VARIATIONAL THEORY FOR TWO-FLUID HYDRODYNAMIC
MODES IN TRAPPED FERMI GASES
by
Edward Taylor
A thesis submitted in conformity with the requirements
for the degree of Doctor of Philosophy
Graduate Department of Physics
University of Toronto
c Edward Taylor, 2008
Copyright For Sandy
Abstract
Variational Theory for Two-Fluid Modes In Trapped Fermi Gases
Doctor of Philosophy, 2008
Edward Taylor
Graduate Department of Physics
University of Toronto
Strongly interacting two-component Fermi gases have become a powerful tool for studying
many-body physics. The interactions between the components can be varied widely, allowing
the full crossover from BCS to BEC superfluids to be explored. In the strong interaction
region at unitarity, the behaviour of the system becomes independent of microscopic details
of the interaction. Hence, these Fermi gases can be used as a testing ground for microscopic
theories of strongly-interacting Fermi matter.
In this thesis, we study the Landau two-fluid hydrodynamic modes of a trapped Fermi
gas superfluid, including the “out-of-phase” oscillations of the superfluid and normal fluid
components. These modes are the analogue of second sound in superfluid 4 He, and have so
far not been observed in trapped Fermi gas superfluids. An important part of this thesis
is the development of a new variational method of solving the two-fluid equations for gases
confined in a harmonic trap. The solution of the resulting algebraic equations for the twofluid mode frequencies requires detailed knowledge of the local thermodynamic quantities in
a trap.
In contrast to “classical” BCS and BEC superfluids such as 3 He and 4 He, both fermionic
and bosonic thermal excitations contribute significantly to the thermodynamics close to
ii
unitarity. The fact that these excitations are coupled at finite temperatures means that one
cannot write down simple expressions for thermodynamic quantities in terms of the spectrum
of these excitations. Our calculations of thermodynamic quantities involve essentially ab
initio calculations based on the microscopic theory developed by Nozières and Schmitt-Rink,
applied to a nonuniform trapped gas using a local density approximation.
We propose that the out-of-phase two-fluid modes can be excited and measured in a
two-photon Bragg scattering experiment, which measures the density response function.
We investigate the structure of the two-fluid density response function at unitarity within
our variational approach. The out-of-phase modes are found to have appreciable weight
at intermediate temperatures in the superfluid phase and should be observable in Bragg
scattering experiments.
iii
Acknowledgements
I would like to start by thanking my supervisor Allan Griffin. His continued guidance
and support during my years as a doctoral student has been invaluable. His enthusiasm
for our work has both inspired and motivated me. Allan has always demanded the highest
standards and I am very proud of the work we have done together and feel honoured to be
his last student.
I have been very fortunate during my doctoral work to collaborate with some of the best
minds in ultracold atom physics. Yoji Ohashi was in many ways my surrogate supervisor
and I have benefited greatly from his expertise. I am grateful for his willingness to answer
my many late-night email queries over the years. Hui Hu and Xia-Ji Liu are world experts
in solving the demanding numerical problems of the BCS-BEC crossover. Their work is
of the highest quality and I have been extremely fortunate to have had the opportunity to
collaborate with them. I owe a special debt of gratitude to Eugene Zaremba, who introduced
me to the world of Bose-Einstein condensation and guided me through my first two years of
life as a physicist.
Last, but not least, I am deeply indebted to my committee members, Arun Paramekanti
and Joseph Thywissen. Their comments and questions over the past few years have contributed greatly to my development as a physicist and clarified my thinking about the work
in this thesis.
My doctoral work has been supported by fellowships from the Natural Science and Engineering Research Council (NSERC) of Canada and the Graduate School at the University
of Toronto.
The fact that I have a Ph.D. is due in large part to the unwavering help and support of
iv
my wife Sandy and my parents-in-law, Ferit and Virginia. I owe them more than I could
possibly tell them.
v
Contents
Abstract
ii
Acknowledgements
iv
1 Introduction
1.1
1
Summary of chapters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Two-fluid hydrodynamics: a review
12
14
2.1
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
2.2
First and second sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
3 Thermodynamics through the BCS-BEC crossover: a microscopic model 24
3.1
Thermodynamic potential for a uniform Fermi superfluid . . . . . . . . . . .
26
3.2
The gap and number equations . . . . . . . . . . . . . . . . . . . . . . . . .
36
3.3
Thermal excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
3.3.1
BCS quasiparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
3.3.2
Collective modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
4 The superfluid density in the BCS-BEC crossover
51
4.1
Definition of the superfluid density . . . . . . . . . . . . . . . . . . . . . . .
52
4.2
The thermodynamic potential for a current-carrying superfluid . . . . . . . .
55
vi
4.3
Superfluid density in the BCS-BEC crossover . . . . . . . . . . . . . . . . . .
58
4.4
The normal fluid density in the BEC limit . . . . . . . . . . . . . . . . . . .
62
4.5
Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
5 Variational formulation of the Landau two-fluid equations
71
5.1
Zilsel’s variational formulation . . . . . . . . . . . . . . . . . . . . . . . . . .
72
5.2
Action for linearized Landau two-fluid hydrodynamics . . . . . . . . . . . . .
80
5.3
Examples of hydrodynamic modes . . . . . . . . . . . . . . . . . . . . . . . .
