study the physics of this, we then consider the dynamics of a dark
soliton described within the stochastic Gross-Pitaevskii equation. Due to propagation
within a thermally excited background density, we observe a range of soliton trajectories
despite an identical means of creation between different runs. A clear distribution of
soliton lifetimes is observed, which should be directly observable within experiments.
11
Chapter 1. Introduction
In Chapter 6, we consider a comparison between two stochastic approaches to finite
temperature BECs, comparing the stochastic Gross-Pitaevskii equation to a number
conserving Bogoliubov approach [117, 118, 119]. We consider a range of the equilibrium
properties in an interacting, trapped Bose gas at various temperatures beneath the
transition temperature. We also compare the condensate statistics predicted by the
grand canonical SGPE, to those due to the model of Svidzinsky and Scully [120] derived
within the canonical ensemble. Finally, we consider the non-equilibrium damping of
oscillations in response to a trap perturbation, as described within each method.
1.4
Collaborations
We highlight here aspects of the work presented in this thesis that was carried out as
part of a collaborative effort. This will also be clearly highlighted in the text where
appropriate.
Part of the work of Chapter 5 is presented in a paper written in collaboration
with Dimitri Frantzeskakis (Department of Physics, University of Athens, Greece),
Theodoros Horikis (Department of Mathematics, University of Ioannina, Greece), Panayotis Kevrekidis (Department of Mathematics and Statistics, University of Massachusetts,
USA), Hector Nistazakis (Department of Physics, University of Athens, Greece) and
my supervisor within the School of Mathematics and Statistics at Newcastle University, Nick Proukakis. In particular, Eq. (5.4) of Chapter 5 was derived by Dimitri
Frantzeskakis and co-workers, while Hector Nistazakis produced Figure 5.11. Eq. (5.4)
and some of the numerical work of Chapter 4 and Chapter 5 is presented in [121], and
also the follow-on work [122].
Again, alongside my supervisor, the results presented in Section 3.5 and Chapter 6 are due to work carried out in collaboration with Carsten Henkel (Institute for
Physics, Potsdam University, Germany) and Antonio Negretti (Ulm University, Institute for Quantum information processing). In particular Antonio Negretti provided the
number conserving Bogoliubov initial states used in the comparison of Chapter 6 and
the data for the condensate statistics results of Svidzinsky and Scully, given by Eq. 5
of [120], also used in Chapter 6. A number of the tests carried out in Chapters 3 and
6 were initiated through discussions with Carsten Henkel, partly undertaken during a
research visit to Potsdam, who also helped in the subsequent analysis of these results.
This work is summarised in [123].
12
Chapter 1. Introduction
Parts of this thesis are presented in the following publications:
The Stochastic Gross-Pitaevskii Equation and some Applications
S.P. Cockburn and N.P. Proukakis
Laser Physics 19 558-570 (2009) (Special Issue on Physics of Cold Trapped
Atoms)
Matter-wave dark solitons: stochastic vs. analytical results
S.P. Cockburn, H.E. Nistazakis, T.P. Horikis, P.G. Kevrekidis, N.P. Proukakis,
D.J. Frantzeskakis
Accepted, Physical Review Letters (2010)
arXiv:0909.1660v2
13
Chapter 2
Theoretical Background
Many theories have been devised in order to describe both the static and dynamical
properties of weakly interacting, ultracold, atomic Bose gases [124, 125]. A crucial
feature of these systems, is that even beneath the temperature signifying the onset of
Bose-Einstein condensation, a condensate coexists with a noncondensed component.
Repulsive interatomic interactions lead to a depletion of atoms from the condensate,
and for weak interactions, it is possible to use a Bogoliubov transformation to move
to a dressed basis in which we instead deal with a system of non-interacting, quasiparticles [126, 127]. At finite temperatures, thermal effects also lead to the promotion
of atoms from the lowest (dressed) energy state, therefore an accurate description of
the properties of partially condensed Bose gases is important.
A mean field treatment is often valid in describing low temperature condensates, in
which there is a well-established average contribution to the condensate order parameter. Beyond this, fluctuations about the mean field can be both quantum and thermal
in nature. Quantum fluctuations are associated with the zero point field energy, the
effects of which have been the subject of a number of studies in the context of weaklyinteracting Bose-Einstein condensates, for example [119, 128, 129, 130, 131, 132, 133,
134].
Our focus in this Thesis however is on thermal effects. For temperatures close to
the transition point, critical fluctuations are of crucial importance and hence essential
in describing the condensate growth process. A theoretical treatment for the dynamics of a partially condensed Bose gases describing these effects is therefore of central
importance. In this chapter, we discuss two approaches to the finite temperature equilibrium state and dynamics of finite temperature Bose gases, which, while incorporating
different physics, additionally share some similar themes. As a basis for our discussion of finite temperature models, we begin with a summary of the zero temperature
Gross-Pitaevskii description of a Bose-Einstein condensate.
14
Chapter 2.
2.1
Theoretical Background
The Gross-Pitaevskii equation and elementary excitations
A mean field model for the dynamic and static properties of a pure Bose-Einstein condensate is provided by the Gross-Pitaevskii equation. This is a zeroth-order equation
in fluctuations about the condensate, and can be obtained as follows:
The Hamiltonian for a many-body system of bosons, is written in second quantised
form as
~2 ∇2
Ĥ = dx Ψ̂ (x, t) −
+ V (x) Ψ̂(x, t)
2m
Z
Z
1
dx dx′ Ψ̂† (x, t)Ψ̂† (x′ , t)Uint (x, x′ )Ψ̂(x, t)Ψ̂(x′ , t),
+
2
Z
†
(2.1)
where Ψ̂(x, t) (Ψ̂† (x, t)) is a Bose field operator which annihilates (creates) a particle
at position x at time t, Uint (x, x′ ) is the interaction potential between particles at x
and x′ , and V (x) is the external trapping potential. The term involving a Laplacian
is the kinetic energy operator. The Heisenberg equation of motion for the Bose field
operator under this Hamiltonian is
i~
i
∂ Ψ̂(x′ , t) h
= Ψ̂(x′ , t), Ĥ .
∂t
(2.2)
Due to the diluteness of typical atomic BECs, interactions can be described by a single
parameter, the s-wave scattering length. Replacing the general potential Uint (x, x′ ) by
a two-body δ-function interaction of the form
Uint (x, x′ ) =
4π~2 a
δ(x − x′ ) ≡ g3d δ(x − x′ )
m
(2.3)
and making use of the Bose commutation relations,
h
i
Ψ̂(x, t), Ψ̂† (x′ , t) = δ(x − x′ )
h
i h
i
Ψ̂(x, t), Ψ̂(x′ , t) = Ψ̂† (x, t), Ψ̂† (x′ , t) = 0,
15
(2.4)
(2.5)
Chapter 2.
Theoretical Background
we have
i ~2 ∇2
+ V (x) Ψ̂(x, t)
Ĥ Ψ̂(x , t) = dx Ψ̂(x , t)Ψ̂ (x, t) − δ(x − x ) −
2m
Z
h
i
g3d
dx Ψ̂† (x, t) Ψ̂(x′ , t)Ψ̂† (x, t) − δ(x − x′ ) Ψ̂(x, t)Ψ̂(x, t).
+
2
2 2
Z
~ ∇
′
†
+ V (x) Ψ̂(x, t)
= dx Ψ̂(x , t)Ψ̂ (x, t) −
2m
Z
g3d
+
dx Ψ̂(x′ , t)Ψ̂† (x, t)Ψ̂† (x, t)Ψ̂(x, t)Ψ̂(x, t)
2
2 2
~ ∇
† ′
′
+ V (x) + g3d Ψ̂ (x , t)Ψ̂(x , t) Ψ̂(x′ , t).
− −
2m
Z
′
h
′
†
′
(2.6)
We then have that
~2 ∇2
† ′
′
Ĥ Ψ̂(x , t) = Ψ̂(x , t)Ĥ − −
+ V (x) + g3d Ψ̂ (x , t)Ψ̂(x , t) Ψ̂(x′ , t).
2m
′
′
(2.7)
and substitution into the Heisenberg equation of motion gives
2 2
~ ∇
∂ Ψ̂(x, t)
†
= −
+ V (x) + g3d Ψ̂ (x, t)Ψ̂(x, t) Ψ̂(x, t).
i~
∂t
2m
(2.8)
Replacing the operator with a classical field representing the mean field of the condensate, Ψ̂(x, t) → φ(x, t), yields the well known T = 0 Gross-Pitaevskii equation (GPE)
[135, 136]
2 2
∂φ(x, t)
~ ∇
2
i~
= −
+ V (x) + g3d |φ(x, t)| φ(x, t)
∂t
2m
(2.9)
≡ HGP φ(x, t).
To describe the dynamics of highly elongated systems, an effective one-dimensional
Gross-Pitaevskii equation can be used. This takes the same form as Eq. (2.9) with
interactions described instead by Eq. (1.8) [137]. This is based upon the assumption
that transversely the system can be regarded as restricted to the ground state.
The stationary ground state solution to Eq. (2.9), which we denote φ0 (x), is found
by solving the nonlinear eigenvalue problem for the ground state energy eigenvalue µ,
HGP φ0 (x, t) = µφ0 (x, t).
(2.10)
In the limit that interactions dominate over kinetic energy effects, this gives the Thomas-
16
Chapter 2.
Theoretical Background
Fermi solution [1],
|φT F (x)|2 =
1
(µ − V (x)) ΘH (R − |x|),
g3d
(2.11)
valid at very low temperatures and sufficiently high densities. Here R is the zero temperature Thomas-Fermi radius, which is the boundary of the condensate in this limit.
Beyond this the Thomas-Fermi solution is zero, as shown by the Heaviside function,
ΘH (R−|x|). For a harmonic trapping potential, V (x) = (1/2)m ωx2 x2 + ωy2 y 2 + ωz2 z 2 ,
the Thomas-Fermi density then takes the form of an inverted parabola.
2.1.1
The zero temperature Bogoliubov equations
Excitations on top of the condensate can be considered by making the substitution
[124]
φ(x, t) = e−iµt [φ0 (x) + δφ(x, t)] ,
(2.12)
where the quantity δφ(x) in square brackets represents fluctuations about the condensate ground state. A linearised equation for these excitations can be found through
substitution of this into Eq. (2.9),
2 2
∂
~ ∇
2
i~ δφ(x, t) = −
+ V (x) + 2g3d |φ0 (x)| − µ δφ(x, t) + g3d [φ0 (x)]2 δφ∗ (x, t).
∂t
2m
(2.13)
To solve this linearised equation, we can use the ansatz
δφ(x, t) =
X
i
ui (x)e−iωi t + vi∗ (x)eiωi t ,
(2.14)
where i labels different modes and ωi their respective frequencies. Doing so leads to
two coupled equations for the Bogoliubov amplitudes ui (x) and vi (x) which can be cast
into matrix form,
L̂(x)
M̂ (x)
−M̂ ∗ (x) −L̂∗ (x)
!
ui (x)
vi (x)
!
= ~ωi
ui (x)
vi (x)
!
(2.15)
with the definitions
~2 ∇2
+ V (x) + 2g3d |φ0 (x)|2 − µ,
2m
M̂ (x) = g3d [φ0 (x)]2 .
L̂(x) = −
17
(2.16)
(2.17)
Chapter 2.
Theoretical Background
For a uniform system, the Bogoliubov amplitudes are plane waves of the form
u(x) = ueik·x ,
v(x) = veik·x .
(2.18)
The solution of Eq. (2.9) for a uniform system, for which V (x) = 0, is µ = g3d n0 , where
n0 = |φ0 |2 . The T = 0 Bogoliubov excitation spectrum [126], in the uniform case is
then given by
~ω =
s
~2 k2
2m
~2 k2
+ 2g3d n0 .
2m
(2.19)
This shows that there are two regimes of behaviour which occur depending upon the
wavelength of the excitation relative to the system healing length, defined in an interacting uniform system as
ξ=√
~
.
mng3d
(2.20)
Physically, this corresponds to the length scale on which the condensate density naturally responds to perturbations. For wavelengths λ ≫ ξ, Eq. (2.19) reduces to the
p
linear dispersion associated with phonons ~ω = c~k, where c = (g3d n0 )/m. For
higher energies, the dispersion relation is that of a free particle, with ~ω = ~2 k2 /(2m).
Within harmonic trapping potentials, a numerical solution to Eqs. (2.9) and (2.15)
leads to a discrete set of excitation frequencies [138, 139]. Long wavelength excitations
are now those for which λ ∼ R, since R characterises the spatial extent of trapped
BECs. These low energy excitations correspond to collective oscillations of the entire
condensate which correspond, for example, to breathing mode or centre of mass oscillations. These can be excited by an instantaneous change in the trap frequency or a
translational shift in the trapping potential, respectively. An analysis based upon Eq.
(2.9) and Eq. (2.15) has been found to yield good agreement with experimentally observed oscillation frequencies of such collective modes, at sufficiently low temperatures
[138]. For short wavelengths, such that λ ≪ R, the condensate can be considered as
locally homogeneous and excitations again take the form of sound waves, as observed
in experimental studies [140].
Though derived independently by Gross and Pitaevskii around half a century ago,
the creation of trapped nonuniform Bose-Einstein condensates in 1995 spawned a vast
amount of studies based upon the GPE, starting from the important work of Ruprecht
et al [141]. One reason for the widespread use of the GPE as a model, is the relative
ease with which it can be solved numerically. Studies employing this model have looked
at the low temperature physics of Bose-Einstein condensates in variety of settings [142].
While its formulation suggests a fairly strict limit on the temperature regime in
which the GPE should be applicable, this approach seems to capture much of the im18
Chapter 2.
Theoretical Background
portant low-temperature physics of these systems, up to T . 0.5Tc [138]. In addition to
the aforementioned agreement with experimental values for collective mode excitation
frequencies, other features captured by the GPE include processes such as vortex reconnection [143], linked for example to quantum turbulence, and the interaction between
nonlinear excitations such as solitons and the background sound field [144]. These
are examples of features which arise due to the nonlinear nature of the interacting
condensate, a characteristic which persists therefore at all temperatures. There are of
course regimes in which a description based upon the GPE alone is not appropriate. It
is inappropriate for describing dissipative effects due to finite temperatures, which are
known to lead to, for example, the damping of collective modes [145] and soliton motion
[146]. To treat situations in which the dynamics of both the thermal and condensate
components should be taken into account, an equation of motion is necessary for each
of these components. We now discuss one approach which addresses this problem from
the viewpoint that the condensate and thermal cloud are subject to such an intrinsic
split.
2.2
Coupling the Gross-Pitaevskii and Boltzmann equations
The GPE represents the simplest microscopic model for a Bose-Einstein condensate,
which nevertheless captures much of the low temperature physics. We might expect,
therefore, that a finite temperature equation of motion should be of a similar form,
which reduces to the GPE in a low temperature limit. One approach to the dynamics of
both a condensate and thermal cloud is to couple such a Gross-Pitaevskii-like equation
to a Boltzmann equation describing the thermal cloud. Various such models have been
put forward by Proukakis et al. [147, 148, 149], Walser et al. [150, 151] and by Griffin,
Zaremba and collaborators [152, 153, 154], extending the early work by Kirkpatrick and
Dorfmann [155, 156], and also Eckern [157], who first derived a closed set of kinetic
equations for an inhomogeneous Bose gas beneath the transition temperature. Due to
the physically intuitive nature of the equations of motion and their proven widespread
applicability, the first model we discuss is the that of Zaremba, Nikuni and Griffin
(ZNG) [154].
2.2.1
The Boltzmann equation
Introducing a mean field to represent the condensate relies upon considering the finite temperature atomic system as inherently split into condensate and thermal cloud
components. The thermal cloud may be described by a phase space distribution, the
19
Chapter 2.
Theoretical Background
Wigner distribution f (x, p, t), in terms of which the thermal density can be defined as
nt (x, t) =
Z
dp
f (x, p, t).
(2π~)3
(2.21)
If collisions between particles of the gas were entirely negligible, then this distribution
would obey Liouville’s equation
df (x, p, t)
= 0.
dt
(2.22)
This is a mathematical statement that probability is conserved along trajectories in
phase space, where it is clear that f (x, p, t) represents a density within phase space.
The total derivative for a gas within a potential V (x) is [158]
df (x, p, t)
∂f (x, p, t) ∂x
∂f (x, p, t)
=
+
· ∇f (x, p, t) + F ·
,
dt
∂t
∂t
∂p
(2.23)
where F = −∇V . If collisions do occur within the gas, then the probability is no longer
constant along phase space paths. This leads to a modification to Eq. (2.22) such that
df (x, p, t)
= C [f (x, p, t)] ,
dt
(2.24)
where C [f ] is a functional of the distribution function. Eq. (2.24) is refered to as a
Boltzmann equation [158].
The thermal cloud dynamics within a trapped Bose gas may be represented by
a quantum Boltzmann equation, which describes the evolution of the single particle
Wigner function, f (x, p, t). The quantum Boltzmann equation of the ZNG approach
[154, 159] is
∂f (x, p, t)
p
+
· ∇f (x, p, t) − ∇U (x, t) · ∇p f (x, p, t) = C12 [f ] + C22 [f ].
∂t
m
(2.25)
The terms to the left of the equals sign are the ‘streaming’ terms, and are the terms
which remain in the collisionless case, and for which the phase space distribution satisfies Liouville’s equation at equilibrium. Here, the energy of a thermal particle is defined
in the Hartree-Fock limit as
|pi |2
+ V (x) + 2g3d |φ(x, t)|2 + nt (x, t) ,
2m
|pi |2
+ U (x, t),
≡
2m
ǫi (x, t) =
(2.26)
and in this semi-classical picture, the thermal density can then be obtained from Eq.
(2.21). The potential U (x, t) is the contribution to the thermal energies at the Hartree20
Chapter 2.
Theoretical Background
Fock level, meaning the mean field potential experienced by the thermal atoms due to
the density of atoms within the trap is accounted for.
We are in general interested in the effect of collisions. Those which transfer a
condensate particle into the thermal cloud, together with the reverse process, are represented by the collisional integral
4π 2
g |φ|2
C12 [f ] =
~ 3d
Z
dp2
(2π~)3
Z
dp3
(2π~)3
Z
dp4
(2π~)3
× (2π~)3 δ(mvc + p2 − p3 − p4 ) × δ(ǫc + ǫ2 − ǫ3 − ǫ4 )
(2.27)
× (2π~)3 [δ(p − p2 ) − δ(p − p3 ) − δ(p − p4 )]
× [f2 (f3 + 1)(f4 + 1) − (f2 + 1)f3 f4 ],
where fi = f (xi , pi , t). The condensate energy is given by the expression
ǫc (x, t) = −
~2 ∇2 |φ(x, t)|
+ V (x) + g3d |φ(x, t)|2 + 2nt (x, t) .
2m|φ(x, t)|
(2.28)
The influence of collisions involving two thermal cloud atoms is described by
C22 [f ] =
4π 2
g
~ 3d
Z
dp2
(2π~)3
Z
dp3
(2π~)3
Z
dp4
(2π~)3
× (2π~)3 δ(p + p2 − p3 − p4 ) × δ(ǫ + ǫ2 − ǫ3 − ǫ4 )
(2.29)
3
× (2π~) [δ(p − p2 ) − δ(p − p3 ) − δ(p − p4 )]
× [f f2 (f3 + 1)(f4 + 1) − (f + 1)(f2 + 1)f3 f4 ].
The δ-functions appearing in Eq. (2.27) and Eq. (2.29) represent the conservation of
energy and momentum, whereas the combinations of Wigner functions represent the
scattering amplitudes for various collisions. For example, the rate at which particles
scatter into the condensate from the thermal cloud, Γin ∝ |φ|2 (f2 + 1)f3 f4 , where the
appearance of distribution functions fi represent the bosonic enhancement of scattering
into a populated state, and the 1 represents the contribution of spontaneous scattering
events. So, Γin describes the probability of two particles scattering out of the thermal
cloud, with one scattering back into a thermal level while the other is transferred into
the condensate. The amplitude for the condensate is accounted for by the density |φ|2 ,
before the integral signs, from which we note also that spontaneous processes are not
accounted for, in considering scattering into the condensate. This reflects the classical
representation of this quantity. The net scattering between the condensate and thermal
cloud is then given by Γin − Γout , where Γout ∝ |φ|2 f2 (f3 + 1)(f4 + 1).
21
Chapter 2.
2.2.2
Theoretical Background
Incorporating collisional particle exchange into the condensate
dynamics
We now describe a generalisation which is made to the GPE in coupling the condensate
dynamics to that of the thermal atoms. We will again adopt the ZNG notation. This
approach adopts an explicit symmetry breaking viewpoint, within which the Heisenberg
equation of motion can be written as [159]
2 2
∂ Ψ̂(x, t)
~ ∇
i~
= −
+ V (x) Ψ̂(x, t) + ζ(x) + g3d Ψ̂† (x, t)Ψ̂(x, t)Ψ̂(x, t).
∂t
2m
(2.30)
The symmetry breaking term ζ(x) is introduced as a perturbation to the Hamiltonian
ĤSB = lim
ζ→0
Z
h
i
dx ζ(x)Ψ̂† (x, t) + ζ ∗ (x)Ψ̂† (x, t)
(2.31)
so that the Bose field operator can have a non-zero average. This is defined to be
hΨ̂(x, t)i = φ(x, t).
Taking the average of Eq. (2.30) gives the equation of motion
2 2
~ ∇
∂φ(x, t)
= −
+ V (x) φ(x, t) + ζ(x) + g3d hΨ̂† (x, t)Ψ̂(x, t)Ψ̂(x, t)i.
i~
∂t
2m
(2.32)
The following decomposition can then be made for the full Bose field operator
Ψ̂(x, t) = φ(x, t) + ψ̃(x, t)
(2.33)
where ψ̃(x, t) is the operator for the thermal component, such that hψ̃(x, t)i = 0. Using
this leads to an expression for the three operators
h
i2
Ψ̂† (x, t)Ψ̂(x, t)Ψ̂(x, t) =|φ(x, t)|2 φ(x, t) + 2ψ̃(x, t)|φ(x, t)|2 + ψ̃(x, t) φ∗ (x, t)
+ ψ̃ † (x, t) [φ(x, t)]2 + 2ψ̃ † (x, t)ψ̃(x, t)φ(x, t)
+ ψ̃ † (x, t)ψ̃(x, t)ψ̃(x, t),
(2.34)
and averaging gives
hΨ̂† (x, t)Ψ̂(x, t)Ψ̂(x, t)i =nc (x, t)φ(x, t) + m̃(x, t)φ∗ (x, t) + 2nt (x, t)φ(x, t)
+ hψ̃ † (x, t)ψ̃(x, t)ψ̃(x, t)i,
(2.35)
with the following definitions for the condensate density nc (x, t) ≡ |φ(x, t)|2 , the noncondensate density nt (x, t) ≡ hψ̃ † (x, t)ψ̃(x, t)i and the anomalous average m̃(x, t) ≡
hψ̃(x, t)ψ̃(x, t)i.
22
Chapter 2.
Theoretical Background
The equation of motion for the condensate mean field is then
2 2
~ ∇
∂φ(x, t)
= −
+ V (x) + g3d (nc (x, t) + 2nt (x, t)) φ(x, t)
i~
∂t
2m
∗
(2.36)
†
+ g3d m̃(x, t)φ (x, t) + g3d hψ̃ (x, t)ψ̃(x, t)ψ̃(x, t)i.
The terms hψ̃ † (x, t)ψ̃(x, t)ψ̃(x, t)i and m̃(x, t) are both of order g3d [159], meaning they
vanish in the case of an ideal gas. The anomalous average m̃(x, t) is associated with
effects due to the fact that scattering events do not occur in vacuo, but rather involve
states which may already be occupied. Neglect of the term m̃(x, t) is referred to as
the ‘Popov’ approximation1 [152], which is the limit taken in deriving the the coupled
Gross-Pitaevskii-Boltzmann equations above. Interaction effects are retained to first
order in g3d in the chemical potential and excitation energies, however to second order
in the collisional integrals [154].
To order g3d , this product of three noncondensate operators is found to be imaginary
and plays the role of a source term within a generalised Gross-Pitaevskii equation
[148, 107, 154, 150]
i~
2 2
∂φ(x, t)
~ ∇
= −
+ V (x) − iR(x, t) + g3d |φ(x, t)|2 + 2nt (x, t) φ(x, t). (2.37)
∂t
2m
This source term, denoted by −iR(x, t), describes scattering processes affecting the
condensate population and is related to the collisional integral via
~
R(x, t) =
|φ(x, t)|2
Z
dp
C12 [f (x, p, t)].
(2π~)3
(2.38)
The g3d |φ(x, t)|2 + 2nt (x, t) terms of Eq. (2.37), plus the ∇U (x, t)·∇p f (x, p, t) terms
of Eq. (2.25), represent the mean field coupling between the condensate and thermal
cloud. This provides a dissipative effect, which has for example been seen to lead to
Landau damping of collective modes of a BEC [161, 162, 163], even in the case where
C12 = C22 = 0. Additional dissipative effects are incorporated on inclusion of these
collisional terms however, and as a whole the method describes self-consistently, both
the condensate and thermal cloud dynamics, treating each within the Hartree-Fock
approximation. It has also been shown to perform well in numerical applications, with
several comparisons to experimental work showing good agreement. In addition, this
formalism can be shown to reduce to the Landau-Khalatnikov two fluid model, with
the source term and C12 collisions essential in establishing a local equilibrium [164].
This approach was also used by Bijlsma, Zaremba and Stoof in addressing the problem
of condensate growth in trapped gases [165], within the ergodic approximation (see
1
This name is perhaps not universally considered as appropriate however [160].
23
Chapter 2.
Theoretical Background
Chapter 3). Subsequent work by Jackson and Zaremba implemented a Monte Carlo
scheme for the thermal cloud, which was treated as a collection of test particles, and
which lifted the requirement of ergodicity [166].
It seems then, that the Gross-Pitaevskii-Boltzmann scheme successfully describes a
large segment of finite temperature BEC behaviour. As a mean field method however,
it implicitly adopts a symmetry broken picture, in which the condensate is represented
by a classical field of definite phase. In this sense, the growth of a condensate cannot be
described within this theory, as once the condensate field is zero, it remains so. In addition critical fluctuations which become important near to the transition temperature
in three-dimensional systems, and over an even broader range within low dimensional
systems, are not fully accounted for within such an approach. Moreover, it is only
the average dynamics which are captured in this description, so it cannot account for
shot-to-shot variations as found in experimental realisations of finite temperature Bose
gases.
To model these aspects of a finite temperature Bose gas requires the addition of
stochastic effects into the system evolution. Hence, we now turn to a discussion of
methods including such effects, however initially, we look in a broader context at the
first model of this type, introduced to explain the curious nature of Brownian motion.
2.3
Stochastic effects in physical systems
The discovery due to a Scottish Botanist, Robert Brown, that small particles undergo an agitated, perpetual motion when suspended within a fluid [167], has had
profound consequences. This motion, termed Brownian motion by Einstein [168], is
now employed in mathematical models of many physical processes, ranging from financial markets [169] to molecular chemistry [170]. Langevin equations feature in many
diverse areas of physics, often arising when incorporating dissipation into quantum
mechanics, as for example in the approach of Caldeira and Leggett [171].
Before turning to a description of a Langevin model applicable to the description
of trapped Bose gases, we consider first the prototypical example describing Brownian
motion. The aim is that this should serve as an illustration of fluctuation-driven processes via a Langevin equation, and also the mapping to a Fokker-Planck equation. We
borrow particularly from the descriptions given by Risken [172] and Reif [173].
2.3.1
Langevin equation of Brownian motion
We may begin from Newton’s third law, which we write as
m
dv
= −αv
dt
24
(2.39)
Chapter 2.
Theoretical Background
and which represents the deterministic equation of motion for a particle of mass m
in a fluid, subject to a frictional force given by Stokes’ law: FStokes = −αv. The
straightforward solution to Eq. (2.39),
v(t) = v0 e−γt ,
(2.40)
tells us that the motion will dissipate with a characteristic time, τ = 1/γ = m/α. Thus
the deterministic motion damps out, due to collisions with the molecules of the fluid.
However, consider that the velocity of a particle within a liquid at some temperature
T is given, according to equipartition, by the relation
1
1
mhv 2 i = kB T
2
2
(2.41)
r
(2.42)
so the average thermal velocity is
vt =
kB T
.
m
Therefore, for m ≪ 1 or T ≫ 1 the effects of the thermal particles becomes nonnegligible, and Eq. (2.39) no longer fully describes the situation. If the mass of the
particle is still large, although small enough that the thermal motion cannot be neglected, then Eq. (2.39) can still be expected to capture most of the physics. To
improve this, we can consider additionally modifying the equation of motion to incorporate the random thermal background motion. This can be done with the addition of
a noise term:
dv(t)
= −γv(t) + Γ (t)
dt
(2.43)
where we have introduced the Langevin force [174], Γ (t), which is a fluctuating force
per unit mass.
As we want the new equation to represent the findings of the original on average,
then we require that the noise term satisfies the relations
hΓ (t)i = 0
hΓ (t)Γ (t′ )i = qδ(t − t′ ),
(2.44)
where the strength of the Langevin force is given by q. So, we regain the fact that
hv(t)i = 0 as before, and also that the correlation between two Langevin forces over
times longer than some characteristic collision time is zero; the delta function arises
in the limit that this time tends to zero. Physically, this form for the noisy term
is justifiable if we consider that the background thermal motion of the liquid is not
expected to lead to a net force in any particular direction, and therefore the effect of
many realisations of this should average to zero. Likewise, for this reason the subsequent
25
Chapter 2.
Theoretical Background
realisations of the noise should not be correlated in time either. If we calculate the
spectral density of this noise,
S(ω) = 2
Z
∞
dτ e−iωτ q δ(τ )
(2.45)
−∞
= 2q,
then we see why noise satisfying these requirements earns the label of white noise, as
it has a uniform power spectrum.
It is immediately possible to solve for the average change in the velocity by writing
the time derivative of Eq. (2.43) in discrete form, and taking the ensemble average
v(t) − v(0)
τ
= − hγv(t)i + hΓ (t)i ,
(2.46)
which trivially yields
hv(t) − v(0)i
= −γhv(t)i.
τ
(2.47)
This is the mean velocity, or first moment of the velocity distribution, which we will
use shortly.
We will also need the mean square velocity, which we solve for next, initially in order
to make contact with the equipartition result. Given the initial condition v(0) = v0 ,
the solution to Eq. (2.43) is,
v(t) = v0 e−γt +
Z
t
e−γt Γ (t′ )dt′ .
(2.48)
0
The two time velocity correlation is given by
hv(t1 )v(t2 )i = v02 e−γ(t1 +t2 ) +
= v02 e−γ(t1 +t2 ) +
Z
Z
t1
0
Z
t2
0
min{t1 ,t2 }
0
′
′
e−γ(t1 +t2 −t1 −t2 ) qδ(t′1 − t′2 )dt′1 dt′2 (2.49)
′
e−γ(t1 +t2 −2t1 ) qdt′1 .
(2.50)
which results in,
hv(t1 )v(t2 )i = v02 e−γ(t1 +t2 ) +
i
q h −γ|t1 −t2 |
e
− e−γ|t1 +t2 | ,
2γ
(2.51)
for the velocity correlation between two times. If we now consider the long time limit,
defined such that t1 , t2 ≫ γ −1 , then
hv(t1 )v(t2 )i →
q h −γ|t1 −t2 | i
.
e
2γ
26
(2.52)
Chapter 2.
Theoretical Background
Making use of this now in the equipartition relation, Eq. (2.41)
1
1
mh[v(t)]2 i = kB T
2
2
(2.53)
we now find for the strength of the noise,
2γkB T
.
m
q=
(2.54)
This is the fluctuation-dissipation relation [173], which specifies the relation between
the strength the of noisy force and the magnitude of the damping. The effect of the
former is to provide a minimum level of agitation to the system, while the damping
dissipates such effects, hence we can imagine how an equilibrium can be reached upon
these two effects balancing one another. This is in fact a fundamentally important
relation, which must be satisfied if thermal equilibrium is to be reached at long times.
Before discussing the characteristic times for this system, we may now proceed to
solve straightforwardly for the mean square displacement h(x(t) − x0 )2 i at equilibrium.
This is given in terms of the velocity as
2
h(x(t) − x0 ) i =
t
Z
(2.55)
hv(t1 )v(t2 )i dt1 dt2
(2.56)
v(t1 )dt1
0
t
Z
v(t2 )dt2
0
which can be written
h(x(t) − x0 )2 i =
=
Z tZ
0
0
t
qt
,
γ2
(2.57)
where we make use of Eq. (2.43) and the fact that at equilibrium v̇ = 0. This defines
the diffusion constant, as famously found by Einstein [168], to be
D=
q
kB T
=
.
2γ 2
γm
(2.58)
Solving for this in a more detailed fashion, we can obtain the two modes of behaviour
in such systems, and their characteristic times. If we multiply Eq. (2.43) by x, we then
wish to solve
d(xẋ)
− ẋ2 = −γxẋ + xΓ (t),
dt
(2.59)
which upon averaging, we can write as
d (xẋ)
dt
=
kB T
− γhxẋi.
m
27
(2.60)
Chapter 2.
Theoretical Background
This can immediately be solved to give
hxẋi =
kB T
1 − e−γt ,
γm
(2.61)
and integrating once more yields the desired result
hx2 i =
2kB T
t − γ −1 (1 − e−γt ).
γm
(2.62)
There are two limits of particular interest now. Firstly, t ≪ γ −1 for which
hx2 i =
kB T 2
t .
m
(2.63)
This implies that the particle then behaves as a free particle, whereas for t ≫ γ −1 ,
hx2 i =
2kB T
t,
γm
(2.64)
the particle behaves like a particle undergoing a random walk, for which hx2 i ∝ t. This
also gives the diffusion coefficient defined via hx2 i = 2Dt, so we again have
D=
2.3.2
kB T
.
γm
(2.65)
Fokker-Planck equation of Brownian motion
We now consider the problem in a more probabilistic sense, with the aim to show
the mapping between the simple Langevin equation considered and a Fokker-Planck
equation, which describes the motion of a probability distribution for the problem.
Consider the probability P (v, t)dv that the velocity of a particle undergoing Brownian motion is between v and v + dv. If we make the Markov approximation, then we
assume the history of the particle is not important, and that its evolution depends only
upon the current value. We can construct the conditional probability:
P dv = P (vt|v0 t0 ) dv,
(2.66)
which is the probability that the velocity of a particle undergoing Brownian motion is
between v and v + dv at time t, if its velocity at t = t0 is v0 . If we let s = t − t0 , then
P (vt|v0 t0 ) dv = P (vs|v0 ) dv
28
(2.67)
Chapter 2.
Theoretical Background
mβ − β mv2
e 2
dv
2π
(2.68)
and for s → ∞, then we expect
P (vt|v0 t0 ) dv →
r
which physically speaking, means that we expect the system to take on the Boltzmann
distribution of velocities, on equilibrating at some temperature. Over any small time
increment, the probability can be expected to satisfy the following relation,
∂P
dv τ = −
∂s
Z
v1
P (v, s|v0 ) dv · P (v1 , τ |v) dv1 +
Z
v1
P (v1 , s|v0 ) dv1 · P (v, τ |v1 ) dv,
(2.69)
which is in fact equivalent to a general master equation. Note that the integrals are
over all possible velocities v1 , and that
Z
P (v1 , τ |v) dv1 = 1.
(2.70)
Eq. (2.69) also resembles a Boltzmann equation, however while atoms or molecules
in a gas can undergo large changes in their velocities due to collisions, over the time
increment we consider here, the velocity of the macroscopic particle we consider can
change by only a small amount. This implies that the probability distribution will be
highly peaked in a small velocity range |ξ| = |v − v1 |. Thus, rewriting Eq. (2.69),
∂P
τ = −P (v, s|v0 ) dv +
∂s
Z
∞
−∞
P (v − ξ, s|v0 ) P (v, τ |v − ξ) dξ,
(2.71)
it is justified then to Taylor expand P (v, s|v0 ) P (v + ξ, τ |v) in powers of the small
velocity range, ξ. The expansion then yields [173]
Z ∞
∞
X
∂P
(−1)n ∂ n
n
τ = −P (v, s|v0 ) +
dξ ξ P (v + ξ, τ |v) . (2.72)
P (v, τ |v − ξ)
∂s
n! ∂v n
−∞
n=0
This leads to the expression in terms of the moments of the velocity increment, over a
time τ , defined as
Mn ≡
=
and finally gives
Z
1 ∞
dξ ξ n P (v + ξ, τ |v)
τ −∞
h[v(τ ) − v(0)]n i
,
τ
(2.73)
(2.74)
∞
X (−1)n ∂ n
∂P
=
[Mn P (v, s|v0 )] .
∂s
n! ∂v n
n=1
29
(2.75)
Chapter 2.
Theoretical Background
However, it is important to note that the quantity h[v(τ ) − v(0)]n i → 0 more quickly
than τ , if n > 2. As the time increment is macroscopically small, however large
compared to the correlation times of the Langevin force, is is possible then to neglect
all but the two lowest order terms in Eq. (2.75), which leads to the Fokker-Planck
equation
∂P
∂
1 ∂2
= − [M1 P ] +
[M2 P ] .
∂s
∂v
2 ∂v 2
(2.76)
We can now make contact with the Langevin method for the same problem via the
first two moments, which we calculated also in that approach. These were given by
M1 =
h[v(τ ) − v(0)]n i
= −γv
τ
(2.77)
from Eq. (2.47), and from Eq. (2.54),
M2 =
2γ
h[v(τ ) − v(0)]n i2
=
.
τ
mβ
(2.78)
Therefore, for the example of Brownian motion, the Fokker-Planck equation is
∂P
∂
kB T ∂ 2
=γ
[vP ] + γ
[P ] .
∂s
∂v
m ∂v 2
(2.79)
Following this background discussion of the Langevin and Fokker-Planck equations,
we now turn to the application of such an approach to a unified description of ultracold
trapped, dilute Bose gases.
2.4
Stoof ’s approach to a stochastic generalisation of the
Gross-Pitaevskii equation
Methods based on the concept of a condensate mean field, assume the existence of a
well formed condensate, hence they are unable to describe fundamental non-equilibrium
aspects of the Bose gas in the regime of critical fluctuations. Fluctuations about the
mean field are important in describing for example condensate growth from a thermal
cloud, where the system must obviously spend time in the critical regime, and where
such fluctuations can be of the size of the condensate order parameter itself.
To treat the fluctuations inherent to non-equilibrium Bose-Einstein condensates,
and in particular the condensation process itself, a non-equilibrium theory is required.
At this point, we highlight that there are a number of approaches that might be taken
in addressing the problem of non-equilibrium Bose gases, many of which were reviewed
recently in [124, 125]. Here, we focus our attention to the approach of Stoof [107,
108, 175]. This approach is based upon a Fokker-Planck equation which describes in
30
Chapter 2.
Theoretical Background
a unified manner the probability distribution for the order parameter of a trapped
dilute Bose gas [128, 107, 175], derived within the Keldysh non-equilibrium formalism
[176, 177]. Applying a Hartree-Fock type ansatz, Stoof split this probability distribution
as P [Φ∗ , Φ; t] = P0 [ψ ∗ , ψ; t]P1 [φ′∗ , φ′ ; t], where this descriptions refers to the fact that
P0 and P1 are probability distributions for the fields ψ and φ′ , which represent the
condensate and noncondensate respectively. This leads to a distribution function for
the high-lying thermal modes, which by integrating out the condensate contributions,
yields a quantum Boltzmann equation of the form
∂f
+ (∇p ǫ) · (∇f ) − (∇ǫ) · (∇p f ) = C12 [f ] + C22 [f ].
∂t
(2.80)
where the collisional integrals have a similar, but distinct, form to those of the ZNG
theory, and are given by
4π 2
C12 [f ] =
g |ψ|2
~ 3d
Z
dp2
(2π~)3
Z
dp3
(2π~)3
Z
dp4
(2π~)3
× (2π~)3 δ(p2 − p3 − p4 ) × δ(ǫc + ǫ2 − ǫ3 − ǫ4 )
(2.81)
3
× (2π~) [δ(p − p2 ) − δ(p − p3 ) − δ(p − p4 )]
× [(f2 + 1)f3 f4 − f2 (f3 + 1)(f4 + 1)],
and
4π 2
C22 [f ] =
g
~ 3d
Z
dp2
(2π~)3
Z
dp3
(2π~)3
Z
dp4
(2π~)3
× (2π~)3 δ(p + p2 − p3 − p4 ) × δ(ǫ + ǫ2 − ǫ3 − ǫ4 )
(2.82)
× [(f + 1)(f2 + 1)f3 f4 − f f2 (f3 + 1)(f4 + 1)].
These integrals represent energy and momentum conserving collisions. Those processes
which scatter atoms between the low-lying system modes and the thermal cloud are
described by C12 , while scattering events which occur between two thermal atoms are
represented by C22 . A crucial difference in this and the ZNG formalism, is that the order
parameter ψ represents a number of low energy modes, as against just the condensate
mode. The dynamics of such modes is described in this theory by a Fokker-Planck
31
Chapter 2.
Theoretical Background
equation for an order parameter, given by [107, 175]
i~
∂
P [ψ ∗ , ψ; t] =
∂t
2 2
Z
~ ∇
δ
2
+ V (x) − iR(x, t) + g3d |ψ(x, t)| − µ(t) ψ(x, t)P [ψ ∗ , ψ; t]
−
− dx
δψ(x)
2m
2 2
Z
δ
~ ∇
2
+ dx
+ V (x) + iR(x, t) + g3d |ψ(x, t)| − µ(t) ψ ∗ (x, t)P [ψ ∗ , ψ; t]
−
δψ ∗ (x)
2m
Z
1
δ2
−
~Σ K (x; t)P [ψ ∗ , ψ; t].
dx
2
δψ(x)δψ ∗ (x)
(2.83)
where δ/δψ represents a functional derivative with respect to the complex field ψ and
R(x, t) is given by
R(x, t) =
2
2πg3d
Z
dp2
(2π~)3
Z
dp3
(2π~)3
Z
dp4
(2π~)3 δ(p2 − p3 − p4 )
(2π~)3
(2.84)
× δ(ǫc + ǫ2 − ǫ3 − ǫ4 )[f2 (f3 + 1)(f4 + 1) − (f2 + 1)f3 f4 ].
The distribution functions fi again represent Wigner functions for the thermal cloud,
however with energies defined now as
ǫi =
|pi |2
+ V (x) + 2g3d h|ψ(x, t)|2 i.
2m
(2.85)
The Keldysh self-energy is the quantity which determines the strength of fluctuations
within this approach. It is defined by a similar integral over momenta,
K
~Σ (x, t) =
2
−4πig3d
Z
dp2
(2π~)3
Z
dp3
(2π~)3
Z
dp4
(2π~)3 δ(p2 − p3 − p4 )
(2π~)3
(2.86)
× δ(ǫc + ǫ2 − ǫ3 − ǫ4 )[f2 (f3 + 1)(f4 + 1) + (f2 + 1)f3 f4 ].
Notice that at equilibrium, the combination of distribution functions in Eq. (2.84) can
cancel, whereas the combination of such functions in the self-energy integral are never
zero, for fi > 0. This illustrates the dynamical equilibrium which is reached within
this theory, when the scattering and fluctuating effects balance one another.
Eq. (2.83) equation can, in principle, be mapped to an equivalent representation
as a Langevin field equation. However, the energy ǫc which appears in the self-energy
expression, and the expression for the damping term R(x, t), is the energy associated
with removing an atom from the condensate, which in this theory is given by the
operator [107, 108]
ǫc = −
~2 ∇2
+ V (x) + g3d |ψ(x, t)|2 .
2m
32
(2.87)
Chapter 2.
Theoretical Background
As pointed out by Duine and Stoof [175], the fact that ǫc is an operator dependent upon
ψ leads to a complicated stochastic equation with multiplicative noise. Referring back
to the simpler Langevin equation of Brownian motion, multiplicative noise refers to a
case where the noise magnitude q of Eq. (2.43) is dependent upon v(t), when there arises
a distinction between interpretations based upon the Ito calculus and Stratonovich
calculus. For additive noise, or q independent of v(t), then these approaches however
coincide.
In order to move towards a numerically tractable equation, as considered previously
by Stoof and co-workers, we may assume that the thermal cloud is in equilibrium,
which implies that we can replace the Wigner functions, f (x, p, t), with Bose-Einstein
distributions, which we henceforth denote by N (x, p, t). This is equivalent to treating
the thermal cloud as a heat bath, as the Bose-Einstein distribution can be shown to
solve the equation C22 [N ] = 0. Hence, we assume collisions leading to thermalisation
of the higher modes happens on a much more rapid timescale than the dynamical
processes being modelled.
In doing so, it is then possible to rewrite the expression for the damping term in
terms of the Keldysh self-energy as [107, 175],
1
iR(x, t) = − ~Σ K (x, t) [1 + 2N (ǫc )]−1 .
2
(2.88)
This is the fluctuation-dissipation relation for the system. It describes the relationship between the magnitude of fluctuations, here parameterised by ~Σ K (x, t), and the
damping, due to the source term iR(x, t), necessary if the correct equilibrium solution is to be obtained. This relation depends upon the mode populations, given by
the Bose-Einstein distribution plus a contribution representing an extra half particle
per mode, on average. These represent stimulated and spontaneous contributions to
the scattering rate, respectively, which points to the inherently quantum origins of the
theory. The latter contribution stems from the symmetrisation associated to the underlying Wigner approach [178]. As the thermal cloud is assumed to be close to thermal
equilibrium, this form for the fluctuation-dissipation relation can be expected to be
valid in the regime of linear response, applicable to excitations which do not strongly
perturb the thermal cloud.
Treating the heat bath in this way, leads to the Langevin equation
i~
2 2
∂ψ(x, t)
~ ∇
= −
+ V (x) − iR(x, t) + g3d |ψ(x, t)|2 − µ ψ(x, t) + η(x, t) (2.89)
∂t
2m
which can be identified as a stochastic generalisation to the usual T = 0 GrossPitaevskii equation. This is similar in form to the modified Gross-Pitaevskii equation
of ZNG, Eq. (2.37), though note that in contrast to the ZNG picture, there is only
33
Chapter 2.
Theoretical Background
the order parameter within the nonlinear term. The order parameter represents the
lowest system modes, as is consistent with the Hartree-Fock ansatz used in splitting
the system distribution function, prior to integrating out the noncondensate degrees
of freedom. This is in contrast to the separate condensate and thermal cloud densities within the nonlinearity of the ZNG modified Gross-Pitaevskii equation, where the
wavefunction represents the condensate alone. Here η represents the dynamical noise
term, which we shall discuss shortly.
For high temperatures, or close to equilibrium, β(ǫc − µ) is small and we are able
to Taylor expand the Bose-Einstein distribution of Eq.(2.88) in terms of this variable.
The result of this is to replace the fluctuation-dissipation relation with its classical
counterpart, based upon the Rayleigh-Jeans distribution, which yields
−iR(x, t) =
β
~Σ K (x, t) (ǫc − µ) .
4
(2.90)
The Fokker-Planck equation for the system then takes the form [175]
i~
∂
P [ψ ∗ , ψ; t] =
∂t
2 2
Z
β
δ
~ ∇
2
K
−
+ V (x) + g3d |ψ(x, t)| − µ(t) ψ(x, t)P [ψ ∗ .ψ; t]
dx ~Σ (x; t)
−
4
δψ(x)
2m
2 2
Z
δ
~ ∇
β
2
K
+ V (x) + g3d |ψ(x, t)| − µ(t) ψ ∗ (x, t)P [ψ ∗ , ψ; t]
dx ~Σ (x; t) ∗
−
−
4
δψ (x)
2m
Z
1
δ2
−
P [ψ ∗ , ψ; t].
dx ~Σ K (x; t)
2
δψ(x)δψ ∗ (x)
(2.91)
Making use of Eq. (2.87), the corresponding Langevin equation can be re-written as
the stochastic nonlinear Schrödinger equation [175]
∂ψ(x, t)
=
i~
∂t
2 2
β
~ ∇
K
2
1 + ~Σ (x, t) −
+ V (x) + g3d |ψ(x, t)| − µ ψ(x, t) + η(x, t),
4
2m
(2.92)
with the assumption that the thermal cloud and condensate share a common chemical
potential, again consistent with the Hartree-Fock type treatment of the mode energies
in Eq. (2.85). Thus we can consider a picture such that for ǫc < µ the gas is in the
condensed phase, whereas for ǫc > µ, the gas is in the normal phase. As the trapping
potential is non-homogeneous, and so ǫc varies with position, then we can expect the
coexistence of both phases within a harmonic trap at temperatures beneath the critical
temperature. In referring to Eq. (2.92), we will adopt the terminology Stochastic
Gross-Pitaevskii equation (SGPE) [109].
With the classical form for the fluctuation-dissipation relation, we can see analogies
34
Chapter 2.
Theoretical Background
between the final Langevin and Fokker-Planck equations and those found in the example
case of Brownian motion. In the latter case, the damping rate was given by γ, while
in the Fokker-Planck equation Eq. (2.91) we have a damping term given by
β
i ~Σ K (x, t)(ǫc − µ) ≡ γ(x, t)(ǫc − µ),
4
(2.93)
which similarly arises from the single derivative terms of the Fokker-Planck equation. In the Brownian motion Langevin equation, the noise correlations were given
by hΓ (t)Γ (t′ )i = (2γkB T /m)δ(t − t′ ), and here we have
hη ∗ (x, t)η(x′ , t′ )i = 2γ(x, t)kB T ~δ(x − x′ )δ(t − t′ ),
(2.94)
which, like in the former case, is associated with the second derivative term of Eq.
(2.91), the so-called diffusion term. It is this term which leads to the incorporation of
irreversibility in the model.
The first application of Eq. (2.92) was by Stoof and Bijlsma [108], in which it
was shown that Eq. (2.92) indeed leads to the correct classical equilibrium. They
also numerically modelled the reversible growth experiment of Stamper-Kurn et al.
[179], carried out at MIT. In this experiment, a condensate was reversibly grown and
evaporated away by adiabatically increasing and decreasing the phase space density.
This was achieved by the introduction of a tight harmonic trap, the minimum energy
of which could be tuned so as to be lower then the chemical potential of the thermal
gas, leading to condensate growth. In the paper of Bijlsma and Stoof, they found a
numerical solution of the one-dimensional SGPE to recover the experimentally observed
lagging behind of the three-dimensional condensate growth, relative to the adiabatic
switching of the system parameters [108].
Simulations solving Eq. (2.92) were undertaken in a paper addressing the problem
of mean-field theories in low dimensional Bose gases [180], where a comparison to the
modified Popov discussed therein showed good agreement. Numerical implementations
of this form of the SGPE have also been applied to the dynamics of a quasi-condensate
in an atom chip setting [181], in which the usually elongated systems require an approach incorporating the effects of phase fluctuations, and also the study of coherence
in one-dimensional Bose gases [182, 183].
This formalism is also amenable to variational calculations, as was applied to modelling low-lying collective modes at finite temperature [175], motivated by the JILA
experiment of Jin et al. [145], an application which has proved a test-bed for numerous
finite temperature theories2 . A stochastic description of the growth and collapse of an
attractive condensate was also obtained [175]. Additionally, a variational approach was
2
For a summary of many of these see Figure 14 of [124].
35
Chapter 2.
Theoretical Background
pursued in studying the motion of a vortex within a partially condensed Bose gas, and
was found to lead to a stochastic equation of motion for the vortex core [184].
In a general context, system-plus-bath models arise in descriptions of many areas
of physics. This is due in part to the difficulty in applying the canonical approach
to quantisation to dissipative systems, which can lead to Fokker-Planck and Langevin
descriptions [185, 171, 186]. Caldeira-Leggett models address this problem through the
coupling of the system to a bath of some form, which often takes the form of a bath
of harmonic oscillators. This allows an explicit mechanism for dissipation, through the
introduction of non-unitary evolution.
Relative to the widely used T = 0 GPE, studies of the SGPE are relatively fewer in
number. Recently, this technique was used to study the Kibble-Zurek mechanism in the
context of soliton generation at the phase transition in low dimensional Bose systems
[187, 188]. It has additionally been applied to the study of vortex formation during
the condensate phase transition [189], vortex lattice formation from a rotating thermal
cloud [190], and very recently to the decay of quantum vortices in finite temperature
BECs [191]. The latter set of investigations were undertaken using a (projected) SGPE
formalism derived from a quantum optics perspective. While differing in the mathematical origins, the equations which are solved take a very similar form, which we
discuss briefly now.
2.4.1
The stochastic (projected) Gross-Pitaevskii equation of C. W.
Gardiner and co-workers
The Langevin equation which we solve in this thesis is very similar to the stochastic
equation of C. W. Gardiner, and co-workers [109, 110, 192], which is more recently
termed the stochastic projected Gross-Pitaevskii equation (SPGPE) [125]. This is
derived from the perspective of quantum optics, using a master equation approach, as
against the non-equilibrium Keldysh formalism employed by Stoof. The formulation
of the S(P)GPE combined the kinetic theory of C. W. Gardiner and co-workers [193,
194, 195] with the finite temperature Gross-Pitaevskii equation, which was developed
by Davis and co-workers [196, 197, 198]. Like in the approach of Stoof, high-lying
modes are eliminated to give an effective interaction, and the remaining energy range
is split further into a condensate band and noncondensate band. In the Gardiner-Davis
approach, however, this elimination is carried out in two steps, initially eliminating the
highest modes in order to obtain the effective interaction, and then in splitting the
remaining system modes into thermal and condensate bands [110]. The process of
obtaining a master equation for the condensate band is in this way seperate from that
of obtaining the effective interaction.
Practically, the principal difference between the Gardiner and Davis method to the
36
Chapter 2.
Theoretical Background
approach we apply here, is the explicit use of a projector in the former. The role of
this is to restrict the evolution described by the order parameter equation of motion
strictly to modes beneath a certain cut-off energy. While a cutoff is implicitly applied
through the numerical solution of a discretised equation, for a harmonic trap, the
energy cutoff is spatially dependent, so a more natural basis may be provided by the
harmonic oscillator basis [199]. The splitting between low- and high-lying modes in the
Stoof theory occurs at the chemical potential, due to the Hartree-Fock nature of the
ansatz used in splitting the full probability distribution. The energy cut-off, Ecut , of the
Gardiner and Davis theory is determined by a requirement that the mode occupations
of low-lying modes be above some minimum level, typically around N (Ecut ) ∼ 3.
The full stochastic differential equation of Gardiner et al. [110, 192], which reduces
to the so-called ‘simple growth SGPE’, contains extra noise terms associated with
scattering processes. These are in addition to the terms discussed which lead to growth
of the condensate in Eq. (2.92). The approach we apply here was demonstrated to be
of the same form as the ‘simple growth SGPE’ [124], subject to the use of a projector
and exact nature of the growth rates. In practice however, these additional scattering
terms are not currently implemented, and do not affect the stationary solution [110].
The simple growth equation was first given in [109], and is the equation which has been
made use of to date in the studies mentioned.
2.5
Related finite temperature approaches
Various aspects of finite temperature Bose gases can be captured by approaches which
are closely related to the SGPE. We discuss two of these below.
2.5.1
Dissipative Gross-Pitaevskii equation as a limit of the SGPE
Neglecting the noise term of Eq. (2.92) leads to a dissipative Gross-Pitaevskii equation
(DGPE),
∂φ(x, t)
i~
=
∂t
2 2
~ ∇
2
1 − iγ(x, t) −
+ V (x) + g3d |φ(x, t)| − µ φ(x, t).
2m
(2.95)
This will be applied in this thesis as a description of the dynamics of the condensate
ground state in the presence of a static thermal cloud. As an initial condition, the
wavefunction will be given by φ(x, t = 0) = φ0 (x), where φ0 (x) is the Gross-Pitaevskii
ground state solution to Eq. (2.10).
The addition of a damping coefficient into the usual zero temperature GPE was
first proposed by Pitaevskii [111, 112] and implemented numerically by Choi et al.
[113]. In each case this was based on a phenomenological value for the damping. A
37
Chapter 2.
Theoretical Background
similar approach was adopted by Tsubota et al. [200], where damping was used in
order to induce relaxation and the formation of a vortex lattice in a rotating system.
It was discussed that such structures could only occur with the addition of dissipation
of some kind. This is to allow for the system to reach the new minimum in the free
energy created on imparting a rotation, for which the equilibrium configuration is the
Abrikosov lattice of vortices. An equation of this type was also applied to the dissipative
dynamics of solitons and vortices [201, 202, 203].
In contrast to phenomenological approaches, the damping within the DGPE which
is obtained from the SGPE formalism is derived from microscopic considerations, and
is ultimately based upon the the Keldysh self-energy. It is given by
γ(x, t) =
iβ
~Σ K (x, t),
4
(2.96)
where ~Σ K can be calculated from the integral Eq. (2.86). An approximate damping
based upon microscopic considerations was used to model vortex dynamics by Penckwitt et al. [204]. In terms of the dimensionless quantity γ of Eq. (2.95), the rate formula
used in [204] gives
γ ≈κ×
4m a 2
.
πβ ~
(2.97)
It was found that to match to most experimental growth rates κ ≈ 3; Eq. (2.97) is
the γ value which corresponds to the rate termed as the ‘bare’ rate in [125] (see also
Chapter 3).
While this ‘bare’ rate approximates the damping to be spatially independent, due
to the presence of a harmonic trapping potential and the mean field repulsion due to
the density of the trapped gas, at equilibrium most thermal atoms reside in the outer
trap regions. Hence, it might be expected that there should be a spatial variation in
the damping. This effect was first considered in the Thomas-Fermi limit by Duine and
Stoof [175], and has also been applied to the stochastic dynamics of dark solitons [121].
We discuss the calculation of γ(z), taking account of these effects, in Chapter 3. The
dark soliton results of [121] are described in Chapter 5.
A dissipative equation of motion of the form of Eq. (2.95), was also employed in the
paper of Stoof and Bijlsma [108], in an attempt to model the MIT reversible growth
experiment [179]. This approach was however unable to describe the experimental
findings, as during the condensate growth and evaporation cycle, the system was at
times in the critical region where fluctuations are known to be important. In this case,
the inclusion of these effects proved essential to modelling the experimental findings.
38
Chapter 2.
2.5.2
Theoretical Background
Thermal classical field simulations
The classical approximation to the SGPE, Eq. (2.92), may also be viewed as a means
to obtain an ensemble of initial conditions for use within a finite temperature classical
field method. We use this term to denote a description of the dynamics of highly
populated modes of a finite temperature Bose gas with the Gross-Pitaevskii equation
[118, 205, 206, 207, 208]. This approach is based on the idea that a description of
a finite temperature Bose-Einstein condensate can be made by the Gross-Pitaevskii
equation, with addition of suitable statistical initial conditions, as first suggested by
Svistunov [205, 209, 210]. It is also applied in other fields, where the infrared modes
give a dominant contribution to the physics, for example in modelling the electro-weak
phase transition in the early universe [211]. For BECs, in contrast to the T = 0
GPE, the initial state in this approach is set to some random initial conditions, with
a particle number and total energy corresponding to a partially condensed Bose gas
in thermal equilibrium. Allowing this state to evolve is indeed found to lead to the
correct, classical, equilibrium [197].
There are a number of implementations of this idea to modelling Bose gases [118,
125, 205, 206, 207, 208]. A notable difference is in the approach implemented by Davis,
Blakie and co-workers, who introduce a projector into the equation of motion [207],
following the observation that the classical field treatment is only appropriate to highly
occupied system modes. The role of the projector in this implementation, termed the
projected Gross-Pitaevskii equation (PGPE), is to formally restrict the equation of
motion to describing only these modes. Starting from some random initial state and
propagating to equilibrium with the GPE gives one realisation of the system. As the
GPE provides an ergodic description of the system, then it is possible to replace ensemble averaging with time averaging. Conversely, solution of the SGPE to equilibrium
provides a grand canonical ensemble of classical fields, which under time evolution via
Eq. (2.92) approach equilibrium on a time scale determined by the quantity |β~Σ K |.
2.6
Finite temperature schemes related to the SGPE applied in this thesis
To describe the dynamics of a finite temperature Bose system, beyond the point of
equilibrium, we implement a number of methods within this thesis. We summarise
these below, however for some applications of these methods and a discussion of their
relation to the SGPE, see also [212].
1. Stochastic simulations using a ‘self-consistent’ Keldysh self-energy:
In this approach, we retain the noise terms and damping terms in Eq. (2.92),
39
Chapter 2.
Theoretical Background
which we shall at points refer to as heat bath terms, during the dynamics. The
Keldysh self-energy is calculated based upon the equilibrium system density prior
to any perturbations. This gives the rate of scattering between the low and high
modes and strength of fluctuations at equilibrium, so providing the perturbation
does not affect the density too strongly, should remain a good approximation as
the new system equilibrium is approached. An additional feature here is that
particle number is not explicitly conserved, due to the static treatment of the
thermal modes, and perturbing the system strongly will lead to a growth or decrease in number for a fixed chemical potential. This is however dependent upon
the new equilibrium configuration: For example, opening up the harmonic trap
would lead to a new particle number, with all other parameters fixed, due to a
lowering in the trap energies, relative to the chemical potential. Note also that
this in not a fully self-consistent treatment, as the self-energy should generally
be time dependent, whereas we calculate this only once, at equilibrium.
2. SGPE equilibrium as initial states for classical field simulations:
Alternatively, taking the T → 0 limit, reduces Eq. (2.92) to an equation of the
form of the GPE, Eq. (2.9). For the SGPE, this was first implemented numerically
by Proukakis et al. [181].
We have φ → {ψ}, where {ψ} is a set of noisy, multi-mode classical fields. In this
way, conventional ensemble averages are possible over {ψ}, without recourse to
time averaging to obtain the correlation functions of interest. For example, for a
set of M classical fields, the average density is given by
M
1 X ∗
n(x, t) = hψ (x, t)ψ(x, t)i ≡
ψi (x, t)ψi (x, t).
M
∗
(2.98)
i=1
This is also the case for method 1.
3. Dissipative simulations using a ‘self-consistent’ damping:
Neglecting noise effects both during the dynamical evolution and in the initial
state, we can consider instead the purely dissipative evolution described by Eq.
(2.95). The damping is calculated ‘self-consistently’ using again the Hartree-Fock
contribution due to the equilibrium solution to the SGPE, as in method 1.
A description of the system dynamics using method 2 listed above, will be termed
SGPEeq , which denotes the fact that the SGPE is used only to obtain an ensemble of
equilibrium initial states, the dynamics of which, where probed, will be described by
the GPE.
40
Chapter 2.
2.7
Theoretical Background
Interpretation of stochastic simulations: single run
vs. averaged results
We conclude this Chapter with an illustration of the features captured within single runs
and the averaged results of the SGPE. An appealing feature of the SGPE description
of Bose gases is the close analogy which can be drawn to the experimental procedure
of averaging over successive runs. This is fundamentally pleasing, as this procedure
lies at the heart of almost all experimentally measured phenomena that we might
aim to model or predict. Fluctuations due to independent noise realisations lead to
shot-to-shot differences between numerical experiments, in a similar way to physical
experiments carried out within the laboratory.
An example of a spontaneous process captured through solution of Eq. (2.92) is
illustrated in Figure 2.1. This shows the creation of vortices as the BEC phase transition
is crossed in a two-dimensional Bose gas, taken from Cockburn and Proukakis [212].
In Eq. (2.92), this corresponds to an abrupt change in the chemical potential, which
controls the crossover from the normal phase to condensed phase. The top row of Figure
2.1 shows density snapshots from an individual realisation of the noise. Time increases
to the right in these images, and we see a number of holes in the density, which we
associate with vortices. In support of this interpretation, the middle row shows images
of the phase of the stochastic order parameter, which shows discontinuities at points
coinciding with the density holes. In moving left to right in the top two rows, we
see two vortices initially close near the centre of the image (leftmost images), which
come together (central images), and then finally annihilate (rightmost images). This
suggests that individual runs capture not only the effects of fluctuations, but also
coherent dynamics as well.
The bottom row of Figure 2.1 shows the averaged density at the same times as the
snapshots of the first two rows were taken. The dynamics of the vortex-antivortex
annihilation, or similar processes which occur in other realisations contributing to this
average, are however washed out. This leads to the question as to which data best
represents reality. Formally, correlations of the noisy field should be obtained by ensemble averaging, like in the example given above for the density, Eq. (2.98). In the
usual classical field approach, this is often replaced by time averaging over a single
randomly seeded field realisation, however in the SGPEeq approach, an ensemble of
states is retained.
In a 3d study of spontaneous vortex formation, using the SPGPE [189], numerical
single run results were found to produce findings very similar to those found experimentally, suggesting that single realisations capture a lot of the physics relevant to
41
Chapter 2.
Theoretical Background
Figure 2.1: Spontaneous vortex production during the growth of a two-dimensional BoseEinstein condensate. Shown is the single run density (top row) and phase middle row and
the averaged density (bottom row). Time increases in these images from left to right; note that
the colours represent different values in different rows. The parameters are ωx,y = 2π × 200 Hz
and ωz = 2π × 4 kHz; N ≈ 6 × 105 23 Na atoms. This data was presented originally in [212];
note the colour scale differs between the density profiles of each column.
42
Chapter 2.
Theoretical Background
experiments. There is also the fact that the trajectories through phase space which
are sampled in this way, are weighted by a probability measure. This means that it is
most likely to see an individual run which is close to the average trajectory, and hence
something close to the likely outcome of an experiment. This is why the GPE, where
fluctuations about the classical trajectory are neglected entirely, describes so well many
experimental features [125]. It is clear from Figure 2.1 also that some details must be
extracted from single runs, prior to averaging. This is the case if, for example, the
number of spontaneous vortices present in each sample was the observable of interest.
The ZNG approach captures only the average condensate dynamics, albeit in strongly
non-equilibrium states, so cannot describe such features either. In addition, while the
split between condensate and thermal cloud makes it easy to identity each component in ZNG calculations, this is further from experimental reality than the unified
representation of low energy modes within the SGPE order parameter.
2.8
Chapter summary
We have discussed in this chapter theoretical approaches amenable to modelling BoseEinstein condensates, beginning with a review of the well-known Gross-Pitaevskii equation and Bogoliubov excitations at T = 0. We then gave a discussion of the currently
most advanced method for finite temperature Bose gases, based on the concept of a
condensate mean field method, though including additionally scattering between condensate atoms and the thermal cloud atoms, as derived by Zaremba, Nikuni and Griffin
[154]. Following this, we motivated the need to incorporate fluctuations into modelling
such a system, which naturally leads to a stochastic description. In this thesis, this
point is pursued using the formalism of Stoof, which we summarised by giving a review
of the theory described within several key papers [107, 108, 175]. We also drew upon
the simpler system of Brownian motion of a massive particle, in order to illustrate the
mapping from a Fokker-Planck to a Langevin equation.
Having discussed the origins of the stochastic Gross-Pitaevskii equation, and its
context within the wider scheme of dynamical models for finite temperature Bose gases,
we turn in the remaining chapters to an application of this formalism to equilibrium
and non-equilibrium Bose gases.
43
Chapter 3
Numerical analysis of the
stochastic Gross-Pitaevskii
equation in one-dimension
The Fokker-Planck equation derived by Stoof, Eq. (2.83), describes in principle the
entire system dynamics for a degenerate trapped Bose gas; upon splitting the system
into high and low energy modes, the probability distribution function for the former
is treated by a quantum Boltzmann equation, whereas the low-lying modes may be
described by means of a stochastic generalisation to the GPE, Eq. (2.89).
To obtain a numerically tractable solution of the SGPE we follow the approach of
Stoof and Bijlsma [108], and consider the SGPE within the so-called ‘classical’ approximation, as described in Chapter 2. In this Chapter we illustrate the consequences of
this approximation in the context of a one-dimensional trapped Bose gas, paying particular attention to the dependence of various system properties upon the numerical
grid spacing used in discretising the equation of motion. Initially, we introduce various
equilibrium system properties and examine their sensitivity to grid changes, and in
some pertinent cases also to the magnitude of the self-energy. We then discuss two
approaches to calculating the scattering rate between the low-lying atoms and thermal
cloud.
3.1
Growth to equilibrium
As the work in this thesis is concerned with situations close to equilibrium, we shall
first demonstrate how this equilibrium is reached, and what it depends upon. The
equilibrium in the SGPE, unlike in ZNG, is obtained by a full dynamical evolution
to equilibrium (numerical details are provided in Appendix A). The noise term of
44
Chapter 3. The stochastic Gross-Pitaevskii equation in one-dimension
the SGPE is essential in seeding the growth of atoms into the low-lying modes of the
system [108]. Before characterising the equilibrium state, we first illustrate the manner
in which this growth to equilibrium occurs.
The growth process is illustrated in Figure 3.1, where we plot snapshots of the
ensemble averaged density and particle number, taken at various times during the
growth to equilibrium (see also Cockburn and Proukakis [212]).
1
1
gn(0)/µ
(a)
0.8
(b)
0.8
0.6
0.4
0.6
0
0
0.1
0.2
t
0.4
0.2
0.3
0.4
0.3
0.4
-1
[ωz ]
(c)
1
N(t)/Neq
gn(z)/µ
0.2
0.8
0.6
0.4
0.2
0
-20
-10
0
20
10
z [lz]
0
0
0.1
0.2
-1
t [ωz ]
Figure 3.1: Growth to equilibrium shown by (a) density snapshots of the low-lying modes
at various times during growth; (b) the central density against time with those points corresponding to density snapshots of (a) marked by a square of corresponding colour; (c) The
total particle number in the simulation versus time. The parameters here are ωz = 2π × 9Hz,
ω⊥ = 2π × 32Hz, µ = 22.1~ωz and the temperature is T = 370nK.
The initial state here is ψ(z, t = 0) = 0, and it is the action of the noise term which
seeds the growth in density. After some time, the total density profiles show the gradual
formation of an inverted parabola shaped density around the trap centre in Figure
3.1(a), associated with the condensed fraction of particles. These reside in the central
trap region at equilibrium and form a much narrower density profile compared to the
broad thermal wings of the density distribution. Also shown in Figure 3.1(b) is the
evolution of the peak density during growth. The central density points corresponding
to times at which the density snapshots of Figure 3.1(a) were taken, are highlighted by
squares of the corresponding colour. The growth in particle number with time is also
shown in 3.1(c). The levelling out in these latter quantities indicates that growth has
stopped and that the system has reached equilibrium, however due to the continued
45
Chapter 3. The stochastic Gross-Pitaevskii equation in one-dimension
action of the noise and dissipation terms, they in fact fluctuate about some equilibrium
value, as is seen most clearly in the central density of Figure 3.1(b).
We next illustrate the role of γ(z) in the rate at which this growth occurs.
20000
N
15000
10000
5000
0
0
100
50
Time
150
200
-1
[ωz ]
Figure 3.2: Variation in the particle number growth with time, for a harmonically trapped
system simulated using the SGPE with γ = 0.05 (blue diamonds), γ = 0.005 (red squares), and
γ = 0.0005 (black circles). The brown dashed lines indicates the equilibrium particle number,
N ≈ 20000 for the chosen parameters. The temperature of the equilibrium state for these
simulations is T = 20nK.
The growth rate is given by the net scattering rate between the condensate and higher
modes, given in the classical approximation by
−iR(z, t) = −iγ(z, t) [ǫc − µ] .
(3.1)
The variation in the growth is illustrated in Figure 3.2 which shows the growth in the
total particle number as a function of time. For simplicity, in this initial discussion,
we consider various spatially constant values of γ, as this does not noticeably affect
the equilibrium result. The temperature is fixed by the heat bath temperature and γ
parameterises the coupling between the system and this bath. We will subsequently
introduce a self-consistent spatial dependence into this quantity, when this becomes
important, as is the case in modelling non-equilibrium dynamics [121].
Figure 3.2 shows that the bosonic enhancement of growth takes over after a period of
relatively much slower growth, the length of which increases as γ is reduced. Stimulated
growth is dependent upon the occupation of the lowest modes, through ǫc in Eq. (3.1),
which is the operator for the energy associated with the gain or removal of an atom from
the condensate. As these modes are populated only very sparsely, with spontaneous
growth initiated by the noise, significant growth does not proceed until these modes
become appreciably populated. The rate of scattering into these modes is directly
proportional to γ, as is the noise, hence the lowest γ result exhibits extremely slow
46
Chapter 3. The stochastic Gross-Pitaevskii equation in one-dimension
growth (the black curve of Figure 3.2), with little enhancement evident on the time
scale shown.
The variation in the growth becomes more apparent when plotted on a log-scale, as
in Figure 3.3(a). Here, the time axis is plotted on a logarithmic scale for two reasons:
the difference in growth times is very different between the γ values simulated, so there
is a large range of times to incorporate into the figure, but more importantly, because
it illustrates that the growth curves are very similar when plotted in this way, bar an
additive shift in time. As an additive shift in log space corresponds to a multiplicative
change in the time, we find this can be scaled out by scaling the time in each simulation
by the γ chosen. This is shown in Figures 3.3(b)-(c), and shows the variation in the
growth curves for the system particle number can be largely accounted for in this way.
20000
(a)
N
15000
10000
5000
0
1
10
100
1000
-1
Time [ωz ]
20000
20000
(b)
15000
N
N
15000
10000
10000
5000
5000
0
(c)
0
0.1 0.2
0.3 0.4
0
0.5
0
0.1 0.2 0.3 0.4 0.5
-1
-1
Time [γωz ]
Time [γωz ]
Figure 3.3: (a) Same data as Figure 3.2 shown on a log scale. The growth curves plotted as a
function of scaled time, γt, showing a comparison between γ = 0.0005 (black) and (b) γ = 0.005
(red) or (c) γ = 0.05 (blue).
Moreover, carrying out this scaling makes it clear that the equilibrium particle numbers
are consistent across the range of γ values considered.
3.2
Influence of the classical approximation
The classical nature of the Langevin equation which we consider here is reflected, in
particular, in a rewriting of the fluctuation-dissipation relation Eq. (2.88), instead as
47
Chapter 3. The stochastic Gross-Pitaevskii equation in one-dimension
Eq. (2.90), which entails making the approximation
1
N (E) +
2
=
≈
1
1
+
−1 2
e(E−µ)β
1
+ O ((E − µ)β) .
β (E − µ)
(3.2)
(3.3)
In doing so, the thermal atoms are then treated as being distributed according to
the Rayleigh-Jeans distribution, rather than the Bose-Einstein distribution. This approximation is valid so long as the exponent in the denominator of the Bose-Einstein
distribution is small, which is most readily fulfilled for low-lying modes close to equilibrium or at high temperatures. Consistent with the classical nature of this limit,
each of these situations typically leads to occupation numbers N (E) ≫ 1. In this case,
quantum fluctuations, represented by the extra half on the left hand side of Eq. (3.3)
in addition to the Bose distribution, can be considered as small by comparison and
justifiably neglected.
The Rayleigh-Jeans distribution was originally derived from classical arguments
to describe radiation emitted from a black body. However, because of the prediction
that the total system energy should be partitioned with equal probability amongst the
available system modes, was found to lead to divergences at high energies. This flaw,
labelled as the ‘ultraviolet catastrophe’ at the time, was remedied by a distribution
later derived empirically by Planck. A key point is that that the Rayleigh-Jeans law
can be obtained as the low energy limit of Planck’s improved distribution. Analogously,
the classical approximation employed here is the low energy limit of the Bose-Einstein
distribution, which, unlike the former, properly accounts for the exponential drop off
in occupation probability at high energies, as in the Planck distribution.
In making approximation (3.3), the aim is at finding an equation to describe dynamically only the lowest modes of the system. In the quantum degenerate regime, the
classical distribution is well applied to a dynamical treatment of these modes, as the
behaviour for such highly occupied states is largely classical in nature. Nonetheless, the
assumption of equipartition of energy within the system introduces an additional issue
as to where the most appropriate point for the separation between high and low modes
lies. A reasonable assumption is that the optimum placement for this is at an energy
such that the Rayleigh-Jeans and Bose-Einstein distributions given similar predictions
for mode occupations.
Figure 3.4 illustrates the approximation (3.3) between the classical and quantum distribution functions, as a function of energy. We can see from this that Eq. (3.3) appears
to be a good approximation for energies up to at least E ≈ kB T . It is interesting
48
Chapter 3. The stochastic Gross-Pitaevskii equation in one-dimension
log(N(E,β))
10
N(E,β)
8
6
100
10
0.02
4
0.04
0.06
βE
0.08
0.1
2
0
0.2
0.4
βE
0.6
0.8
1
Figure 3.4: Plot illustrating the validity of the approximation (3.3) as a function of scaled
energy, βE. Shown are the Rayleigh-Jeans (green) and Bose-Einstein (black, dot-dashed)
distributions, together with the Bose-Einstein distribution function plus the extra one half
contribution (red, dashed).
to note also that it is only for lower energies, E < 0.1kB T , that the Bose-Einstein
occupation numbers without the half particle contribution match the Rayleigh-Jeans
result to a similar level of agreement. The extra half stems from the fact that the full
quantum theory of Stoof is formulated in terms of Wigner functions[107]. Averaging
over complex fields in this representation returns a result equivalent to an average over
symmetrised products of operators, so there is a correction which must be made in finding the desired normally ordered average. This is discussed in more detail in Chapter
6, however see also the discussions in [178, 213, 125]. The half is usually interpreted as
the vacuum quantum noise contribution, though it arises also as the next term beyond
Rayleigh-Jeans in the Taylor series of the Bose-Einstein distribution.
Due to ultraviolet divergences, we have seen that the classical approximation may
become a problem if we wish to solve the SGPE in the continuum limit. However, in
practical terms, the numerical solution employed throughout this work relies upon a
discretised form of the SGPE, and as such a cutoff is implicit in all the calculations
undertaken. It is then important to assess the effect of introducing a high momentum
cutoff to the results produced by the SGPE. We consider this in the remainder of
this Chapter, concentrating on the equilibrium properties of both an ideal and then a
weakly interacting bosonic system.
3.3
Equilibrium properties: an ideal thermal Bose gas
Embodied in the fluctuation-dissipation relation is the assurance that solving the SGPE
leads to the correct classical equilibrium state for a weakly interacting Bose gas at a
49
Chapter 3. The stochastic Gross-Pitaevskii equation in one-dimension
given temperature [175]. As a classical solution, this necessitates a consideration of
the high energy behaviour, so we now assess the cutoff dependence for both ideal and
interacting Bose gas.
As a first measure we consider the ideal Bose gas to show the cutoff dependence of
the equilibrium density profiles obtained via the SGPE. The non-interacting case is also
useful as a check on our numerical solution, as it is possible to calculate the expected
result for the density analytically, as considered in [108], and which we reproduce for
illustrative purposes here.
For a three-dimensional system, the thermal particle density is given by the semiclassical expression [1]
nt (x) =
1
(2π~)3
Z
dp N (x, p)
(3.4)
where N (x, p) is the probability distribution function. Within the classical approximation, the density in the one-dimensional case is given by
1
n(z) =
2π~
Z
∞
0
2
−1
p
.
2dp β
+ V (z) − µ
2m
(3.5)
Introducing the variable x = βp2 /2m, this can be rewritten as
1
n(z) =
π~
r
m
2β
Z
∞
dx
0
x−1/2
.
x + β (V (z) − µ)
(3.6)
It is now instructive to introduce a cutoff in x, so we have
1
n(z) =
π~
r
m
lim
2β Λ→∞
Z
Λ
dx
0
x−1/2
,
x + β (V (z) − µ)
(3.7)
which evaluates to
1
nΛ (z) =
π~
r
s
#
"
1
2m
Λ
p
lim arctan
β
β (V (z) − µ)
β (V (z) − µ) Λ→∞
(3.8)
where Λ = βEmax , and Emax = p2max /2m. In practice this expression is useful as it
corresponds to the thermal density for a particular cutoff, Λ, and as we shall see provides
a means by which to determine the maximum energy for a particular numerical grid.
Taking the limit Λ → ∞ gives
1
nicl (z) =
~
r
1
m
p
,
2β β (V (z) − µ)
(3.9)
which is the result for the density of an ideal Bose gas in the continuum limit.
While it is clear that the high energy limit gives a convergent result in the one50
Chapter 3. The stochastic Gross-Pitaevskii equation in one-dimension
dimensional case considered here, it is worth noting that for dimensions greater than
one this is no longer the case, due to the difference in the form of the density of states.
A comparison of the analytic expression for the classical density to the density for
an ideal gas obeying Bose-Einstein statistics is shown in Figure 3.5. It is clear from
Figure 3.5 that Eq. (3.9) agrees well with the equivalent result obtained using the
Bose-Einstein distribution for Emax ≪ kB T . As expected, the agreement is better in
the region of highest density at the trap centre. Also apparent is that the classical
expression over-estimates the density in the wings, relative to the exponential decay
of the Bose-Einstein result as evidenced by the more linear slope of the latter on the
logarithmic scale of Figure 3.5.
Emax=0.1kBT
100
100
10
10
0
100
10
20
0
100
Emax=kBT
10
20
Emax=2.0kBT
-1
n(z) [lz ]
Emax=0.5kBT
10
10
0
10
20
0
10
20
z [lz]
Figure 3.5: Ideal gas densities which result for Bose-Einstein (black, dot-dashed) and RayleighJeans distributions (red, solid). In each sub-plot, the distributions were integrated up to the
same cutoff in energy, Emax , as denoted in the sub-plots.
Having discussed the expected behaviour for the ideal case, we now move to a comparison between the densities predicted by the SGPE for various grid spacings, ∆z, and
the classical ideal gas in the continuum limit.
The variation with grid spacing of the densities obtained by numerical solution of
the SGPE, again for an ideal gas, is shown in Figure 3.6. Also plotted is the analytic
result for the ideal gas with an explicit energy cutoff, Eq. (3.8), where now Λ = βEgrid
with Egrid the energy cutoff due to the numerical grid spacing, ∆z. Here, Egrid is
extracted for each numerical grid by fitting the density to Eq. (3.8) which yields a
value for Λ, and so Egrid . Also shown is the limit as Egrid → ∞, Eq. (3.9). It is clear
51
Chapter 3. The stochastic Gross-Pitaevskii equation in one-dimension
that as we move towards increasingly finer numerical grids, from (a) to (d) in Figure
3.6, the SGPE result approaches the analytic value in the Λ → ∞ limit (the SGPE
result obscures the ideal gas result in (d)). Recovering this behaviour provides a good
first check on the numerical routine employed for the solution of the SGPE (for details
see Appendix A). It should be noted however, that it is a linear stochastic equation
which is solved in the ideal gas case, and as such similar tests for the nonlinear Langevin
equation, representing the interacting case, are discussed in the following Section.