85
5.3.1
Uniform gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
5.3.2
Dipole mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
5.3.3
Breathing modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
Hydrodynamic Theory at T = 0 . . . . . . . . . . . . . . . . . . . . . . . . .
95
5.4
6 Hydrodynamic modes in trapped gases at unitarity
99
6.1
Thermodynamics at unitarity . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.2
Locally isentropic dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.3
Breathing modes at unitarity . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.3.1
In-phase mode at unitarity . . . . . . . . . . . . . . . . . . . . . . . . 109
6.3.2
Out-of-phase breathing mode at unitarity . . . . . . . . . . . . . . . . 111
6.4
Dipole modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.5
Calculation of the isentropic compressibility and superfluid density at unitarity116
7 The two-fluid hydrodynamic density response function
119
7.1
Density response function for a uniform superfluid: review . . . . . . . . . . 120
7.2
The f -sum rule for the density response
7.3
Density response function for trapped superfluid Fermi gases . . . . . . . . . 129
vii
. . . . . . . . . . . . . . . . . . . . 125
7.4
Variational formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
7.5
The contribution to the f -sum rule from two-fluid normal modes . . . . . . . 137
8 Bragg scattering from two-fluid hydrodynamic modes at unitarity
139
8.1
A brief review of two-photon Bragg scattering . . . . . . . . . . . . . . . . . 139
8.2
Dynamic structure factor for hydrodynamic normal modes at unitarity . . . 142
8.2.1
Dipole modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
8.2.2
Breathing modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
8.3
Two-fluid resonances as a function of q and T . . . . . . . . . . . . . . . . . 146
8.4
Direct excitation of normal modes . . . . . . . . . . . . . . . . . . . . . . . . 150
9 Conclusions and future work
153
Appendices
156
A A single-channel model for a Feshbach resonance
156
A.1 A brief review of Feshbach resonance physics . . . . . . . . . . . . . . . . . . 156
A.2 The single-channel model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
B Fluctuations in the gap and number equations: a critical review
164
Bibliography
169
viii
Chapter 1
Introduction
The modern era of condensed matter physics began in 1941 when Landau [1, 2] formulated
the equations for two-fluid hydrodynamics of superfluid 4 He. Landau calculated the thermodynamic functions that arose in his equations in terms of a weakly interacting gas of
Bose quasiparticles, given by the famous phonon-roton spectrum. This theory of superfluid
4
He still plays a key role in our modern understanding of superfluidity (Hohenberg and
Martin [3]). Since the 1960s, the microscopic basis for Landau’s two-fluid hydrodynamics
has been understood as arising from an underlying Bose condensate of Helium atoms. In
a generic sense, these hydrodynamic two-fluid equations also describe the dynamics of all
Bose superfluids, whether they are gas or liquid, when the system is in a state of local thermodynamic equilibrium. This latter condition requires short collision times for the thermal
quasiparticles making up the normal fluid.
This thesis is concerned with using Landau’s two-fluid hydrodynamics to study the dynamics of a trapped two-component gas of Fermi atoms. Using a Feshbach resonance, Fermi
superfluids in the BCS-BEC crossover region allow one to deal with an adjustable interaction
between atoms in different hyperfine states. One can thus achieve hydrodynamic local equilibrium by working close to unitarity where the s-wave scattering length becomes very large.
The chief difficulty in working with the Landau two-fluid equations in trapped Fermi gases,
1
1. Introduction
2
however, is the inherent spatial nonuniformity of these systems. In this thesis, we develop a
new variational approach to solve the two-fluid equations in a spatially nonuniform gas. This
allows one to determine the frequencies of the two-fluid hydrodynamic modes through the
BCS-BEC crossover as a function of the temperature. In addition, we extend this variational
approach to calculate the density response function for trapped Fermi superfluids, which is
directly involved in the two-photon Bragg scattering cross-section (Ozeri et al. [4]).
In his 1941 paper, Landau argued that a quantized version of the hydrodynamics for
classical fluids led to a new superfluid degree of freedom. This superfluid was assumed to
carry no entropy and flow with an irrotational velocity field [∇×vs (r) = 0]. The normal fluid
thus carried all the entropy of the system and behaved as a conventional Navier-Stokes fluid.
That is, its hydrodynamics was the same as any classical fluid. From these simple but very
subtle ideas, Landau was led to a set of closed equations which described the hydrodynamics
of the coupled superfluid and normal fluid components (this theory is reviewed in Chapter 2).
Classical hydrodynamics predates the microscopic description of a fluid in terms of interacting atoms (or molecules). It works completely in terms of a coarse-grained picture where
the fluid is described by macroscopic variables such as the local fluid velocity v(r, t) and the
mass density ρ(r, t). In microscopic theories, we know that hydrodynamics describes the low
energy, long wavelength dynamics of liquids when the system is in a state of local thermodynamic equilibrium. Local thermodynamic equilibrium occurs when the collisions between
the molecules are sufficiently rapid that the mean time between collisions is much smaller
than the intrinsic time scale describing the slowly-varying dynamics of the fluid. In simple
terms, hydrodynamics1 is the correct description of the collective oscillations of frequency ω
1
Throughout this thesis, we use the term “hydrodynamics” to mean the state of local thermodynamic equilibrium brought on by collisions, except where stated otherwise. This is in contrast to recent literature which
describes the theory of Bose superfluids at T = 0 as hydrodynamic. This language is used since all the atoms
in the condensate are described by two variables, the density n(r, t) and the velocity
vc (r, t) ≡ ∇φ(r, t)/m,
p
which characterize the macroscopically occupied single-particle state Ψ(r, t) = n(r, t) exp[iφ(r, t)].