400
400
(a)
300
300
200
200
100
100
0
-20
400
-10
0
20
10
0
-20
400
(b)
-10
0
-1
n(z) [lz ]
(c)
(d)
300
300
200
200
100
100
0
-20
-10
0
10
20
10
20
0
-20
-10
0
10
20
z [lz]
Figure 3.6: Ideal gas density profiles for T=500nK, µ = −5~ωz , and N∼ 6000 particles (as
Λ → ∞) for which T1d = 276nK. The plots are for grid spacings of (a) ∆z = 0.512 (Λ = 0.0127),
(b) ∆z = 0.256 (Λ = 0.021), (c) ∆z = 0.128 (Λ = 0.331) and (d) ∆z = 0.016 (Λ = 21.2). Shown
in each graph are the ideal continuum limit result (brown, dash-dotted), the SGPE result for
each grid spacing (black, solid) and the ideal continuum result, with an explicit energy cutoff
(red, dashed).
3.4
Equilibrium properties: an interacting, partially condensed Bose gas
Despite the weakly interacting nature of the atomic species often considered in ultracold
Bose gas experiments, the addition of interactions changes significantly the behaviour
of the many-body system. It is therefore important to also consider the behaviour of
the SGPE with grid spacing (or energy cutoff) for the physically relevant case of the
interacting Bose gas.
52
Chapter 3. The stochastic Gross-Pitaevskii equation in one-dimension
3.4.1
Density profiles
An experimental system usually involves confinement within a non-homogeneous trapping potential, often well approximated as being harmonic. Therefore, different terms
in the Hamiltonian of Eq. (2.92) dominate in different spatial regions. Near the trap
centre, where for a harmonic trap the density is largest, the interaction and potential
terms are most influential. Conversely, far from the trap centre, the kinetic energy term
in the Hamiltonian becomes relatively more important, as only higher energy thermal
particles tend to reside in the wings.
For the partially condensed gas density, the central region will typically take on an
inverted parabola-like profile, as in the Thomas-Fermi solution to the T = 0 GrossPitaevskii equation, found by neglecting the kinetic energy term entirely in the GPE.
For T 6= 0, the Thomas-Fermi radius can be considered as a temperature dependent
quantity [214], as the condensed fraction is a function of temperature, which approaches
zero for high temperatures. In the wings of the distribution, the higher trapping potential ensures the density is much lower, and so interactions play a smaller role here.
To a good approximation then, the gas is effectively non-interacting in this region, a
feature also made clear in the Bogoliubov dispersion relation Eq. (2.19).
With this two component picture in mind, we consider in Figure 3.7 a density
for a partially condensed Bose gas, obtained from solving the SGPE to equilibrium.
Shown also in the Figure are two mean field results, valid in different spatial regions
of the trap: in the central region, the Thomas-Fermi solution to the GPE is shown
for the same value of the chemical potential as used in the SGPE simulations, which
corresponds to the T = 0 result. In the wings, the ideal gas result Eq. (3.8) is plotted.
The dotted vertical lines show the boundary at which V (z) = µ, corresponding to the
T = 0 Thomas-Fermi radius, R. We see that for µ > 0, the ideal gas result within a
harmonic trap diverges as |z| → R from above, so cannot describe the system at the
trap centre. In contrast, the Thomas-Fermi result is valid only for |z| < R. The SGPE
density however shows a smooth crossover between these two regions, describing in
a unified manner the central trap region, where the condensate resides, as well as the
thermal wings of the distribution. Consistent with the idea of the highest energy atoms
being effectively non-interacting, the variation of the density in the wings displays the
same spatial dependence as in the ideal gas. The finite temperature SGPE density is
well matched to the ideal gas density expression Eq. (3.8) with a cutoff Λ dependent
upon the numerical grid used.
We now consider in a more quantitative manner, the dependence of the equilibrium
density as the grid spacing is varied, for both the central (z < R) and outer (z > R)
trap regions.
53
Chapter 3. The stochastic Gross-Pitaevskii equation in one-dimension
1
gn(z) / µ
0.8
0.6
0.4
0.2
0
-20
-15
-10
0
-5
5
10
15
20
z [lz]
Figure 3.7: Density profile for a partially condensed Bose gas of about 2000087Rb atoms at
T=370nK (black, solid). Also shown are mean field results for the condensate and thermal
densities at the same chemical potential, given by the Thomas-Fermi (blue, dashed) and classical
ideal gas (red, dashed) results, respectively. The T = 0 Thomas-Fermi radius is shown by
the vertical dotted lines, and the parameters used are µ = 22.4~ωz , ωz = 2π × 9Hz and
ω⊥ = 2π × 32Hz.
Central trap region
The behaviour of the density in the central region is characterised in Figure 3.8, where
we plot the change in the central density, n(0), scaled to the interaction strength g,
relative to that expected in the Thomas-Fermi limit at zero-temperature, which is just
the chemical potential, µ. We consider this as a function of the dimensionless quantity
βEgrid .
It is apparent from Figure 3.8, that the central density converges to a constant value
as the energy cutoff is increased. For a high energy cutoff, so small grid, the SGPE
result is noticeably lower than the Thomas-Fermi result, due to Landau and Beliaev
effects which are implicitly included [108]. As the energy cutoff is reduced, the central
density of the SGPE becomes much closer to that of the T = 0 Thomas-Fermi result.
Reducing the energy cutoff has the effect of lowering the number of modes represented in the simulation. The low cutoff behaviour is consistent with the picture that
as the number of modes in the SGPE, within the classical approximation, is reduced,
the equilibrium density tends towards the interaction dominated single mode result,
given by the Thomas-Fermi solution to the GPE.
The high energy cutoff trend of Figure 3.8 is suggestive of the fact that there is a
54
Chapter 3. The stochastic Gross-Pitaevskii equation in one-dimension
g<n(0)>/µ
1.00
(a)
0.99
0.98
0.97
0.96
0
1
2
3
βEgrid
4
5
6
g<n(0)>/µ
1.00
(b)
0.99
0.98
0.97
0.95
0.94
0
0.5
1
1.5
βEgrid
2
2.5
3
Figure 3.8: Variation in the central density of the SGPE equilibrium density profiles with grid
energy. Data is plotted for (a) T = 185nK and (b) T = 370nK; in each case the dotted vertical
line indicates the chemical potential on this scale, βµ, and the red dashed line indicates the
line of best fit through the data as a guide to the eye.
minimum energy that must be retained in simulations, if the correct equilibrium is to be
reached. That is, there is a limit of how little of the total system should be described by
the order parameter if we are to describe correctly the equilibrium thermal state. From
this analysis, a reasonable minimum cutoff can be taken as Λ & 0.5. These findings are
also in agreement with those of [125]: in making the classical approximation, the SGPE
is able to describe fluctuations which are classical in nature. Critical fluctuations occur
in the infrared modes, and these must be included in the order parameter (termed
the c-region in [125]) if we are to accurately describe the bosonic system we wish
to simulate. This implies a minimum cutoff on the order of the interaction energy
E ∼ gn(z) ≈ gn(0) ≈ µ. This energy is indicated for each temperature in Figure 3.8
by the vertical dot-dashed lines. We see that the transition to a consistent density
profile occurs only a little above this energy for our system at the two temperatures
considered.
Outer region
The inclusion of interaction effects, means that g 6= 0 and so (2.92) becomes a nonlinear
Langevin equation. It is important then to examine the dependence upon the grid
spacing now for the interacting case, as shown in Figure 3.9. Here, the analytic solution
for the ideal gas density is valid only valid in the wings, so in a way analogous to the
55
Chapter 3. The stochastic Gross-Pitaevskii equation in one-dimension
analysis performed in experiments to determine the temperature of atomic samples, we
therefore fit the wings of the density profiles for different grid spacings to Eq. (3.8).
From this we can extract the dependence of Λ on ∆z.
3000
10
2500
βEgrid
Egrid [hωz/2π]
1
2000
1500
0.1
0.01
1000
0.01
0.1
2πβ / (∆z)
500
1
10
2
0
0
0.2
0.4
∆z [lz]
0.6
0.8
1
Figure 3.9: Variation in the maximum grid energy due to cutoff imposed by the numerical
grid spacing, measured from fitting to the classical thermal density in the wings. The dotted
curve is ∝ ∆z −2 . Inset: logscale plot of the measured dimensionless energy cutoff versus the
theoretical formula in terms of ∆z, and a straight line with gradient 1 for comparison.
The results of this procedure are summarised in Figure 3.9. The main plot shows the
grid energy versus the grid spacing, and we find the density in the thermal wings of
the trap to follow the scaling Egrid ∝ (∆z)−2 , shown by the dotted line. Moreover, we
find the constant of proportionality to be 2π, as is shown in the inset, which shows the
fitted values of Λ extracted from simulations, plotted against the scaled grid energy.
The dashed line has a gradient of 1, showing the proposed relation appears to hold
over a range of about three orders of magnitude in grid energies. This behaviour is
also consistent with the Λ values obtained in fitting to the non-interacting densities of
Figure 3.6.
3.4.2
Relation between grid spacing and energy cutoff
Having measured the cutoff imposed by the choice of grid, good agreement is found
between the analytic expression for the ideal gas density and the numerical results of
the SGPE if the energy is related to the grid spacing by
Egrid =
2π ~2
.
(∆z)2 m
56
(3.10)
Chapter 3. The stochastic Gross-Pitaevskii equation in one-dimension
Taking the ratio of this numerical result for the grid energy to the average thermal
energy, gives
Egrid
2π
~2
=
,
kB T
(∆z)2 mkB T
(3.11)
and making use of the definition for the thermal de Broglie wavelength Eq. (1.1), we
can rewrite Eq. (3.11) in the more succinct form:
Λ=
λdB
∆z
2
.
(3.12)
Hence, the numerical findings for the cutoff energy due to the grid spacing, suggests
that the dimensionless energy cutoff Λ is given by the ratio of the grid spacing to the
thermal wavelength associated to a given temperature.
This finding may be put onto a more physical footing if we consider that the noise
term in Eq. (2.92) represents kicks due to interactions with thermal atoms at some
temperature, T , so the average momentum kick then should have a magnitude ∼ 1/λdB .
On a numerical grid, the maximum momentum is related to the highest wavenumber,
which is turn is inversely proportional to the grid spacing, so kgrid ∝ 1/∆z. The
maximum grid energy is then Egrid ∝ 1/(∆z)2 , whereas the thermal energy can be
considered as Etherm ∝ 1/λ2dB . Taking also Etherm = kB T , then the cutoff Λ is just the
ratio of these two quantities, as is now clear from Eq. (3.12).
A sensible cutoff in energy might be chosen at the point where quantum effects
become comparable to behaviour which is well represented classically, which might be
expected to correspond to an energy ∼ kB T . Eq. (3.12) implies that to achieve an
energy cutoff equal to this value, given by Λ = 1, the grid spacing should be chosen so
∆z = λdB .
3.5
Some properties of the low-lying modes
The SGPE order parameter represents a number of system modes, up to the cut-off due
to the grid choice, which for an appropriately chosen grid includes both condensed and
non-condensed particles. A key temperature dependent property, however, is the fraction of condensed particles within the system and we now discuss a means of recovering
this information, from the unified representation of the lowest system modes.
3.5.1
Penrose-Onsager definition of the condensate mode
To define a condensate fraction, we use spatial coherence as a measure by employing the
Penrose-Onsager criterion for Bose-Einstein condensation [55, 207, 215]. This states
that condensation has occurred when an eigenvalue of the one-body density matrix
57
Chapter 3. The stochastic Gross-Pitaevskii equation in one-dimension
takes on a macroscopic value. The eigenmode corresponding to this is then identified
as the condensate mode, while the eigenvalue provides information as to the number of
condensed atoms. We can apply this idea to the ensemble of classical fields generated
in solving the SGPE, each corresponding to different realisations of the noise, by constructing the one-body density matrix, approximated for our one-dimensional system
as
ρ(z, z ′ ) = hΨ̂† (z)Ψ̂(z ′ )i ≈ hψ ∗ (z)ψ(z ′ )i.
(3.13)
It is then possible to numerically diagonalise this matrix and identify the condensate
mode as that corresponding to the largest eigenvalue, assuming we are at a temperature
such that this is macroscopic in size. For a spatial grid with M grid points, this
diagonalisation yields M complex mode functions, ϕi (z), and complex eigenvalues, ci ,
where i is the mode index. The order parameter is then decomposed in its diagonal
basis as
ψ(z) =
M
−1
X
ci ϕi (z),
(3.14)
i=0
where the modes are numbered such that increasing mode index corresponds to increasing energy, and hence decreasing occupation numbers, at equilibrium. The average
occupation of the condensate mode, at the time the density matrix was constructed
from the order parameter, is then given by N0 = |c0 |2 .
From this procedure, we can also obtain the average condensate density, defined as
n0 (z) = |c0 |2 |ϕ0 (z)|2 . Densities corresponding to the different phases of matter present
within a partially Bose gas are shown in Figure 3.10. Due to thermal depletion, not
all of the atoms in the system are are condensed. Instead, Figure 3.10(a) shows that
the condensed particles reside in the central region of the trap, occupying spatially
the region of lowest energy. The density profile for the non-condensed fraction can be
inferred as those atoms in modes perpendicular to the condensate mode,
nt (z) = h|ψ(z)|2 i − n0 (z).
(3.15)
This is indicated by the dashed red curve of Figure 3.10(a). The density shows that
thermal atoms are largely pushed out from the central region due to the repulsive meanfield potential of the condensate, which results a double peaked thermal density. This
is explained by the mean field potential experienced by the thermal atoms, illustrated
in Figure 3.10(c). This plot also shows the difference between using the ThomasFermi density and the finite temperature SGPE density, each of which is illustrated for
these parameters in Figure 3.10(b), in calculating the Hartree-Fock mean-field potential
58
Chapter 3. The stochastic Gross-Pitaevskii equation in one-dimension
1.5
1.2
(a)
gn(z) / µ
1
0.6
0.4
0.2
0
-12
-9
-6
-3
0
3
6
1.5
1
0.5
(c)
0
-6
-3
0
3
6
-12
9
12
2
0
-9
9
z [lz]
2.5
0.5
Energy / µ
gn(z) / µ
(b)
1
0.8
-9
-6
-3
0
3
6
9
12
z [lz]
z [lz]
Figure 3.10: Plot illustrating the Penrose-Onsager means of identifying a condensate fraction.
Shown is: (a) The total atomic density (black, solid), condensate mode density (blue, dotdashed), and the non-condensed fraction (red, dashed) plus the harmonic trapping potential
(brown, dotted) in units of the chemical potential; (b) Comparison between the Thomas-Fermi
condensate density profile (orange, dashed), finite temperature condensate mode and the total density as in (a) for the same µ; (c) The Hartree-Fock potential seen by the thermal
atoms, where the density mean-field contribution is based on the Thomas-Fermi profile (orange, dashed) and the SGPE density (black, solid). Also indicated are relevant energies µ
(green, dot-dashed) and the trapping potential. Parameters as in Figure 3.7.
59
Chapter 3. The stochastic Gross-Pitaevskii equation in one-dimension
experienced by the non-condensed particles,
U (z) = V (z) + 2g(nc (z) + nt (z)).
(3.16)
In the Thomas-Fermi approximation, the contribution to the energy is small due to the
thermal density, so making use of Eq. (2.11), we can write
U (z) =

2µ − V (z)
V (z)
µ > V (z)
(3.17)
µ ≤ V (z).
By contrast, we see that using the finite temperature density smooths the sharp turning
points in the Hartree-Fock potential, which has a discontinuous first spatial derivative
for the Thomas-Fermi case. This implies a divergence in the force given by the gradient
of this potential. Using the finite temperature density also moves the non-condensate
density peaks in towards the trap centre, as is consistent with the reduction in the size
of the condensed density at non-zero temperatures.
3.5.2
Extracting a quasi-condensate fraction
The quasi-condensate fraction, representing those atoms for which density fluctuations
are largely suppressed, can be characterised using [216],
nqc =
p
2h|ψ(z)|2 i2 − h|ψ(z)|4 i
(3.18)
This measure has also been considered for the one-dimensional SGPE in [183, 212] and
the PGPE in two-dimensions [215]. Shown in Figure 3.11 is a comparison between the
densities for the quasi-condensate and Penrose-Onsager condensate mode.
It is evident that for these parameters, far more of the total system atoms are within the
regime of suppressed density fluctuations, shown by the green, solid curve in Figure
3.11 relative to those in the more coherent state of having additionally suppressed
phase fluctuations, shown by the blue, dot-dashed curve. This is consistent with the
temperature for which the data was taken, as T ≪ Td , though T ≈ 0.5Tφ . Also
shown by the maroon dot-dashed curve, is the density profile which is inferred from the
subtraction of the quasi-condensate density from the total density. Given the larger
quasi-condensate component, obviously this is smaller than in the Penrose-Onsager
case, however it is also noticeably flatter, which is due to the quasi-condensate density
matching far more closely the total density shape. The quasi-condensate density profile
retains the roughly inverse parabolic shape of the total density, whereas the PenroseOnsager density displays a more gradual decay as the density edge is approached.
60
Chapter 3. The stochastic Gross-Pitaevskii equation in one-dimension
1
gn(z) / µ
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10
z [lz]
Figure 3.11: Density profiles for the Penrose-Onsager condensate mode (blue, dot-dashed) and
resulting non-condensate density (red, dashed) plus the quasi-condensate profile which results
from the definition Eq. (3.18) (green, solid) and the remaining fraction of the total density
(maroon, dot-double dashed). The total density is indicated by the dotted black line.
This yields a relatively more pronounced double peak structure to the non-condensate
density, shown by the dashed, red curve.
Penrose-Onsager condensate mode grid dependence
Figure 3.12 shows the relatively limited effect of the grid spacing, and so the energy
cutoff, on the Penrose-Onsager method of extracting a condensate mode. Shown in
Figure 3.12(a) are condensate mode densities for an extremely low cutoff, βEgrid = 0.02,
and for a higher, more physically reasonable cutoff, βEgrid = 4.8. Given the vast
difference in grid choice, the resultant densities are fairly similar, with the lower cutoff
result slightly overestimating the condensate number, relative to the larger cutoff result.
This is in agreement with the findings of Figure 3.8, since the chemical potential on
this scale is βµ = 0.05, and so the grid for the lower energy cutoff, βEgrid = 0.02, might
well be expected to poorly represent the system properties.
Shown in Figure 3.12(b)-(c) are the density in the wings and central trap region
respectively, for two further cutoff choices: βEgrid = 1.0 and βEgrid = 9.6. Good agreement is demonstrated in the central region between these results, with the differences
manifest only in the wings of the density distributions, where the behaviour however
matches the expected ideal gas result given by Eq. (3.8), and shown by the dotted
curves. We see then that for the higher energy grid cutoffs, or smaller grid spacings, a
consistent behaviour emerges. It is also apparent that the condensate mode density at
least, is fairly robust to the choice of numerical grid.
61
Chapter 3. The stochastic Gross-Pitaevskii equation in one-dimension
0.9
0.015
gn(z) / µ
(a)
gn(z) / µ
0.75
(b)
0.012
0.009
0.006
0.6
0.003
0
16
24
32
40
48
0.45
0.8
gn(z) / µ
0.3
0.15
0
56
z [lz]
-6
-4
-2
0
(c)
0.6
0.4
0.2
0
-8
z [lz]
-4
0
4
8
z [lz]
Figure 3.12: (a) Penrose-Onsager condensate mode densities for two grid choices, Λ = 0.02
(blue) and Λ = 4.8 (black); (b) zoomed profiles of the density in the wings for data with
Λ = 1.0 (red) and Λ = 9.6 (green); dotted lines here show the ideal gas result for Λ = 1.0 (lower
curve) and Λ = 9.6 (upper curve); (c) Condensate and thermal fractions for the data of (b).
Trap parameters are as in Figure 3.7.
3.5.3
Condensate number statistics
A practical advantage of the Penrose-Onsager approach to identifying a condensate
mode, is that it identifies a condensate wavefunction, and not just the density profile
as in the quasi-condensate definition, Eq. (3.18). This allows for more insight into
the system properties, as in addition to the density, requiring only knowledge of the
magnitude of the wavefunction, we also retain information on the phase.
The condensate mode is one of M orthonormal (down to numerical error) basis
vectors, obtained upon diagonalising ρ(z, z ′ ). Projecting the order parameter on to this
mode provides a further method by which to obtain the occupation of the condensate
mode. However, this is distinct from the number obtained as the largest eigenvalue of
ρ(z, z ′ ). The eigenvalue obtained from the density matrix is the average condensate
number across the ensemble of noisy states, whereas projecting the order parameter as,
c0,j =
Z
dz φ∗0 (z)ψj (z),
(3.19)
yields the condensate number Nj = |c0,j |2 , for the j-th realisation of the noise, {η}j .
Repeating this projection for all states within the ensemble, then gives a set of condensate numbers {Nj }, which have a temperature dependent distribution. Taking the
mean, h{Nj }i, however, regains an average condensate number, typically very close to
that found from the density matrix eigenvalue.
62
Chapter 3. The stochastic Gross-Pitaevskii equation in one-dimension
0.0002
P(N0)
0.00015
0.0001
5e-05
0
0
0.2
0.4
0.6
0.8
1
N0/N
Figure 3.13: Normalised histogram showing the distribution of condensate statistics for an
ensemble of 1000 realisations of the noise, calculated at T = 185nK with βEgrid = 4.8.
We show in Figure 3.13 the form of the distribution of condensate numbers across the
ensemble. Results are shown for the particular grid spacing used in the comparison
undertaken in Chapter 6, at this temperature. To test grid effects, the results of two
other, very different, grids which were used are shown in Appendix B. It is shown
that they each reproduce the result of the method of Svidinsky and Scully [120] (see
also Chapter 6) to good accuracy, hence we can be satisfied that this property is not
strongly affected by the choice of numerical grid. We also carried out simulations for a
number of intermediate grids, with the same conclusion. So, the results of these tests
have established that not only the mean condensate number, but also the fluctuations
around this value, are relatively insensitive to the numerical discretisation.
3.5.4
Modes above the condensate
In diagonalising the one-body density matrix, in theory we obtain a diagonal basis for
our system in the trap. We can thus equally well project the order parameter ψ(z)
onto any mode within this basis, and extract the occupation number from the expansion
coefficient which results. More formally, for the i-th expansion coefficient, we calculate
the projection as
ci =
Z
dz φ∗i (z)ψ(z)
(3.20)
from which the occupation of the i-th mode can be calculated, and is given by
Ni = |ci |2 ,
as for the condensate mode.
63
(3.21)
Chapter 3. The stochastic Gross-Pitaevskii equation in one-dimension
-1
Probability density [lz ]
0.24
0.21
0.18
0.15
0.12
0.09
0.06
0.03
0
-8
-4
-6
-2
0
2
4
6
8
z [lz]
Figure 3.14: The first (black, solid), third (red, dashed), fifth (blue, dot-dashed) and seventh
(green, dot-double dashed) trap modes, which are obtained upon diagonalising ρ(z, z ′ ).
As an illustration of the structure of the modes above the condensate in this diagonal
basis, the first 4 odd mode densities, extracted from a simulation of an ensemble of
1000 runs, are shown in Figure 3.14. As expected they take the form of harmonic
oscillator modes with features characteristic of this: For example, the i−th mode has
i + 1 turning points, and the higher energy modes show an increasing spatial extent as
the energy is increased, corresponding to the classical turning point in the harmonic
potential moving outwards with increasing energy.
Given the mode functions, we might also wish to extract an associated energy, which
can be done using the Gross-Pitaevskii energy functional
E [φi (z)] =
Z
dz
"
#
~2 ∂φi (z) 2
+ V (z)|φi (z)|2 + g|φi (z)|4 .
2m ∂z (3.22)
In order to accurately determine the behaviour of the discretised Laplacian of the
SGPE as closely as possible, we do not evaluate the first derivative of Eq. 3.22 directly.
Instead, we write this as a second derivative, making the identification
2
∂φi (z) 2
→ 1 Re ψ ∗ ∂ ψ
∂z 2
∂z 2
(3.23)
which is obtained from the kinetic energy term of Eq. (3.22) by integrating by parts,
and where Re denotes the real part1 . The kinetic and potential terms which result
from evaluating Eq. 3.22, are shown in Figure 3.15. This forms a useful check on the
behaviour of the energy extracted, as we can check the results against the predicted
behaviour from the virial theorem within a harmonic potential, that hEk i = hEp i.
1
Practically, we may extract the result of approximating the second derivative when using the
Numerov method, by solving a matrix equation, as described in Appendix A.
64
Chapter 3. The stochastic Gross-Pitaevskii equation in one-dimension
1.25
βE
1
0.75
0.5
0.25
(a)
0
0
100
200
300
400
500
ln[N(Ek)]
Mode number, i
12
10
8
6
4
2
0
-2
-4
(b)
-2
0
2
4
6
8
ln(Ek)
Figure 3.15: (a) The kinetic energy (black circles) and potential energy (red squares) of the
modes due to diagonalising ρ(z, z ′ ), scaled to the thermal energy. (b) Logarithm of the mode
occupations versus the logarithm of the kinetic energy of each mode.
In Figure 3.15(a), we see that the kinetic and potential energy curves agree well for a
large number of the system modes.
Given that the occupation numbers and energies of higher energy modes are readily
obtainable, we now consider the form of the distribution of particles amongst the trap
modes, as a function of energy. Shown is a plot of the particle number versus kinetic
energy in Figure 3.15(b). At equilibrium, due to the underlying Rayleigh-Jeans distribution, we expect a linear dependence of the occupation numbers with energy, and the
straight line fit on this plot of ln(N ) against ln(EK ), yields a gradient of −1.01.
We now compare how well different grid spacings represent the expected results, as
shown in Figure 3.16. Plotted here is ln (βN (Ei )) versus ln(Ei ), where Ei is the energy
found by upon applying the functional Eq. (3.22) to the i−th mode. For an ideal,
Rayleigh-Jeans distributed system, plotting these quantities should display a straight
line behaviour with gradient −1, which is the behaviour observed in Figure 3.16 for
energies greater than µ. We would certainly expect to see this trend for E > µ, as these
modes would not feel the presence of a condensate, so ought to display the free-particle
behaviour. For energies less than the chemical potential, we find that the occupation
numbers display a rather different behaviour. This shows the influence of the condensed
fraction upon the modes immediately above the ground state, with E . µ.
65
Chapter 3. The stochastic Gross-Pitaevskii equation in one-dimension
15
ln(µ)
ln(kBT)
ln(N)
10
5
0
-5
1
2
3
4
5
6
7
8
ln(E)
Figure 3.16: The natural logarithm of the density matrix eigenmode occupation numbers
against total energy, for the following grid energy cutoffs: βE = 1 (red squares) and βE = 4.8
(black circles). The vertical dashed and dotted lines show µ and kB T on the logarithmic scale.
The green, dashed line shown has a gradient of -1.
It is worth mentioning at this point differences to the microcanonical ensemble
considered in the GP-classical field approach. In this method, energy is fixed in the
initial condition so increasing the cutoff leads to the inclusion of more modes. Retaining
the same total energy, if the initial state is unchanged as the cutoff is varied, leads to a
change in temperature, as the same amount of energy is shared with equal probability
amongst more modes. Of course, it is then possible to vary the initial condition to
maintain a fixed temperature.
This scenario should be compared to the grand canonical ensemble of the SGPE,
where more energy and particles are included if the cutoff is increased at fixed temperature, due to the action of the noise term. This seeds growth into modes according
to the Rayleigh-Jeans distribution, though obviously a high enough cutoff will yield
empty modes, as the probability of occupation falls off, albeit algebraically, with increasing energy. However, in the SGPE approach, the downside to a more controllable
temperature, is that the particle number is then cutoff dependent. This can be remedied by variation of the chemical potential, which could be carried out systematically
in one-dimension by matching it to that obtained from the modified Popov theory of
Andersen et al. [217], for example.
3.6
Calculation of noise and damping terms within the
ergodic approximation
We first discuss the introduction of the ergodic approximation, which allows for a
simplification to the integrals which must be performed in calculating the coupling to
the thermal modes2 , iR(x, t). In doing so, we retrace the steps presented in [165], and
2
For generality we discuss the three-dimensional case, though later we will apply this only in one
dimension.
66
Chapter 3. The stochastic Gross-Pitaevskii equation in one-dimension
on arriving at this simplified equation, we then discuss two approaches to its calculation,
contrasting the expression of Bijlsma et al. [165] to the scattering rate of the closely
related approach of Gardiner et al. [218].
The phase space integrals appearing in Eq. (2.89) for the damping term of the
SGPE are complicated by the fact they are in general integrals over momenta. As
vector quantities, this leads to a large number of degrees of freedom. One means by
which to reduce this number, is to appeal to the ergodic approximation, through which
we assume that states of equal energies equilibrate on a much shorter time scale than
states of different energies. This is equivalent to binning quantum states, specified
in a spherical basis by quantum numbers (n, ℓ, m), according only to their principal
quantum number, n, which is certainly the case in equilibrium. Additionally, it has
been found that following depletion of one of three degenerate levels of a dilute Bose
gas, equilibrium is reached within approximately the mean collision time for the gas
[219], indicating this to be a reasonable approximation, at least close to equilibrium
[220].
In our case, the ergodic approximation is most clearly expressed as [165],
ρ(E, t)g(E, t) ≡
dxdp
δ (E − ǫ (x, p, t)) f (x, p, t)
(2π~)3
Z
(3.24)
where the distribution functions in terms of energy are denoted as
g(E, t) = g(ǫ(x, p, t), t) ≡ f (x, p, t),
(3.25)
and the density of states is given by
ρ(E, t) =
Z
dxdp
δ (E − ǫ (x, p, t)) .
(2π~)3
(3.26)
This is perhaps more usefully expressed as a mapping from the distribution functions
of Eq. (2.84),
f (xi , pi , t) =
Z
dEi δ(Ei − ǫi ) g(Ei , t).
(3.27)
Applying this to the definition of R(x, t) gives
g2
R(x, t) = 3d 5
(2π)
Z
dk1
Z
Z
dk2
dE1
Z
Z
dE2
Z
dE3 δ (E2 + E3 − E1 − ǫc ) ×
[g1 (1 + g2 )(1 + g3 ) − (1 + g1 )g2 g3 ] ×
dk3 δ (k2 + k3 − k1 ) δ (E1 − ǫ1 ) δ (E2 − ǫ2 ) δ (E3 − ǫ3 ) . (3.28)
where we can note the simplification which arises in considering the distribution func67
Chapter 3. The stochastic Gross-Pitaevskii equation in one-dimension
tions as functions of energy alone. The problem now is to evaluate the integral over
momentum. Considering these in isolation, we wish to evaluate
K≡
Z
dk1
Z
dk2
Z
dk3 δ (k2 + k3 − k1 ) δ (E1 − ǫ1 ) δ (E2 − ǫ2 ) δ (E3 − ǫ3 )
(3.29)
which, substituting −k1 → k1 , can be written as
K=
Z
3 Z
dξ Y
dki eiξ·ki δ (Ei − ǫi )
(2π)3
(3.30)
i=1
using the Fourier representation for the Dirac δ-function:
δ(x) =
Z
dk eik.x .
(3.31)
We can compute now one of the terms within the product of Eq. (3.30),
Z
dk eiξ·k δ (E − ǫ)
2 ′2
Z
Z
~ k
′ ′2
iξk ′ cos θ
= 2π dθ dk k sin θe
δ
− (E − U ) .
2m
P ≡
(3.32)
(3.33)
Evaluating the k′ -integral yields a Heaviside step function, ΘH . Writing the exponential
in terms of trigonometric functions, and expanding as a Taylor series, we find
P =
Z π
2πm
(kξ)2
2
cos θ + . . . sin θ+
ΘH (E − U )k
dθ
1−
~2
2!
0
Z π 2πm
(kξ)3
3
i
dθ kξ cos θ −
cos θ + . . . sin θ, (3.34)
ΘH (E − U )k
~2
3!
0
where we have defined
k=
r
2m
(E − U ).
~
(3.35)
Since the integrand in the imaginary term is an odd function over [0, π], there is only
a contribution from the real part, to give
P =
4πm
~2
ΘH (E − U )k
Z
π
2
dθ cos(kξ cos θ) sin θ.
(3.36)
0
The spherical Bessel function of order ν can be defined by the integral relation
jν (z) ≡
(z/2)ν
Γ (ν + 1)
Z
π/2
dt cos(z cos(t)) sin2ν+1 (t),
0
68
(3.37)
Chapter 3. The stochastic Gross-Pitaevskii equation in one-dimension
and so we can write (3.36) in terms of this,
P =
4πm
~2
kΘH (E − U )j0 (kξ).
(3.38)
The spherical Bessel function j0 can also be written in terms of trigonometric functions,
j0 (z) =
so we can then write
P =
4πm
~2
sin(z)
,
z
ΘH (E − U )
(3.39)
sin(kξ)
.
ξ
(3.40)
The result is that we now have,
)
( 3 Y 4πm 3
sin(k1 ξ) sin(k2 ξ) sin(k3 ξ)
dξ
ΘH (Ei − U )
K=
(2π)3
~2
ξ3
i=3
Z
dξ sin(k1 ξ) sin(k2 ξ) sin(k3 ξ)
4πm 3
Θ
(E
−
U
)
=
H
min
~2
(2π)3
ξ3
Z
(3.41)
(3.42)
where Emin = min {E1 , E2 , E3 }.
The products of sine-functions can be regrouped to yield,
K=
4πm
~2
3
1
ΘH (Emin − U ) 2
8π
Z
dξ {sin [(k1 + k2 − k3 )ξ] + sin [(k1 − k2 + k3 )ξ]
− sin [(k1 + k2 + k3 )ξ] − sin [(k1 − k2 − k3 )ξ]} . (3.43)
Moreover, we note that the following can be shown to be true [221],
Z
∞
0
sin(λx)
1
dx = sgn(λ)π
x
2
(3.44)
for λ 6= 0, and we therefore have
K=
2m
~2
3
π 2 ΘH (Emin − U )S(k1 , k2 , k3 ),
(3.45)
where
1
S(k1 , k2 , k3 ) = {sgn(k1 + k2 − k3 ) + sgn(k1 − k2 + k3 )
2
−sgn(k1 + k2 + k3 ) − sgn(k1 − k2 − k3 )} .
69
(3.46)
Chapter 3. The stochastic Gross-Pitaevskii equation in one-dimension
In which case, we have that
Z
Z
Z
2 2g3d
m 3
dE1 dE2 dE3 ΘH (Emin − U )δ(E2 + E3 − E1 − ǫc )×
R(x, t) =
(2π)3 ~2
[g1 (1 + g2 )(1 + g3 ) − (1 + g1 )g2 g3 ] S(k1 , k2 , k3 ). (3.47)
It is convenient to transform to a shifted energy variable [165]
Ē ≡ E − U (x, t),
(3.48)
so,
Z
Z
Z
2 2g3d
m 3
dĒ1 dĒ2 dĒ3 δ(Ē2 + Ē3 + U (x, t) − Ē1 − ǫc )
R(x, t) =
(2π)3 ~2
× [ḡ1 (1 + ḡ2 )(1 + ḡ3 ) − (1 + ḡ1 )ḡ2 ḡ3 ] S(k̄1 , k̄2 , k̄3 ), (3.49)
and the integration range is now fixed between
0 ≤ Ē < ∞. In addition, we now have
q
2m
Ē . Performing the integral over the Ē1
that ḡi = g(Ēi + U (x, t), t) and k̄i =
~2 i
variable then leaves us with
R(x, t) =
Z
Z
2 2g3d
m 3
dĒ3 S(k̄1 , k̄2 , k̄3 )
d
Ē
2
(2π)3 ~2
(3.50)
× [ḡ1 (1 + ḡ2 )(1 + ḡ3 ) − (1 + ḡ1 )ḡ2 ḡ3 ] ,
where the delta function has imposed that Ē1 = Ē2 + Ē3 + U (x, t) − ǫc . It is convenient
now to re-introduce the Keldysh self energy, ~Σ K (x, t), which following the ergodic
approximation becomes,
~Σ K (x, t) = − i
Z
Z
2 4g3d
m 3
dĒ3 S(k̄1 , k̄2 , k̄3 )
d
Ē
2
(2π)3 ~2
(3.51)
× [ḡ1 (1 + ḡ2 )(1 + ḡ3 ) + (1 + ḡ1 )ḡ2 ḡ3 ] .
If it is assumed that we can treat the thermal modes as a heat bath, by which
they are taken to form a reservoir with fixed thermodynamic parameters µ and T , the
ḡ-functions can then be replaced with Bose distributions:
h
i−1
Ni = N (Ei ) = eβ(Ei −µ) − 1
.
(3.52)
For these distributions the following identity proves useful
N (x) ≡ − (N (−x) + 1)
70
(3.53)
Chapter 3. The stochastic Gross-Pitaevskii equation in one-dimension
as we can then show
N1 (1 + N2 )(1 + N3 ) ± (1 + N1 )N2 N3 = e(ǫc −µ)β ± 1 (1 + N1 )N2 N3 .
(3.54)
Using this, in combination with Eq. (3.49) and Eq. (3.51), it is straightforward to see
that the fluctuation-dissipation relation Eq. (2.88) is obtained.
Upon making the classical approximation to the quantum fluctuation-dissipation
relation, we find the SGPE within the classical approximation, with a damping which
can be written as
Z
Z
m 3
2
g3d dĒ2 dĒ3 S(k̄1 , k̄2 , k̄3 )(1 + N̄1 )N̄2 N̄3
γ(x, t) = 2β
2π~2
(3.55)
where we have assumed that we are close to equilibrium, as is consistent with the
static thermal cloud approximation, and the linear response form for the fluctuationdissipation relation.
As this is the manner in which we solve numerically, and for completeness, we can
write Eq (2.92) in terms of this
∂ψ(x, t)
=
i~
∂t
2 2
~ ∇
2
1 − iγ(x, t) −
+ V (x) + g3d |ψ(x, t)| − µ ψ(x, t) + η(x, t),
2m
(3.56)
and so the noise correlations then are
hη ∗ (x, t)η(x′ , t′ )i =
2~γ(x, t)
δ(x − x′ )δ(t − t′ ).
β
(3.57)
0.02
γ(z)
0.016
0.012
0.008
0.004
0
-30
-20
-10
0
10
20
30
z [lz]
Figure 3.17: Illustration of γ(z) within a harmonic trapping potential with ωz = 2π × 10Hz
and ω⊥ = 2π × 2500Hz, for µ = −50~ωz and U (z) = V (z) only. The curves are for T = 150nK
(red, dashed) and T = 300nK (black, solid).
71
Chapter 3. The stochastic Gross-Pitaevskii equation in one-dimension
Ideally, Eq. (3.55) would reduce to an analytic expression, however this does not
appear immediately possible without further approximations. One route is to proceed
numerically, and this integral can be evaluated using, for example, a Gauss-Legendre
quadrature. The results of this for some example parameters are shown in Figure
3.17. The distribution is peaked in the centre, as in this case we have used only the
harmonic trapping potential contribution to U (x, t) in calculating the integrals of Eq.
(3.55). This makes physical sense however, as we would expect the scattering from
a heat bath into an empty harmonic trap to occur with the highest rate at the trap
centre, where the energy is lowest.
Another approach is to neglect the S(k̄1 , k̄2 , k̄3 ) term, which yields an expression in
terms of the Lerch transcendent [218], which has the practical advantage that it can
be quicker to calculate numerically.
3.6.1
Lerch transcendent expression of C. W. Gardiner et al.
The integral of Eq. (3.55) is similar to that considered in the quantum kinetic theory
of Gardiner et al [218], if we neglect the S(k̄1 , k̄2 , k̄3 ) term. Explicitly writing out the
energy dependencies, under the assumption that we are close to equilibrium, then we
wish to evaluate,
h
Z
Z
m 3
i−1 h
i−1
(Ē2 +U (x,t)−µ)β
(Ē3 +U (x,t)−µ)β
2
e
−
1
e
−
1
g
d
Ē
d
Ē
3
2
3d
2π~2
h
i−1 h
i−1 h
i−1 (Ē2 +Ē3 +2U (x,t)−2µ)β
(Ē2 +U (x,t)−µ)β
(Ē3 +U (x,t)−µ)β
+ e
−1
e
−1
e
−1
.
γ(x, t) = 2β
(3.58)
Taking the first term, this can be rewritten by expanding the Bose-Einstein function
as a geometric series, which due to the definition of the Γ -function
Γ (z) =
and the polylogarithm,
Z
∞
tz−1 e−t dt
(3.59)
0
∞
X
zm
Lis (z) =
,
ms
(3.60)
m=1
gives
Z
dĒ2
Z
dĒ3
h
i−1 h
i−1
e(Ē2 +U (x,t)−µ)β − 1
e(Ē3 +U (x,t)−µ)β − 1
i2
1 h = 2 ln 1 − eβ(µ−U (x,t))
.