1. Introduction
3
such that ωτR ≪ 1, where τR describes the time it takes for a nonequilibrium state to reach
local equilibrium (Huang [5]).
In experiments involving Bose-condensed atomic gases after the discovery of BEC in
1995, the typical collisional cross-section and the achievable densities were not sufficient to
reach local equilibrium required for two-fluid hydrodynamics to be valid. This thesis will be
concerned with trapped Fermi gases with a Feshbach resonance which we argue can be used
to realize a phase where Landau’s two-fluid theory describes the low frequency dynamics.
In the last five years,2 many experiments have studied trapped two-component gases of
atomic fermions, initially with the goal of realizing a BCS superfluid of Cooper pairs. The
basis of the original Bardeen, Cooper, and Schrieffer (BCS) [7] theory of superconductivity
was that in the presence of a weak attractive interaction, fermions can pair up to form
Cooper pairs, which then Bose-condense. This Bose-condensate of Cooper pairs can flow
without resistance. This is the mechanism for superconductivity when the Cooper pairs
are pairs of charged electrons, and superfluidity when the Cooper pairs are made up of
neutral Fermi atoms, such as in superfluid 3 He. It is now generally accepted that BCS
superconductivity is a consequence of what is essentially a Bose-condensation of Cooper
pairs. This point was first clarified by Leggett [8] in his seminal discussion of the “BCS-BEC
crossover”. The crossover idea was originally discussed by Eagles [9] and later explored in
much more detail by Leggett [8]. Leggett pointed out that at T = 0, the many-body BCS
wave function describing Cooper pairs evolved smoothly to a ground state describing a BEC of
dimer molecules, as the attractive interaction between the fermions was steadily increased.3
At finite temperatures, this was later extended to include the effects of particle-particle
fluctuations around the generalized BCS mean field theory of Leggett in a seminal paper by
2
3
A trapped degenerate (T < TF ) Fermi gas was first created at JILA in 1999 using 40 K [6].
For a modern discussion of the BCS-BEC crossover in the context of superconductivity, see Randeria [10].
1. Introduction
4
Nozières and Schmitt-Rink (NSR) [11]. This crossover between the “weak-coupling” BCS
state and a “strong-coupling” BEC region has attracted enormous interest in the recent
work on ultracold Fermi gases (see the recent review articles by Chen, Stajic, Tan, and
Levin [12], Giorgini, Pitaevskii, and Stringari [13], Grimm [14], and Bloch, Dalibard, and
Zwerger [15]). The strong-interaction region at unitarity, where the s-wave scattering length
|as | becomes infinite, is of special interest. While the crossover between the BCS and BEC
regimes is smooth (i.e., there is no phase transition), with the size of the bound fermion pairs
becoming progressively smaller as one goes from the BCS to BEC regions, we emphasize that
the nature of the bound states of two atoms differs in the two limits. In the BCS region,
Cooper pairs only arise as a many-body effect [7], and are dependent on the existence of
a Fermi surface. Cooper pairing does not occur for two fermions in vacuum with a weak
attractive interaction. In contrast, in the BEC region, the bound states are a two-body
effect. A single dimer can exist in vacuum.
The key development in ultracold Fermi gases was that the s-wave scattering length that
characterizes the interactions between atoms in different hyperfine states in a dilute Fermi
gas could be made arbitrarily large by using a Feshbach scattering resonance [16]. At resonance, the s-wave scattering length between fermions in different hyperfine states diverges,
leading to a strongly-interacting Fermi gas superfluid (Timmermans, Furuya, Milonni, and
Kerman [17]; Holland, Kokkelmans, Chiofalo, and Walser [18]; Ohashi and Griffin [19]).
A remarkable feature of the Feshbach resonance is that the sign of the scattering length
can be tuned continuously from negative to positive, by changing a small external magnetic
field (see Fig. 1.1). Two-component Fermi gases (the analogue of spin up and down electrons
in metallic superconductors) with a Feshbach resonance can thus be used to produce both
BCS-type pairing between fermions (as < 0) as well as a molecular Bose-Einstein condensate
5
1. Introduction
scattering length (ao)
3000
2000
1000
0
-1000
-2000
-3000
215
220
225
B (gauss)
Figure 1.1: The s-wave scattering length as between
magnetic field (from Regal et al. [20]).
40
230
K atoms as a function of the applied
(BEC) of dimer molecules (as > 0), and the region between. Thus by using a Feshbach
resonance to tune the value of as , one can study the entire BCS-BEC crossover, including
the strong-interaction region close to unitarity.
To describe the microscopic physics of a two-component Fermi gas with a Feshbach
resonance, in this thesis we use a single-channel model. This model describes two species of
fermions (the “open channel” fermions prepared in two different hyperfine states) interacting
via a “tunable” s-wave interaction. It is completely equivalent to the model discussed by
Leggett [8] and Nozières and Schmitt-Rink [11] in their treatment of the BCS-BEC crossover.