β
(3.61)
The second term can be evaluated in by starting in a similar manner, and rewriting
72
Chapter 3. The stochastic Gross-Pitaevskii equation in one-dimension
the Bose-Einstein distributions as geometric sums,
Z
dĒ2
h
i−1 h
i−1
e(Ē2 +U (x,t)−µ)β − 1
dĒ3 e(Ē2 +Ē3 +2U (x,t)−2µ)β − 1
h
i−1
× e(Ē3 +U (x,t)−µ)β − 1
=
Z
Z
∞ X
∞ X
∞ (2µ−2U (x,t))βp
X
e
−β(p+q)Ē2
dĒ2 e
dĒ3 e−β(p+r)Ē3 .
2
β
p=1 q=1 r=1
Z
(3.62)
Computing the energy integrals, and making use of the definition of the Lerch transcendent,
Φ(z, s, α) =
∞
X
n=0
zn
,
(n + α)s
(3.63)
we arrive at the final expression for γ(x, t), in the approximation that the effects of the
function S(k̄1 , k̄2 , k̄3 ) are small,
4m a 2
γ(x, t) =
πβ ~
h
(2µ−2U (x,t))β
+e
i2
ln 1 − eβ(µ−U (x,t))
∞ p h
i2 X
(µ−U (x,t))β
(2µ−2U (x,t))β
Φ(e
, 1, p + 1)
e
.
(3.64)
p=1
3.6.2
Comparison between the integral and Lerch expressions for γ(z)
For a one dimensional Bose gas, a comparison between the result of using Eq. (3.55)
and Eq. (3.64) in calculating γ(z) is shown in Figure 3.18.
In these calculations, by contrast to the example of Figure 3.17, the Hartree-Fock
density contribution to the potential U (z) in the thermal energies has been included,
using the equilibrium density from the SGPE. This creates a double peaked structure
to the self-energy, and hence γ(z), compared to the single peaked distribution when
only the energy due to the trapping potential was included. The physical picture now,
is that due to the presence of a condensed fraction, the higher energy thermal particles
are pushed to the wings, and so the scattering between condensate and thermal cloud
is maximum at the condensate edge. The vertical dotted lines indicate the T = 0
Thomas-Fermi radius for the chemical potential used in these calculations, and the
γ(z) which results on using the T = 0 Thomas-Fermi density is also shown by the blue,
dashed curve. This illustrates the divergence at the condensate edge in the ThomasFermi limit, as pointed out by Duine and Stoof [175]. Using the equilibrium finite
temperature density of the SGPE removes this divergence, and results in moving the
peaks of γ(z) in towards the trap centre. This is consistent with the idea that the
73
Chapter 3. The stochastic Gross-Pitaevskii equation in one-dimension
0.035
0.03
γ(z)
0.025
0.02
0.015
0.01
0.005
0
-30
-20
0
-10
10
20
30
z [lz]
Figure 3.18: Profiles for γ(z) calculated using Eq. (3.55) (red) and Eq. (3.64) (black) with the
SGPE equilibrium density. The T = 0 Thomas-Fermi radius is shown by the vertical dotted
lines, and the γ(z) obtained using the Thomas-Fermi density is shown by the dashed, blue lines.
The trapping potential is ωz = 2π × 10Hz and ω⊥ = 2π × 2500Hz, and at T = 300nK with
µ = 395 ~ωz , this gives N ≈ 20000 87 Rb atoms at equilibrium.
condensate fraction is reduced as we increase temperature for a fixed particle number,
and with the introduction of a temperature dependent Thomas-Fermi radius, as the
condensate fraction shrinks [214].
3.7
Chapter Summary
In this Chapter we have discussed a selection of the equilibrium properties of both
the ideal and interacting Bose gas within a harmonic trap. As numerical simulations
form an important part of the work reported in subsequent Chapters, we assessed the
sensitivity of these properties to the numerical discretisation, with the finding that grid
spacings ∆z . λdB yield consistent results across the tests carried out.
We additionally assessed the role of γ(z), which under the assumption of a static
thermal cloud, is not represented in a time dependent way. Considering several spatially
constant values for γ, it was found that the difference in growth curves could be largely
accounted for by scaling the time to the γ used.
While the thermal cloud is treated in a static fashion, for perturbations which do
not strongly change the density, we can closely approximate the form of γ(z) during
the non-equilibrium evolution, through the inclusion of the Hartree-Fock potential in
calculating the self-energy integral.
In this chapter we also discussed some practical aspects of implementing the SGPE,
in particular the calculation of the Keldysh self-energy which is of central importance
74
Chapter 3. The stochastic Gross-Pitaevskii equation in one-dimension
within Stoof’s theory. To summarise, the approximations made in arriving at the
equation which we solve numerically are:
• The high-lying thermal modes are in static equilibrium so that they act as a heat
bath to the low-lying modes;
• The quantity (ǫc − µ) β is small;
• The modes represented dynamically are sufficiently populated that the classical
approximation may be used in writing the fluctuation-dissipation relation as Eq.
(2.90);
• The ergodic approximation is valid, such that thermalisation between states of
the same energy happens much more rapidly than that between non-degenerate
states.
In particular, this chapter has summarised parts of various papers of Stoof, and collaborators, in particular making use of results from two excellent papers by Duine and
Stoof [175] and Bijlsma, Zaremba and Stoof [165].
In the forthcoming two Chapters, we discuss the dynamics of dark solitons within
this model, considering both dissipative and stochastic effects.
75
Chapter 4
Finite temperature matter waves
In this Chapter, we discuss the application of the dissipative Gross-Pitaevskii equation, Eq. (2.95), to the study of dark solitons at finite temperature. We begin with
some introductory material on solitons in general, before motivating the first of the
finite temperature models for soliton dynamics considered in this thesis. Initially, we
discuss a means of obtaining an appropriate initial condition for both the dynamically
represented order parameter and its coupling to the thermal modes. The damping of
our dissipative mean-field approach is based upon a scattering rate calculated from
microscopic considerations. In addition, a comparison is made between the dynamical
results obtained for a range of approximations which can be made in choosing a particular form for the damping. We also assess the validity of the 1d model considered in
this and the following Chapter, through a comparison to cylindrically symmetric, 3d
simulations.
4.1
Introduction
Solitons are fundamental solutions to the equations describing many nonlinear systems
and as such appear frequently as solutions to mathematical models devised to describe
the physics of these. Of great interest, soliton-like excitations are known to arise
spontaneously at phase transitions [187, 188], and appear in diverse fields such as fluid
mechanics, plasma physics, condensed matter physics and biophysics; see for example
[222] or [223].
The first recorded observation of a soliton was made by John Scott Russell in 1834
[224], who reported having observed a ‘wave of translation’ propagating from the hull
of a boat along the Union canal in Scotland. A modern description characterises a
soliton as a localised solitary wave which maintains its form as is it propagates, and
which also emerges unchanged from collisions, up to a shift in phase [225].
Russell’s discovery regained prominence in the 1960s when exact solitonic solutions
76
Chapter 4.
Finite temperature matter waves
of the Korteweg-de Vries equation, describing weakly nonlinear water waves, were discovered via inverse scattering techniques [222]. This equation belongs to a larger group
of classical partial differential equations which are termed as being integrable. For a
Hamiltonian system integrability means it admits the maximum possible number of
constants, or integrals, of motion. That solitons are not dissipated during collisions,
can be understood from the fact that an integrable nonlinear equation describes a system with an infinite number of conserved quantities. This includes energy, therefore
physically there is no mechanism for decay. For a decay mechanism to manifest, this
integrability must be lifted in some manner.
A second prototypical example of such an equation is the one-dimensional nonlinear
Schrödinger equation (NLSE), as used for example in nonlinear optics to describe media with Kerr nonlinearity [226]. Physically, solitons arise as solutions to this equation
due to a cancellation of the dispersive effects represented by the kinetic energy term
with the focusing, or defocusing, effect of the nonlinear term; a focusing nonlinearity
represents attractive interactions, whereas a defocusing nonlinearity represents a repulsive interaction. Due to the many similarities between optical systems and atomic
BECs, it is perhaps not surprising that the lowest order equation of motion for a condensate should take a similar form to the NLSE, and therefore also support solitary
wave solutions.
4.1.1
Solitons in BECs
As intriguing realisations of quantum objects on a macroscopic scale, the fact that the
GPE predicts solitons to be supported within atomic BECs has led to much experimental and theoretical work in this area. In BEC physics, there are two types of solitons
which can occur: bright solitons and dark solitons, and while the dynamics of the latter
will be the main focus of our attention, we introduce both of these in the context of
BECs briefly now.
Bright solitons
As their formation requires a focusing nonlinearity, bright solitons occur in atomic
systems with attractive interactions, in which the interactions balance the effects of
wave packet dispersion [227]. Appearing as a wave in the usual sense of a localised peak
in the condensate density, bright solitons were first realised experimentally in 2002 [228,
229], within a Bose-condensed sample of 7 Li atoms. Despite the seemingly aggressive
means of generation, in which interactions were rapidly changed from repulsive to
attractive using a Feshbach resonance, bright solitons were observed to propagate over
macroscopic scales (∼ 1.1mm) following their release into a tightly confining atomic
waveguide [228]. Also exploiting the Feshbach technique, Strecker et al. were able to
77
Chapter 4.
Finite temperature matter waves
observe the formation of a ‘soliton train’, in which bright solitons found to preserve
their form following numerous collisions [228, 229]. Cornish et al. also studied the
production of bright solitons during the collapse of a condensate of
85 Rb
atoms [230],
though in a more three-dimensional geometry. They again saw the formation of robust
bright solitons in a train-like structure, with neighbouring solitons found to exhibit
repulsive interactions with one another.
Dark solitons
In contrast to bright solitons, dark solitons in BECs instead appear as a notch in the
condensate density, the depth of which, nd , is dependent upon its speed, v. For a
static soliton, the density of the condensate reaches zero at the soliton centre; this is
often termed as being a black soliton by analogy to those first observed in nonlinear
optics [231], which appeared as dark shadows. For v > 0, the soliton is referred to as
being grey, as the density does not reach zero, but instead dips to a velocity dependent
minimum. As well as a density dip, there is a characteristic phase shift across a dark
soliton, the extent of which depends also upon the soliton speed.
0.5
(a)
S / 2π
0.4
0.3
0.2
0.1
0
-0.2
-0.15
-0.1
0
-0.05
0.05
0.1
0.2
0.15
z [lz]
1
(b)
gn(z)/µ
0.8
0.6
nd
0.4
0.2
0
-0.2
-0.15
-0.1
0
-0.05
0.05
0.1
0.15
0.2
z [lz]
Figure 4.1: Shown is (a) the phase and (b) the density profile ±4ξ about
of a harmonically trapped BEC, following the introduction of solitons with
v = 0.1c (red, dotted), v = 0.25c (black, solid), v = 0.5c (green, dashed), and
dot-dashed). The soliton depth, nd , is indicated for the v = 0.25c example
length, ξ = 0.05lz .
78
the trap centre
initial velocities
v = 0.75c (blue,
and the healing
Chapter 4.
Finite temperature matter waves
The speed of a soliton is given in terms of the depth of the density dip, nd , through
the expression
nd i1/2
v h
,
= 1−
c
n
(4.1)
where c is the Bogoliubov speed of sound. However, v can also be linked to the phase
slip across the soliton, S, via
v
= cos
c
S
.
2
(4.2)
Hence S = π for a static, black soliton, and a smaller change in phase corresponds to
an increase in speed, or correspondingly a smaller density minimum, as illustrated in
Figure 4.1.
The wavefunction for a dark soliton at the trap centre is given by the function
ψsol = ζtanh
where ζ =
p
ζz
ξ
+i
v c
e−iµt/~ ,
(4.3)
p
1 − (v/c)2 and ξ = ~/ mn(0)g is the healing length. The number of
particles within a soliton, or perhaps more properly, displaced by the soliton, is given
by
M = 4n(z)Aξm,
(4.4)
where A is the transverse area of the condensate, perpendicular to the axis of soliton
propagation. Due to this removal of particles, to first order, the soliton can be treated
as a classical particle of negative effective mass [232, 233], with a momentum given by
p = −M ż and therefore which experiences an acceleration in the opposite direction to
any applied force. For small velocities, the equation of motion is
d2 zsol
ω2
= − z zsol
2
dt
2
(4.5)
√
which implies an oscillation frequency of ωsol = ωz / 2 within a harmonic trapping
potential with axial frequency ωz [232, 233, 234]. A negative mass also coincides with
the counter-intuitive property that as the soliton decays, the soliton speed in fact
increases, meaning that the depth decreases, from Eq. (4.1). In fact, the non-topological
nature of dark solitons means it is possible for the wavefunction to continuously deform
to the equilibrium state of the system in a trap. Of course, the precise nature of this
state depends upon the physics of the equation being solved, as we shall see later in
this and the following Chapter.
As soliton decay is to be an important feature in this work, we now focus on the
various instabilities which offer avenues for decay, affecting soliton lifetimes in trapped
atomic BECs.
79
Chapter 4.
4.1.2
Finite temperature matter waves
Dark soliton experiments
Most BEC experiments take place usually within a trapping potential of some kind. The
introduction of a spatially inhomogeneous background potential lifts the integrability
of the system which, in principle, means solitons are no longer protected from decay
by energy conservation, as in the integrable homogeneous case, at T = 0.
Instabilities of trapped dark solitons
Busch et al. considered the longitudinal stability of a dark soliton under various trapping potentials and concluded that solitary waves should not decay in the special case
of a harmonic confining potential [232]. In fact, from Eq. (4.5) solitons are found to
oscillate within the trap, where it is additionally found that they periodically emit and
reabsorb sound waves, in a state of dynamical equilibrium [144].
A further important practical point is that experiments are never truly one- dimensional. For isotropically trapped three dimensional systems, solitons are not dynamically stable, and instead decay to more stable excitations in the form of vortices and
phonons [50]. It has been shown that in sufficiently elongated BECs, however, solitons
can be rendered dynamically stable against decay induced by transverse excitations
[235]. The criterion for stability is given by gn(0)/~ω⊥ . 2.5, the reason being that in
this regime the transverse snaking instability, via which solitons decay to vortices in
more isotropic systems, is suppressed [236].
While it is possible to engineer a harmonically trapped system such that a soliton is
dynamically stable, thermal effects can also lead to the decay of a soliton. A mechanism
for this was proposed by Fedichev et al. where reflection of thermal excitations, or
phonons, was found to lead to the gradual decay of a soliton such that the condensate
wavefunction gradually re-attains the equilibrium form [233, 232]. This is in contrast to
topological excitations such as vortices, for which it is not possible for the wavefunction
to continuously deform to the ground state. Instead, due to the associated topological
charge, vortices can only decay by moving to the edge of the condensate. Nevertheless,
solitons are still often considered in some sense as the one-dimensional counterpart to
the vortex.
Synopsis of dark soliton experiments
Experimentally, dark solitons are typically created in BECs by phase engineering techniques as proposed in [237], in which a phase shift is imprinted across an atomic sample
by a far detuned laser beam [146, 238, 50, 115]. They may also be created in a controlled manner using density engineering techniques [239, 240, 241] or combining two
condensates from a double well trap [116]. Focussing on the phase imprinting method,
80
Chapter 4.
Finite temperature matter waves
imposing a phase gradient of ∼ π or larger, over a region smaller than the healing
length subsequently leads to the excitation of a soliton as the density adjusts to the
perturbation in the phase [146, 238, 242]. This technique was reported in the experiment of Denschlag et al. carried out at NIST [238], and also Burger et al. in Hannover
[146]. The NIST group produced images of dark solitons moving at speeds noticeably
less than the speed of sound, and found a bending effect due to the differential speed
along the extent of the soliton within the relatively isotropic trapping potential. While
the system of Burger et al. was somewhat more elongated, solitons were found to decay
rather rapidly on reaching the condensate edge, an effect attributed to thermal decay
[146, 243]. Dark solitons have also been produced in spherical traps, within which it
was possible to observe the onset of the transverse snake instability, whereby a soliton
first bends with the formation of waves along its length, before ultimately decaying to
form vortices [236]. As expected, the subsequent soliton decay led to the formation of
other more dynamically stable nonlinear structures such as vortex rings [50].
Following these early experiments, solitons have since been observed with much
longer lifetimes. It is now experimentally possible to study the dynamics of dark matter
wave solitons over timescales sufficiently long that an average oscillatory pattern can
be traced. For example, solitons were seen to exist for times on the order of seconds
following phase imprinting [115], and multiple oscillations of the density notch were
clearly visible in the experimental data.
As further evidence of the level of experimental control possible within BEC systems, head on collisions between solitons were studied by Stellmer et al [244], their
results confirming the particle-like nature of soliton collisions. In experiments at Heidelberg, Weller et al. were also able to study the dynamics of multiple solitons [116],
excited through the merger of two separately prepared condensates within a double
well potential. In this case, the solitons were seen to undergo many oscillations within
the trapping potential. They reported that reproducible averages allowed following the
soliton evolution for around 100 ms, and a subsequent analysis of the interactions was
found to further corroborate the particle model of soliton interactions.
One interesting feature of the experiments of Becker et al., was the shot to shot
variation reported: as in any experiment, repeated tests were performed, and an average
taken over these findings. However, solitons were found to oscillate for variable amounts
of time before their signal was lost. It was found that single experimental runs could
yield soliton lifetimes of up to 2.8 seconds, whereas an averaged result was obtainable
for only a tenth of this time [115]. This effect was attributed to preparation errors in
the introduction of the soliton into the atomic sample. However, a similar effect might
be expected to occur in introducing a soliton into a fluctuating background, and is an
effect likely to be enhanced at finite temperatures due to thermally induced fluctuations
81
Chapter 4.
Finite temperature matter waves
in the phase and density.
Numerical simulations of dark soliton dynamics
There are a number of relevant features which should be incorporated in simulating
the finite temperature dynamics of dark solitons. In first instance, a model must
include the nonlinear effects inherent to the existence of these structures, shown to
be well captured within the GPE framework. For low temperature experiments, the
observed average soliton oscillations have been shown to agree well with numerical
predictions of the GPE [115, 116]. Simulations of the three-dimensional GPE can be
computationally expensive, however the one-dimensional non-polynomial Schrödinger
equation has also been employed successfully in simulating experimentally observed
soliton dynamics. This approach approximately accounts for the transverse features
of the system through the inclusion of higher order nonlinear terms in the equation of
motion.
The study of solitons within finite temperature BECs, requires the inclusion of
additional effects beyond those captured by the terms in the T = 0 GPE. The decay
seen in the Hannover experiments of Burger et al., and later theoretical modelling
of this within the ZNG theory [243], shows the important role of thermal dissipative
effects. Similarly, the variation between run to run results of Becker et al. suggests
fluctuations also have a strong influence on the physics of these systems. Hence, in the
following two Chapters, we will discuss methods by which to simulate these features
numerically.
The interaction between a soliton and thermal excitations was considered in [233,
235] by means of a Fokker-Planck equation, under the assumption that the momentum
transfer per soliton-excitation interaction is much smaller than the soliton momentum
(or M ≫ m). Given the picture of a heavy soliton oscillating within a background of
lighter thermal particles, undergoing many scattering events, it is tempting to draw
on the analogy to the Brownian motion of a particle moving within a fluid of lighter
particles. This is suggestive that a Langevin approach is well suited to the study of
such systems, having been originally conceived for just this purpose [168, 174]. As
a Langevin equation for trapped gases, the SGPE captures both the dissipative and
fluctuating dynamics inherent to finite temperature BECs, additionally satisfying the
required balance between these two factors.
It is to be expected that both randomness and dissipation will influence soliton
dynamics in experiments. Initially, we choose to decouple these effects, and consider
first the dissipative behaviour alone through mean field simulations using the DGPE.
A crucial ingredient in these calculations is the form of the damping term which we
will derive from recourse to the SGPE. We will however suggest alternative approaches
82
Chapter 4.
Finite temperature matter waves
to the calculation of this quantity, based on more straightforward means.
4.1.3
Choice of system parameters
So that both the dissipative and fluctuating nature of low dimensional systems are
relevant in our study, we choose the following parameters:
• Axial trapping frequency, ωz = 2π × 10 Hz;
• Transverse trapping frequency, ω⊥ = 2π × 2500 Hz;
• Chemical potential, µ = 395 ~ωz ;
• Particle number, N ≈ 20000
87 Rb
atoms;
• Soliton velocity scaled to the sound speed, v/c = 0.25 (unless otherwise indicated).
These parameters result in the following values for some relevant quantities,
• Tφ ≈ 25nK;
• T1d ≈ 900nK;
• Healing length, ξ ≈ 0.05 lz ≈ 0.0018R;
• Soliton mass, M ≈ 50m.
We choose to consider soliton dynamics for temperatures in the range T = 100nK300 meaning we work in the phase fluctuating regime Tφ ≪ T ≪ T1d . The T ≫ Tφ
requirement is to enhance the statistical spread of soliton behaviour observed already
in experiments, and to assess whether such fluctuations offer a mechanism for this
feature. The coherence length, Lφ is still 0.1R < Lφ < 0.2R for this combination of
temperatures and particle number, so relative to the soliton size, large regions of spatial
coherence can be expected to exist. We wish for the solitons to undergo a number of
oscillations prior to decaying, in order to study the dynamics, so require additionally
T ≪ T1d . However, we also want to see an appreciable amount of decay and therefore
need T not to be too small. Also, while the soliton is very thin relative to the overall
system size for these parameters, it still maintains the condition justifying a FokkerPlanck treatment in [233], that the soliton mass be much larger than the atomic mass,
M ≫ m.
A comparison of the parameters used here and those for various dark soliton experiments carried out to date are shown in Table 4.1. This shows that we consider
a realistic temperature range, with a particle number number and geometry chosen
83
µ/~ω⊥
T /Tφ
Tφ /Tc
T /Tc
Hannover
7
< 0.7
0.7
∼ 0.5
Hamburg
4
< 0.3
1.4
. 0.2
Chapter 4.
Finite temperature matter waves
Heidelberg
2
∼ 0.06
1.4
< 0.1
Our simulations
2
6 < T /Tφ < 11
0.03
0.15 − 0.35 (based on T1d )
Table 4.1: Table showing the parameters used in numerical simulations of soliton dynamics and
a comparison to those of dark soliton experiments carried out to date. Note the main difference
between experiments and our simulations is in the temperature relative to Tφ .
in order to push the system further into the phase fluctuating regime (T > Tφ ) than
previously experimentally probed. Transversely, the soliton stability is comparable to
that of all the experiments of Table 4.1.
4.2
Numerical simulations of dissipative dark soliton dynamics
In the following sections, our aim is to discuss modelling the dynamics of dark matter
waves, assessing the effect of temperature and variations in the spatial form of the
dissipative term of Eq. (2.95).
4.2.1
Choice of initial state
The dissipative term in the DGPE introduces propagation in time such that the time
unit vector no longer points exclusively along the real axis, but instead has an imaginary
component too. This allows the system to relax to its ground state configuration,
damping out excitations above the ground state at a rate dependent upon γ(z).
For similar reasons, a stationary initial state for the GPE can be found by propagating the GPE entirely in imaginary time, as can be understood from the following:
We wish to solve the stationary eigenvalue problem for the GPE
HGP φi = εi φi ,
(4.6)
for i = 0, for which the eigenvalue is ε0 = µ, in order to obtain the ground state.
Starting from the time dependent GPE,
i~
∂φ
= (HGP − µ) φ,
∂t
84
(4.7)
Chapter 4.
Finite temperature matter waves
making the substitution t → −it yields
~
∂φ
= − (HGP − µ) φ.
∂t
(4.8)
Therefore, starting from some initial solution, φ(0) = ϕ, and iterating in time, there
are three possibilities:
(i) HGP ϕ > µϕ, such that ϕ̇ < 0 and solutions above the ground state are damped
out, since HGP φi > µφi for i > 0;
(ii) HGP ϕ < µϕ, which results in ϕ̇ > 0 and so growth;
(iii) HGP ϕ = µϕ which gives ϕ̇ = 0.
Hence, propagation in imaginary time solves the nonlinear eigenvalue problem as required, since the evolution is always towards the stationary solution ϕ = φ0 , satisfying
(iii), with the solutions for modes above the chemical potential damped out. The imaginary time equation takes the form of a heat equation, the long time solution to which
is also a stationary state of the GPE. In fact, the application of heat-type equations
to eigenvalue problems for the Laplacian operator, and the long time behaviour of the
heat kernel in particular, is interesting in itself [245].
As the DGPE dissipates excitations, an initially multi-mode finite temperature
initial state, given by the solution to equilibrium of the SGPE for example, would
be damped to the ground state under this dissipative evolution. Hence, we choose to
consider simulations which describe the dynamics of the condensate mode only, subject
to a damping due to the higher energy excited modes, parameterised by γ(z). So, as an
initial condition, we represent the lowest system mode by the imaginary time solution
of the GPE, which is readily calculated as described.
In principle, increasing the temperature reduces the condensate fraction for a fixed
total particle number. In higher dimensions, one could account for this by solving
iteratively for the condensate plus thermal modes in a Hartree-Fock manner [175],
which results in a self-consistent single chemical potential for the full condensate plus
thermal component system. This is not possible in one-dimension however, due to
infrared divergences which appear [246]. An alternative might be to use the modified
Popov approach of Andersen et al. [217], from which a temperature dependent chemical
potential can be obtained. A further means of obtaining a consistent condensate mode
would be to solve the SGPE to equilibrium, in order to obtain the one-body density
matrix. Diagonalisation of this would yield the equilibrium condensate wavefunction
at any temperature. However, as a first approximation, here we use the same chemical
potential to generate the GPE ground state for all temperatures. This is because,
firstly, we find the variation in SGPE densities is not so great for the temperatures
85
Chapter 4.
Finite temperature matter waves
probed, and secondly, we wish later to compare to the SGPE results, maintaining the
chemical potential as a fixed parameter in performing this comparison.
Upon obtaining an equilibrium initial condition, a dark soliton of specified velocity
v, can be introduced at the trap center by multiplying the wavefunction by the soliton
wavefunction Eq. (4.3). The initial density which results following the introduction
of a soliton with v = 0.25c, is shown in Figure 4.2(a). For the parameters chosen,
the background density profile is very close to the Thomas-Fermi result of an inverted
parabola, also shown by the thick brown curve.
1
(a)
gn(z)µ
0.8
0.6
0.4
0.2
0
-28
-21
-7
-14
0
7
14
21
28
z [lz]
gn(z)/µ
1
(b)
0.8
0.6
0.4
0.2
0
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
z [lz]
Figure 4.2: (a) Comparison between the density profile of the GPE imaginary time solution
(black, thin) and the Thomas-Fermi density profile (brown, thick), for the parameters given in
the text. The GPE solution shows also the soliton initial condition for a speed v = 0.25c; (b)
the zoomed density profile for the soliton within the GPE imaginary time solution (black) and
corresponding signal (red) used by the soliton tracking algorithm.
4.2.2
Tracking a soliton
In addition to the density profiles, there are a number of other properties of interest,
although to extract these, we require knowing the soliton position with time. This
means we must be able to track the position of the soliton. Numerically, this is achieved
using the following algorithm:
1. Store the unperturbed density, nbkg (z).
2. Store the (initial, if t = 0) soliton position, zsol .
3. After some time ∆τ , for the region zsol ±∆τ v, compute nlocal (z) = nbkg (z)−n(z).
86
Chapter 4.
Finite temperature matter waves
4. The soliton position is given by the maximum in nlocal (z)1
In this study, a typical search range was zsol ± 5lz (±100ξ). An example of the signal
which we follow is shown by the red curve in Figure 4.2(b), together with the soliton
it represents, shown in black. Recording the position of the maximum in the signal at
each time yields a trajectory for the soliton centre over time.
It is clear from Figure 4.2(b) that the initially very deep soliton is obviously visible
in the signal followed by the tracking routine, as it is by far the largest density perturbation. While this is the case for the smooth background density of the GP ground
state, we will see later that the situation is different for the noisy background encountered once thermal fluctuations are included. Over the smooth background, however,
we have verified that the routine is able to track the soliton to very small depths, at
which the density perturbation is barely visible in the full density profile.
4.2.3
Calculating an appropriate damping term
In Section 3.6, we discussed the form of the self-energy, ~Σ K . It is this quantity
which determines the damping upon which the DGPE depends. To calculate the selfenergy, an appropriate choice for the potential term U (z, t) must be made, ultimately
depending upon the density present within the trap. In experiments, solitons are
typically introduced into a system with a well formed condensate, otherwise in thermal
equilibrium at some temperature, T . Therefore, in addition to the trapping potential,
the thermal heat bath atoms see a repulsive potential due to the density of atoms
occupying the low-lying modes. The effect of this density can then be included in the
self-energy calculation through retaining the Hartree-Fock contribution to U (z, t). In
the results which follow we use this fact and the Lerch transcendent expression, Eq.
(3.64), in calculating the damping2 [218].
For a sufficiently large atom number, such that we are well within the Thomas-Fermi
regime [64], the effect of introducing a soliton can be justified as a small perturbation to
√
the density: Since the healing length is ξ ∝ 1/ µ in the Thomas-Fermi limit, and the
√
Thomas-Fermi radius is ∝ 2µ, for large µ, the extent of the soliton (which is ∼ O(ξ))
is very small relative to the size of the condensate. It is therefore reasonable to treat
the density of the low-lying modes as unchanged by the introduction of a soliton, in
this regime. So, to good approximation, it is possible to take the equilibrium density,
prior to the introduction of a soliton, and use this in calculating the Hartree-Fock
contribution to self-energy. Moreover, under the assumption of a static thermal cloud,
1
2
In FORTRAN, the intrinsic functions maxval and maxloc are useful in implementing this procedure.
This method was applied here for numerical convenience, due to the large grid sizes employed.
87
Chapter 4.
Finite temperature matter waves
0.015
γ(z)
0.01
0.005
0
-20
0
20
z [lz]
Figure 4.3: Spatial profile of the dissipation γ(z) = iβ~Σ K (z, T )/4 for T=150nK (black, solid),
200nK (red, dashed), and 300nK (blue, dot-dashed). The zero temperature Thomas-Fermi
radius is shown by the brown, vertical dotted lines.
this seems the most consistent approach, as the linear response treatment is valid
for perturbations about the equilibrium state, so the thermal populations should be
calculated taking into account the presence of a well formed condensate, rather than
calculating the self-energy for an empty trap.
Besides the explicit temperature dependence shown in Figure 4.3, the Hartree-Fock
density contribution to the Bose-Einstein distribution energies leads to an additional
temperature dependence. This is due to the changing density profile as the condensate
fraction varies with temperature, leading to a temperature dependent Thomas-Fermi
radius [180]. To account for the density variation, we use the equilibrium result of the
SGPE at each temperature which incorporates this additional dependence into γ(z).
The result of this is shown in Figure 4.3 for three temperatures. Relative to the T = 0
Thomas-Fermi radius, the effect of increasing temperature is to reduce the extent of the
flattened region around the trap centre. This is consistent with the increasing thermal
energy and so a reduction in the condensate fraction as temperature is increased. In
addition, the peaks around the condensate edge increase with temperature, as the
occupation of higher energy modes is enhanced.
4.3
DGPE simulation results
We now move onto a description of the finite temperature dynamics of a soliton of fixed
initial velocity, v/c = 0.25, at various temperatures, using the DGPE. We choose this
velocity, as it yields a fairly deep soliton, meaning a longer lifetime prior to decay, yet
is not so close to being black (v = 0) that simulations take an impractical amount of
time.
88
Chapter 4.
4.3.1
Finite temperature matter waves
Dynamics with damping due to γ(z)
To model the thermal cloud effect in first instance, the full spatial form for the damping
is used. Therefore, the mean-field repulsion due to the density of the condensate and
lowest system modes, has been taken into account in calculating the damping rate.
1
1
(a)
0.6
0.4
0.2
0
-28 -21 -14 -7
0.6
0.4
0.2
0
7
0
-28 -21 -14 -7
14 21 28
z [lz]
1
7
1
(c)
14 21 28
(d)
0.8
gn(z)/µ
gn(z)/µ
0
z [lz]
0.8
0.6
0.4
0.2
0
-28 -21 -14 -7
(b)
0.8
gn(z)/µ
gn(z)/µ
0.8
0.6
0.4
0.2
0
7
14 21 28
z [lz]
0
-28 -21 -14 -7
0
7
14 21 28
z [lz]
Figure 4.4: Density profiles for an initially identical soliton solution to the GPE following
evolution under the GPE (red, dashed) and DGPE with γ(z) calculated at T=200nK (black,
solid). The snapshots are shown for the following times: (a) t = 0ωz−1 , (b) t = 12ωz−1 (c)
t = 33ωz−1 (d) t = 39ωz−1 .
Relative to evolution under the conservative GPE, we expect here to see the density
with the soliton gradually deform to the background equilibrium density, as its velocity
tends towards the speed of sound [233]. In Figure 4.4, density profiles are shown
at several times, showing the evolution of a dark soliton of speed |v| = 0.25c for
both the GPE and DGPE at 200nK. It is evident that while the initial conditions are
identical, it does not take long for decay due to the presence of a static thermal cloud to
affect the soliton, with the soliton-sound interaction perturbed by the damping term.
Under dissipative evolution, the soliton is seen to ‘anti-damp’ [232] as its oscillation
amplitude grows towards the condensate edge. This is due to the decrease in depth of
the soliton, which pushes outwards from the centre the position of the turning point in
the oscillatory motion. This illustrates an interesting peculiarity in soliton behaviour:
as it is characterised by absence of particles, it can also be considered as an effective
classical particle of negative mass [232, 236]. As such, it attains increasingly larger
velocities as it decays and so is able to probe regions of higher energy within the trap
upon successive oscillations.
89
Chapter 4.
Finite temperature matter waves
Figure 4.5, shows the time dependent position of the soliton for T = 0 and for
various non-zero temperatures. For the T = 0 results, we observe uniform oscillations
√
at the predicted oscillation frequency, ωz / 2. Relative to this, the frequency appears
shifted downwards for the finite temperature results. It is clear that dissipation effects
are more prominent as the temperature rises, illustrated also by the enhanced antidamping of the trajectories as the soliton decays. Trajectories such as these are also
able to be extracted in the laboratory [115, 116] (in an averaged manner), so a direct
comparison between theory and (averaged) experiment is possible.
Visibilty(zsol)
nsol / n(0)
zsol [lz]
30
15
(a)
0
-15
-30
1
0.8
0.6
0.4
0.2
0
1
0.8
0.6
0.4
0.2
0
0
(b)
(c)
5
10
15
20
25
30
35
40
45
-1
Time [ωz ]
Figure 4.5: (a) Position, (b) depth (denoted as nsol for t > 0), and (c) visibility for a dark
soliton with initial speed v0 /c = 0.25, predicted by the DGPE with γ(z) calculated using the
SGPE equilibrium density at T = 150nK (black), T=200nK (red), T=300nK (blue), plus the
T = 0 solution of the GPE (brown). In (a) the Thomas-Fermi radius, RT F , is also shown (grey,
dashed). The trapping frequencies are ωz = 2π × 10Hz and ω⊥ = 2π × 2500Hz and the chemical
potential, µ = 395 ~ωz which corresponds to a particle number, N ≈ 20000.
Soliton decay can also be illustrated in the time dependence of the soliton depth,
since this quantity is dependent upon the velocity of the soliton, through the relation
Eq. (4.2), the depth is expected to decrease to zero as the soliton decays. This is also
plotted in Figure 4.5. As well as the decay which is evidenced by the decreasing depth,
an additional feature of note is the oscillation in the depth. This is best resolved in
the 300nK results, due to the shorter time scale for dissipation. The oscillation is due
to soliton-sound interactions, as the soliton periodically emits and reabsorbs phonons,
which cause the depth to vary as the soliton gains and loses energy. In contrast to the
90
Chapter 4.
Finite temperature matter waves
conservative GPE simulation, where the sound energy remains within the trap, in the
dissipative case phonons are emitted which are not necessarily reabsorbed. This leads
to a net decrease in the soliton energy, and hence a reduction in the soliton depth.
Another experimentally relevant quantity is also the visibility of the soliton, which
can be defined as [247],
Vis(z) =
nsol (z)
,
2n(z) − nsol(z)
(4.9)
where nsol denotes the soliton depth for t > 0. The visibility can also readily be
extracted from simulations, and a plot of its behaviour with temperature is shown in
Figure 4.5(c). The points at which the visibility reaches unity are the turning points
in the soliton motion, which occur when the soliton depth is equal to the density of
the background condensate. Again, the higher temperature results quickly become out
of phase with the results of the T = 0 GPE simulation. This quantity may be related
to experiments, as it is closely related to how well a soliton can be resolved within
experimental images [248].
4.3.2
Comparison between γ(z) and spatially independent damping
A spatially constant value of the damping coefficient has been used in a number of
numerical studies using variants of the DGPE [113, 204, 200, 202, 203]. There are a
number of possible ways in which to extract a spatially constant value for the damping
from the ab initio γ(z), and we consider two such possibilities here: the central trap
value, γ0 = γ(0) and the average value of the damping across half the T = 0 ThomasFermi radius about the trap centre,
γ̄ =
Z
R/2
γ(z) dz.
(4.10)
−R/2
A comparison between these quantities and the ‘bare’ rate [125, 204]
γbare = κ ×
4m a 2
πβ ~
(4.11)
is given in Figure 4.6. In matching to experiment, for example to recover experimentally measured growth rates, the ‘bare’ rate is typically augmented with an additional
pre-factor, which we denote here as κ, with a value in the range 1 − 10. This is to
account for the neglect of the logarithmic and Lerch transcendent terms of Eq. (3.64).
Consistent with this, we find the ‘bare’ rate to match the various forms for the damping
employed, for 1 < κ < 5. It is clear, however, that the pre factor itself should also
have a temperature dependence, suggesting that matching α to fit an experiment at
one temperature, is not necessarily sufficient to fit all temperatures.
91
Chapter 4.
Finite temperature matter waves
Damping coefficient, γ
0.025
0.02
0.015
0.01
0.005
0
100
200
300
400
500
600
700
800
T [nK]
Figure 4.6: Temperature dependence of two forms for the spatially constant damping coefficient,
γ̄ (blue squares) and γ(0) (red crosses) with a comparison to the ‘bare’ damping coefficient used
in [204] for the pre-factor κ = 1 (brown, dotted) and κ = 4 (brown, dashed). Also shown in
green are fits to a function ∝ T 3 for the results which are shown by the dot-double dashed line
in the γ̄ case and dot-dashed for the γ(0) case.
For the system considered here, we find γ ∼ T 3 , as indicated by the fits in Figure 4.6.
This means that the damping rate, which is given by |Σ K |, actually varies ∼ T 4 , since
|Σ K | =
4γkB T
.
~
(4.12)
This temperature dependence is consistent with the result for a particle within a quantum dissipative system, where the decay mechanism is the scattering of excitations
[249].
As we use the spatially dependent γ(z) calculated already, each of the spatially
constant damping terms are only obtained once a suitable finite temperature density
has been determined, in order to account for the Hartree-Fock contribution to the high
lying thermal particle energies. To avoid the need to solve the SGPE for this purpose, an
estimate may be obtained by calculating only 2gn(0), or the equivalent average across
R/2 about the trap centre, for γ0 or γ̄, respectively. This could be done by making
an estimate for the condensed fraction from ideal gas relations, for example, and using
the Thomas-Fermi profile corresponding to the appropriate condensate number. The
divergence in the self-energy shown in Figure 3.18 at the Thomas-Fermi radius, would
then be unimportant as we would not be interested in γ(z) in this region. Hence, it is
useful to know if the results for γ(0) and γ̄, yield similar findings to γ(z), given it is
possible to more easily estimate these.
The dynamics which result from the use of each form for γ can be seen in Figure
4.7. Although dependent upon the observable in question, the agreement is reasonable
92
Chapter 4.
Finite temperature matter waves
for the temperature shown. The soliton trajectories in Figure 4.7(a) agree well in all
cases, while the soliton depths are more sensitive to variations in the spatial form for
the damping. Initially, the γ(z) data lies above the γ̄ data in figure 4.7(b), but the
curves cross as the soliton begins to probe the outer regions of the trapping potential,
where the thermal cloud largely resides at or close to equilibrium, and the damping is
enhanced due to the increase in |γ(z)| in this region. Thus, the spatially dependent
γ(z) seems more consistent with the underlying mechanism of dissipation being that of
scattering due to thermal excitations, as this is expected to increase at the edge of the
condensate region, where the majority of thermal atoms reside. This behaviour is also
demonstrated in the visibility, shown in 4.7(c). The data for γ0 displays a noticeably
slower damping, indicating that taking this value would lead to an overestimate of
the lifetime of a soliton, while the data for γ̄ and the full form γ(z) produce similar
Visibilty(zsol)
nsol / n(0)
zsol [lz]
predictions for this observable.
30 (a)
20
10
0
-10
-20
-30
1
0.8
0.6
0.4
0.2 (b)
0
1
0.8
0.6
0.4
0.2
0
(c)
0
15
30
45
60
75
-1
Time [ωz ]
Figure 4.7: Comparison between observables for the dynamics of a dark soliton at T = 200nK
where the damping coefficient is given by γ(z) (black), γ̄ (red) and γ(0) (blue). Shown are (a)
position, (b) depth and (c) visibility of the soliton versus time.
Soliton lifetimes
One means of quantifying the comparison between different forms for the damping
terms is to look at the rate at which a soliton decays. This can be characterised by
looking at the soliton lifetime, which we consider here by measuring the soliton half-life,
93
Chapter 4.
Finite temperature matter waves
τ1/2 . We have chosen to define this as the time it takes for a soliton to decay to half its
initial depth, as this is a relevant quantity for experiments. To show how the soliton
dynamics compare within each form for the damping as temperature is changed, these
lifetimes are shown in Figure 4.8. The spatially dependent γ(z) results display a very
similar behaviour to the results obtained using the integrated damping, γ̄, however the
agreement is less good between these, and the simulations in which the damping was
set equal to the value at the trap centre used.
180
150
-1
τ1/2 [ωz ]
120
90
60
30
0
100
150
200
250
300
T [nK]
Figure 4.8: Soliton half-life, τ1/2 , versus temperature. Shown are the results for the spatially
dependent damping coefficient γ(z) (green diamonds), and spatially constant damping coefficients γ̄ (blue squares) and γ(0) (red crosses). Also shown is a fit to the data for γ(z) to a
function f (T ) = a T b , which gives the values a = 1.8 × 1010 and b = −3.83.
Following a least squares fit of each set of data to a polynomial function of temperature with a variable power, f (T ) = a T b , the half-life data gives the power laws shown
in the following table. This suggests that the soliton lifetime varies with temperature
a
b
γ(z)
1.8 × 1010
−3.835
γ̄
1.6 × 1010
−3.828
γ(0)
1.8 × 109
−3.343
Table 4.2: Table showing the results of a least squares fit to the τ1/2 data, where a is the
coefficient and b represents the power law dependence upon temperature.
in a way that is close to a quartic variation in the damping rate, as we would expect the
lifetime to be inverse to the damping rate. That this is recovered to good accuracy, is a
nice feature and also a good test of the numerics. Furthermore, as would be expected,
in the limit as T → 0, this relation for the half-life leads to the conclusion that the
94
Chapter 4.
Finite temperature matter waves
soliton becomes stable. Practically, this should be so, as we would reduce to solving
the GPE which we know leads to undamped oscillations. Physically, this must be the
case as there is no mechanism for the production of thermal excitations in this limit.
Interestingly, similar ∼ T 4 behaviour has also been found in studies on dissipation
within quantum systems [114], in the limit where the long-wavelength physics dominates. In a BEC context, it has also been shown that in the quantum regime, where
kB T ≪ µ [233, 250], this is the expected behaviour, while in the classical regimes of
kB T ≫ µ it is expected that the lifetime of a soliton should decrease linearly with
temperature [235, 250]. For the parameters chosen here, we are in the regime where
0.65 < kB T /µ < 1.6, and so we find that the low-temperature behaviour is also apparent in the intermediate regime where kB T ∼ µ. It is interesting that this should be
the case, as in [250] it is argued that for decay via two vertex processes, i.e. Raman
processes, it is ultimately only long wavelength phonons which participate within the
dissipative dynamics, and that phonons with wavelengths shorter than the soliton size
practically do not interact with the soliton. For the kB T ∼ µ regime, we might expect
that short wavelength physics may start to play a role. For the parameters considered
however, the characteristic thermal length scale, λdB , compared to the scale relevant
to a soliton, the healing length, is
2<
λdB
< 3.2.
ξ
(4.13)
Therefore, relative to the soliton, the thermal excitations may still be considered as
long wavelength and so expected to interact appreciably with the soliton.
4.4
Three-dimensional analysis of dissipative soliton dynamics
As discussed, both thermal and transverse degrees of freedom break the integrability of
the GPE, even in the homogeneous case. Within a trap, each disrupts the soliton-sound
interaction, which for the one-dimensional harmonically trapped case, otherwise leads
to a dynamical equilibrium of stable, periodic soliton oscillations [232, 236]. Having
considered the dissipative dynamics in one-dimension, we now extend the analysis to
an equivalent three-dimensional setting. In doing so, we aim to validate the onedimensional study undertaken in the previous section, by determining the influence of
transverse effects on the soliton.
The issues we wish to address are, firstly, whether the full 3d simulations reveal
a damping of the soliton even at zero temperature, and secondly, whether the threedimensional dissipative dynamics are well represented in one-dimension. Before pro-
95
Chapter 4.
Finite temperature matter waves
ceeding with these issues, we discuss first the issue of translating our 1d parameters
into the most appropriate 3d system.
4.4.1
Initial state
To compare the one- and three-dimensional results, we first require the appropriate
parameters to match the axial wavefunctions. With the same trapping potential, and
other parameters fixed to those used in the one-dimensional study, this is done by
varying the 3d chemical potential, µ3d , until the correct density profile is obtained.
The parameters which provided the best match were µ3d = 1.425µ = 563~ωz ; in 3d
this yields a particle number of N ≈ 20950, which is within 5% of the one-dimensional
particle number used.
1
1
(a)
0.8
n(r)/n(0)
n(z)/n(0)
0.8
0.6
0.4
0.6
0.4
0.2
0.2
0
(b)
-600 -400 -200
0
200 400 600
0
0 0.05 0.1 0.15 0.2 0.25
r [lz]
z [ξ]
Figure 4.9: Plots showing: (a) axial density profiles for the 1d (blue, solid) and 3d (orange,
dashed) simulations; (b) radial density profiles for the 3d case, the lowest transverse harmonic
oscillator state density (blue, solid) and the fit to density for the function n⊥ (r) (green, dotdashed) defined in the text.
A comparison between the 1d and 3d densities is shown in Figure 4.9(a), which shows
a good match can be obtained in the manner described. Figure 4.9(b) shows the
transverse density profile for these parameters, and a comparison to the transverse
ground state density n⊥ (r) = exp −(ω⊥ /ωz )r 2 /σ 2 , shown by the blue curve of Figure
4.9(b). Here, ω⊥ is the same as that used in calculating an effective scattering for the 1d
simulations through Eq. (1.8) and r is in units of lz . For the true transverse harmonic
oscillator ground state, σ = 1lz , and fitting n⊥ (r) to the transverse density from the
3d simulations, shown by the orange dashed curve, gives σ = 1.5lz . The transverse
wavefunction is then close, but not exactly in the ground state. This is likely due to
the fact that µ/~ω⊥ ≈ 1.6, and to further freeze out modes above the transverse ground
state requires µ/~ω⊥ to be smaller.
96
Chapter 4.
4.4.2
Finite temperature matter waves
Transverse stability
Based on the criterion of Muryshev et al. [235], solitons are expected to be dynamically
stable for the geometry and particle number chosen. For this to be true at all soliton
speeds, it was shown in this work that that the system must satisfy
gn(0)
< 2.5,
~ω⊥
(4.14)
with the critical value for this ratio increasing with soliton velocity. That is faster
solitons are more stable transversely. For the parameters used here, the ratios are
gn(0) = 1.92~ω⊥ and gn(0) = 1.56~ω⊥ , for the 3d and 1d cases respectively.
We have numerically verified the stability of the pertinent case of a soliton with
zsol [ξ]
speed v = 0.25c in 3d, via direct GPE simulations as shown in Figure 4.10.
200
150
100
50
0
-50
-100
-150
-200
(a)
0
15
30
45
60
75
60
75
-1
Time [ωz ]
nsol/n(0)
1
(b)
0.9
0.8
0
15
30
45
-1
Time [ωz ]
Figure 4.10: Soliton (a) position and (b) depth relative to the condensate peak density for the
1d (blue, solid) and 3d (orange, dashed) GPE simulations.
It is clear from these results that both the trajectory and soliton depth do not show
signs of the soliton decaying. The inclusion of transverse dynamics does lead to an
additional oscillation in the soliton depth however, relative to the pure 1d results,
which leads to a shift in phase between the 1d and 3d soliton trajectories, as shown in
Figure 4.10(a).
97
zsol(t) [ξ]
Chapter 4.
25
20
15
10
5
0
-5
-10
-15
-20
-25
0
20
40
Finite temperature matter waves
60
t
80
100
-1
[ωz ]
Figure 4.11: Soliton position for 1d DGPE with γ̄ and T = 150nK (blue, dot-dashed) and 3d
GPE (orange, solid) simulations. The black, dashed lines indicate the amplitude of the first
oscillation of the 3d GPE soliton.
In total, the 3d soliton was allowed to execute around 11 full oscillations, as shown in
Figure 4.11. The soliton oscillation amplitude was found to increase by only 7%, which
should be compared to an increase of around 180% for a similar number of oscillations
using the 1d DGPE, even at the relatively very low temperature of T = 150nK, shown
by the blue, dot-dashed curve of Figure 4.11. Hence we can be confident that coupling
to transverse modes will have little impact on the soliton decay times in 3d, which we
now also consider.
4.4.3
3d vs. 1d DGPE results
The soliton half-life is a good quantity on which to base a comparison between the
3d and 1d dissipative results, as the soliton depth is naturally scaled to the central
density, which differs between the 1d and 3d cases. We again track the soliton during
its evolution and obtain the half-life as the time taken to reach half its initial depth.
Error bars are obtained by the time taken to reach depths 0.45 and 0.55 times the
original depth, as previously considered by Jackson et al. [243] and in Figure 4.8.
The damping considered in the 3d case at each temperature is γ̄, so the 3d results
should agree most closely with the 1d results for the corresponding damping. In Figure
4.12 this is the data shown by blue squares (1d) and orange triangles (3d). Shown in
Figure 4.12, is a log-log plot of the soliton half lives versus temperature. The 3d data
matches well the results obtained through the various 1d simulations, as expected.
For completeness, also plotted is a straight line with gradient -4, together with the
half-lives found for other forms of the damping. We see all of the decay times display a
gradient close to that of this line, particularly for low temperatures. The temperature
corresponding to the chemical potential, µ/kB = 189nK, is shown by the vertical dashed
line. A crossover to the predicted high temperature behaviour, in which the damping
98
Chapter 4.
Finite temperature matter waves
should be ∼ T might be expected to occur at kB T ≈ µ, and the gradient in Figure
4.12 certainly appears to be becoming more shallow for the higher temperature points.
More data would be required to make a more solid statement on this trend however.
5.5
5
4.5
ln(τ1/2)
4
3.5
3
2.5
2
1.5
4.8
5
5.2
5.4
5.6
5.8
ln(T[nK])
Figure 4.12: Log-log plot of τ1/2 vs. temperature for the 1d and 3d DGPE simulations. The
results for 1d γ(z) are shown by the green diamonds, 1d γ̄ by the blue squares, 1d γ(0) by
the red crosses and the 3d γ̄ results are shown by the orange triangles. The diagonal dashed
line has a gradient of −4 and the vertical line indicates the temperature corresponding to the
chemical potential on this scale.
4.4.4
Comparison to ZNG
We conclude this section with a comparison to soliton half-lives obtained from simulations using the ZNG theory for finite temperature Bose-Einstein condensates [154, 159].
These results are extracted from Figure 4 of Jackson et al. [243]. As this data was
from 3d simulations, scaled to the 3d critical temperature, we scale the 1d results obtained for the DGPE with γ(z) to the characteristic quasi-condensation temperature,
Eq. (1.10). On performing this scaling, we find the results shown in Figure 4.13.
As in the original figure of Jackson et al., the filled triangles in Figure 4.13 represent
data from collisionless simulations, meaning the scattering terms in the ZNG Boltzmann
equation are both set to zero, C12 = C22 = 0. The open triangles correspond to the
half-lives for collisional simulations in which C12 6= 0, C22 6= 0. We see that the DGPE
results lie somewhere between the results of these simulation types, at the temperatures
for which a comparison was possible. Arguably the DGPE data follows more closely the
behaviour of the collisionless results (filled triangles), which is as might be expected,
since both approaches include the dissipative effect of the thermal cloud in a mean field
way in this limit.
99
Chapter 4.
Finite temperature matter waves
50
-1
τ1/2 [ωz ]
40
30
20
10
0
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
T/Tc
Figure 4.13: Dark soliton half-lives for the 1d DGPE with γ(z) (green, diamonds) and 3d ZNG
simulations, taken from Jackson et al. [243]. The filled (hollow) triangles represent data from
collisionless (collisional) simulations. The red, dashed line indicates the fit (∼ T −4 ) to the 1d
DGPE data. For the 1d data, temperatures are scaled to the characteristic temperature of
Ketterle and van Druten, Eq. (1.10), whereas for the 3d ZNG results, Tc is given by Eq. (1.4).
A comparison to the SGPE results would be desirable also, however as we shall
see in the next Chapter, it is not possible to extract a half-life at all temperatures
within the SGPE simulations, as the magnitude of the thermal background noise varies
strongly with temperature. To make such a comparison, we would need to analyse the
original ZNG data by extracting new decay times, where the decay depth would be
defined in a temperature dependent manner.
4.5
Chapter Summary
To summarise, neglecting the noise in the SGPE leads to the DGPE, which can describe
the dissipative nature of dynamics within a BEC at finite temperature. In this Chapter
we applied this as a model for soliton decay within a condensate surrounded by a static
thermal cloud. We discussed various possible forms for the damping term, γ(z), and
compared spatially varying and homogeneous approximations for this, based upon the
ab initio damping rate obtained from the SGPE fluctuation-dissipation relation. We
find that in the regime µ ∼ kB T , the soliton half-life behaves as
τ1/2 ∼ T −3.83 ,
(4.15)
which is close to that for a system in which the principle decay mechanism is the
scattering of long wavelength thermal excitations (∼ T 4 ). This should be compared
to the findings of previous studies in complementary parameter ranges, kB T ≫ µ and
kB T ≪ µ, with our findings in agreement with the latter.
100
Chapter 4.
Finite temperature matter waves
As a check on the validity of our 1d simulations, we performed also 3d, cylindrically
symmetric, simulations of the DGPE with a spatially constant damping, γ̄. Good
agreement was found between the soliton half-lives predicted in each method.
Finally, we made a comparison of the 1d DGPE results to 3d simulations, carried out
using the ZNG method for equivalent parameters. The soliton half-lives were found
to be comparable between the two approaches, with the DGPE results found to lie
between the collisional and collisionless ZNG data, though more closely following the
trend of the latter.
In addition to dissipative effects, we now look to simulations which satisfy the
fluctuation-dissipation relation, considering in the next Chapter the effects of fluctuations upon soliton dynamics.
101
Chapter 5
Stochastic effects in finite
temperature matter waves
The motion of particles within a gas at finite temperatures has both dissipative and
fluctuating characteristics. In this Chapter, we wish to address the role of each of these
on the motion of a dark soliton within a finite temperature Bose-Einstein condensate.
We start with some introductory comments motivating the stochastic approach we implement and placing our study in context within the wider field of dissipative quantum
systems.
Beyond this, we discuss extracting meaningful data from soliton simulations, which
again raises the issue of single run versus averaged data, though offers strong parallels
to experiments. To characterise the soliton dynamics, we consider initially the motion
of individual trajectories, and the effects of propagation within a background sea of
thermal excitations. We then move to a fuller characterisation of the distribution of
decay time statistics for the ensemble as a whole, comparing also the average SGPE
results to those of the DGPE. Finally, we consider dark solitons within thermal classical
field simulations using the GPE as the equation of motion, and the SGPE to generate
an initial ensemble of states. In this case, we find the average trajectory to coincide
with that of the zero temperature GPE, even for the finite temperature initial states.
5.1
Introduction
Highly elongated trapping potentials are necessary if a soliton is to be stable against
transverse instabilities. A stochastic approach is then particularly suited to the study
of these essentially one-dimensional objects, as the inclusion of phase and density fluctuations is likely to prove essential in capturing all the salient aspects of the necessarily
low dimensional systems in which soliton experiments must be performed. Moreover,
in a general sense, the fluctuation-dissipation relation embodies the fact that dissi102
Chapter 5.
Stochastic effects in finite temperature matter waves
pative effects are accompanied by fluctuating forces which should also be taken into
consideration. In retaining only the dissipative contribution to the dynamics beyond
Gross-Pitaevskii level, this relation is not satisfied.
The link between the quantisation of classical theories permitting soliton solutions,
and dissipative quantum systems was discussed in [114, 249, 251]. It was highlighted
that just as the motion of a classical particle within a viscous environment has both a
damped and fluctuating component, the situation is the same in the quantum case. The
motion of such a classical particle can then be characterised by two system properties,
a damping due to the systematic force applied to the particle and a diffusion related
to this interaction [173]. This was shown to be true also in the quantum case [171],
in which the dissipation manifests instead as a damping of the particle wavepacket
centre of motion, whereas diffusive effects lead to a spreading of the wavefunction for
the particle [249]. For the case of solitons, the former would lead to a damping of
the motion of the soliton centre, while the latter to an increased uncertainty in the
soliton position. For soliton solutions to integrable, classical one-dimensional theories,
in which case dissipation has no role, the propagation in space is undamped and the
soliton dynamics is entirely captured by knowledge of the soliton centre. However, in
the quantised field-theory, at finite temperature not all degrees of freedom ‘collaborate’
in the formation of the soliton [249] and the result is a residual interaction which is
shown to lead to a Brownian type motion of the soliton. Therefore, the dynamics is
no longer entirely captured by knowledge of the centre of mass alone, as the diffusive
nature also becomes important.
In [249], the Brownian nature of the soliton motion is related to the coupling of the
soliton to the other system modes and excitations due to the presence of the soliton.
The damping and diffusion in the quantum dissipative system must be temperature
dependent, since the excitations which scatter from the soliton are thermally activated.
Returning to the BEC context, there is an obvious analogy between this work and
a soliton propagating within a finite temperature BEC. In the previous section, we
considered the purely dissipative dynamics, and found the soliton to decay, with a halflife which varies approximately ∼ T −4 . This behaviour has been found in studies of a
one-dimensional, homogeneous system of bosons [233, 250], in the latter of which the
dominant decay process was proposed to be interactions via two phonon processes. It
was also noted that a full treatment of the soliton dynamics should satisfy a fluctuationdissipation theorem, neglected in only retaining the damping aspect of the viscous
thermal background. Similarly, this low temperature dependence was found in a study
on polarons [251], and also in the more general case of the ‘quantum impurity problem’,
applied in the setting of a heavy particle within a Luttinger liquid [252].
Experimentally, there is justification for modelling beyond the deterministic dissi-
103
Chapter 5.
Stochastic effects in finite temperature matter waves
pative dynamics too, as solitons detected in experiments have been seen to exist for
times much longer than a reproducible average trajectory could be produced [115, 116],
which is also suggestive that fluctuations are important. In addition, the necessity for
repeated runs in the stochastic formalism has a strong link to the experimental approach of averaging over successive realisations of any experiment, so incorporates this
effect naturally.
In this section we consider the scenario of a soliton propagating in a fluctuating
background density, within an experimentally relevant harmonic trapping potential.
First, we will discuss the relevant initial state and damping used in the stochastic
simulations before examining the variation in the initial speed of the soliton, due to
its introduction into a fluctuating background. We then discuss various means of
extracting data from the stochastic simulations, in which we find a consistent method
coincides with the idea of the soliton centre as a true quantum dynamical variable
[249]. We then consider the decay of a dark soliton within a fluctuating background
density, and characterise the statistical form of the resulting distributions of decay
times. Finally, we compare the SGPE results to the DGPE findings, and also assess
the importance of the heat baths terms of Eq. (2.92) to the soliton behaviour.
5.2
Numerical simulations of stochastic dark soliton dynamics
As in the dissipative simulations of the previous section, it is important to choose
an appropriate initial condition and form for the damping, in solving the stochastic
equation Eq. (2.92). To allow for a straightforward comparison to the DGPE results
of Chapter 4, we retain the same system parameters, listed below for completeness:
• ωz = 2π × 10 Hz
• ω⊥ = 2π × 2500 Hz
• µ = 395~ωz
• N ≈ 20000
87 Rb
atoms
The form for the damping is also unchanged, with the full spatially varying form used
for γ(z) in the SGPE simulations.
5.2.1
Initial condition and damping term γ(z)
While the system parameters are unchanged, a difference does present in the nature
of the initial condition. Rather than the GPE ground state, we use the analogous
104
Chapter 5.
Stochastic effects in finite temperature matter waves
stationary state for the Langevin equation Eq. (2.92). This is given by the classical
equilibrium of the low-lying modes represented by ψ, which differs in a few ways to
the GPE ground state. Firstly the stochastic order parameter represents many modes,
which are populated due to the action of the noise term in Eq. (2.92). This seeds growth
into modes above the ground state, up to the energy cutoff due to the numerical grid.
Furthermore, the equilibrium state varies with temperature even for a fixed chemical
potential, due to the temperature dependence of the thermal occupation of modes above
the condensate. For the GPE case, the system number is controlled by the chemical
potential alone. In addition, the SGPE requires a new initial condition per realisation
of the noise, so simulations become more computationally intensive. A means by which
we address this is to use the Condor high throughput system, as discussed in Appendix
A.
1.5
1.5
(a)
(b)
1.25
1
0.75
1
0.5
0.25
0
0
10
20
30
gn(z)/µ
gn(z)/µ
1.25
gn(z)/µ
z [lz]
0.75
0.5
(c)
0.2
0.25
0.1
0
20
30
25
35
0
-0.5 -0.25
0
0.25
0.5
z [lz]
z [lz]
Figure 5.1: (a) Density profiles of the GPE ground state (blue, dashed) and the SGPE equilibrium solution for T = 150nK, each solved for the same value of µ. For the latter, the result
for a single realisation of the noise is shown by the solid, red noisy curve and the averaged
result is shown by the green, solid curve; (b) Zoomed density of a soliton in the same single
run stochastic wavefunction (red, solid) and GP ground state (blue, dashed); (c) as (a), but
focussed about the T = 0 Thomas-Fermi radius. The grid spacing here is ∆z = 0.00625lz .
To undertake the SGPE simulations, we must first calculate the damping γ(z) as
outlined in the previous section. This requires the averaged, equilibrium density profile
from the SGPE at each temperature. However, the equilibrium which results on solving
Eq. (2.92) is unaffected by the choice of γ(z), so the density can be calculated more
rapidly through the use of an artificially high value (typically ∼ 0.01). To obtain an
105
Chapter 5.
Stochastic effects in finite temperature matter waves
initial condition for the stochastic simulations, the SGPE is again solved to equilibrium
with an artificially enhanced value for γ(z), before switching to the form calculated ab
initio using Eq. (3.64). Though the equilibrium should be unaffected by this change,
prior to introducing the soliton, the system is allowed to propagate unperturbed for a
period with the self-consistent damping, to negate the effect of any transient behaviour
due to abruptly changing the value of γ(z).
A comparison between a soliton in the GP ground state and the SGPE stationary
solution is shown in Figure 5.1. The thermal effects included in the stochastic approach
are apparent in the noisy density of the single realisation, though the averaged result
is seen to yield a much smoother density and more closely resembles the T = 0 density.
The main difference between the latter two profiles is seen at the trap centre, where
the finite temperature peak is lower, due to the inclusion of Landau and Beliaev effects
[108], and also in the wings of the averaged stochastic result where higher energy
thermal particles, above the condensate mode, reside.
5.3
SGPE simulation results
The results presented in this Section are based upon the introduction of an identical soliton over many repeated tests (typically 200), using the SGPE to model the
influence of thermal fluctuations upon dark solitons. We begin by looking at the effect such fluctuations have on the soliton upon its introduction into an equilibrium
quasi-condensate.
5.3.1
Dependence of soliton speed on thermal fluctuations
We now discuss the influence of fluctuations on the soliton initial state. To generate
solitons experimentally, the technique of phase-imprinting is often used [237], whereby
a phase gradient is imparted along an atomic sample in order to excite a soliton.
However, in order to remove additional variables from our study a soliton is introduced to the system by multiplying the (stochastic) wavefunction by the soliton solution
Eq. (4.3), as for the DGPE case studied in the previous Chapter. Since quantifying the
effect of phase fluctuations is a primary aim in using the stochastic approach, a consistent manner of soliton creation should ensure that differences between the dissipative
and stochastic dynamics can be associated purely to this effect.
To determine the initial speed, the soliton depth is extracted upon introduction to
the system, and the speed inferred from Eq. (4.2). In calculating this, the variation
in n(0) with temperature is also accounted for, by scaling each soliton depth to n(0)
for that temperature. Despite the identical nature in which the soliton is introduced
in each run, a spread in the initial soliton speeds is found across the ensemble of
106
Chapter 5.
Stochastic effects in finite temperature matter waves
trajectories. This spread is illustrated by the histograms in Figure 5.2(a), where the
lower temperature result displays a narrower width relative to the higher temperature
Probability density
data.
15
(a)
10
5
0
0.1
0.2
0.15
0.3
0.25
0.35
v0/c
0.3
<v0/c>
0.28
(b)
0.26
0.24
0.22
0.2
0.18
100
150
200
250
300
T [nK]
Figure 5.2: (a) Normalised histogram showing the initial velocity spread for solitons created
with an identical initial condition, corresponding to v0 /c = 0.25, obtained from SGPE simulations. The data shown is for temperatures T=150nK (black, 1000 samples) and T=300nK
(red, 1000 samples), with other parameters as in figure 4.5. (b) The mean velocity, based upon
the measured initial depth, versus temperature.
Since the means of adding the soliton is identical from run to run, the observed variations can be attributed to the fluctuating background density to which the soliton is
added. From Figure 5.2(b), we can see that the mean velocity is approximately equal
to the input value of v = 0.25c, though it is typically slightly below this value. The
data shown in Figure 5.2 suggests that the speeds are not evenly distributed about the
mean, as one might expect, and instead we find the data shifted and to have a slight
positive skewness. The distribution has a longer tail towards lower speeds, indicating
that the solitons in some cases will have an initial velocity which is far less than the
mean, yet relatively high initial speeds occur more rarely. In terms of the expected
soliton lifetimes, we might expect to see long lived solitons purely on the basis of this
alone. However, as the coupling to the heat bath is maintained throughout the soliton
lifetime, interactions during this period will certainly have an effect also. We now move
on to an analysis of the influence this has on soliton decay.
107
Chapter 5.
5.3.2
Stochastic effects in finite temperature matter waves
Extracting soliton data: averaged vs. single run densities
As shown by the density profiles in Figure 5.1, a soliton can be identified with a notch
in the ambient background atomic density, which is smooth when given by the T = 0
GPE ground state. By contrast, a single density realisation of the SGPE is noisy, and
only after sufficient averaging over the noise do we obtain a relatively smooth density
profile. We now consider how this averaging process affects the density profile of a
soliton.
Stochastic effects on a moving soliton
A typical density which results upon averaging is shown in Figure 5.3(a)-(b). The
densities shown are the result of taking the ensemble average h|ψ(z, t)|2 i at several
times.
1
1
(a)
gn(z)/µ
gn(z)/µ
t=t3
0.8
0.8
0.6
0.4
t=t2
0.6
t=t1
0.4
0.2
0.2
0
0
(b)
g<nsol>/µ
t=0
1
0.8
0.6
0.4
0.2
0
0
-20
0
20
(c)
t=t1
0
2
z [lz]
z [lz]
t=t2
t=t3
0.1
0.2
0.3
-1
Time [ωz ]
Figure 5.3: SGPE density profiles following the introduction of a soliton for T = 300nK. The
density profiles are obtained following averaging over 200 runs; shown are (a) total and (b)
zoomed ensemble averaged density snapshots at times t = 0, t1 = 0.0405ωz−1, t2 = 0.081ωz−1
and t3 = 0.304ωz−1; (c) shows the ensemble averaged soliton depth versus time. The times
corresponding to the density snapshots of (a) and (b) are labelled by arrows.
However, due to the stochastic nature of the soliton evolution, averaging over the ensemble of densities does not yield the expected oscillatory behaviour. Instead, because
of the slight difference in the position of the soliton between runs, we find that information due to the soliton part of the wavefunction is washed out by the averaging process.
108
Chapter 5.
Stochastic effects in finite temperature matter waves
A similar effect has also been observed in studies on vortex formation from a rotating
thermal cloud, using a related SGPE formalism [190].
Rather than an oscillating soliton, we observe the soliton filling up over a very short
time scale, prior even to completing one quarter of an oscillation. This behaviour is
shown in the densities of Figure 5.3(a) and Figure 5.3(b). Considering Figure 5.3(c)
however, we see that the ensemble average of the soliton depth, hnsol (t)i, does not
show any such rapid decay, with the depth remaining approximately constant over the
short timescale shown. This implies that the information due to the soliton cannot be
extracted by averaging over the density constructed from the total matter field.
So, to clarify, the disparity highlighted in Figure 5.3 is that while the soliton is lost
within the average density profile, extracting information on the soliton from individual
runs shows that a soliton still exists within all of these runs. If the soliton decayed
within a number of the individual cases, we would see the average depth decrease over
time in Figure 5.3(c). At least, this would certainly be the case if the solitons decayed
as fast as the notch in the averaged density does in Figures 5.3(a)-(b). Therefore, extracting information on the soliton from individual runs, and then afterwards ensemble
averaging this observable, appears to preserve to the relevant physical content within
the simulations.
Stochastic effects on a static soliton
To isolate translational effects from the effect of the stochastic background, we instead
consider a black soliton situated at the trap centre. This is again introduced once
equilibrium is reached, and the behaviour which follows due to interactions with the
thermal heat bath atoms is investigated.
Like the v > 0 case, we also see a filling of the soliton, despite the fact that now
the soliton is introduced with zero velocity, as shown in Figure 5.4(a). Note also the
similarity in appearance between this behaviour and the soliton solutions of differing
speed shown in Figure 4.1.
The central density is found to rapidly tend towards the equilibrium value, as shown
in Figure 5.4(b), and the growth is found to be a good fit to the function
n(0, t) = α [1 − exp (−Γ t)] ,
(5.1)
with α ≈ 0.86 and Γ ≈ 20.68. The rate Γ gives a characteristic growth time of
Γ −1 ∼ 0.05ωz−1 for T = 300nK. This has the same form as the expression describing
the final stages in the growth of a condensate under strong cooling, applied in [253],
and proposed in [220].
109
Chapter 5.
Stochastic effects in finite temperature matter waves
gn(z) /µ
1
0.8
0.6
0.4
(a)
0.2
0
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
z [lz]
1
gn(0)/µ
0.8
0.6
0.4
0.2
0
(b)
0
0.2
0.1
0.3
-1
Time [ωz ]
Figure 5.4: (a) Evolution of the average density profile of a quasi-condensate following introduction of a static soliton at the trap centre. Density snapshots are shown for t = 0 (black,
solid), t = 2.03 × 10−2 (red, dashed), t = 4.05 × 10−2 (blue, dot-dashed), t = 0.101 (brown,
dot-dot-dashed), and t = 0.304 (green, dot-double dashed); (b) Growth of hn(0, t)i with time
(black, solid) and a fit to the function f (t) = α [1 − exp (−Γ t)] (orange, dashed) as described
in the text. T = 300nK in these simulations.
This filling process is similar to that considered by Dziamaga [254], although there
it was quantum effects at T = 0 that were found to lead to a diffusion in the soliton
position, and so a filling of the hole which appears in the diagonal of the particle
density matrix. This is however very much analogous to the diffusion due to thermal
fluctuations, which fills the density notch in Figure 5.4(a).
While the average density suggests a soliton decays rather rapidly, a question remains as to the fate of the solitons within single runs which contribute to the averaged
results of Figure 5.4. We consider this in Figure 5.5, where we plot the density profiles
for a representative single realisation of the noise.
In this case, we see that following the introduction of an initially stationary soliton,
thermal fluctuations are sufficient to dislodge the soliton from the energy minimum at
the trap centre. This shows energy is transferred to the soliton, through interactions
with the background medium alone. The density and phase profiles of Figure 5.5 are
shown for the same duration over which the average soliton density was shown to fill
up almost completely, however the soliton in this case is still very deep at this time.
This is true of all the solitons within the ensemble, and it is only the uncertainty in
position which makes the soliton appear to have decayed in the averaged density. As
110
Chapter 5.
Stochastic effects in finite temperature matter waves
gn(z)/µ
1.5
1.2
0.9
0.6
0.3
0
-1
(b)
(c)
-0.8
-0.6
-0.4
(a)
-0.2
0
0.2
0.4
0.6
0.8
1
0.2
0.4
0.6
0.8
1
Phase angle
z [lz]
3
2
1
0
-1
-2
-3
-1
(a)
(c)
(b)
-0.8
-0.6
-0.4
-0.2
0
z [lz]
Figure 5.5: The density (upper plot) and phase (lower plot) profiles for a static soliton introduced at the trap centre. Density and phase snapshots are shown for (a) t = 0 (black), (b)
t = 1.01ωz−1 (red), and (c) t = 0.304ωz−1 (green), with coloured arrows/letters highlighting the
phase shift (≈ π) associated with the soliton centre at each time. The temperature here is
T = 300nK.
pointed out by Dziarmaga et al. a photograph of many separate realisations of the
system would reveal a soliton with a random position in each case [255].
5.3.3
Tracking a soliton in a fluctuating background
As an oscillatory soliton trajectory cannot be extracted, we find the averaged density is
not a good observable by which to measure the soliton dynamics. We instead consider
the position of the soliton centre, zsol (t), as a good quantum observable in a particle like
description of a soliton [249]. While this cannot be extracted from the averaged density,
it can be measured in individual runs using the algorithm described in Chapter 4. An
interesting issue relating to this means of extracting information from the simulations
is that now, instead of meaningful results only being obtained following averaging
over correlations of products of the wavefunction like Eq. (2.98), we attribute meaning
to the dynamics occurring within a single realisation. This is in close analogy to the
procedure in experiments, where many repeated tests are carried out, before the results
are averaged. It is tempting then, to adopt the viewpoint that each realisation of the
noise corresponds potentially to the results of a single realisation of such an experiment.
In both the theoretical and experimental setting, a particular realisation may be viewed
like one element within a grand canonical ensemble, since in both the particle number
and energy fluctuate between realisations.
111
Chapter 5.
Stochastic effects in finite temperature matter waves
1.8
(a)
gn(z)/µ
1.5
1.2
0.9
0.6
0.3
0
-5
-4
-3
-2
0
-1
1
2
3
4
5
z [lz]
1
(b)
gn(z)/µ
0.8
0.6
0.4
0.2
0
-5
-4
-3
-2
0
-1
1
2
3
4
5
z [lz]
Figure 5.6: (a) stochastic (red) versus mean field (black) density for a soliton of speed v = 0.25c;
(b) the signal followed by the soliton tracking algorithm for both the stochastic (red) and mean
field (black) simulations. The realisation of the noise shown is the red curves is for a temperature
T = 300nK.
In Figure 5.6(b) is an illustration of the signal followed, shown by the red curve, and the
noise associated with this signal. The soliton centre can be tracked within the stochastic
simulations, using the algorithm described, only until its depth becomes comparable to
the background noise. The algorithm used naturally loses the soliton at this point, as
it is no longer able to distinguish the soliton density notch over the background. The
time at which this occurs in simulations is defined as the decay time for that run. Since
the magnitude of the noise is temperature dependent, and microscopic in origin, this
can be considered as an effect likely to feature in experiments, and so can be considered
as a strength of the model.
Stochastic effects on a low speed soliton
A feature of Langevin models is the existence of two characteristic behaviours, that
of diffusion and that of evolution as a free particle, as described in our discussion
of Brownian motion in Chapter 2. To illustrate this in our system, we consider the
evolution of initially identical solitons with a low initial speed, v/c = 0.1. We consider a
fairly low temperature, T = 150nK, in order to follow the dynamics for an appreciable
timescale, however note that this is still within the phase fluctuating regime (Tφ =
25nK).
The soliton solution, Eq. (4.3) is multiplied into each stochastic wavefunction within
112
Chapter 5.
Stochastic effects in finite temperature matter waves
an ensemble of 200 noise realisations. Figure 5.7 shows the soliton trajectories for three
such realisations, with the DGPE result for the same γ(z), shown by the green, dashed
line. The blue curve shows a trajectory from the ensemble which follows fairly closely
the purely dissipative dynamics represented within the DGPE. This is the evolution we
might expect for a ‘free’ particle, which is relatively unaffected by the noisy background.
20
15
zsol[lz]
10
5
0
-5
-10
-15
-20
0
8
24
16
Time
32
40
-1
[ωz ]
Figure 5.7: Trajectories for solitons introduced with an initial speed v/c = 0.1 modelled using
the DGPE (green, dashed) and SGPE (solid blue, black and red curves). The different SGPE
curves represent simulations for different realisations of the noise.
In contrast to this, the black curve shows a soliton which does not initially undergo well
defined oscillations, as expected for a soliton within a harmonic potential, but instead
takes a more diffusive path prior to adopting this motion. The red curve shows the
path of a soliton even more strongly affected by the background noise, for which the
initial oscillation is actually in the opposite direction to the initial velocity with which
it was introduced. This illustrates the importance of noise effects upon the evolution
of dark solitons within such systems, even at relatively low temperatures. We find that
behaviour such as reversals in the trajectory direction happens more in the case of low
velocities, for which the background thermal motion is then relatively larger.
Rather than concentrating upon individual cases, we now move on to characterising
the behaviour of the ensemble of solitons as a whole.
5.3.4
Statistical spread of soliton trajectories
Tracking the solitons across the ensemble is found to yield a spread in trajectories, as
shown in the upper plot of Figure 5.8. Like the soliton initial velocity distribution, we
also find a skewed distribution for the decay times. From the individual trajectories, a
well-defined average trajectory can also be obtained, which is also shown in Figure 5.8
(lower plot). Since the mean trajectory is obtained as the average over a set of individual
113
Chapter 5.
Stochastic effects in finite temperature matter waves
trajectories, the duration for which this trajectory can be extracted is limited by the
shortest decay time within this sample set. To obtain the averaged data for Figure 5.8,
we considered 10 randomly chosen realisations, as this is comparable to the number of
tests that might be carried out in an experiment.
It is notable that often single run trajectories taken from the mean bin of the decay
time histogram, match well the averaged trajectory result. This displays a similarity
then between the result of ensemble averaging over single trajectories, and a trajectory
which is associated with the distinct process of averaging over the decay times from the
ensemble. By the nature of the binning procedure, a trajectory from the mean bin has
a decay time close to the mean, which is longer than the typical shortest decay time
for a set of around 10 trajectories. Therefore, a trajectory from the mean bin extends
the duration for which a meaningful trajectory can be obtained to around the mean
decay time, hτdecay i, given by
M
1 X
hτdecay i =
τi ,
M
(5.2)
i
where τi is the decay time of the i-th run, out of a total of M realisations.
0.024
(b)
0.016
(c)
0.008
(a)
0
0
zsol [lz]
30
25
50
100
75
125
-1
Decay time [ωz ]
(c)
(b)
(a)
15
0
-15
-30
0
15
30
45
Time
-1
[ωz ]
60
75
Figure 5.8: Histogram showing the distribution of soliton decay times for 200 realisations (top)
and soliton trajectories (bottom). The coloured curves and letters indicate the histogram bin
for each trajectory; the vertical dashed lines show the times at which the soliton was lost to the
background noise. The black circles show the 10 trajectory average, while the green, dashed
curve is the DGPE result for the same γ(z), as used in the SGPE results. The results presented
are based on T = 175nK.
Figure 5.8 also shows that the DGPE result (green) for the same γ(z) as that used in
114
Chapter 5.
Stochastic effects in finite temperature matter waves
the SGPE simulations, gives good agreement with both the set averaged (black circles)
and mean bin (red) trajectories. So, as might be expected, we see that on average the
stochastic soliton dynamics matches well the dynamics obtained from solving only the
deterministic part of the SGPE.
A comparison to characterise the variation in decay with temperature between the
stochastic and dissipative simulations, can be made more quantitative by comparing the
averaged decay time, hτdecay i, to an appropriately scaled decay time. This is discussed
in the following Section.
5.3.5
Comparison to the DGPE
To compare the soliton decay times within the SGPE and DGPE simulations, it is necessary to account in some way for the additional physics incorporated in the fluctuating
background of the stochastic approach. Because the resultant decay times in the SGPE
are dictated by the fluctuating background density, we must scale out the difference
between identifying a soliton propagating on the smooth background of the GP initial
state, as in the DGPE simulations, and the noisy initial state which is input into the
SGPE. This is important because the magnitude of density fluctuations is temperature
dependent, while there is no such variation for the DGPE initial state. We therefore
analyse the results of the DGPE in a manner so as to avoid spurious differences when
comparing the temperature dependence between each method.
We attempt to negate such effects by firstly measuring the magnitude of density
fluctuations about the equilibrium average density in the SGPE for each temperature.
Calculating a time series over a spatial region equal in size to that probed by the search
algorithm, ±100ξ, we record,
δn (z, t, T ) = max |n(z, t) − hn(z, teq )i| ,
(5.3)
which gives a representation of the depth at which a soliton might be lost at each time
in the SGPE. This is because, to find the new whereabouts of the soliton following some
time, the noisy instantaneous density is compared to the smoother average equilibrium
density obtained for that temperature. We assume that the soliton is lost when its depth
has decreased so as to be comparable to the average size of the maximum background
density fluctuations. A decay time can be extracted from the DGPE results, as the
time at which the soliton decays to a depth hδn iz,t (T ), where the angle brackets hiz,t
denote an average over a spatial region and over some time.
A complication is that the size of density fluctuations is not uniform across the trap,
due to the inhomogeneity of the density, and therefore the quantity δn varies spatially
as well as temporally. While the search algorithm (see Chapter 4, Section 4.2.2) used
115
Chapter 5.
Stochastic effects in finite temperature matter waves
to track the soliton samples a region of fixed size, this region moves with the soliton,
and hence searches for the soliton in a background with fluctuations of varying size.