In Appendix A we derive the single-channel model from a two-channel model (originally
discussed by Timmermans et al. [17] and Holland et al. [18]) that describes the coupling
between open-channel Fermi atoms and dimer bound states (Feshbach molecules) in an
atomic gas with a Feshbach resonance. The Feshbach resonance is a scattering resonance
for two “free” fermions (in different hyperfine states ↑, ↓) to form a tightly bound dimer
molecule of two fermions in a new set of hyperfine states ↑′ , ↓′ .4 The interaction between the
4
For a review, see Duine and Stoof [21].
1. Introduction
6
fermions in the ↑, ↓ states (“open channel”) is different from that in the ↑′ , ↓′ states (“closed
channel”), the latter supporting a deeply bound state, the Feshbach molecule. For a broad
Feshbach resonance (Diener and Ho [22], Romans and Stoof [23]), only a minuscule fraction
of the fermions get converted into these Feshbach molecules over the range of magnetic fields
typically probed in experiments that model the BCS-BEC crossover (see Partridge, Strecker,
Kamar, Jack, and Hulet [24]). However, weakly bound dimer molecules of fermions in the
open channel can form on the BEC side of resonance (but not too far from unitarity) as
a result of a strong attractive interaction mediated by virtual Feshbach molecules. This
interaction also leads to Cooper pairing between open channel fermions on the BCS side
of resonance. Consequently, the “pairs” (Cooper pairs, dimer molecules) discussed in this
thesis are always pairs of fermions in the open channel. Moreover, when we discuss the “BEC
region” in this thesis, we are referring to a Bose condensate of open-channel dimers, and not
closed-channel Feshbach molecules.
By virtue of the strong interactions between the atomic fermions in different hyperfine
states close to unitarity, two-component Fermi gases with a Feshbach resonance can be
used to probe a collisionally hydrodynamic regime at finite temperatures in the region close
to unitarity. We argue that the dynamics of such gases are described by Landau’s twofluid hydrodynamics. This is the basis of this thesis and was first discussed by Taylor and
Griffin [25]. In Fig. 1.2 we plot a schematic of the BCS-BEC crossover region in terms of the
standard dimensionless interaction parameter5 (kF as )−1 and temperature. This plot shows
the superfluid region and the region close to unitarity [(kF as )−1 = 0] where we expect strong
interactions lead to a collisionally hydrodynamic normal fluid. The overlap between these
two regions gives the region where we expect two-fluid hydrodynamics to apply. Note that
5
kF = (3π 2 n)1/3 is the Fermi wavevector, expressed in terms of the totoal density n = n↑ + n↓ of the
Fermi atoms.
7
1. Introduction
T
111111111111111
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Collisionless 111111111111111
Collisionless
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normal fluid 111111111111111
normal
fluid
Collisionally
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(molecules) 111111111111111
hydrodynamic
(fermions)
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normal fluid
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Two−fluid
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region
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BCS superfluid
BEC superfluid
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as > 0
0
as < 0
−1/(k F a s)
Figure 1.2: Schematic diagram of the BCS-BEC crossover showing the region where twofluid hydrodynamics is expected to describe the dynamics of the superfluid. Unitarity, where
|as | = ∞, corresponds to 1/(kF as ) = 0. Note than in the BEC region, non-condensed dimer
molecules dominate the normal fluid when kB T ≪ Eb , where Eb is the dimer binding energy
(adapted from Wright et al. [26]).
for trapped gases, the density decreases with increasing temperature, meaning that the gas
will not be collisionally hydrodynamic at high temperatures, in contrast to a uniform gas.
The first demonstration of superfluidity in trapped two-component Fermi gases was reported by Jin and coworkers in 2004 [27], quickly followed by work from the research groups
of Salomon [28], Ketterle [29], Grimm [30, 31], and Thomas [32]. Almost immediately, experiments were carried out to study the collective modes (Bartenstein et al. [33]; Kinast et
al. [34]) in the BCS-BEC crossover. These results were successfully explained by T = 0
theories for the breathing modes (Stringari [35]; Heiselberg [36]; Hu, Minguzzi, Liu, and
Tosi [37]), based on the equation of motion for a pure Fermi superfluid. Since it corresponds
to an oscillatory compression and rarefaction of the gas with a single displacement node at
the trap center, such breathing modes can be excited by oscillating the frequency of the
harmonic trap potential.
1. Introduction
8
The experiments by Kinast, Turlapov, and Thomas [34] showed that the breathing mode
had a frequency which was essentially independent of temperature, unchanged well into
the normal phase. One of the major results of this thesis (see Chapter 6) is that in the
two-fluid region at unitarity, there is an “in-phase” breathing mode which is completely
independent of temperature. We also argue in Chapter 8 that the breathing modes studied
in all recent experiments only deal with this in-phase mode. This explains the puzzling fact
that theoretical studies at T = 0 were so successful in predicting the frequency of this mode.
The most spectacular prediction of Landau two-fluid hydrodynamics is the existence of
an out-of-phase oscillation of the normal and superfluid components, in addition to the inphase mode (discussed above) where the two components move together. In uniform Bose
superfluids, these in-phase and out-of-phase modes are called first and second sound, respectively (for recent reviews, see Pethick and Smith [38] and Pitaevskii and Stringari [39]).
These modes have been extensively studied theoretically and experimentally in superfluid
4
He [2]. The analogous modes in dilute Bose-condensed gases have been discussed theoreti-
cally by Griffin and coworkers [40, 41]. Unlike the in-phase breathing and dipole modes, the
frequency of the out-of-phase modes in trapped gases depend sensitively on the temperature.
We argue that the detection of these out-of-phase modes at finite temperatures in the BCSBEC crossover will provide a very sensitive probe for testing current microscopic theories of
Fermi gases at unitarity, such as the predictions of “universal thermodynamics” (Ho [42]).
Such strongly interacting Fermi gases have been the subject of much recent research in the
context of quark-gluon plasmas (Schaefer [43]).
The study of two-fluid hydrodynamic modes in a trapped superfluid Fermi gas has two
major difficulties. First, in contrast to a uniform system where the solutions are plane waves,
it is very difficult to directly solve the Landau two-fluid equations in trapped gases. They are
1. Introduction
9
coupled differential equations with coefficients involving thermodynamic quantities that are
spatially nonuniform, making a reliable “brute-force” numerical calculation very challenging
(the two-fluid equations have been solved for a Bose-condensed gas by Shenoy and Ho [44]
and for a two-component Fermi superfluid by He, Chen, Chien, and Levin [45]). Second,
the interesting out-of-phase hydrodynamic modes are not easy to excite experimentally. The
usual methods employed to excite normal modes in trapped gases involve some type of
modulation of the harmonic trap potential. However, in the region close to unitarity where
the Landau two-fluid equations are expected to be valid, these methods can be shown to
only excite in-phase modes (see Chapter 8).
The major accomplishment of this thesis is to provide solutions to both these problems
using a powerful new variational approach. We use this to generate approximate (but probably quite accurate) solutions of the Landau two-fluid equations using a variational ansatz for
the superfluid and normal fluid velocity fields. Our work (Taylor and Griffin [25]) is based
on a phenomenological hydrodynamic action first proposed by Zilsel in 1950 [46] to give an
alternative formulation of Landau’s two-fluid equations of motion. Considering fluctuations
of this action about an equilibrium state, we introduce a simple but physically motivated
variational ansatz for the velocity fields corresponding to the dipole and breathing modes,
following the work of Zaremba, Nikuni, and Griffin [47]. The final result is that the two-fluid
hydrodynamic modes are given by the solutions of a closed set of a few algebraic equations for
the variational parameters describing the normal modes, analogous to the equations for two
coupled harmonic oscillators. The effective masses and spring constants of the two oscillators
are given explicitly in terms of spatial integrals over various static equilibrium thermodynamic functions and their derivatives. We believe this approach gives a simpler and more
effective method of calculating the two-fluid mode frequencies in trapped gases than a brute-
1. Introduction
10
force numerical solution of the two-fluid differential equations. Our variational method also
allows us to write down simple expressions for the mode frequencies (see Chapters 6 and 8)
which give insight into the nature of the two-fluid hydrodynamic modes.
Our variational solution to Landau’s equations is only useful if we have reliable results for
the thermodynamic properties of the superfluid as a function of the temperature, including
the superfluid density ρs . The same quantities are also needed, of course, in solving Landau’s
equations in a uniform Bose superfluid. In both cases, one needs a microscopic model for
the thermally important excitations, i.e., the equivalent of the phonon-roton quasiparticles
in superfluid 4 He. In Chapters 3 and 4, we calculate the thermodynamic potential using an
extended NSR model, limiting ourselves to a uniform superfluid Fermi gas. This model is
essentially a generalized BCS mean-field theory in which the value of the s-wave interaction
is treated as an adjustable parameter (tuning through a Feshbach resonance). In addition,
following NSR, we include fluctuations past the mean-field BCS description, which describe
the dynamic effect of particle-particle interactions and the formation of bound states.
The evaluation of thermodynamic functions such as the superfluid density as a function of
temperature and interaction strength involves demanding numerical calculations, including
the self-consistent evaluation of the chemical potential µ and the BCS gap function ∆0 .
The numerical results we report in this thesis were done by Naoki Fukushima and Yoji
Ohashi (Chapters 3 and 4) based on earlier numerical work (Ohashi and Griffin [48]), and
also by Hui Hu and Xia-Ji Liu (Chapters 6 and 8), who developed numerical codes in
earlier papers [49, 50] to carry out calculations based on the NSR theory. These numerical
results were then used within a local density approximation (LDA) to work out the effective
superfluid and normal fluid masses as well as the effective spring constants which arise in
our variational solution (as described above). We note that the NSR type theory we use
1. Introduction
11
has been shown by Hu, Liu, and Drummond [50] to give excellent agreement with ab-initio
Monte Carlo calculations (Bulgac, Drut, and Magierski [51]) at finite temperatures.
In Chapter 7, we derive a variational expression for the density response function for
a nonuniform superfluid. We believe this is a significant result, since the imaginary part
of the density response function can be measured using two-photon Bragg scattering. This
technique has been used with great success in studying the collisionless excitations of Bosecondensed gases [4]. In such experiments, two counter-propagating laser beams are applied to
the trapped gas. The momentum q ≡ k1 − k2 and frequency Ω ≡ ω1 − ω2 difference between
the two laser beams determines the momentum and energy transferred to the superfluid in
a two-photon process. The trapped gas can absorb a photon (k1 , ω1 ) from one beam and
coherently re-emit a photon (k2 , ω2 ) into the other. The values of Ω and q can be tuned
continuously and if these values coincide with the energy and momentum of an excitation of
the system, there will be a strong scattering resonance in the Bragg scattering cross-section.