0.05
(a)
0.04
0.03
0.02
0.01
0
-30
-20
0
-10
20
10
30
z [lz]
0.015
(b)
0.012
0.009
0.006
0.003
0
0.3
0.45
0.6
0.75
0.9
gδn(t) / µ
1.05
1.2
Figure 5.9: (a) Distribution of soliton decay positions (black circles), and a fit to a Gaussian
(red, dashed). Data obtained from all temperatures considered was combined to give the
distribution shown by the black circles. (b) Normalised histograms of the density fluctuations
about the ensemble average density for T = 150nK (black), T = 175nK (red), T = 200nK
(blue), T = 250nK (green) and T = 300nK (brown). This is scaled to the chemical potential,
which is the same at each temperature.
So there are two issues which should be addressed in matching the dissipative and
stochastic results: firstly, the variation of the fluctuations in time, and secondly, the
position at which the averaging should take place due to the spatial variation in the
magnitude of density fluctuations. The former can be addressed by averaging over
δn (z, t, T ) for a significant number of time-steps, while the latter we now discuss further.
A measure of the most pertinent spatial region to sample δn (z, t, T ) can be gained
by considering the distribution of locations at which solitons decayed across all temperatures within the stochastic simulations. An equal number of trajectories from each
temperature is considered, and from this total distribution, we can extract a spatial
weighting for the likelihood of decay. From this, we decide upon the most relevant
region to measure the background density fluctuations. Figure 5.9(a) shows the result
of binning the decay positions for all temperatures considered, where the black circles
represent spatially the probability for decay. The distribution that results is well fitted
by a Gaussian with a mean of approximately zero, showing that the highest number of
solitons are lost in the central trap region. This may seem counter-intuitive, given the
116
Chapter 5.
Stochastic effects in finite temperature matter waves
form of the damping in Figure 4.3, however the density fluctuations are larger in this
region than at the edges, making the soliton more likely to be lost.
We therefore base the noise scaling upon sampling over a fixed region ±5lz (±100ξ)
about the trap centre. This corresponds in size to the region probed by the search
algorithm. The distribution which is formed from a time series of δn (±0.1785R, t, T ) is
shown in Figure 5.9(b) for various temperatures. We again see a skewed distribution
with a long tail extending to large density fluctuations.
The tendency is for the distribution of density fluctuations to narrow and approach
zero for smaller temperatures, as is expected if the Gross-Pitaevskii mean-field model
is to be recovered for T → 0. For higher temperatures, at which solitons are found to
survive for only short times, the density fluctuations are of a similar scale, in terms of
energy, to the chemical potential.
-1
Decay time [ωz ]
100
80
60
40
20
0
150
200
250
Temperature [nK]
300
Figure 5.10: Mean soliton decay times as a function of temperature from 1d SGPE simulations
(red circles), DGPE with γ(z) (green diamonds), and γ̄ (blue hollow squares). Also shown
are the 3D DGPE results corresponding to the same γ̄ (orange, filled squares). The DGPE
data shows the time for the soliton generated to decay to a depth comparable to the average
background density fluctuations, with error bars corresponding to depths within the standard
deviation of these fluctuations about the mean. The dotted horizontal line indicates the GPE
prediction for one oscillation time.
Based on the above considerations, the decay times between the dissipative and stochastic methods can be compared in a systematic manner, and plotted in Figure 5.10 are
the results based on the scaling described. The data for the SGPE includes error bars,
which show the standard deviations of the decay time distributions obtained for each
temperature. The error bars for the DGPE results are given by the time it took for
the soliton to decay to a depth equal to hδn (z ± 100ξ, t, T )i, plus or minus the standard
deviation of δn (z ± 100ξ, t, T ). Physically, that is the time to reach a depth equal to
the average maximum value of the fluctuations in the density about the mean; errorbars are obtained from the time to reach this mean depth, plus or minus the standard
117
Chapter 5.
Stochastic effects in finite temperature matter waves
deviation of δn (z ± 100ξ, t, T ). Therefore, due to the counter-intuitive nature of soliton
decay, the upper error bound for the decay times was given by the time for the soliton
to reach the smallest depth.
It is apparent that the DGPE results lie beneath the stochastic results for all cases.
This seems consistent with Figure 5.2 and the result that the SGPE initial velocities
are on average lower than in the DGPE case, for which the initial speed is always fixed.
A further effect here, and possibly the primary effect, is due to the use of the average
maximum value of the background noise in scaling the decay times. Any other choice of
scaling could only push the DGPE results upwards, as the maximum noise corresponds
to a deeper soliton, and therefore an earlier decay time. Nevertheless, this appears
to capture the general trend and at worst over-estimates the discrepancy between the
dissipative and stochastic results.
To illustrate further the agreement between the SGPE and DGPE soliton trajectories, we plot in Figure 5.11 soliton oscillation data recorded at a low temperature,
T = 150nK, in order that we can follow the dynamics for an appreciable number of
soliton oscillations.
: DGPE
: ODE
: ODE (linear)
z [lz]
20
10
0
-10
: SGPE (single)
: SGPE (average)
-20
0
20
-1
t [ωz ]
40
60
Figure 5.11: Soliton trajectories for the DGPE with γ̄ (black), SGPE 10 trajectory average
(stars) and SGPE histogram mean-bin (circles) compared to the solution to Eq. (5.4) (red) and
the linear version of this (green, dashed). For these trajectories, T = 150nK.
Initially we focus on the results for the DGPE (black) and also two trajectories from
the SGPE simulations: a 10 trajectory average (stars) and a trajectory from the mean
bin of the decay time histogram (circles). We see that the agreement is very good
between each of these, so the average stochastic dynamics of the soliton appears to
recover the behaviour found for the dissipative simulations. This is true for both the
ensemble average, and also a different average, based upon the decay time histogram.
118
Chapter 5.
Stochastic effects in finite temperature matter waves
In Cockburn et al. [121], an equation of motion for the soliton centre was presented,
based upon perturbation theory for solitons [142]. In terms of the velocity, v = dzsol /dt,
this takes the form
#
"
i
h
Ω 2
dv
2
zsol 1 − (v)2 ,
=
γµv − √
dt
3
2
(5.4)
where Ω = ωz /ω⊥ . Linearising this, under the assumption that |v| is small, the mo-
tion of the soliton centre is that of a classical particle undergoing damped, harmonic
oscillations,
zsol (t) =
v0 γµt/3
e
sin (ω0 t) .
ω0
(5.5)
Here v0 is the initial soliton velocity and
Ω
ω0 = √
2
s
1−
γ2
,
2
γcr
(5.6)
√
with γcr = (3/µ)(Ω/ 2). The dynamics for the soliton centre for both the linear and
nonlinear cases are shown by the green (dashed) and red lines respectively in Figure
5.11.
5.3.6
Analysis of the soliton decay time distribution
We now turn to an analysis beyond consideration of the mean soliton decay times,
and consider characterising the distribution of decay times from across the statistical
ensemble.
A question might then be how do stochastic effects manifest in this dissipative
model? Some insight might be gained by considering if the behaviour of the stochastic
trajectories can be regained, by considering the solution for the dissipative trajectories
with the damping γ replaced by a random variable; this should obey some statistical
distribution, such that hΓ i = γ.
We start by rewriting Eq. (5.5), so the position of the soliton centre is instead given
by
zsol (t) =
v0
S(t) sin (ω0 t) ,
ω0
(5.7)
such that the amplitude of the oscillatory soliton motion, with frequency ω0 , is time
dependent and denoted by S(t). Assuming also that this amplitude grows exponentially
as the soliton decays, then the behaviour of the envelope of oscillations can be regarded
as given by the solution to
dS(t)
= Γ S(t).
dt
(5.8)
Here Γ is some rate at which the soliton oscillation envelope increases as the soliton
119
Chapter 5.
Stochastic effects in finite temperature matter waves
anti-damps.
If we assume Γ to be a stochastic quantity, and re-write (5.8) in a discrete form,
we have
Sn+1 − Sn
= Γn S n ,
δt
(5.9)
where time is discretised so tn = nδt and Γn is the value of the damping at the nth
time step. Rearranging gives,
Sn+1 = Sn + δtΓn Sn
= (1 + ǫn ) Sn ,
(5.10)
(5.11)
where ǫn = δtΓn and |ǫn | ≪ 1. This implies that at a time tN , where N ≫ 1,
SN =
N
−1
Y
(1 + ǫn ) S0 .
(5.12)
n=0
Taking the logarithm of this gives
ln
SN
S0
=
≈
N
−1
X
n=0
N
−1
X
ln (1 + ǫn )
(5.13)
ǫn .
(5.14)
n=0
If we assume that the soliton undergoes many scattering events during its lifetime, then
the central limit theorem states that ln (S/S0 ) will approximate a normal distribution,
which means that S/S0 must approximate a lognormal distribution.
A lognormal model is often applied to a system in which decay is caused by random
events, which causes decay at a rate proportional to the amount already present, so in
other words a runaway process. Kolmogorov suggested that for anything which decays
in this multiplicative way, referred to as multiplicative degradation [256], the time to
failure should follow a lognormal distribution [257]. As the soliton decay time is defined
as the time it takes for the soliton to reach a depth comparable to the thermal density
fluctuations, this motivates an analysis of the decay times data based on a lognormal
fit. A similar idea has been applied in describing the collisions between optical solitons
[258], where it was found that these collisions could be described by a NLSE perturbed
by stochastic parameters obeying strongly non-Gaussian statistics; interestingly, in this
case the soliton amplitude was also found to be lognormally distributed.
Moreover, further justification can be drawn from the property of self-similarity
inherent to lognormally distributed quantities: if A and B are two random variables,
120
Chapter 5.
Stochastic effects in finite temperature matter waves
and if A is lognormal with
B = rAd
(5.15)
then B is also lognormal. The parameter r is known as the scale factor and d is the
fractal dimension. Kolmogorov used this property in his argument on the fragmentation
of grains [257], to argue that if the distribution of grain sizes is lognormal, then so are
the grain volumes retained in sieves of different mesh size. Similarly, here we might
argue that the soliton amplitude and decay times fulfil a similar relation.
Since the soliton depth determines how far from the trap centre the turning point
of the motion is, then it is directly related to the amplitude of the oscillations S(t).
In turn, as the decay time is determined by reaching a certain depth, then we expect
the distribution in this variable to display a similarly lognormal distribution, under the
action of a stochastic damping.
0.08
(a)
0.06
0.04
0.02
0
0
30
15
60
45
τ
0.03
75
-1
[ωz ]
(b)
0.025
0.02
0.015
0.01
0.005
0
0
40
80
τ
120
160
-1
[ωz ]
Figure 5.12: Fitting a lognormal distribution to the decay time data for (a) T = 175nK and
(b) T = 250nK.
With this in mind, we proceed by fitting the distributions of decay times to the following
lognormal probability density function (pdf),
"
− (ln τ − m)2
P (τ ; m) =
exp
2σ 2
τ σ 2π
1
√
#
(5.16)
where m is the mean and σ the standard deviation of ln τ . To fit the data, a histogram
(normalised to 1) was constructed at each temperature, to which Eq. (5.16) was then
121
Chapter 5.
Stochastic effects in finite temperature matter waves
fitted. An example of the fit which results is shown for two temperatures, in Figure
5.12. It is clear that the fit matches the trend of the data well, suggesting that the
underlying nature of decay is that of multiplicative degradation.
We compare the results of fitting the decay times for all temperatures considered
in Figure 5.13, from which we can extract the dependence of the fitting parameters
m and σ. We see from Figure 5.13(a) that the distributions of decay times become
increasingly shifted towards the origin, as the dissipative effects reduce the soliton
lifetimes. We also would expect these distributions to narrow as T → 0, and thermal
fluctuations are no longer present1 . Shown in Figure 5.13(b), is the mean of ln(τ )
which decreases with a roughly linear dependence upon temperature, and a linear fit is
plotted for comparison. As is to be expected in the presence of increased fluctuations,
the variance of ln(τ ) increases with temperature, and we again find a linear fit to match
the data well.
0.14
(a)
0.12
0.1
0.08
0.06
0.04
0.02
0
0
30
90
60
τ
5
(b)
0.6
(c)
0.5
0.4
3
0.3
2
1
150
σ
m
4
120
-1
[ωz ]
0.2
150
200
250
300
150
T [nK]
200
250
300
T [nK]
Figure 5.13: (a) Fitted lognormal probability distribution functions for T = 150nK (black,
solid), T = 175nK (red, dashed), T = 200nK (blue, dot-dashed), T = 250nK (green, dotdouble dashed) and T = 300nK (brown, dotted); (b) m and (c) σ extracted from the fits,
versus temperature.
1
Note that our method ignores quantum fluctuations which would ultimately become important at
extremely low temperatures and large g/N .
122
Chapter 5.
5.3.7
Stochastic effects in finite temperature matter waves
Comparison to different forms of γ(z)
While retaining the Hartree-Fock term in calculating the self-energy integral captures
the major features expected of scattering within a trap with a well formed condensate,
we have considered so far the damping due to a coefficient calculated based on the
Lerch formula Eq. (3.64). We now compare to the distribution of decay times which
result if, rather than this form, we retain the S (k1 , k2 , k3 ) function in calculating γ(z).
This means we must use instead the integral method which involves directly evaluating
Eq. (3.55) numerically. In first instance, the Lerch method is quicker, and hence useful
for large grid sizes, however the integral method is in principle more accurate, and we
aim now to indicate the effect of making this refinement.
0.02
0.08
0.015
γ(z)
PDF
0.06
0.04
0.01
0.005
0
0.02
-30
-20
-10
0
10
20
30
z [lz]
0
0
10
20
30
40
-1
50
60
70
Time [ωz ]
Figure 5.14: Dark soliton decay times at T = 250nK for damping due to γ(z) calculated using
the Lerch formula (black), by evaluating the integral of Eq. (3.55), retaining the contribution
of the S (k1 , k2 , k3 ) function (green), and also γ̄ for the latter (blue). Fits of the decay times
histograms to the lognormal pdf Eq. (5.16) are shown in each case. Inset: Form of γ(z)
contributing to the decay time distributions; the colours correspond to the curves in the main
plot.
Shown in Figure 5.14 is a plot of the distribution of soliton decay times for T =
250nK, for each of the methods to calculate γ(z). As might be expected, the generic
features are the same in that the distribution of decay times appears close to lognormal
for both damping forms, meaning the distribution is again skewed, with a long tail
for increasing lifetimes. An important point to be taken from this also, is that the
underlying mechanism leading to a lognormal distribution is unaffected by the precise
form of the damping.
Connection to geometric Brownian motion
The prevalent model used in the prediction of the price of stocks is that of geometric Brownian motion. The notion that Brownian motion could be representative of
123
Chapter 5.
Stochastic effects in finite temperature matter waves
stock market behaviour was first used by Bachelier in 1900 to model the Parisian stock
markets [169]. Later, in the 1960s, Samuelson introduced instead the use of geometric
Brownian motion, due to the desirable feature that the associated probability distribution function is lognormal, meaning it satisfied the physical requirement of predicting
only positive stock values. Indeed, Black and Scholes (1973) developed their famous
approach to the pricing and hedging of options in the context of this model.
Like stock prices, the amplitude of soliton oscillations is strictly positive or zero,
and we have seen that the decay times which parameterise the amplitude of oscillations,
appear also to be lognormally distributed. This is then suggestive of a potential link
to an underlying geometric Brownian motion [259].
If we again refer to the equation for soliton motion, Eq. (5.7), we may introduce the
stochastic, time-dependent, soliton oscillation amplitude S(t). To make the stochastic
nature of the damping more concrete, we now propose the time dependence of this
variable to be given by
dS = m̃Sdt + σdW,
(5.17)
where dW is the derivative of a Brownian motion Wiener process [259]. It is clear that
a result for the oscillation amplitude of the same form as that found in Eq. (5.7) can be
recovered if σ → 0, which is equivalent to neglecting any fluctuations about the mean.
Making use of Ito’s lemma, the logarithm of the amplitude is
d(ln S) =
σ2
m̃ −
2
dt + σdW
(5.18)
which is a Wiener process with drift m̃ − σ 2 /2 and variance σ 2 . Defining X = ln S,
then the probability distribution for X at some time t is given by
 h


X
−
X
−
m̃ −
0
1
f (X) = √
exp −

2σ 2 t
σ 2πt

σ2
2
i2 

t 
Given that the probability distribution for S, P (S), must satisfy
P (S)dS = f (X)dX
then we find that
P (S) =
which gives
1
f (X)
S
 h
i2 


S


ln
−
m
t
S0
1
P (S) = √
exp −
,


2σ 2 t
σ 2πtS


124
.
(5.19)


(5.20)
(5.21)
(5.22)
Chapter 5.
Stochastic effects in finite temperature matter waves
where we have used the definition m = m̃ − σ 2 /2. Thus we see that the lognormal
fitting function Eq. (5.16), is the same as that for geometric Brownian motion, if
the time is absorbed into the mean and variance such that mt → m(t) and σ 2 t →
σ(t). In this case, the logarithm of the soliton amplitude is then normally distributed,
with a mean and variance which are proportional to time, hence justifying the name
geometric. Moreover, this seems a fitting generalisation to the exponentially increasing
amplitude of the deterministic case, which there is due to the time dependent growth
being proportional to the size at each time. The analogous situation in this stochastic
model leads similarly to a distribution with a mean and variance which grow in time
with increments relative to the present values.
Interestingly, the generalisation from the deterministic to stochastic evolution, dS =
mSdt → dS = mSdt + σSdW , displays nicely the difference in moving from the DGPE
to the SGPE. To illustrate this point, the expectation value of S, which is found from
evaluating
S̄ =
Z
∞
dS S P (S),
(5.23)
0
yields
S̄ = S0 exp (mt) ,
(5.24)
which is reflective of the fact that the average stochastic soliton trajectories at each
temperature match well the underlying DGPE results at each temperature.
5.3.8
Role of the heat bath in soliton dynamics
As the system is brought into equilibrium with the heat bath prior to the introduction
of a soliton, it is interesting to consider the role the heat bath plays in the soliton
dynamics which follow. This can be modelled by removing contact with the heat bath
at the point at which soliton is introduced, so the dynamics are described by the GPE
alone. This corresponds to the SGPEeq method, described at the end of Chapter 2,
and was first applied to quasi-condensate dynamics on an atom chip [181].
This approach is similar in spirit to the truncated Wigner method [213]. The
premise of the truncated Wigner approach within BECs is that quantum noise effects
can be incorporated approximately into the condensate equation of motion, by the
addition of appropriately sampled noise to the initial condition. Thermal effects can
also be incorporated approximately, through suitable generalisations. The method is
derived upon neglecting third order derivative terms which appear in the Fokker-Planck
equation for the probability distribution function [213, 124, 125], which provides an
equivalence between this and a stochastic differential equation of the same form as the
GPE. We consider a comparison between the SGPE used in this way, and a number
conserving Bogoliubov approach to the ensemble of stochastic initial conditions, in
125
Chapter 5.
Stochastic effects in finite temperature matter waves
Chapter 6.
In equilibrating to a classical equilibrium prior to removing the heat bath, the SGPE
equilibrium is such that
h(HGP − µ) ψi = iγh(HGP − µ) ψi
(5.25)
which can only be solved non-trivially if
HGP hψi = µhψi.
(5.26)
Hence hψi is a stationary solution to the GPE also.
This also has obvious similarities with other classical field approaches for BECs
[118, 205, 207, 208], which also use the (projected)GPE as an equation of motion. This
is justified as the GPE describes non-perturbatively, those modes whose behaviour
is predominantly classical, that is modes with occupations N (E) ≫ 1. This is a
generalisation of the situation in which the order parameter represents only the ground
state, as in the T = 0 GPE approach. In such methods, growth into higher modes is
seeded typically using a random initial condition. Due to nonlinear mode mixing, this
initial condition will ultimately equilibrate to a steady state, and for the same cutoff
and total energy per particle, should be identical to the equilibrium solution of Eq.
(2.92), for a single realisation.
To perform the ensemble averaging required to construct correlation functions of
interest, time averaging can be justified by ergodicity of the GPE [260]. A notable
advantage of solving the SGPE to obtain an initial state, is the action of the heat bath
as a thermostat, which means the temperature can be decided a priori. In addition,
there is no recourse to time averaging as an ensemble is provided by repeating the
process for many noise realisations. A key difference is the statistical ensemble of
each approach; whereas the SGPE is formulated within the grand canonical ensemble,
classical field theories which generalise the GPE to many, highly occupied modes, can
be considered as micro canonical in nature, due to the conservation of particle number
and energy.
An interesting point in modelling the dynamics in this way is that the soliton
evolution is ergodic under the GP equation. This is true once contact is removed with
the heat bath, which is equivalent to taking the limit γ → 0 in Eq. (2.92). On removing
the dissipation and noise terms from the SGPE, the resulting equation of motion then
conserves particle number and energy. The only factor lifting the integrability of the
equation of motion is then the harmonic trap, so it is interesting to consider how
moving the system closer to integrability affects the soliton dynamics within a noisy
background density.
126
Chapter 5.
Stochastic effects in finite temperature matter waves
Two competing effects on the random nature of the soliton trajectories are the initial fluctuations in the background density which cause a variation in the initial soliton
speed, and also interactions with thermal excitations as the soliton propagates. To
compare the relative effects of each of these, we plot in Figure 5.15 a scatter plot of the
decay times, versus initial soliton velocity, represented on the vertical and horizontal
scales respectively. Both the decay times and initial velocities are scaled to their respective mean values. Each point corresponds to one realisation of the noise, and maps
the correlation between initial velocity and soliton lifetime for each trajectory.
5
5
5
(b)
(a)
4
4
3
3
3
2
2
2
1
1
1
0.75
1
1.25
5
0
1.5 0.5
0.75
1
1.25
1.5
5
4
4
3
1
1.25
1.5
(f)
0.5
2
1
0.75
0.6
3
2
0
0.5
0.7
(e)
(d)
σ/µ
0
0.5
τdecay / <τdecay>
(c)
4
0.4
0.3
0.2
1
0.1
0
0.5
0.75
1
1.25
0
1.5 0.5
0.75
1
vsol / <vsol>
1.25
1.5
0
100 150 200 250 300 350
T [nK]
Figure 5.15: Scatter plots showing the spread in decay times and initial velocities, scaled to
their respective mean values. The plots (a)-(e) show the results for T = 150nK, T = 175nK, T =
200nK, T = 250nK and T = 300nK respectively with the standard deviations indicated by red,
dashed lines. Plot (f) shows the standard deviation, σ, scaled to the mean, µ for the distribution
of decay times (green, squares) and initial velocities (blue, diamonds), versus temperature.
It is clear that the low temperature results of Figures 5.15(a)-(e) are more localised
about the (1,1) point, which represents the mean value of each quantity, than the higher
temperature results. Also shown by the red dashed lines are the standard deviations
in the quantities at each temperature. For the decay times, there are very few points
which lie beneath hτdecay i − σ, indicted by the lowest horizontal red lines in each plot.
There are many more cases of decay times well above the average, though, as expected
for a lognormally distributed quantity.
From Figure 5.15(f), we see that the relative variation in the decay times (green
squares), parametrised by the ratio of the standard deviation to the mean, is far greater
127
Chapter 5.
Stochastic effects in finite temperature matter waves
than that of the initial velocities (blue diamonds). The standard deviation in the decay
times is between around 30 − 60% of the mean value, whereas for the velocities this is
only 5 − 15% Also, there appears to be little correlation between a high (low) initial
speed and a short (long) lifetime, as one might have expected. Were long decay times
associated with low velocities, this would have given a diagonal line of gradient −1 in
Figures 5.15(a)-(e). That these quantities are not correlated indicates that interactions
with the heat bath during evolution greatly influences the soliton dynamics.
We now compare the dynamics of single trajectories when the heat bath terms are,
or are not, retained following equilibration and the introduction of a soliton. Beyond
this point, the dynamics were simulated using the GPE and are compared to the
corresponding trajectories when the noise and damping terms are retained in Eq. (2.92).
In these simulations, identical initial conditions and noise realisations are compared,
so the ensembles were identical until the time at which the soliton was introduced. We
also compare these stochastic results to the dissipative results discussed in the previous
section.
In summary then, we now consider the soliton dynamics via evolution of the full
SGPE, i.e. including dissipation and fluctuations, the GPE with stochastic initial conditions, so conservative evolution beyond equilibrium which we refer to as SGPEeq , and
the DGPE, which is the case of purely dissipative evolution. These results are summarised in Figure 5.3. The soliton paths shown are grouped according to the behaviour
observed for the particular realisation of the noise, under the SGPE: Plot (a) shows
a realisation in which the soliton had a short lifetime, Plot (b) shows the case where
the decay time was around the mean, and Plot (c) shows the case of an above average
lifetime.
The first point to note is that there appears to be no damping of the soliton oscillations when the heat bath terms are neglected (red curves). The evolution is very
similar to that of the GPE (brown), but with a different oscillation amplitudes due to
the different initial soliton velocities between runs. This variation in initial speeds, as
in the full SGPE simulations, means trajectories are not identical in each realisation.
Correspondingly, the filling in of the soliton notch with time, found in the averaged
soliton densities in Figure 5.3, still occurs.
While the soliton is still propagating within a noisy background, its motion appears to be qualitatively the same as that for a soliton propagating within a smooth
background density. That is, equivalent to an ensemble of GPE simulations with a
distribution of initial speeds. The harmonic trap still breaks the integrability of the
underlying equation of motion, so we still expect the emission of sound waves as the
soliton oscillates. The absence of dissipation suggests that the dynamical equilibrium
128
Chapter 5.
zsol [lz]
20
Stochastic effects in finite temperature matter waves
(a)
10
0
-10
zsol [lz]
-20
20
(b)
10
0
-10
zsol [lz]
-20
20
(c)
10
0
-10
-20
0
50
-1
Time [ωz ]
Figure 5.16: Comparison between trajectories for solitons introduced once contact to the heat
bath was removed (red, dashed), and when contact was maintained (blue, thin solid). Also
shown as a reference is the DGPE result (green, dot-dashed) at the same temperature, T =
150nK, and the GPE result (brown, thick solid). Outcomes are shown for realisations which,
when the heat bath terms were retained, the solitons decayed at times (a) τ < hτ i + σ , (b)
τ ∼ hτ i and (c) τ < hτ i − σ.
129
Chapter 5.
Stochastic effects in finite temperature matter waves
between emission and re-absorption of phonons is maintained in these simulations. This
is consistent with the findings of [261], in which a soliton was found to oscillate without
decay for the case of a harmonic trap plus sufficiently tight optical lattice potential.
For a high enough lattice period (≪ ξ), a soliton is unable to heal to such short range
density changes in the density, and so in that case was found to oscillate as though
within a harmonic trap with depth shifted upwards by the lattice depth.
Also of interest in these simulations is the fact that when the system is able to
undergo undamped centre-of-mass oscillations, the soliton too performs undamped oscillations. When the condensate oscillations are damped, as in the DGPE and SGPE
simulations, the soliton motion becomes damped also. This is best illustrated in Figure
6.17 of Chapter 6, where the centre of mass is shown to undergo undamped oscillations
for evolution of both the ground state solution φ, and a noisy thermal ensemble of states
{ψ}, under the GPE. By comparison, for the DGPE case and the full SGPE evolution
of the same set of states {ψ}, we see similarly damped centre of mass oscillations. We
might point to the static representation of the thermal cloud as one cause for this, as in
the former approach for {ψ}, all approach thermal modes included in the description,
that is the lowest system modes, are represented dynamically. In the latter case, higher
lying modes are represented statically in addition.
However, simulations of the ZNG method for this soliton system, similarly revealed
damped oscillations for the solitons, even when the thermal cloud was dynamically
modelled (C22 6= 0), hence we can exclude the additional damping of the condensate
centre of mass motion as the dissipative mechanism. Moreover, since we have N ≈
20000 and the number of particles displaced by the soliton M ≈ 50, then oscillations
of the soliton will transfer little momentum into the background density.
We now consider the specific cases, (a)-(c), in Figure 5.16 individually. For the
example realisation in which the soliton decayed relatively early, Figure 5.16(a), all
three trajectories start off very close, although the stochastic trajectory quickly diverges
from the other two, highlighting the influence of the heat bath terms on the soliton
dynamics. In this realisation, as expected for a soliton whose decay time was shorter
than the mean, the DGPE trajectory decay is relatively much slower. For the SGPEeq
trajectory, the amplitude of oscillations is unchanged for the time shown. This is
also found to be the case at other temperatures, showing that soliton decay has no
dependence on temperature within this model, which seems to predict an infinite soliton
lifetime, as in the harmonically trapped GPE with the ground state initial condition.
Figure 5.16(b) shows the results for an SGPEeq soliton with decay time close to
the mean. As discussed, the DGPE and SGPE results match well in this case, whereas
the SGPEeq instead follows closely the GPE result shown by the thick brown solid
line. The SGPEeq results of Figure 5.16(a) and (c) do not follow the GPE result so
130
Chapter 5.
Stochastic effects in finite temperature matter waves
closely, with the difference due entirely to the initial speed imparted to the soliton.
This also demonstrates that there is not necessarily a correlation between long lived
solitons and the initial soliton speed in the SGPE simulations. The SGPE soliton in
Figure 5.16(c) seems to have started with an above average speed, shown by the large
initial oscillations relative to the DGPE, however lived considerably longer than the
average decay time.
Plots (a) and (c) also demonstrate the different regimes of soliton transport, namely
diffusion or mobility dominated [233]. These regimes are dictated respectively by dominance of either the noise or damping terms in the SGPE. The former corresponds to
Plot (a) in which the decay is dominated by the prevailing frictional force and quickly
anti-damps, whereas the latter is shown in Plot(c), in which the soliton damping does
not respond to the thermal frictional force as in the deterministic case, instead taking
on a more diffusive nature. Note also in Plot (a) the two stochastic paths are very
close for the first half oscillation, whereas in Plot (c) they already differ significantly,
showing the impact of retaining the heat bath terms on the trajectory.
Removing these terms before introducing the soliton leads to undamped soliton
oscillations. Since the the initial speeds across the ensemble are stochastic quantities,
we now look at the average trajectory predicted by these simulations, and whether this
behaviour varies with temperature.
28
21
zsol [lz]
14
7
0
-7
-14
-21
-28
0
10
20
30
40
-1
Time [ωz ]
Figure 5.17: Soliton paths for T = 150nK and T = 300nK from the DGPE, shown by the
green (dot-dashed) and blue (dotted) curves respectively. The averaged results from the SGPE
simulations, in which contact with the heat bath is removed prior to introducing the soliton,
are shown by red (dashed) for T = 150nK and black (thin, solid) for T = 300nK.
To illustrate this, plotted in Figure 5.17 is the averaged soliton trajectory for the
SGPEeq simulations carried out at T = 150nK and T = 300nK, together with the
131
Chapter 5.
Stochastic effects in finite temperature matter waves
corresponding DGPE results. In each case, the average trajectory is well matched by
the T = 0 GPE result. This shows that a noisy initial condition, followed by evolution
under the GPE, does not predict the soliton to decay. This implies also that other
classical field methods in which the GPE is the equation of motion, should not predict
soliton decay. A similar effect was observed in recent work on vortices also [191].
Solitons were however found to decay in the truncated Wigner simulations of Martin
and Ruostekoski [262, 263], in which simulations were undertaken for smaller particle
numbers, where quantum effects are expected to be more important.
5.4
Chapter summary
In this Chapter, we have considered the effect of phase and density fluctuations on the
dynamics of dark matter wave solitons within a harmonic trap, retaining the HartreeFock contribution in calculating the damping as in the DGPE simulations of Chapter
4.
Different observables were probed, with ensemble averaged features due to the soliton found to be quickly washed out. In contrast, monitoring solitons within individual
realisations allowed us to extract the expected oscillatory behaviour of the soliton.
We considered the distributions of both initial velocities and decay times which result
due to background fluctuations. A clear distribution in decay times was found, which
proved to be well represented by a lognormal fit. We then considered the dark soliton
oscillating within a finite temperature Bose gas in the context of a model based upon
geometric Brownian motion.
The DGPE was found to represent well both the average stochastic results, and
single stochastic trajectories with a decay time close to the ensemble mean. The latter
of these average quantities extends the time over which mean stochastic trajectories
can be meaningfully extracted; unlike the ensemble average, it is not restricted to the
shortest decay time within the ensemble, which, by definition, is always less than the
mean decay time.
Finally, we considered modelling solitons using the SGPEeq approach, based on
SGPE growth to equilibrium and subsequent removal of the damping and noise terms
beyond equilibrium. This approach was found to produce single run trajectories with
random amplitudes, but undamped soliton dynamics on average. These were found to
be extremely close to the oscillations predicted by the T = 0 GPE, for a soliton with
the same initial depth as that of the stochastic simulations.
In the following Chapter, we will discuss a comparison between this final approach
and a number conserving Bogoliubov method for generating finite temperature, stochastic classical fields.
132
Chapter 6
Comparing the stochastic
Gross-Pitaevskii equation and
number conserving Bogoliubov
approach
In this Chapter, we consider a comparison between two stochastic approaches to modelling the dynamics of partially condensed Bose gases: the SGPE as implemented here
[107, 108, 175], within the classical approximation, and the truncated Wigner method
[213], with thermal effects included into the initial condition derived using the number
conserving Bogoliubov formalism of Sinatra et al [117, 118, 119, 264]. This work was
carried out in collaboration with Carsten Henkel of Potsdam University and Antonio
Negretti of Ulm University, as detailed in Chapter 1.
The comparison focuses on the differences in equilibrium properties predicted by
the initial states generated within each of the methods. Where possible, the predictions of other, distinct, theories are used as independent tests of such properties, in
order to benchmark our findings, and quantify the relative merits of each approach.
We also employ these alternate approaches as an independent means of thermometry
within the simulations as appropriate. Finally, we consider a comparison between the
system dynamics as represented within each approach, considering the response to an
instantaneous trap perturbation.
6.1
Motivating a stochastic approach
As the typical number of particles within BEC experiments is N ∼ 103 −109 , theoretical
models based upon following the dynamics of each particle individually is quickly seen
133
Chapter 6. Stochastic Gross-Pitaevskii and number conserving Bogoliubov methods
as an insurmountable task, as Hilbert spaces grow rapidly with increasing particle
and mode numbers. Progress may be made however, by following an approach often
adopted in the field of quantum optics, where the behaviour of a large collection of
photons turns out to be well described by the classical Maxwell’s equations. Just as a
large collection of photons is well represented by a classical field, we equally well expect
this concept to apply to a bosonic system of trapped atoms. In one limit, this line of
thought leads us to the GPE, which is the result of taking the classical, mean field limit
of the Heisenberg equation of motion for the bosonic field operator. If we assume that
the operator can be decomposed such that
Ψ̂ = hΨ̂i + Ψ̂ − hΨ̂i ,
= φ + δ̂
(6.1)
(6.2)
so the operator is first split into classical field, φ, and operator, δ̂, components. Taking
a further step, in which we say the mean field φ represents only the condensate mode
and δ → 0, then we effectively neglect all fluctuations about the mean field and we
obtain the GPE from Heisenberg’s equation. This is valid strictly only at T = 0 and
in the limit in which this mode has a large occupation, so quantum effects become
negligible. Practically, in numerical simulations, this is equivalent to using the ground
state in the trap as the initial state to the GP equation of motion, which is the lowest
energy stationary solution to the GPE (see Chapter 4 for details).
There are however higher energy solutions to the GPE, associated with finite temperatures. Considering instead a redefinition of the elements of the decoupled Bose
field operator, where rather than mean field and operator components, we replace the
single mode φ with a complex field ψ, which we view as representing the predominantly
classical, coherent modes of the system. This generalises the GPE equation to a multimode formulation, in which the picture of coherent dynamics is no longer restricted to
just the ground state, but extends to any highly occupied modes which are collectively
represented by a classical field ψ [205, 118, 206, 207, 208].
This is the manner in which we apply the GPE in what follows, i.e. as the equation
of motion for a classical field, which represents many modes of a trapped, partially condensed Bose gas. It turns out that the inclusion of noise effects is crucial in describing
the physics of more than the ground state with the GPE, which can be achieved through
a suitable random initial condition, or in a slightly different setting through the methods we now compare. The main difference between the methods in this comparison lies
in the nature of the initial state generation, as we now discuss.
134
Chapter 6. Stochastic Gross-Pitaevskii and number conserving Bogoliubov methods
6.2
Summary of methods
We give here a summary of the most pertinent aspects of the two methods which
we consider in this Chapter; for a more in depth discussion of numerous elements of
each, and the relation with other finite temperature methods, see the recent reviews
[124, 125].
6.2.1
Truncated Wigner plus number conserving Bogoliubov
Phase space methods use the idea of averaging over trajectories in phase space, in
order to follow the quantum dynamics of a system. A typical procedure is to write the
density matrix as an expansion over a coherent state basis, with some quasi-probability
distribution. There is some freedom in the choice of representation for the density
matrix at this point, with common choices including the Glauber-P representation, the
so-called Q representation or the Wigner representation [178]. These approaches differ
on several counts, for example in the amount of noise added within each and differences
which arise in operator ordering. On making one such choice, the master equation for
the density matrix can then be written in terms of this distribution function, leading to a
Fokker-Planck equation [213]. The Fokker-Planck equation, under certain restrictions,
can then be mapped on to a set of classical stochastic field equations. In choosing
the Wigner function as the quasi-probability distribution, however, such a restriction
is encountered for the second quantised Hamiltonian which leads to HGP at T = 0,
in that third order derivatives appearing in the Fokker-Planck equation do not allow
for a simple mapping to an equivalent Langevin description. One means by which
to progress is then to truncate the Fokker-Planck equation at second order, leading
to the truncated Wigner approximation. Upon making this truncation, the equation
of motion which results is formally equivalent to the GPE, however the wavefunction
whose evolution it describes represents now many system modes. The third order
term neglected in arriving at this equation, represents the contribution of quantum
noise to the system evolution. However in neglecting this term, some noise effects
are retained, and persist in the stochastic initial conditions which sample the initial
time Wigner function. Indeed, it was shown by Polkovnikov [265] that expanding
perturbatively about the classical phase space trajectories yielded the GPE at lowest
order and the truncated Wigner approximation to the next order; the GPE captures
the essentially classical dynamics, while at the next order, spontaneous quantum effects
are incorporated in the initial conditions, although the evolution remains classical.
So, to summarise briefly, the truncated Wigner approach incorporates quantum
effects into the system dynamics through the addition of appropriately sampled noise
to the initial conditions, which are then propagated according to the GPE. As a classical
135
Chapter 6. Stochastic Gross-Pitaevskii and number conserving Bogoliubov methods
limit, this approach is generally applicable to system modes which are highly populated.
While the GPE is inherently a theory of stimulated processes, spontaneous processes
are modelled approximately as weakly seeded stimulated processes [125], introduced
through fluctuating initial conditions which sample the initial Wigner function. As the
equation of motion is classical, however, the extra noise mimicking quantum effects
will ultimately thermalise to a classical equilibrium after a sufficiently long time. This
is due to the action of nonlinear mode mixing, described by the nonlinearity of the
GPE, which models the scattering between particles in different modes, and leads to
an ergodic evolution in phase space. This places a limit on the time for which the
dynamics are likely to be representative of the quantum evolution, due to the classical
nature of the GPE.
Wigner corrections
From the stochastic wavefunction, it is possible to construct expectation values of
products of operators, which represent the physical properties we will be interested in.
However, averaging over the complex fields of the Wigner formalism returns symmetrically ordered operator products, that is, the average over all ways of ordering the
operators [178]. For example, the average of the symmetric ordered quantity
1
hâ↠+ ↠âi =
2
Z
d2 α|α|2 W (α, α∗ ),
(6.3)
where W (α, α∗ ) is the Wigner function and α is a complex valued classical field (c-fields,
in the terminology of [125]). The Wigner function is defined as1
1
W (α, α ) = 2
π
∗
Z
d2 β Tr(eiβ
∗ a+iβa†
ρ)e−iβ
∗ α−iβα∗
.
(6.4)
Of practical interest in numerical calculations is the correction which should be applied
to obtain the average corresponding to normally ordered operators of the form
O(â, ↠) =
X
cn,m â†n âm .
(6.5)
n,m
This correction is given generally by
∗
OS (α, α ) =
X
n,m
cn,m
β∗
∂
+
∂(iβ)
2i
n β
∂
+
∗
∂(iβ ) 2i
m
eiβ
∗ α+iβα∗
β ∗ =β=0
.
(6.6)
For the properties which we consider in what follows, two particularly relevant
1
We follow the notation of [178]; in Eq. 6.4 and Eq. 6.6 β is a complex field, however at all other
points in this thesis, it represents the inverse temperature, as usual.
136
Chapter 6. Stochastic Gross-Pitaevskii and number conserving Bogoliubov methods
operator correspondences are given by
O(â, ↠) = ↠â;
and
O(â, ↠) = a† a† aa;
1
OS (â, ↠) = |α|2 − ,
2
(6.7)
1
OS (â, ↠) = |α|4 − 2|α|2 + .
2
(6.8)
Number conserving Bogoliubov initial state
In our study, the finite temperature initial state for the truncated Wigner simulations
is obtained according to the prescription given in [117, 118, 119], which here was implemented by Antonio Negretti, who calculated the ensembles of initial states for this
method.