This cross-section is directly related to the imaginary part of the density response function
χρρ (q, Ω) (see Zambelli, Pitaevskii, Stamper-Kurn, and Stringari [52]).
The specific system we concentrate on in this thesis is the BCS-BEC crossover in twocomponent superfluid Fermi gases, with special emphasis at unitarity. However, we note
that our variational formulation of the Landau two-fluid equations for trapped gases is valid
for any Bose superfluid with strong enough collisions to produce local equilibrium. Thus
our general formalism can also be applied to trapped atomic and molecular Bose-condensed
gases, assuming that local equilibrium conditions can be reached. This extension is not
discussed in this thesis.
A simple understanding of the two-fluid hydrodynamics of Landau is not easy, although
many authors have tried to do this (see, for example, Ch. 3 in the new book by Leggett [53]).
1. Introduction
12
Landau’s original discussion was essentially a marriage of classical hydrodynamics with a new
quantum degree of freedom associated with the superfluid. In this connection, we note that
in a dilute Bose gas, Zaremba, Nikuni, and Griffin (ZNG) [47] successfully derived Landau’s
equations starting from a generalized Gross-Pitaevskii theory for a Bose condensate plus a
kinetic equation describing the atoms in the thermal cloud. These two equations of motion
were coupled via mean fields and atomic collisions. After a somewhat lengthy and subtle
analysis, the Landau two-fluid equations were shown to emerge in the limit of very short
collision times which lead to local equilibrium. While limited to a dilute Bose gas (in which
the superfluid could be identified with the condensate and the normal fluid identified with
the thermal cloud atoms), the ZNG derivation gives an explicit microscopic basis for the
Landau two-fluid hydrodynamic equations. It removes some of the mystery attached to
these generic equations.
1.1
Summary of chapters
We now give a brief outline of the chapters in this thesis. In Chapter 2, we review Landau’s
theory of two-fluid hydrodynamics, as originally developed for uniform Bose superfluids. In
Chapter 3, we discuss the microscopic model we use to calculate the thermodynamic functions
for a uniform Fermi superfluid which enter the linearized two-fluid equations. We use a
functional integral version (Sá de Melo, Randeria, and Engelbrecht [54, 55]) of the pairing
fluctuation theory of Nozières and Schmitt-Rink [11]. Of special interest is our calculation
of the superfluid density in Chapter 4, based on this NSR model. This was the first detailed
study of the superfluid density across the entire BCS-BEC crossover, reproducing well-known
analytic results in the BCS and BEC limits from a microscopic model (see Taylor, Griffin,
Fukushima, and Ohashi [56] and Fukushima, Ohashi, Taylor, and Griffin [57]).
1. Introduction
13
In Chapter 5, we review Zilsel’s hydrodynamic action and then use it to construct a
variational theory of the two-fluid normal modes. We illustrate our approach by working
out explicit expressions for the in-phase and out-of-phase branches of the breathing and
dipole modes. In Chapter 6, we consider the special case of two-fluid modes at unitarity,
where the absence of any length scale apart from the mean interparticle spacing means
that thermodynamics is expected to be “universal”, as first emphasized by Ho [42]. Using
this universality at unitarity, we show how our variational expressions for the two-fluid
mode frequencies can be simplified considerably. We also prove that the in-phase breathing
mode at unitarity corresponds to a locally isentropic mode with a frequency independent of
temperature. Numerical results for the more interesting out-of-phase modes at unitarity are
also given (see Taylor, Hu, Liu, and Griffin [58]).
In Chapter 7, we extend our variational method to obtain explicit expressions for the
two-fluid hydrodynamic density response function for trapped superfluids. In Chapter 8, we
work out the density response function for the breathing and dipole modes at unitarity in
an isotropic trap. Our results show that the out-of-phase mode shows up in the spectrum
of the response function at intermediate temperatures, but this is very dependent on the
momentum transfer q (see Taylor, Hu, Liu, and Griffin [59]). This response function can be
probed in a two-photon Bragg scattering experiment, which we discuss in Chapter 8.
In Appendix A, we review the basic physics of the Feshbach resonance and give a new
derivation of the single-channel model (used in our discussion of thermodynamics in Chapters 3 and 4) from the microscopic two-channel model.
Chapter 2
Two-fluid hydrodynamics: a review
In this chapter, we review Landau’s two-fluid hydrodynamics that describes the finite temperature dynamics of Bose superfluids with strong interactions. While the central topic of
this thesis is the calculation of the two-fluid modes in trapped gases, we will defer a discussion of the two-fluid modes in this nonuniform system until Chapter 5. In this chapter,
we concentrate on Landau’s theory for uniform superfluids. We describe the basic physical
features common to all strongly interacting Bose superfluids that motivated the two-fluid
theory of superfluid 4 He. The same physics will also be valid in the two-fluid hydrodynamics
of trapped Fermi gases, to be discussed in later chapters.
In our discussion of the dynamics of trapped Fermi gas superfluids in this thesis, we will
only consider the Landau two-fluid equations without dissipation associated with transport
coefficients. In the limit where collisions are sufficiently rapid to ensure local thermodynamic
equilibrium over any other dynamic time scale, the viscosity and other transport coefficients
of the normal fluid can be neglected and there is no damping in the two-fluid equations.
In this limit, the normal fluid reduces to a so-called “ideal fluid” (see Ch. 1 of Landau and
Lifshitz [60]), its dynamics governed by a simple Euler equation, coupled to the equation of
motion for the superfluid. Recently, the idea that a Fermi gas close to a Feshbach resonance
might constitute such an ideal fluid has generated significant interest, both theoretically
14
2. Two-fluid hydrodynamics: a review
15
(Schaefer [43]) and experimentally (Kinast et al. [34]). In particular, high-energy theorists
have become interested in such a “perfect fluid” as a description of recent experiments
involving strong-coupling quark-gluon plasmas (for further references, see Ref. [43]). The
recent experimental work by Clancy, Luo, and Thomas [61], and Wright, Riedl, Altmeyer,
Kohstall, Sanchez-Guajardo, Hecker-Denschlag, and Grimm [26], have given strong evidence
that the normal part of Fermi superfluids close to unitarity are collisionally hydrodynamic
and flow with very little dissipation. We should emphasize that so-called extended LandauKhalatnikov two-fluid equations which do include damping from transport processes are also
well known (see Khalatnikov [2]), but are not treated in this thesis.
The classic account of Landau’s two-fluid hydrodynamics in superfluid 4 He is given in
the book by Khalatnikov [2], one of Landau’s students. The recent books by Pethick and
Smith [38], and Pitaveskii and Stringari [39], have discussed two-fluid hydrodynamics in
the context of uniform ultracold Bose gases. Putterman [62] provides a particularly detailed analysis of the two-fluid hydrodynamic equations, including many of the steps in the
sometimes arduous algebra that is involved in manipulating these equations.
While it is not our goal to derive the equations of two-fluid hydrodynamics for trapped
Fermi superfluid gases from a microscopic model in this thesis, we recall that such a microscopic derivation for a trapped Bose-condensed gas is given by Zaremba et al. [47]. Bogoliubov [63] gave the first detailed microscopic derivation of the Landau equations for superfluid
4
He starting from the existence of a Bose broken symmetry. This work was later extended
to deal with a uniform weak-coupling BCS superfluid by Galasiewicz [64]. A forthcoming
book [65] gives a detailed derivation and historical description of Landau’s two-fluid theory
from a modern viewpoint (i.e., starting from Bose broken symmetry), including the effect of
transport coefficients. The present chapter largely follows Ch. 15 of Ref. [65].
16
2. Two-fluid hydrodynamics: a review
2.1
Overview
We start by writing down the Landau two-fluid equations. The first two equations are
familiar from ordinary fluid dynamics1 . These are the continuity equation
∂ρ
+ ∇ · j = 0,
∂t
(2.1)
and an equation describing momentum conservation,
∂ji
∂Πij
+
= 0,
∂t
∂xj
(2.2)
where the index j = x, y, z is summed over. Equation (2.1) describes the conservation of
mass, where j is the mass current. The second equation, (2.2), is just a statement of Newton’s
second law, since the first term is proportional to the rate of change of the momentum of
a fluid element. The second term, which involves the momentum current tensor Πij , is the
force acting on an element of the fluid. For a classical (single component) fluid, the mass
current is given by the product of the fluid density ρ and the fluid velocity v:
j = ρv.
(2.3)
The force acting on the fluid per unit volume is given by the gradient of
Πij = P δij + ρvi vj ,
(2.4)
where P is the local pressure. Using this in (2.2), the latter reduces to Euler’s equation for
a fluid without viscosity,
ρ
∂v
+ v · ∇v + ∇P = 0.
∂t
(2.5)
In addition to these two equations, which follow from elementary mechanical considerations, Landau posited that the motion of a “quantum” fluid could be split into two
1
For an excellent review of ideal fluid hydrodynamics, see Ch. 1 in the book by Landau and Lifshitz [60].
17
2. Two-fluid hydrodynamics: a review
components: a normal fluid ρn that carries the thermal excitations of the system and a
superfluid ρs that accounts for the new quantum degree of freedom (now associated with
the appearance of a Bose-condensate). At T = 0, the system is in its ground state with
no thermal excitations and where by definition, the normal fluid vanishes. Above a certain
transition temperature Tc , the superfluid component vanishes. The key defining dynamical
identity involving the superfluid and normal fluid components is that the total mass current
j is given by [compare with (2.3)],
j = ρs vs + ρn vn .
(2.6)
Here, vs and vn are the velocities of the superfluid and normal fluid components, respectively,
with mass densities ρs and ρn . The sum of the superfluid and normal fluid densities equals
the total mass density ρ:
ρ = ρs + ρn .
(2.7)
A major achievement in this thesis is the calculation of the superfluid density ρs (T ) that
includes the contribution from bosonic excitations in addition to the mean-field Fermi quasiparticles in the BCS-BEC crossover (Chapter 4).