The aim is to arrive at a finite temperature state which contains quantum effects,
in the shape of quantum noise, and thermal effects, which are incorporated through
population of the Bogoliubov modes according to Bose-Einstein statistics, all within a
number conserving approach. Obviously, this is not an easy task, however, a means to
do this was given by Sinatra and co-workers [117, 118, 119]. In terms of notation, we will
√
√
find a noisy initial state ψ = N0 φ′ (z)+ψ⊥ (z), where N0 φ′ (z) is the temperature dependent condensate mode, and ψ⊥ (z) is the wavefunction representing non-condensate
particles, necessarily occupying orthogonal modes to that of the condensate.
The numerical procedure which derives from this approach is heuristically summarised as follows [266] 2 :
1. Define a spatial grid with M grid points.
2. Solve the GPE in imaginary time to obtain the stationary solution for N atoms,
φ0 .
3. Solve the eigenvalue problem for the Gross-Pitaevskii operator (see also Eq.
(2.15))
LGP =
HGP + g|φ0 |2 − µ
−g [φ∗0 ]2
g [φ0 ]2
−(HGP + g|φ0 |2 − µ)
!
(6.9)
where the ground state wavefunction is normalised to the number of atoms,
Z
dz |φ0 (z)|2 = N.
(6.10)
4. Select the (M − 1) eigenmodes with ǫ > 0, i.e. remove the Goldstone mode.
2
We are grateful to Antonio Negretti for providing the explanation that this is based upon.
137
Chapter 6. Stochastic Gross-Pitaevskii and number conserving Bogoliubov methods
5. Determine the (M −1) eigenmodes |uj i,|uj i of the L operator defined in [117, 119],
to which we refer the reader for further details.
6. Calculate the number of condensed atoms N0 , which is a stochastic quantity [119].
7. Calculate the correction φ(2) (z) to the Gross-Pitaevskii imaginary time solution,
φ0 (z). This is orthogonal to the GPE state and is can be expanded over the
M − 1 modes of L
φ(2) (z) =
X
cj (t)uj (z) + c∗j (t)vj∗ (z).
(6.11)
j>0
The complex expansion coefficients cj (t), must then be solved for in imaginary
time.
√
8. The corrected condensate mode is then defined as: φ′ (z) = φ0 (z)/ N +φ(2) (z)/N ,
which is normalised to unity.
9. The total matter field can then be defined as ψ(z) =
√
N0 φ′ (z) + ψ⊥ (z). The
average norm of this field is then N +M/2, which is the particle number for which
the GPE was solved, plus a contribution reflecting the addition of quantum noise.
This forms the initial state for propagation via the GPE.
Due to the finite temperature formalism used here, we refer to this approach as the
truncated Wigner plus number conserving Bogoliubov approach (TWNCB). The truncated Wigner part of this name refers to the inclusion of quantum noise in the initial
conditions, due to the sampling of the Wigner function to provide stochastic initial conditions for propagation with the GPE. Therefore, strictly, we consider the truncated
Wigner aspect of this method only in dynamical tests, whereas in static equilibrium
tests, we are only examining the number conserving Bogoliubov method of initial state
preparation. For simplicity, however, we use a single term here.
In constructing the initial state in this manner, the physical picture is that we begin
with the ground state solution to the GPE, at some specified particle number N . The
Bogoliubov spectrum is then calculated, based upon this condensate number, and the
thermal modes populated, on average, according to the Bose-Einstein distribution plus
an additional half particle per mode representing quantum noise. To make the theory
number conserving, there is then a correction to the condensate mode, to account for
atoms depleted into the higher energy modes 3 . This preparation procedure is repeated
many times to provide an ensemble of initial states, however total particle number is on
average conserved, so it is the canonical ensemble in which we work. While generally
3
There are also other common finite temperature truncated Wigner approaches which are not number conserving, see for example [125].
138
Chapter 6. Stochastic Gross-Pitaevskii and number conserving Bogoliubov methods
a property associated to the grand canonical ensemble, as the Langrange multiplier
for the variation of particle number with respect to energy, the chemical potential
appearing in the TWNCB approach should be identified instead with the nonlinear
eigenvalue of the GPE, corresponding to the fixed total system norm. In the SGPE
approach, the chemical potential has the usual grand canonical interpretation.
6.2.2
Stochastic Gross-Pitaevskii plus classical approximation
While the full theory of Stoof is formulated in terms of Wigner functions, the method
applied numerically here, following also the approximation introduced by Stoof [107,
108, 175], considers the Langevin equation of this theory in the classical limit. In this
case the pertinent thermal distribution function is the Rayleigh-Jeans distribution, as
discussed in Chapters 2 and 2. In addition, quantum noise effects are neglected in
the approximate form for the fluctuation-dissipation theorem used when applying the
SGPE in this limit. Hence, operator averages are considered in the classical field limit
hΨ̂† (z)Ψ̂(z)i ≈ hψ ∗ (z)ψ(z)i.
(6.12)
So, solving the SGPE to equilibrium for a number of realisations of the noise, results in
an ensemble of noisy thermal states. The thermodynamics of these is then described by
the Rayleigh-Jeans distribution which nonetheless approximates well the single particle
Wigner function, or Bose-Einstein distribution, for sufficient mode occupations.
Upon reaching equilibrium, if we then consider the γ(z) → 0 limit in Eq. (2.92),
which is equivalent to the physical process of removing contact between the system
and heat bath, the equation of motion also reduces to a Gross-Pitaevskii equation for a
noisy wavefunction, representing a number of the lowest system modes. On removing
the noise and damping terms of Eq. (2.92), the grand canonical ensemble is reduced
to a microcanonical ensemble, as the energy and particle number are then fixed; as
mentioned in Chapter 2, we denote this approach by SGPEeq . This distinction only
becomes relevant when dynamics are considered and we remove contact to the heat
bath (as in the soliton simulations at the end of Chapter 5), so when this is not the
case we will retain the abbreviation SGPE.
Parameters
We choose to compare the methods using the following trap parameters:
• ωz = 2π × 9 Hz;
• ω⊥ = 2π × 32 Hz;
• µ = 22.4 ~ωz ;
139
Chapter 6. Stochastic Gross-Pitaevskii and number conserving Bogoliubov methods
• N ≈ 1 × 103 - 4 × 104
87 Rb
atoms.
In the following test, we consider N ≈ 20000 unless otherwise stated. For this number,
these parameters result in the following values for some relevant quantities,
• Tφ ≈ 384nK;
• T1d ≈ 812nK;
• ξ ≈ 0.21 lz .
6.3
Equilibrium state comparison
The distinct ensembles of initial states, which form the stochastic fields to be input
into the GPE equation of motion, constitute the major difference between the methods
we apply here. Accordingly, in this section we present a detailed comparison between
the equilibrium properties which result for each approach.
6.3.1
Density profiles
In embarking on a comparison between the methods, we first illustrate the different
statistical distributions underlying the models. Also, we consider this as an example of
the correction necessary in calculating normally order operator averages in the TWNCB
method. These points are demonstrated in Figure 6.1, which shows the density in the
wings of a partially condensed Bose gas.
1/2
gn(z)/µ
0.1
z=(2kBT)
0.01
0.001
0.0001
1
2
3
4
5
6
7
8
z/R
Figure 6.1: Thermal density profiles of TWNCB prior to (green, dashed) and following the
Wigner correction to the ensemble averaged density (red). The SGPE data is shown by the
black, solid line together with the density for an ideal gas obeying Bose-Einstein statistics,
shown by the blue, dashed line. The vertical dotted line indicates the point at which βV (z) = 1,
which is where the trap energy is equal to the thermal energy kB T .
140
Chapter 6. Stochastic Gross-Pitaevskii and number conserving Bogoliubov methods
A first point of note is that the TWNCB density matches well the result for an ideal gas
obeying Bose-Einstein statistics, once the Wigner correction has been subtracted. Prior
to this, the averaged TWNCB density still includes the contribution due to an extra half
particle per mode, shown by the green, dashed line. However, we see that the classical
SGPE thermal density follows this curve closely, as expected from the Taylor series
expansion for the Bose-Einstein distribution. This is until the trap energy becomes
√
larger than the average thermal energy, which occurs at a position z = 2kB T lz , and
the SGPE density begins to fall off more rapidly than the uncorrected TWNCB result.
We now move to consider a more complete comparison between the densities at
different temperatures. Shown in Figure 6.2, is the total density representing the initial
state for each method, at three temperatures.
It is clear that there is very little difference between the results for the lowest two
temperatures, shown in Figure 6.2(a)-(b), whereas for the higher temperature case Figure 6.2(c), there is strong disagreement between the density profiles. As an independent
reference in this case, also shown in Figure 6.2(c) is the density predicted by the modified Popov theory of Andersen et al., in which the infrared divergences appearing in
the Bogoliubov approach applied to low dimensional systems was remedied, by taking
proper account of the contributions due to spurious phase fluctuations [217]. For the
chosen system parameters and no additional free parameters, the SGPE results show
good agreement to the findings of this independently derived method.
The densities of Figure 6.2(c) suggest that the condensate mode produced by the NCB
method is not adjusted correctly to the presence of the thermal cloud for the equilibrium
thermal occupation at this temperature. The underlying reason for this may be that
in calculating the Bogoliubov modes, the system is assumed to be entirely condensed,
so N0 = N . The thermal modes are then populated in accordance with Bose-Einstein
statistics, and a correction to the initial over estimate for the ground state occupation is
made. This procedure yields good agreement to the SGPE at the lower temperatures,
however fails to do so as the level of thermal depletion is increased, suggesting that
this first order correction to the ground state is no longer sufficient to account for the
higher level of thermal depletion.
The fourth plot of Figure 6.2, shows the total density which results on allowing the
TWNCB initial state at T = 185nK to evolve unperturbed, via the GPE, i.e. truncated
Wigner evolution. It is known that the ergodic nature of GPE propagation, leads to
thermalisation at a classical equilibrium [119], and this is what we see occurring in this
system. Following a period of around 400 ωz−1 , we find that the TWNCB total density
profile is already close to the SGPE equilibrium result, however the data of Figure
141
Chapter 6. Stochastic Gross-Pitaevskii and number conserving Bogoliubov methods
1
0.8
0.6
0.4
0.4
0.2
0.2
-5
0
5
1
10
15
(c)
0.8
0
-15 -10
1
0.6
0.4
0.4
0.2
0.2
-5
0
5
10
-5
0
5
15
0
-15 -10
10
15
(d)
0.8
0.6
0
-15 -10
(b)
0.8
0.6
0
-15 -10
gn(z)/µ
1
(a)
-5
0
5
10
15
z [lz]
Figure 6.2: Total density profiles for the TWNCB (red) and SGPE (black) approaches at (a)
T = 20nK, (b) T = 60nK and at (c) T = 185nK, where the modified Popov result is also shown
in green, dot-dashed; (d) The TWNCB state initially calculated at T = 185nK following ergodic
evolution to a time terg = 4000ωz−1 (blue, dashed), with the SGPE density at T = 185nK again
shown for comparison.
6.2(d) was obtained following an evolution time of 4000 ωz−1 . We see the state has now
evolved towards one with a density close to that of the SGPE result.
To analyse this further, we consider the proportion of the system in the condensed
and non-condensed phase, in each case. We approach this problem by extracting the
condensate mode based upon the Penrose-Onsager criterion [55], as described in Chapter 3. The densities which result are shown in Figure 6.3. As might be expected from
the total densities, the lowest temperature results display good agreement of Figure
6.3(a), with both the condensate and thermal fractions practically indistinguishable
between the approaches. There is a slight disparity in the results at the intermediate
temperature probed, shown in Figure 6.3(b), largely only at the trap centre, whereas
a much stronger disagreement arises in the highest temperature case of Figure 6.3(c).
In Figure 6.3(c), the condensate fractions are clearly different, even though the axial
extent of the condensate modes remains similar. Many more of the TWNCB atoms are
in modes orthogonal to the condensate mode, suggesting that at this temperature the
correction to the initial T = 0 wavefunction results in an over estimation in the level
of thermal depletion. This is likely due to a distortion of the Bogoliubov spectrum due
to its calculation based on the presence of too large a condensate.
142
Chapter 6. Stochastic Gross-Pitaevskii and number conserving Bogoliubov methods
Surprisingly, while matching the SGPE total density quite well following a long
period of equilibration via the GPE, the ergodic TWNCB result does not match the
SGPE condensate mode in Figure 6.3(d). We again find a reduced condensate fraction,
relative to the SGPE initial state, suggesting that the initial state of the TWNCB is
in fact at a higher temperature than that specified, as expected [119].
1
1
(a)
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
-15 -10
1
-5
0
5
15
(c)
0.8
gn(z)/µ
10
0
-15
1
0.6
0.4
0.4
0.2
0.2
-5
0
5
10
-10
-5
0
5
15
0
-15
10
15
(d)
0.8
0.6
0
-15 -10
(b)
0.8
-10
-5
0
5
10
15
z [lz]
Figure 6.3: Density profiles showing the condensed and non-condensed phases within the trap,
for each method. Shown are: The Penrose-Onsager condensate mode for the SGPE (black,
solid) and the TWNCB (red, dashed), and the density representing modes perpendicular to
this for the SGPE (brown, solid) and the TWNCB case (blue, dashed). As in the previous
figure, the plots represent (a) T = 20nK, (b) T = 60nK, (c) T = 185nK and (d) T = 185nK
for GPE evolution of the TWNCB state to terg = 4000ωz−1.
This analysis also allows us to extract the level of thermal depletion at each temperature, as summarised in Table 6.1. Here, the number of non-condensate atoms, Nt , is
obtained by integrating the non-condensate density profiles of Figure 6.3.
T [nK]
20
60
185
Nt /N in TWNCB
0.05
0.165
0.48 (→ 0.64 after terg = 4000ωz−1 )
Nt /N in ‘classical’ SGPE
0.05
0.148
0.38
Table 6.1: The number of non-condensate atoms, Nt , as a fraction of the total particle number,
N , for the TWNCB and SGPE methods at the temperatures considered.
143
Chapter 6. Stochastic Gross-Pitaevskii and number conserving Bogoliubov methods
6.3.2
Spatial correlation functions
Continuing the assessment of equilibrium properties, we now look at measures by which
to probe the level of coherence within our test-bed system. To do so, we make use of the
correlation functions introduced by Glauber [267], the first few of which are typically
sufficient to fully characterise the coherence properties of a system. While a study of
these functions was carried out in the context of a one-dimensional Bose gas, using
the SGPE in [183], such measurements are also able to be performed experimentally
[28, 27, 77, 80, 78], making this an interesting system property to explore.
We examine this at equilibrium, denoted by the time teq . We look first to measure
the phase coherence within the numerical systems under investigation, which we approach by comparing the normalised first order spatial correlation function, evaluated
as
g(1) (0, z; teq ) =
q
≈
p
hΨ̂† (z; teq )Ψ̂(0; teq )i
(6.13)
hΨ̂† (z; teq )Ψ̂(z; teq )ihΨ̂† (0; teq )Ψ̂(0; teq )i
hψ ∗ (z; teq )ψ(0; teq )i − {1/(2∆z)}
(6.14)
.
[h|ψ(z; teq )|2 i − {1/(2∆z)}] [h|ψ(0; teq )|2 i − {1/(2∆z)}]
As is clear from the above definition, this measures the correlations between off-diagonal
elements of the density matrix. The formation of spatial order in such correlations
signifies the phase transition to the Bose condensed state [54, 55], making this an
important measure of coherence.
We also consider density-density correlations, as parameterised by the normalised
second order correlation function
g(2) (z; teq ) =
≈
hΨ̂† (z; teq )Ψ̂† (z; teq )Ψ̂(z; teq )Ψ̂(z; teq )i
hΨ̂† (z; teq )Ψ̂(z; teq )i2
h|ψ(z; teq )|4 i + −2h|ψ(z; teq )|2 i/(∆z) + 1/(2∆z 2 )
(h|ψ(z; teq )|2 i − {1/2∆z})2
(6.15)
.
(6.16)
In the definitions of g(1) (0, z; teq ) and g(2) (z; teq ), quantities in curly brackets denote
the Wigner correction used to obtain normally ordered operator products. These corrections are subtracted in the TWNCB case only and are taken as equal to zero in the
SGPE case, since quantum fluctuations are neglected in the classical implementation
of the SGPE employed here.
The normalised first order correlation functions for the TWNCB and SGPE equilibrium states are shown in Figure 6.4. It is again evident that the low temperature
results of each method agree well, yet the high temperature results display rather different behaviour.
144
Chapter 6. Stochastic Gross-Pitaevskii and number conserving Bogoliubov methods
This is mirrored also in the behaviour of the second order correlation function.
Here, rather than a comparison between the spatial variation in the coherence profiles,
we draw instead on the results of Kheruntsyan et al. [268], as a basis for comparison
between the methods. It was found in this work that the second order correlation
function at the trap centre should vary with temperature as
1
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
-10
(1)
g (z,0)
1
(a)
0.8
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-10
-5
0
10
5
0
-10
(b)
-5
0
1
(c)
10
5
(d)
0.8
0.6
0.4
0.2
-5
0
5
10
0
-10
-5
0
5
10
z [lz]
Figure 6.4: Normalised first order correlation function, g (1) (0, z), for the SGPE (black, solid)
and TWNCB (red, dashed) at (a) T = 20nK, (b) T = 60nK and (c) T = 185nK; (d) GPE
evolution of the TWNCB state at T = 185nK to terg = 4000ωz−1 (blue, dashed) plus the SGPE
result of (c).
g
(2)
√
4 2 kB T
.
(0) = 1 +
3 N ~ωz
(6.17)
To gain some insight into the coherence properties of each initial state relative to
this, we compare our findings to this result in Figure 6.5. Shown is the variation
in g(2) (0) for the SGPE, TWNCB and TWNCB following thermalization for terg =
4000ωz−1 . As expected from earlier findings, the SGPE and TWNCB results agree
for lower temperatures, yet disagree in the higher temperature case. The results of
Kheruntsyan et al. appear to favour the findings of the SGPE equilibrium result, with
good agreement found at all three temperatures probed.
The second order correlation function takes in a value close to unity for a coherent
sample, in which one mode dominates, whereas it is equal to twice this value for a
classical multimode gas. Due to the higher values predicted by the TWNCB initial
state, shown by the red curve of Figure 6.5, the nature of the disagreement between the
TWNCB results and the Kheruntsyan formula suggests, like the density profiles, that
145
Chapter 6. Stochastic Gross-Pitaevskii and number conserving Bogoliubov methods
the TWNCB initial state is at a higher temperature than was input initially. To proceed
then, we wish to establish two things: firstly, whether the system is equilibrated, and
secondly, assuming this first point to be true, the temperature at which the initial state
has thermalised to.
1.2
(2)
g (0)
1.16
1.12
1.08
1.04
1
0
50
100
150
200
T [nK]
Figure 6.5: Shown are the values for g (2) (0) against temperature, calculated via simulations of
the SGPE (black circles), TWNCB (red squares) and TWNCB plus ergodic thermalisation to
terg = 4000ωz−1 (blue, triangle).
6.4
Equilibration and thermometry following ergodic thermalization
Having seen qualitatively that the TWNCB initial state undergoes some heating relative to the initial temperature at which it was calculated, we wish to make this
statement more quantitative. One way of doing so, is to extract the new resultant
temperature, although as an equilibrium property, we first must address whether a
new equilibrium has actually been reached. To assess whether the system has reached
a steady state, we consider first the condensate statistics and compare these to the
expected equilibrium behaviour.
6.4.1
Condensate statistics
At equilibrium, the condensate number fluctuates between different realisations of the
system within the statistical ensemble. An analytical form for the expected condensate statistics has been presented recently by Svidzinsky and Scully [120]. Their result
is based upon a matched asymptotic formula, which provides the full distribution of
condensate statistics, working within the canonical ensemble. Taking the Bogoliubov
modes as an input, which are calculated in a very similar fashion to the modes in
146
Chapter 6. Stochastic Gross-Pitaevskii and number conserving Bogoliubov methods
the NCB approach, their theory is however valid at temperatures beyond the limitations of the Bogoliubov spectrum inherent in this initial step, due to the procedure of
asymptotic matching [120].
We show in Figure 6.6, the predicted condensate statistics, together with the result of projecting the SGPE and TWNCB ensemble of stochastic wavefunctions onto
the condensate mode derived from each numerical scheme. Figure 6.6(a) shows the
condensate statistics for the lowest temperature case, with the TWNCB initial state
result closer to the analytic result of Svidzinsky and Scully, shown by the green curve.
The SGPE statistics display a more symmetric distribution, with a notable point being that fluctuations are strong enough to produce cases with the SGPE ensemble for
which the condensate number is larger than the average total particle number. This is
reasonable, as within individual runs, the total particle number can be greater than the
ensemble average, however the condensate fluctuations are not as close to the Scully
theory. This is likely an effect due to the low temperature, meaning few modes will be
highly occupied.
-3
-4
2×10
6×10
(a)
-3
2×10
4×10
-3
-4
1×10
3×10
-4
-4
5×10
1×10
0
17000
18000
19000
-4
0
20000
10000 12000 14000 16000 18000 20000
-4
2.5×10
2.5×10
(c)
-4
2.0×10
(d)
-4
2.0×10
-4
-4
1.5×10
1.5×10
-4
-4
1.0×10
1.0×10
-5
-5
5.0×10
0.0
(b)
-4
5.0×10
0
5000
10000
15000
0.0
20000
0
5000
10000
15000
20000
Nc
Figure 6.6: Normalised histograms of the condensate statistics for the SGPE (black) and
TWNCB (red) simulations at (a) T = 20nK, (b) T = 60nK, (c) T = 185nK and (d) T = 185nK
for GPE evolution of the TWNCB state to terg = 4000ωz−1. The results of Svidzinsky and
Scully are shown for each temperature by the green curve, for N = 2 × 104 .
For the intermediate case of Figure 6.6(b), the behaviour between the stochastic
methods is very similar, each reproducing reasonably well the predicted analytic distri147
Chapter 6. Stochastic Gross-Pitaevskii and number conserving Bogoliubov methods
bution for this temperature. Figure 6.6(c) shows clearly a difference between the initial
states, as found in other system properties. As we have moved to a higher temperature,
the SGPE is closer to its intended regime of applicability, a fact reflected in the close
match between these results and the Svidzinsky and Scully distribution. The TWNCB
initial state for this temperature predicts a higher proportion of low condensate numbers, consistent with the density profiles of Figure 6.3(c). The condensate statistics for
the ergodically evolved TWNCB initial state are shown, following an elapsed time of
4000ωz−1 , in Figure 6.6(d). Compared to the initial time result of Figure 6.6(c), the
distribution is noticeably shifted to the left, corresponding to more members of the
ensemble having lower condensate numbers.
Given the strong deviation in the condensate statistics of the TWNCB approach
relative to the initial condition, following this period of ergodic evolution, it is interesting to consider the evolution in stages, as is shown in Figure 6.7. This displays the
condensate statistics at various intermediate times. This allows us to probe dynamically, how the condensate statistics change as the system equilibrates ergodically, which
is an interesting process in itself.
0.0002
(a) terg=0
(b) terg=1000ωz
-1
(c) terg=2000ωz
-1
0.00015
0.0001
5e-05
0
0
10000
20000
0.0002
(d) terg=3000ωz
0
-1
10000
20000
(e) terg=4000ωz
0
-1
10000
20000
(f) terg=5000ωz
-1
P(N0)
0.00015
0.0001
5e-05
0
0
10000
20000
0
10000
20000
0
10000
20000
N0
Figure 6.7: Condensate statistics obtained from the classical field ensemble of the TWNCB
method following numerical evolution via the GPE. The statistics were collected following
evolution times, terg , as indicated. Also shown is the Svidzinsky and Scully result for T = 185nK
and N = 20000.
As mentioned, the NCB approach to obtain a finite temperature initial condition,
148
Chapter 6. Stochastic Gross-Pitaevskii and number conserving Bogoliubov methods
relies on diagonalising the BdG operator, whose construction is based on the assumption
that all the system particles are within the condensate mode. A correction to this
assumption is then made, while simultaneously conserving particle number, however at
this temperature, we see that initially in Figure 6.7(a), relative to the Svidzinsky and
Scully result, there are far more occurrences of low condensate numbers.
The evolution displayed in Figures 6.7 is suggestive of a scenario where initially,
many of the low energy particles are scattered into higher modes, corresponding to
a quite dramatic shift to the left in the distributions, as shown particularly in distribution for terg = 2000ωz−1 and terg = 3000ωz−1 . Following this, we would expect the
system to begin to equilibrate classically. In this respect, it is tempting to say that the
final distribution in Figure 6.7(f) is approaching a shape similar to that found in the
approach of Svidzinsky and Scully, only at a slightly higher temperature [120]. Further
evolution is required to confirm this however.
So, summarising our findings with regards the comparison so far: we see that while
in the low temperature plots, the statistics of each method matches relatively well the
Svidzinsky and Scully findings, for the higher temperature the TWNCB result is not
so close to this curve. In particular, Figure6.7(d) shows the non-equilibrium nature
of the system at the time the statistics were extracted, suggesting that despite a long
period of thermalization, a new equilibrium has not yet been reached. This slow rate of
equilibration is intimately linked to the fact that the trapped one-dimensional system
we consider is close to integrability, with this lifted only by the harmonic trapping
potential. For higher dimensional systems, we would expect this to be a far quicker
process, as even the homogeneous case is not integrable in this case. To understand
this further, we now look to the dynamics of the condensate mode as a measure of the
new system behaviour.
6.4.2
Oscillation of the Penrose-Onsager mode
Monitoring the temporal behaviour of the condensate number with time, for the unperturbed system, reveals oscillations in this quantity, which increase as temperature is
increased for the TWNCB approach. The condensate mode occupations are extracted
by projecting the stochastic wavefunction on to the equilibrium Penrose-Onsager condensate mode, as described in Chapter 3, and taking the ensemble average at each time
this is sampled.
A difference in the nature of the equilibrium solution provided by each of the methods is clear in this data, as in contrast to the TWNCB behaviour, the SGPE state
generates condensate number which on average appears to be constant in time. In addition, the SGPE predicts a consistently higher condensate fraction than the TWNCB
state at each temperature.
149
Chapter 6. Stochastic Gross-Pitaevskii and number conserving Bogoliubov methods
From Figure 6.8, we also see that there is a periodic transfer of particles between the
condensate mode and non-condensate modes, shown by the oscillation in the population
of the condensed fraction.
N0(t)
20000
15000
10000
(a)
N0(t)
5000
0
20000
20
40
60
80
100
15000
10000
(b)
N0(t)
5000
0
20000
20
40
60
80
100
(c)
15000
10000
5000
0
20
40
60
-1
80
100
Time [ωz ]
Figure 6.8: Temporal behaviour of the occupation of the Penrose-Onsager condensate mode
for the SGPE (black) and TWNCB approach (red) at (a) T = 20nK, (b) T = 60nK and (c)
T = 185nK. The total particle number is N ≈ 20000.
To analyse these oscillations, the Fourier transform was taken of the data available at
the two highest temperatures, which is shown in Figure 6.9. We find that the major
non-zero frequency component corresponds to an oscillation frequency of ωN 0 = 2ωz ,
which is the frequency of the breathing mode of a thermal gas [1]. This seems to be true
for both of the temperatures, as shown by the strong overlap in the Fourier transforms,
suggesting that the underlying frequency of oscillation is not strongly temperature
dependent, which is consistent if it is the thermal modes driving the oscillations.
The behaviour of the system appears to be relatively consistent over the timescales
probed in the previous two Figures, yet ultimately, we would expect some progress
to be made towards a new equilibrium. We therefore consider monitoring the condensate mode upon allowing the TWNCB states to evolve for far longer within the
150
Fourier amplitude [arb. units]
Chapter 6. Stochastic Gross-Pitaevskii and number conserving Bogoliubov methods
2000
1500
1000
500
0
0
1
2
3
4
5
Frequency [ωz]
Figure 6.9: Fourier transform of the condensate number as a function of time for the TWNCB
results at T = 60nK (red, dashed) and T = 185nK (black, solid). The vertical green dashed
line indicates a frequency of ωN 0 ≈ 2ωz .
unperturbed trap, with the aim of estimating the likely final temperature. Plotted in
Figure 6.10(a) are condensate mode densities, extracted from the TWNCB simulations,
following varying periods of ergodic evolution, in addition to the equilibrium SGPE result at the same initial temperature. We use this as a reference, due to the thermostat
inherent in the system plus heat bath approach.
The Penrose-Onsager condensate modes are calculated at different times by constructing the ensemble average over the quantity hψ ∗ (z, t)ψ(z ′ , t)i, defining the density ma-
trix, which can then be diagonalised. The results in Figure 6.10 are taken at times
so as to probe the short and long term behaviour, and we see that the condensate
mode appears to be undergoing oscillations in time, as we might expect given the oscillations in the condensate number. The data of Figure 6.10(b) shows the evolution
of the central density of the Penrose-Onsager mode, plotted alongside the equivalent
quantity extracted from SGPE simulations at various temperatures. Due to the long
equilibration times, we settle here in attempting to place bounds on the temperature
at which the TWNCB will ultimately settle to; based on Figure 6.10 we may estimate
that this is likely to be in the range 2Tinput < T < 4Tinput , where Tinput is the input
temperature for the TWNCB initial state.
6.5
Condensate statistics and number fluctuations
The different ensembles of statistical mechanics leads to differing predictions for condensate statistics within an ideal gas [269, 270, 271, 272] It is well known that fluctuations
in the grand canonical ensemble display anomalous scaling with number in the ideal gas
case; see [273] and references therein. Motivated by this, we consider here the weakly
interacting Bose gas with varying particle number, and all other parameters fixed, in
151
Chapter 6. Stochastic Gross-Pitaevskii and number conserving Bogoliubov methods
1
gn(z)/µ
0.8
t=1000.33 ωz
(a)
t=1001 ωz
-1
-1
0.6
t=1001.5 ωz
0.4
t=4000 ωz
SGPE t=teq
-1
-1
0.2
0
-4
-6
-2
0
2
4
6
8
10
z [lz]
n(0)PO/<n(0)>
0.9
0.8
T=Tinput
(b)
T=2 Tinput
0.7
T=2.7 Tinput
0.6
T=4Tinput
0.5
-1000
0
1000
2000
3000
4000
5000
6000
-1
Time [ωz ]
Figure 6.10: (a) Density profiles for the condensate mode of the TWNCB approach, extracted
from the one-body density matrix at short and long time intervals, as indicated, plus a comparison to the equilibrium SGPE result for T = 185nK. (b) The central density of the PenroseOnsager condensate mode taken from the TWNCB simulations, scaled to the instantaneous
peak density, versus time. As a temperature scale, also shown are the equilibrium SGPE
results at various temperatures; Tinput = 185nK for this data.
152
Chapter 6. Stochastic Gross-Pitaevskii and number conserving Bogoliubov methods
order to determine the role of interactions on this aspect of the system behaviour.
As a measure, we again use the Svidzinsky and Scully result for the condensate
number distribution, which is derived from the contrasting viewpoint of a canonical
ensemble calculation. The fixed number of the canonical approach, has been found to
tame the anomalous behaviour of the grand canonical ensemble [273]. The results of
our comparison are summarised in Figure 6.11, which shows a comparison between the
first three moments of this distribution. The data represented by the black curves and
circles, is obtained from the SGPE simulations. Each data point represents a different
value for the chemical potential in solving Eq. (2.92), which results in a different total
particle number for all other trap parameters held constant. We also fix the ratio
T /T1d , where T1d is the characteristic temperature for the onset of quasi-condensation
Eq. (1.10), though the use of this scaling belies a subtlety: the formula, Eq. (1.10) is
valid in the N → ∞ limit, and hence its applicability varies as we vary the total system
number. An improved calculation might fix the condensate fraction, as the total system
number changes, however this involves a detailed parameter search in the SGPE case,
which is time consuming even in one-dimension. So, in first instance, we make use of
the proposed scaling, which likely explains at least partially the variation in the average
condensate fraction of Figure 6.11(a), as N is increased. Despite this, we see that for
the same system parameters the Svidzinsky and Scully approach, shown by the green
squares, and SGPE results for the mean values, agree well across the whole range of
parameters tested, except perhaps for the lowest total number considered.
The results for the standard deviation of both the SGPE and Svidzinsky and Scully
results are again in good agreement. The final plot, Figure 6.11(c), shows the skewness,
and the ensemble difference between canonical and grand canonical is also hinted in
these results. The main point in Figure 6.11, is the agreement between the SGPE and
Svidzinsky and Scully distributions as the particle number is increased. This shows
that as interactions become more dominant within the system, the effects of ensemble
differences are lessened, at least up to differences in the first three moments of the
condensate number distribution. This suggests that interactions smooth the number
fluctuations which are known to lead to unphysically large fluctuations in the grand
canonical ideal gas case [270, 273].
6.6
Summary of equilibrium results
Some salient points to note in a comparison between these methods are summarised in
Table 6.2. We emphasise however that the properties listed here relate to the classical
153
<N0>/N
Chapter 6. Stochastic Gross-Pitaevskii and number conserving Bogoliubov methods
1
0.9
0.8
0.7
0.6
0.5
0.4
0
(a)
10000
20000
40000
(b)
0.4
0.3
0
σN /<N0>
0.5
0.2
0.1
0
Skew(N0)
30000
N
0
0.5
0
-0.5
-1
-1.5
-2
-2.5
-3
10000
20000
30000
40000
N
(c)
0
10000
20000
30000
40000
N
Figure 6.11: First three moments of the condensate number probability distribution function
against total particle number, for the SGPE (black, circles) and the distribution function of
Svidzinsky and Scully (green, squares; Eq. (5) of [120]) Shown are (a) the mean, (b) the standard
deviation, scaled to the mean and (c) the skewness of the condensate number.
154
Chapter 6. Stochastic Gross-Pitaevskii and number conserving Bogoliubov methods
SGPE model as it is solved numerically here; the full quantum theory derived by Stoof,
does not share many of the properties summarised there.
Optimum temperature regime
Equilibrium thermal distribution
Thermodynamic ensemble
Number conserving
Quantum noise
Stationarity under GPE
TWNCB
T ≪ Tc
Bose-Einstein(+1/2)
Canonical
✓
✓
✗
‘classical’ SGPE
T >0
Rayleigh-Jeans
Grand canonical
✗
✗
✓
Table 6.2: Table summarising the conceptual differences and intended regimes of applicability
of the TWNCB and ‘classical’ SGPE (plus static thermal cloud) methods.
An additional point of note here is that having quantified to a large extent the grid
spacing dependence of the SGPE at equilibrium, in Chapter 3, we can be confident
that our findings are not strongly influenced by the numerical discretisation that we
work with. This is chosen to be fixed between the methods, but is allowed to vary with
temperature, in order that βEgrid ≥ 1 is at least satisfied.
Figure 6.12 shows a summary of our findings from the comparison at equilibrium
between the SGPE and TWNCB approaches. Note that g is held constant in varying
the particle numbers. These, and the results from the earlier parts of the Chapter
suggest the following conclusions:
• For small numbers and low temperatures, the TWNCB method predicts the correct condensate statistics, whereas the SGPE does not.
• The disparity between the SGPE statistics and those of Svidzinsky and Scully is
either due to the classical approximation breaking down, or the use of the grand
canonical ensemble in this approach. Figure 6.12(a) suggests it is the latter, as the
TWNCB statistics calculated replacing the Bose-Einstein distribution with the
Rayleigh-Jeans distribution show consistent results. Therefore, we can conclude
that number fluctuations in the grand canonical ensemble play a large role in this
regime.
• For larger particle numbers and low temperatures, the SGPE and TWNCB results
agree extremely well in all tests. This shows there is an intermediate regime in
which the two methods agree.
• For larger particle numbers, and higher temperatures, the TWNCB method does
not produce a good thermal state.
155
Chapter 6. Stochastic Gross-Pitaevskii and number conserving Bogoliubov methods
• In one-dimension, ergodic evolution leads to extremely slow thermalisation. This
is likely due to the system being close to integrability, with this lifted only by the
harmonic trap.
• The state to which the TWNCB state might eventually thermalise will be at a
higher temperature than that at which the state was originally calculated.
0.01
0.0005
(a) Small N, low T
0.008
0.0004
0.006
0.0003
0.004
0.0002
0.002
0.0001
0
500
1000
750
0
1250
(b) Intermediate regime
8000
12000
0.0002
20000
0.0002
(c) Larger N, higher T
0.00015
0.00015
0.0001
0.0001
5e-05
5e-05
0
16000
N0
N0
0
5000
10000
15000
0
20000
(d) Ergodic NCB for
larger N, higher T
0
5000
10000
15000
20000
N0
N0
Figure 6.12: Condensate statistics results from the SGPE (black), TWNCB (red) and Svidzinsky and Scully approach (green) for: (a) N ≈ 1000, T ∼ 0.68 Tφ , T ∼ 0.21 T1d , also
shown here is the result of using Rayleigh-Jeans rather than Bose-Einstein statistics for the
TWNCB approach (light-blue); (b)N ≈ 20000, T ∼ 0.16 Tφ ,T ∼ 0.07 T1d ; (c)N ≈ 20000,
T ∼ 0.48 Tφ ,T ∼ 0.23 T1d ; (d) as (c) but the TWNCB initial state has thermalised for a time
teq = 5000ωz−1.
6.7
Non-equilibrium Tests
Having discussed various static system properties, we now examine the response to perturbations about equilibrium. Initially, as for the equilibrium properties of the SGPE
within the classical approximation, considered in Chapter 3, we assess the variation in
SGPE results with the grid spacing used. We then compare the results of the TWNCB
and SGPE simulations when the harmonic trapping frequency is perturbed.
156
Chapter 6. Stochastic Gross-Pitaevskii and number conserving Bogoliubov methods
6.7.1
Assessment of grid effects: response to trap opening
In the following, we keep the system size fixed, and vary the number of grid points,
Nz , that we use in representing the system numerically. Therefore, larger Nz implies a
smaller grid spacing. The results presented here are for simulations in which the SGPE
was solved to provide an ensemble of equilibrium states. These states then serve as
stochastic initial conditions for simulations using the GPE as the equation of motion
in studies beyond equilibrium.
The dynamical perturbation we consider as a test case is an instantaneous change in
the trapping frequency: ωz → 0.8 ωz . This stimulates oscillations of the system within
the new trap, as the initial states no longer correspond to the equilibrium solution in
the new trapping potential. Shown in Figure 6.13 are the central density oscillations
which result for various grid spacings. For all grid spacings considered, the system
displays damped oscillations as the new equilibrium is approached.
The feature of most interest at present is the difference in oscillations for different
grid spacings. It is apparent that the coarser grids lead to additional damping relative to the finer grid spacings, showing that the rate at which the new equilibrium is
approached is dependent upon this. A reason behind this trend may be an increased
numerical diffusion associated with a larger grid spacing, increasing the rate at which
equilibration occurs.
gn(0)/µ
1
0.95
0.9
0.85
0
10
20
30
t
40
50
-1
[ωz ]
Figure 6.13: Central density oscillations upon instantaneously changing the trap frequency
ωz → 0.8ωz at T = 120nK for four numerical grids: nz = 128 (blue), nz = 256 (green),
nz = 512 (red), nz = 2048 (black). The system size is fixed in each test, so a larger number of
grid points implies a smaller grid spacing.
While the results for finer grid spacings (Nz ≥ 512) seem qualitatively to agree in
Figure 6.13, we consider this agreement in more quantitative manner, by fitting the
157
Chapter 6. Stochastic Gross-Pitaevskii and number conserving Bogoliubov methods
central density oscillations to the function
a + b cos (ωosc t) exp (−δt) ,
(6.18)
paying particular attention to the damping rate δ and oscillation frequency ωosc . Here
a and b are other fitting parameters.
It is evident from Figure 6.14 that the results of this fitting procedure seem to
plateau as the number of grid points increases. This behaviour suggests that there is
a minimum number of grid points which must be used in order to obtain consistent
result from simulations, corresponding to some minimum energy cutoff. The plateau
in the results of Figure 6.14 occurs around the grid number which would correspond
to a maximum grid energy equal to the thermal energy, kB T , which seems a sensible
result on physical grounds, and also given the findings of Chapter 3.
0.008
δ [ωz]
(a)
0.006
0.004
0.002
0
500
1000
1500
2000
2500
Ngrid
1.58
ωosc [ωz]
(b)
1.56
1.54
1.52
1.5
0
500
1000
1500
2000
2500
Ngrid
Figure 6.14: (a) Damping and (b) Frequency of central density oscillations for the perturbation
considered in Figure 6.13. The vertical dashed line indicates the number of grid points needed
to give a grid energy Egrid = kB T . The system size is fixed, so an increasing number of grid
points means a smaller grid spacing has been used.
6.8
Dynamical test between methods: response to a trap
perturbation
The final comparison we make, is to simulate how the SGPE and TWNCB methods
methods respond to a trap perturbation such that ωz → 0.8ωz .
The simulation procedure for the dynamical test was, for the TWNCB method,
158
Chapter 6. Stochastic Gross-Pitaevskii and number conserving Bogoliubov methods
to obtain an initial state, based on the procedure outlined in Section 6.2.1, at each
temperature4 . The analogous SGPE state was obtained by propagating an ensemble of
states representing the system to equilibrium, again at each temperature to be tested.
Beyond this point, the simulations for each method were identical, with the GPE used
as the equation of motion in each approach. The subsequent evolution then conserves
both particle number and energy, as would be expected. Nonetheless, damping in
the central density oscillations is still observed, as was found in applying the SGPEeq
approach to quasi-condensate dynamics on an atom chip [181].
We show in Figure 6.15 the central density oscillations which follow upon opening
up the harmonic trapping potential as described. Results were collected by preparing
an ensemble of initial states, typically 1−2×103 at each temperature, for each method.
We then propagate each of these states using the GPE for the time shown in Figure
6.15. The ensemble average of the central density oscillations for each initial state is
then calculated.
gn(0)/µ
1 (a)
0.95
0.9
gn(0)/µ
gn(0)/µ
0.85
0
10
20
30
40
50
0.85
0
10
20
30
40
50
1 (c)
0.95
0.9
0.85
0.8
0
10
20
30
40
50
1
(b)
0.95
0.9
-1
Time [ωz ]
Figure 6.15: Central density oscillations obtained upon perturbing the trap frequency so ωz →
0.8ωz . The SGPE results are shown in black, and the TWNCB results by the red, dashed
curve. The ergodic (terg = 1000) TWNCB results are shown by the dot-dashed, blue curve.
The simulations were carried out for input states at (a) T = 20nK, (b) T = 60nK and (c)
T = 185nK.
Consistent with the equilibrium findings, the low temperature states agree well, and
4
As mentioned, these were supplied by Antonio Negretti (Ulm University).
159
Chapter 6. Stochastic Gross-Pitaevskii and number conserving Bogoliubov methods
we see that the TWNCB and SGPE results of Figure 6.15(a)-(b) are practically indistinguishable. The results for the higher temperature are shown in Figure 6.15(c), both
for the initial states given by the NCB approach, and also for the same ensemble of
states which were allowed to evolve under the GPE for a period of 1000 ωz−1 in the
unperturbed trap. These are compared to the results of the SGPE at the same temperature. While the TWNCB initial states show very different dynamics, as might be
expected, the ergodically equilibrated results are much closer to the SGPE results at
short times (t < 10 ωz−1 ). Beyond this point, the SGPE oscillations display less damping, consistent with the picture that the TWNCB initial state has, at least partially,
equilibrated to a state of higher temperature.
To analyse these results more quantitatively, we fit the central density oscillations to
a damped sinusoid, and find a linear variation with temperature, for those temperatures
probed, as shown in Figure 6.16.
Damping rate [ωz]
0.03
0.025
0.02
0.015
0.01
0.005
0
0
100
200
300
400
T [nK]
Figure 6.16: Temperature dependence of the damping rate of the central density oscillations of
Figure 6.15. SGPE data is shown by black circles, the TWNCB data by red squares and the
TWNCB ergodic data by the blue triangle. The errorbars are obtained from the least squares
fitting procedure; the dotted lines indicate the extrapolation to the SGPE fit in assigning a
temperature to the ergodic TWNCB ensemble.
Extrapolating this linear behaviour to higher temperatures, it is possible to infer a
temperature for the ergodically equilibrated TWNCB states. This is shown in Figure
6.16 by the black solid line. The dotted black lines show the temperature range inferred
from the error in fitting the oscillations of Figure 6.15(c), which are shown by the
errorbars of the blue data point. This gives a new temperature somewhere in the
range 340nK < T < 375nK, though again as the system may not be fully equilibrated,
serves only as an estimate as to the likely final temperature. That said, the estimate
we obtain by this dynamical means, is in reasonably good agreement with that based
160
Chapter 6. Stochastic Gross-Pitaevskii and number conserving Bogoliubov methods
upon the behaviour of the Penrose-Onsager mode, which gave the far larger range
370nK < T < 740nK. The last few points of Figure 6.10 might be interpreted as
showing a tendency towards equilibration at T ≈ 2 Tinput , where here Tinput = 370nK.
6.9
Centre of mass oscillations
At the end of Chapter 2, we gave several methods to which the classical approximation
to the SGPE is in some way similar. Here we illustrate a difference between these approaches, through a further dynamical test. We again perturb the trapping potential,
however now test the dynamical response to a translational shift in the trapping potential, such that V (z) → V (z − R/2), where R is the T = 0 Thomas-Fermi radius. This
stimulates the centre-of-mass, or Kohn [274, 275], mode. For a gas in a harmonic trap,
this mode has a frequency equal to the trapping frequency, and according to Kohn’s
Centre of mass [z/R]
theorem [275], the gas should perform undamped oscillations, at this frequency.
(a)
1
0.5
0
0
10
20
30
40
50
30
40
50
-1
Time [ωz ]
22000
(b)
N
20000
18000
16000
0
10
20
-1
Time [ωz ]
Figure 6.17: (a) Position of the centre of mass of the finite temperature system. following
an instantaneous shift in the position of the harmonic trapping potential for the SGPE with
noise at all times (red), the SGPE with contact to the heat bath removed prior to the trap
perturbation (black), the DGPE (blue, dashed) and the GPE (green, dashed). These results
are for an equilibrium ensemble at T = 120nK, prior to the perturbation; the simulations were
performed for a spatially constant self-energy, such that γ(z) = 0.005.
Our aim is to use this to show the difference in retaining or disregarding the dynamical
noise contribution beyond reaching equilibrium, and the effect of a static thermal cloud.
161
Chapter 6. Stochastic Gross-Pitaevskii and number conserving Bogoliubov methods
The results obtained are shown in Figure 6.17. We consider first the SGPE simulations,
which are identical up to the perturbation. Beyond this point, in one case, we solve
the full stochastic equation of motion with additive noise at all times (red curve). In
the other case, the SGPEeq method, we keep the noise term up to the point at which
the trapping potential is shifted, beyond which time the dynamical evolution is via the
GPE (black curve). We see the latter leads to undamped motion, while the former
methods leads to a damping effect. This is due to the approximation in which the
high-lying modes of the thermal cloud are treated as static. Therefore, because we
do not represent these modes dynamically, we see an additional damping effect. This
would not be the case if we were to solve the quantum Boltzmann equation, which was
shown to satisfy Kohn’s theorem [107]. This was also shown to be the case for the
analogous equation within the ZNG approach [154].
From another perspective, the action of the heat bath terms in Eq. (2.92) is to drive
the system towards equilibrium. Hence, we would expect a damping of the oscillatory
motion, as the system heads towards the new equilibrium such that the centre of mass
resides at the new trap centre, z = 0.5R. For comparison, we also show the analogous
mean field results for each of the SGPE approaches mentioned. The T = 0 GPE result
shows the dynamics which are observed if the T = 0 ground state solution is used as
an initial state, rather than the ensemble of classical states generated by the SGPE.
We see that the average dynamics of this ensemble, is in fact the same as that for the
ground state propagated via the GPE. To compare to the full SGPE simulations, we
also show the results of retaining the dissipative term of Eq. (2.92), but neglecting the
noise term, which results in the DGPE. In this case we again start with the ground
state for the system, however in contrast to the GPE case, we see a relaxation towards
the new equilibrium, which is a feature linked to retaining the imaginary source term
in the equation of motion, Eq. (2.95). Also shown in Figure 6.17(b) are the system
norms for each of these simulation methods. This illustrates the perturbation from, and
subsequent approach to, equilibrium for the full SGPE and dissipative cases, whereas
the conservation of particle number for dynamical evolution via the GPE alone.
It is also important also to note that while the equilibrium reached in solving the
SGPE is not strongly dependent upon the choice of γ(z), the system response to a
dynamical perturbation is. This can be seen from the results of Section 3.2, from
which we can see that the rate at which the system grows to a new equilibrium is
strongly dependent upon γ(z). Here we chose an arbitrary value, γ(z) = 0.005, as our
aim was merely to illustrate the differences between the approaches discussed.
This also motivates a next step in the implementation of the SGPE approach to
a dynamical description of non-equilibrium weakly interacting Bose gases, namely the
coupling of the SGPE to a quantum Boltzmann equation [107, 124].
162
Chapter 6. Stochastic Gross-Pitaevskii and number conserving Bogoliubov methods
6.10
Chapter Summary
To summarise, in this Chapter we have compared the physical consequences of the
formal differences between the stochastic Gross-Pitaevskii and truncated-Wigner plus
number conserving Bogoliubov approaches to finite temperature Bose fields.
At equilibrium, we found for small numbers and low temperatures, that the TWNCB
approach led to better agreement with the Svidzinsky and Scully predictions for the
condensate number statistics [120] than the SGPE. We attribute this effect to enhanced number fluctuations within the grand canonical ensemble of the SGPE versus
those within the canonical ensemble of the TWNCB method. For larger particle numbers and low temperatures, we found an intermediate regime in which the classical field
methods showed good agreement. Increasing the temperature however led to disagreement between the initial states of these methods. In the higher temperature regime,
the SGPE was found to agree with the density profiles of the modified Popov theory
of Andersen et al. [217], the second order correlation function results of Kheruntsyan
et al. [268] and also the condensate statistics of Svidzinsky and Scully [120].
Allowing the system to thermalise, we found the total system densities to agree more
closely between the TWNCB and SGPE. However, extracting the Penrose-Onsager
condensate modes for each case revealed the TWNCB method to have thermalised to a
higher temperature, in agreement with the behaviour found in a previous study [119].
Despite a long period of evolution, the system did not fully thermalise. This is likely due
to the one-dimensional system considered being close to integrability, making ergodic
thermalisation inefficient. Nonetheless, we attempted some approximate thermometry
schemes, which suggested that for the parameters tested, the new system temperature
was in the range 2Tinput < T < 4Tinput , with Tinput the temperature of the prepared
initial states.
163
Chapter 7
Conclusions and future work
7.1
Conclusions
In this thesis, we have presented a study of weakly interacting, one-dimensional atomic
Bose gases within harmonic trapping potentials, both in and out of equilibrium. While
the dynamical properties of Bose-Einstein condensates at zero temperature can be well
described by the T = 0 Gross-Pitaevskii equation, which captures the ground state dynamics associated with nonlinear effects due to interatomic interactions, our aim was
to additionally include finite temperature effects. At T > 0, and particularly in highly
elongated geometries, there is an enhancement in the strength of fluctuations in the
density and phase of a quantum degenerate Bose gas. Following a perturbation from
equilibrium, the presence of thermal excitations typically leads to dissipative effects
which can be seen, for example, in the damping of collective modes or dark soliton
oscillations. In addition, thermal fluctuations can be expected to contribute to a shotto-shot variation between experimental realisations within such systems. To model
these finite temperature effects, our method was to numerically solve the stochastic
Gross-Pitaevskii equation [107, 108, 175, 109, 110, 182, 276, 181, 183, 212, 121]. In
addition to the T = 0 Gross-Pitaevskii approach, this equation additionally includes a
damping term and a noise term; respectively, these terms represent coherent and incoherent scattering processes between low-energy modes, which we represent dynamically,
and higher energy thermal modes which are treated as a heat bath.
We initially applied the stochastic Gross-Pitaevskii equation to the equilibrium
properties of a trapped Bose gas, comparing the effect of grid spacing used in the numerical approach adopted [277, 108]. Beyond these tests, we considered the dynamics
of dark solitons at finite temperatures. Dissipative effects are known to manifest as
a damping of soliton motion in these systems, and we modelled this aspect using the
dissipative Gross-Pitaevskii equation. To study features of interest beyond the average, mean-field dynamics, we then considered the dynamics of dark solitons described
164
Chapter 7. Conclusions and future work
within the stochastic Gross-Pitaevskii equation. Finally, in Chapter 6, we considered a
comparison between two stochastic approaches to finite temperature Bose Einstein condensates, comparing the stochastic Gross-Pitaevskii equation to a number conserving
Bogoliubov approach. We considered a range of equilibrium properties of an interacting, trapped Bose gas at various temperatures beneath the transition temperature. Our
findings in each of these areas are summarised in more detail in the following sections.
7.1.1
Dissipative dark solitons
As a first step towards a full description of the dynamics of dark solitons within partially condensed one-dimensional Bose gas, we considered as a model the dissipative
Gross-Pitaevskii equation which arises on neglecting the noise within the stochastic
Gross-Pitaevskii equation. To account for the effect of the Hartree-Fock potential of
the equilibrium system, we used the stochastic Gross-Pitaevskii equation equilibrium
density to calculate an appropriate contribution to the thermal energies of Eq. (3.55).
We also made a comparison between spatially constant approximations to the damping
rate, finding γ̄, the integrated value over −R/2 < z < R/2, to give good agreement
with the results using the self-consistent spatially varying γ(z). The integrated value
depends upon the stochastic Gross-Pitaevskii equation density used in calculating γ(z).
However, given the good agreement between γ̄ and γ(z), and that the Thomas-Fermi
density only leads to divergences at the Thomas-Fermi radius [175] (see also Figure
3.18 of Chapter 4), our findings show that calculating γ̄ based upon the Thomas-Fermi
density is a reasonable approximation in this regime.
We found that the dynamics of dark solitons predicted by simulations of the dissipative Gross-Pitaevskii equation led to damped motion, consistent with that of a particle
within a quantum dissipative system [114, 249]. The observed decay times vary with
temperature such that
τ1/2 ∼ T −3.83 ,
(7.1)
which is close to the T −4 scaling that is expected for a dark soliton interacting with
long-wavelength thermal excitations [233, 250]. While this result was previously found
in the regime µ ≪ kB T , our findings suggest this behaviour also extends to the regime
in which µ ∼ kB T .
7.1.2
Stochastic dark solitons
Having considered the dissipative aspects of dark soliton dynamics within a one-dimensional
Bose-Einstein quasi-condensate, we then considered the propagation of these nonlinear
excitations within a description satisfying the full fluctuation-dissipation relation. Using the stochastic Gross-Pitaevskii equation, we modelled the effects of both dissipation
165
Chapter 7. Conclusions and future work
and fluctuations on a soliton moving through a sea of thermal excitations.
Experimental realisations have reported a shot-to-shot variation in soliton lifetimes.
Motivated by this, we considered the effect of a fluctuating background density on the
lifetime of solitons, concentrating on temperatures within the strongly phase fluctuating
regime. We focused on this scenario in order to enhance the statistical effects on soliton
propagation. We found a rich variety of additional features beyond those captured
within the Gross-Pitaevskii or dissipative Gross-Pitaevskii mean field models, which can
only represent the average dynamics. Thermal fluctuations were found to be sufficiently
strong to cause a diverse range of soliton behaviour, including reversals in the soliton
motion relative to the initial velocity with which they were introduced, in extreme
cases. This type of behaviour was more common for low velocity solitons, which were
strongly affected by fluctuations associated within the turbulent thermal background.
We considered a comparison of the soliton decay time statistics, which displayed
a wide statistical spread. This is despite the fact that within the ensemble of initial
states, the solitons were identically created. We showed that such an identical means
of introduction gave rise to a spread in initial velocities, but also that the coupling to
thermal modes during the soliton evolution had a relatively larger effect on the decay
time statistics. The distribution of decay times was fitted to a lognormal distribution,
which led to the proposal of multiplicative degradation as the underlying decay mechanism. Motivated by this, we additionally suggested a stochastic generalisation to the
analytic formula presented in [121] for the average soliton trajectory, which made a
connection to the model of geometric Brownian motion.
Dark solitons were also modelled using the equilibrium stochastic Gross-Pitaevskii
equation initial states as inputs to classical field simulations. The dynamics in this
case was then described via the Gross-Pitaevskii equation. While the solitons had a
temperature dependent distribution of initial velocities, leading to distinct soliton paths
between individual runs at a given temperature, the ensemble averaged trajectory was
found to recover that predicted by the T = 0 Gross-Pitaevskii equation. This behaviour
was found to be independent of the temperature of the initial states.
7.1.3
Comparison between the stochastic Gross-Pitaevskii and number conserving Bogoliubov methods
Following on from the soliton classical field simulations, we then made a comparison
between the equilibrium thermal state of the stochastic Gross-Pitaevskii equation, and
that predicted within a number conserving Bogoliubov approach.
Through a comparison of equilibrium properties such as the system density, condensate statistics, and two lowest order spatial correlation functions, we found the
stochastic Gross-Pitaevskii equation and number conserving Bogoliubov method to
166
Chapter 7. Conclusions and future work
agree well at low temperatures. This agreement did not persist as the temperature
was increased, with the NCB method found to lead to a higher temperature state than
that specified as an input in its calculation. At this higher temperature, the state was
allowed to thermalise classically through ergodic evolution under the Gross-Pitaevskii
equation. This led to a better agreement between the total densities of the methods,
however the condensate fraction was found to differ still.
Following a long period of evolution, the number conserving Bogoliubov state did
not reach a clear equilibrium, though we made some estimates on the temperature,
which we found to be in the range 2Tinput . T . 4Tinput , where Tinput denotes the
intended input temperature for the simulations.
We also discussed the variation in condensate statistics as the total particle number
was varied within the stochastic Gross-Pitaevskii equation. The first three moments of
the condensate statistics distribution were compared to the formulae of Svidzinsky and
Scully [120]. As a grand canonical approach, this highlighted that the stochastic GrossPitaevskii equation method was unable to match the canonical ensemble statistics
predicted by the method of Svidzinsky and Scully for small system numbers. Our
findings showed however, that these approaches coincide as particle number increases,
and interactions become more important.
7.2
Future prospects
Having concentrated upon the characteristics and excitations of a one-dimensional system in this work, a goal is to progress to studies in higher dimensions. There are
however challenges associated with this, due to the classical nature of the implementation of the SGPE applied here. We discuss now some means by which to address these
challenges, and also some potential applications of interest.
7.2.1
Addressing grid issues in higher dimensions
As expected of a classical field theory, the SGPE implemented within the classical
approximation [107, 108] suffers from ultraviolet divergences. We show in Figure 7.1
some results of preliminary work on solutions to the SGPE in two and three dimensions.
To demonstrate the enhanced divergences in higher dimensions, we also show the results
in one-dimension. The data shown is the particle number, which we have scaled to
the largest particle number measured in that dimension, and which corresponds to
the result of the smallest grid spacing tested. Note that in the one-dimensional case,
however, the particle number has a good continuum limit for increasingly small grids,
as discussed in Chapter 3. For the 2d case, these results were published in [212].
167
1
(a)
1
(b)
0.8
0.8
0.96
0.6
0.6
0.94
0.4
0.4
0.92
0.2
0.2
Scaled <N> (arb. units)
0.9
0
1.2
0.2
0.4
0.6
0.8
1
Grid spacing ∆x [lz]
0
0
0.1
0.2
0.3
(d)
1
0.8
0.6
0.4
0.2
0
1
2
3
4
5
0.4
1.2
0.6
0.5
(c)
1
0.98
Scaled <N> (arb. units)
Scaled <N> (arb. units)
Chapter 7. Conclusions and future work
0
0
0.2
0.4
0.6
0.8
1
5
6
(e)
1
0.8
0.6
0.4
0.2
0
0
1
2
3
4
1/(∆x)
ln(1/∆x)
Figure 7.1: The total particle number in the one-, two- and three-dimensional stochastic GrossPitaevskii equation. Shown are scaled particle numbers against grid spacing for
√ a harmonically
trapped finite temperature Bose gas in (a) 1d (plus fit a to f (∆x) ∝ arctan([ ∆x]−1 )), (b) 2d
(plus a fit to f (∆x) ∝ ln([∆x]−1 )) and (c) 3d (plus a fit to f (∆x) ∝ ([∆x]−1 )); (d) 2d scaled
number versus log([∆x]−1 ); (e) 3d scaled number versus ∆x−1 . In the lower plots, the green
dashed curve is a least squares fit to a straight line. The numbers are scaled to the maximum
particle number recorded for each dimension. Note the difference in grid spacings considered;
in the higher dimensional cases, we considered an isotropic system so the grid spacing was the
same in each direction.
168
Chapter 7. Conclusions and future work
Figure 7.1(e) shows the scaling in 2d of the particle number with log([∆x]−1 ), and
the straight line fit shows the numerical results follow the expected scaling with the
grid spacing. This is similarly true for the 3d case, where we find the number scales
∼ [∆x]−1 . In contrast, we see the 1d results of Figure 7.1(a) show a far more limited
√
dependence upon the grid, which we find is well matched to ∼ arctan([ ∆x]−1 ).
We compare the severity of the problem between simulations in different spatial
dimensionality in Figure 7.2. Here we have taken the fitted functions from Figure
7.1(a)-(c), and scaled these so that they pass through hN i = 1 at the same value of
the grid spacing. We can see clearly from Figure 7.2 the different extent to which
simulations are affected in higher dimensions. While there is up to a 10% change for
the 1d case, across a very large range of grids, fitting to the 3d results suggests around
a 95% change in particle number for the same range. Of course, the effect is always
going to be amplified in higher dimensions, as the number of modes increases with the
momentum cutoff more rapidly than in 1d.
One approach to the problem of ultraviolet divergences is to introduce counter terms
into the equation of motion. These are introduced in order to balance the divergent
behaviour at higher energies. This approach has been discussed for example by Parisi
[278] in the context of the stochastic Ginzburg-Landau model, and has been applied in
renormalising lattice theories in relativistic models of φ4 field theories [279].
Scaled <N> (arb. units)
1
1d
0.8
0.6
2d
0.4
0.2
3d
0
0
0.2
0.4
0.6
0.8
1
Grid spacing [lz]
Figure 7.2: The scaled, average particle numbers in simulations of the SGPE in 1d (black, solid),
2d (red, dashed) and 3d (blue, dot-dashed). The harmonic trapping potential was isotropic (in
the directions simulated) in the 2d and 3d cases, with lz the harmonic oscillator length of the
isotropic trapping frequency. Note that grids approaching zero are limited by long computation
times, whereas grids approaching 1 on this scale would likely be too coarse to represent the
system well.
Alternatively, and certainly more fundamentally pleasing, is to solve the problem of the
whole trapped gas self-consistently, by coupling the SGPE, Eq. (2.89), to the quantum
169
Chapter 7. Conclusions and future work
Boltzmann equation, Eq. (2.80). Describing the high-lying modes dynamically would
mean that they are no longer assumed to be statically distributed. Therefore the
classical approximation to the fluctuation-dissipation relation, Eq. (2.90), is no longer
necessary, and the −iR coupling term in Eq. (2.89) is retained. A more satisfactory
model is obtained in this way, as a description of the dynamical equilibration would be
retained for all modes, rather than only the low-lying modes. This is by no means a
trivial task, even numerically, but represents another avenue for future work.
7.2.2
Modelling applications of ultracold gases
An application of ultracold Bose gases which has received much attention is that of
atom interferometry. This is envisaged as a basis for extremely sensitive measuring
devices, and has advantages over optical interferometry, due, for example, to the range
of atomic species which can be chosen based on their particular physical properties [53].
Unlike photons, atoms offer different characteristics which can therefore be tailored
depending upon the particular measurements to be undertaken. Decoherence between
the splitting and interfering of ultracold atomic samples is of key importance, with
fluctuations in low-dimensions likely to play a large role in such systems. This makes a
stochastic approach, such as the SGPE (with a fully dynamical cloud), a good candidate
for modelling these devices.
Figure 7.3 shows some density snapshots from a preliminary numerical experiment
simulating the various stages of an atomic matter wave interferometry experiment.
The densities of Figure 7.3(a)-(d) show the growth into a single, but elongated trapping
potential. During the growth process, the vertical striation in the density profiles shows
the formation of soliton-like structures during the non-equilibrium growth process.
These are visible as shadows in the first four density profiles, and are seen to retain
their soliton-like form, rather than decaying into vortices, due to enhanced stability
within the elongated geometry. Figure 7.3(e) shows the equilibrium density, which can
then be split using a double well potential, as shown in Figure 7.3(f). This is analogous
to the experimental procedure carried out in [52, 280], albeit in a 2d toy model. The
two separated quasi-condensates are then recombined, as in Figure 7.3(g), which was
performed here rather abruptly, and found to lead to shape oscillations as shown by
the noticeably different density of Figure 7.3(h) compared to the equilibrium density
of Figure 7.3(e).
This illustration shows the results of a simulation for one realisation of the noise.
The analogy to different noise realisations corresponding to physical realisations of
an experiment is an appealing one, as highlighted here in the work of Chapter 5 on
solitons within the SGPE. Correlations of relevance to interferometry can be extracted
by time averaging a single realisation such as the one shown, or by repeated simulations
170
Chapter 7. Conclusions and future work
for distinct noise realisations. A strength of the SGPE approach is the ability to
approximately simulate all aspects of the experiment, from the initial stages of growth
into the trap to the coherent splitting and recombination of a quasi-condensate.
Figure 7.3: Density snapshots from a simulation of the growth, splitting and recombination of
an atomic quasi-condensate performed using the stochastic Gross-Pitaevskii equation. Time
increases from (a)-(h). The results are from a preliminary calculation in 2d, using the SGPEeq
approach. The aspect ratio for the trap in the vertical and horizontal directions is 50.
Unless performed slowly enough so as to be adiabatic, this sort of procedure is likely
to result in periods of highly non-equilibrium behaviour. Therefore, the treatment of
the thermal cloud as static during the entire evolution is difficult to justify, and so
a dynamical description of the thermal cloud dynamics is likely to prove essential in
describing the physics of realistic atom interferometers.
Overall, the SGPE represents a versatile model, applicable to a wide range of the
phenomena observed within current weakly-interacting, atomic Bose Einstein condensate experiments. Present numerical implementations of this theory are restricted to
working within the classical approximation to this equation, however, and therein to
171
Chapter 7. Conclusions and future work
modelling the higher lying thermal atoms in a static manner. To move beyond this,
a logical progression would be to couple the SGPE with a full dynamical description
of the thermal modes, which therefore requires additionally solving the corresponding quantum Boltzmann equation. While not straightforward, this would represent an
extremely good model for trapped ultracold Bose gases at finite temperatures, and a
significant step forward in the theoretical description of such systems.
172
Appendix A
Numerical methods
We summarise here the numerical methods we use in solving the discrete form of the
stochastic Gross-Pitaevskii equation. In addition, we highlight a means by which to
address, in a highly parallel manner, simulations which require repeated runs over large
statistical ensembles.
A.1
Time-stepping and spatial discretisation
A description of the numerical solution to the Gross-Pitaevskii equation serves as a good
model to explain the numerical methods we apply for the dissipative and stochastic
equations, also considered in this thesis.
Starting with Eq. (2.9), we consider for brevity the Gross-Pitaevskii equation in the
form
i~
∂ψ(x, t)
= (HGP − µ) ψ(x, t).
∂t
(A.1)
Defining the time index to be m = 0, 1, 2, . . . , mmax , to propagate forward from a time
m∆t ≡ tm by an amount ∆t the solution is
ψ(x, tm+1 ) = e−i(HGP −µ)∆t/~ ψ(x, tm ).
(A.2)
Since we are solving for a Hamiltonian system, to evolve forwards in time we use
Cayley’s form form for the exponential
exp [−i(HGP − µ)∆t/~] =
1 − i∆t(HGP − µ)/2~
+ O(∆t2 ),
1 + i∆t(HGP − µ)/2~
(A.3)
which provides a unitary operation at each time-step. Therefore this conserves important quantities like the total particle number and energy. It can also be identified as
173
Appendix A. Numerical methods
the first order Padé approximant for the exponential we wish to approximate.
So, dropping the subscript on the Hamiltonian, numerically we must solve
h
i
h
i
1 + i∆t(H m+1/2 − µ)/2~ ψ(x, tm+1 ) = 1 − i∆t(H m+1/2 − µ)/2~ ψ(x, tm ). (A.4)
If H was not dependent upon ψ(x, t), the right hand side would be a known quantity.
However, it is nonlinear in ψ(x, t) and we use the notation
Hm = −
~2 ∇2
+ V (x) + g3d |ψ(x, tm )|2
2m
and
H m+1/2 =
1 m
H + H m+1 ,
2
(A.5)
(A.6)
to denote the self-consistent average of the nonlinear term, which is evaluated at the
mid-point of the time-step. In practice, this is obtained by iterating for the wavefunction at the next time step, so the procedure is to solve for ψ m+1 using ψ m in the
nonlinear term on both sides of Eq. (A.4), before solving again for ψ m+1 , now with
ψ m+1 + ψ m /2 in the nonlinear term. We carry out this procedure once at each time
step.
The second order spatial derivative may be discretised using a second order centered
difference scheme. In one-dimension, the Laplacian can be written as the centered
difference operator
∂2ψ
ψj+i − 2ψj + ψj−1
=
+ O(∆z 2 )
∂z 2
∆z 2
≡ δz2 ψj + O(∆z 2 ).
(A.7)
Insertion of this into the spatially discrete form of Eq. (A.4) gives
i∆t
1+
2~
~2 2
m+1/2
δ + V̄j
−µ
−
2m z
ψjm+1
i∆t
= 1−
2~
~2 2
m+1/2
δ + V̄j
−µ
−
2m z
ψjm
(A.8)
where ψjm = ψ(j∆z, m∆t), ∆z is the spatial grid spacing and j = 1, 2, 3, . . . , Nz . We
have defined also
2
V̄jm ≡ Vj + g3d ψjm ,
(A.9)
as a shorthand for the trapping potential and nonlinear term, to isolate the Laplacian
part of interest. Using the centered difference method gives a set of tridiagonal matrix
equations which can be solved using standard techniques [281].
174
Appendix A. Numerical methods
A more accurate way to implement the spatial discretisation for the GPE, is to make
use of Numerov’s method which, like a centered difference scheme, gives a tridiagonal
matrix form for the Laplacian. Numerov’s method is however accurate to O(∆z 4 ),
whereas the centered difference scheme has an error of O(∆z 2 ).
Eq. (A.7) can be used to to find a better approximation for the Laplacian, since
Taylor expanding the following gives
ψ(z + ∆z) − 2ψ(z) + ψ(z − ∆z) = (∆z)2
∂ 2 ψ ∆z 4 ∂ 4 ψ
+
+ O(∆z 6 ),
∂z 2
12 ∂z 4
(A.10)
and
2
∂4ψ
2∂ ψ
=
δ
+ O(∆z 2 ).
z
∂z 4
∂z 2
(A.11)
Therefore we can write
δz2 ψ =
∂ 2 ψ ∆z 2 2 ∂ 2 ψ
+
δ
+ O(∆z 4 ),
∂z 2
12 z ∂z 2
(A.12)
which rearranged gives
∂2ψ
2
4
= M−1
z δz ψ + O(∆z ),
∂z 2
(A.13)
where the tridiagonal matrix Mz is defined as
Mz ≡
∆z 2
1+
12
δz2 .
(A.14)
The Crank-Nicholson routine for the GPE operator is unitary, so we consider just
the left hand side, as the right hand side is simply the complex conjugate. This is
h
m+1/2
1 + i∆t(Hj
i
− µ)/2~ ψjm+1 =
i
i∆t
1 −1 2 h m+1/2
1+
− Mz δz + V̄j
−µ
ψjm+1
2~
2
(A.15)
which, multiplying through by Mz , gives
h
i
m+1/2
− µ)/2~ ψjm+1 =
Mz 1 + i∆t(Hj
h
i
1 2
i∆t
m+1/2
−µ
− δz + Mz V̄j
ψjm+1 .
Mz +
2~
2
(A.16)
As the multiplication by the matrix Mz occurs on both sides of Eq. (A.4), then this
175
Appendix A. Numerical methods
can be cancelled with that appearing on the right hand side. As we are again left with
a tridiagonal system, efficient routines are available for its solution [281].
To approach the solution of the SGPE, Eq. (2.92), we follow the time-stepping
method outlined in Bijlsma and Stoof [108], though first discussed in [277]. The solution
to this equation can straightforwardly found as in the Brownian motion case considered
in Chapter 2. In a similar way to Eq. (A.2), the solution to this is
ψ(x, tm + ∆t) = exp [−iα(HGP − µ)∆t/~] ψ(x, tm )−
Z tm +∆t
i
′
′
′
dt exp iα(HGP − µ)t /~ η(x, t ) ,
exp [−iα(HGP − µ)tm /~]
~
tm
(A.17)
where HGP is the operator defined in Eq. (2.9) and α = α(x) ≡ 1 − iγ(x).
To evolve the system forward in time, we again use Cayley’s form form for the
exponential
exp [−iα(HGP − µ)∆t/~] =
1 − iα∆t(HGP − µ)/2~
+ O(∆t2 ),
1 + iα∆t(HGP − µ)/2~
(A.18)
which provides a unitary operation at each time-step, in the case that γ = 0. Defining
the noisy field
ξm (x) ≡ exp [−iα(HGP − µ)tm /~]
Z
tm +∆t
tm
dt′ exp iα(HGP − µ)t′ /~ η(x, t′ ), (A.19)
this has correlations given by
∗
hξm
(x)ξn (x′ )i = 2γ(x, t)kB T ~δ(x − x′ )δmn ∆t + O(∆t2 ).
(A.20)
Numerically the problem is then reduced to the solution of
h
i
1 + iα∆t(H m+1/2 − µ)/2~ ψ(x, tm + ∆t) =
h
i
i
m+1/2
1 − iα∆t(H
− µ)/2~ ψ(x, tm ) −
ξ(x, tm ) ,
~
(A.21)
which can be evaluated as in the GPE case. Retaining the α contribution to Eq. (A.21),
but neglecting the noisy field ξ(x, tm ), leads to a numerical scheme for the DGPE.
A.1.1
Calculation of the mode energy in the Numerov method
To calculate the mode energies for modes which result on diagonalising the density
matrix, we use the GPE energy functional Eq. (3.22). If the simulation has used
176
Appendix A. Numerical methods
the centered difference rule for the second derivative, then this is straightforward to
calculate. In simulations for which the Numerov method is used for a given mode ϕi (z)
we make use of Eq. (A.14) to give
(2)
Mz ϕi (z) = δz2 ϕi (z),
(A.22)
(2)
where ϕi (z) denotes the Numerov approximation to the second derivative of ϕi (z).
For a centered difference scheme the numerical representation of kinetic energy on the
grid can be calculated using the operator on the right hand side. To calculate the
kinetic energy term for simulations based upon the Numerov scheme, the Numerov
approximation to the second derivative can be obtained by solving the tridiagonal
(2)
system for ϕi (z). On doing so, the kinetic term of Eq. (3.22) can also be easily
evaluated.
A.2
Condor as a tool for statistical physics
We conclude this section with a description of the highly parallel manner in which
simulations that make use of ensemble averages over stochastic trajectories can be
undertaken. We make use of the Condor high throughput system [282], which is a
mechanism by which to make use of the processors of otherwise idle cluster machines,
as found on any typical university campus. The system is based upon constructing a
‘Classad’ for your job, which contains the specific properties and any requirements, like
to only run on machines with certain memory requirements, for example. This is to be
submitted to the Condor collector, the machine which distributes jobs to machines as
they become available.
The structure of a typical classad is as follows:
#Executable to run:
Executable
= SGPE1d.x
#The vanilla universe means we must specify to Condor which input files
#need to be copied over and which output files need to be copied back
Universe
= vanilla
#Specify the operating system and any machine requirements:
Requirements
#Arguments
= OpSys == "LINUX" && FileSystemDomain == "MATH_STATS"
=
177
Appendix A. Numerical methods
transfer_input_files
=
../Gamma_1d.dat, restart_int.dat
, restart_array.dat
should_transfer_files
= YES
when_to_transfer_output = ON_EXIT_OR_EVICT
notification = never
#Log files:
Error
Output
= SGPE1d.err #Any errors:
= SGPE1d.out #Any output
Log
= SGPE1d.log #A log of the jobs submission history:
#Submit preferentially to the fastest machines in the pool:
Rank = KFlops
As a cluster machine may be used, in the conventional sense, at any point, it is
best for the executable which is submitted via Condor to be restartable. In addition,
the results which are returned then need to be averaged over in some way. This can be
achieved in a straightforward manner using shell scripts which can be used on Linux
based systems, for example.
To illustrate the speed up which can be gained through the use of Condor, let us
consider a realistic example relating to the SGPE: For smooth averaged results, depending of course on the temperature, particle number, and other system specifics, a
typical SGPE ensemble might require use of 2000 trajectories. These must, in general,
each be simulated through the growth process, to equilibrium, prior even to any dynamical tests which might be studied in well formed condensates. On a large numerical
grid, a realistic estimate is that one complete trajectory might take 30 minutes, which
means the entire ensemble will take around 42 days, if addressed in a linear fashion.
The Condor system allows us to split this task over many machines, dependent upon
the size of the pool available, though 200 is easily achievable. Packaging the 2000 trajectories into 200 packages of 10 immediately reduces the execution time by a factor of
200, meaning the same ensemble can be complete instead within 5 hours.
Use of this system has proved essential in the production of much of the data presented in this thesis, in particular in undertaking the stochastic soliton simulations
presented in Chapter 5, for which individual trajectories could take a day or so, due
to the small grid spacing used, and the comparison between stochastic approaches of
Chapter 6. In the case of the latter, we usually considered an ensemble of 2000 trajectories for each temperature, and each method, meaning Condor led to a considerable
speed up.
178
Appendix B
Supplementary results
This appendix contains supplementary results of tests characterizing the SGPE simulations.
B.1
Condensate statistics for different grid spacings
0.0002
(a)
0.00015
0.0001
5e-05
0
(b)
0.00015
0.0001
5e-05
0
(c)
0.00015
0.0001
5e-05
0
0
5000
10000
15000
20000
N0
Figure B.1: Normalised histogram showing the distribution of condensate statistics for an
ensemble of 1000 realisations of the noise, calculated at T = 185nK with (a) βEgrid = 2.9
(blue) (b) βEgrid = 4.8 (black, as shown also in Figure 3.13 of Chapter 3) and (c) βEgrid = 9.6
(red). Shown in green in each plot is the result of Svidinsky and Scully [120], for N = 20000.
The SGPE simulations have a particle number of N ≈ 21000.
179
Appendix B. Supplementary results
The results of testing the sensitivity of the condensate statistics to the numerical grid,
or equivalently energy cutoff, used in solving the SGPE are shown in Figure B.1.
It is clear that the variation tested has little effect upon the condensate statistics.
This was found to be true quite generally, with this trend remaining for tests carried
out at other temperatures and parameters.
B.2
Tests of the effect of varying γ(z)
An important input to the SGPE, γ(z) parameterises the rate of scattering between
the thermal component and low-lying modes. Since the Bose-Einstein distribution is a
solution to C22 = 0 (see Eq. (2.80)), assuming this distribution to describe the thermal
modes implies that they act as a heat bath. In addition, in this form these modes then
act as an entropy sink, or source, so the interaction with the heat bath serves to push
the low energy part of the system towards equilibrium. On physical grounds, since the
dynamics of the thermal modes are neglected, we would expect that the coupling to
these would not be a major factor contributing to thermal equilibrium reached by the
low modes. However, for completeness, we show the results of tests looking at various
equilibrium system properties, as we vary γ.
As equilibrium is reached when a stationary solution to Eq. (2.92) is found, the
final result should be largely independent on the choice of γ(z), subject perhaps to
the introduction of numerical instabilities at extreme values. We stress however that
in studying dynamics which perturb the system about equilibrium, an appropriate
calculation of γ(z) is of utmost importance in modelling the (linear) response to such
perturbations, in the presence of a thermal cloud.
We saw in Chapter 3 that the manner in which equilibrium is approached for different γ(z) can be largely accounted for by a rescaling of the time. The results here show
some more stringent tests on the variation of the equilibrium reached, due to different
values of γ(z).
We first check the total system densities which result for a wide range of dissipative
coefficients. As is clear from Figure B.2, there is no visible variation between the
equilibrium densities for each γ value used. The results shown are representative of
many other tests carried out for intermediate values, which gave the same conclusion
as the data shown in Figure B.2. In this way, we have found that the density profile is
unchanged for 5 × 10−4 ≤ γ ≤ 1, in agreement with the claim that the system norm is
unchanged.
The condensate statistics across the grand canonical ensemble offer a further test
of the nature of the dynamical equilibrium reached, as rather than considering an
averaged quantity, we now consider how the nature of the fluctuations about the mean
180
Appendix B. Supplementary results
are affected by the choice of γ.
1
0.004
0.003
gn(z)/µ
gn(z)/µ
0.8
0.6
0.002
0.001
0
0.4
10
15
20
25
30
35
40
z [lz]
0.2
0
0
2
4
8
6
10
12
z [lz]
Figure B.2: Variation in the total density profiles with the dissipation coefficient; profiles shown
are for γ = 5 × 10−4 (red, dashed) and γ = 1 (black, solid). The inset shows the same data
focusing on the less populated regions in the density wings.
Here we show the results of projecting the 1000 realisations of the stochastic wave
function on to the Penrose-Onsager condensate mode, obtained as described in Section
3.5.1. We consider the effect of varying the dissipative coefficient by simulating the
growth to equilibrium for different values of γ, before carrying out the projection.
1.2e-03
0.06
0.04
γ(z)
Prob. density (N0)
0.05
9.0e-04
6.0e-04
0.03
0.02
0.01
3.0e-04
0
-12
-9
-6
-3
0
3
6
9
12
z [lz]
12000
14000
16000
18000
20000
N0
Figure B.3: Condensate statistics for several values and forms of γ(z): shown are γ = 0.05
(black), γ = 0.5 (red), γ = 1 (green) and also spatially dependent forms for the dissipation, as
shown in the inset where the colours match the histogram in each case.
Figure B.3 shows the equilibrium ensemble condensate statistics, for various γ types,
with all other parameters fixed. As in the case of the density, little variation is evident
between various forms of dissipation.
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