From (2.6), it follows that the momentum current tensor Πij in (2.2) is given by
Πij ≡ P δij + ρs vsi vsj + ρn vni vnj − ρ
∂Vext
,
∂xi
(2.8)
Generalizing Landau’s equations (which were originally developed for superfluid 4 He), we
have included the effect of an external harmonic trapping potential2
1
Vext ≡ (ωx2 x2 + ωy2 y 2 + ωz2 z 2 ).
2
2
(2.9)
Note that this is the harmonic trapping potential divided by the particle mass, as follows from our use
of the mass density ρ, rather than the number density n. For the same reason, in this chapter the chemical
potential µL corresponds to µ/m, i.e., the chemical potential per unit mass.
2. Two-fluid hydrodynamics: a review
18
Landau made two additional physical assumptions to obtain a closed set of hydrodynamic equations. One was that the normal fluid carried all the entropy (since the superfluid
component was quantum in nature). The second assumption was that the superfluid flow
was irrotational,
∇ × vs = 0,
(2.10)
where vs is the superfluid velocity. These assumptions led Landau to the following two
additional equations [1]:
∂s
+ ∇ · (svn ) = 0,
∂t
(2.11)
∂vs
1 2
= −∇ µL + vs ,
∂t
2
(2.12)
where s is the local entropy density and µL ≡ µ/m is the chemical potential per unit mass.
Since we are only considering dissipationless hydrodynamics, the total entropy is conserved
and one expects a continuity equation of the form given by (2.11), where the second term is
the divergence of the entropy current. The fact that the entropy current equals svn quantifies
the assumption that the entropy is completely carried by the normal fluid. Landau argued
that the dynamics of the superfluid velocity vs can be described by (2.12) [1]. We will not
attempt to paraphrase his subtle arguments here. In Chapter 5, we give a derivation of this
equation within a variational formulation of the two-fluid theory.
When the superfluid component vanishes (ρs = 0, vs = 0 and hence, ρn = ρ) above Tc ,
(2.1), (2.2), and (2.8) reduce, as required, to the standard hydrodynamic equations of an
“ideal” (dissipationless) fluid, namely the continuity and Euler equations (see, for example,
Landau and Lifshitz [60]).
In Chapter 6, where we discuss the hydrodynamic modes in a trapped superfluid Fermi
gas at unitarity, it will prove useful to have the entropy conservation equation given by
19
2. Two-fluid hydrodynamics: a review
(2.11) written in a different way. In terms of the local entropy per unit mass defined by
s̄(r, t) ≡ s/ρ, this equation can be written as
∂s̄
s̄
= ∇ · ρs (vs − vn ) − vn · ∇s̄.
∂t
ρ
(2.13)
In arriving at this expression, we have used the mass and entropy continuity equations, (2.1)
and (2.11), respectively, as well as (2.6). In a normal liquid (where ρs = 0), the first term
vanishes and then (2.13) is referred to as the adiabatic equation. Locally isentropic processes
corresponds to the situation where the entropy per unit mass s̄(r, t) ≡ s(r, t)/ρ(r, t) =
S(r, t)/ρ(r, t)∆V does not change in time as the mass element ρ(r, t)∆V moves with the
fluid. Defining the Lagrangian derivative
D
∂
≡
+ v · ∇,
Dt
∂t
(2.14)
locally isentropic hydrodynamics requires
Ds̄
= 0.
Dt
(2.15)
In a superfluid, (2.13) shows that a locally isentropic oscillation requires that vs = vn ≡ v.
This type of condition will be discussed at length in Chapter 6.
2.2
First and second sound
We now consider the linearized Landau equations, that is, we expand the local variables to
first order in small fluctuations around the static equilibrium values. The local equilibrium
entropy density is s(r, t) = s0 + δs, the local equilibrium kinetic pressure is P (r, t) = P0 + δP
and the local chemical potential is µL (r, t) = µL0 + δµL (r, t). We assume that the twofluid components are stationary in equilibrium (vs0 = vn0 = 0), so that δvn = vn and
δvs = vs . In what follows, we restrict ourselves to a uniform superfluid, where ρs0 , ρn0 , s0
2. Two-fluid hydrodynamics: a review
20
and other thermodynamic quantities are all position-independent. Thus, δj(r, t) = j(r, t) =
ρs0 ∇ · vs + ρn0 ∇ · vn . In a uniform superfluid, these linearized Landau equations give rise to
two kinds of hydrodynamic normal modes, called first and second sound [2]. The following
derivation follows the discussion by Nozières and Pines [66].
The linearized two-fluid equations obtained from (2.1), (2.2), (2.8), (2.11), and (2.12),
reduce to
∂δρ
= −∇ · j,
∂t
(2.16)
∂j
= −∇δP,
∂t
(2.17)
∂δs
= −s0 ∇ · vn ,
∂t
(2.18)
∂vs
= −∇δµL .
∂t
(2.19)
and
Combining (2.16) and (2.17), one finds
∂ 2 δρ
= ∇2 δP.
∂t2
(2.20)
Similarly, combining (2.18) and (2.19), using (2.17), gives
∂ 2 δs
s0 2
∇ δP − ρs0 ∇2 δµL .
=
2
∂t
ρn0
(2.21)
In order to rewrite δµL in terms of δT and δP , we use the Gibbs-Duhem equation,
ρ0 δµL = −s0 δT + δP.
(2.22)
s20 ρs0 2
s0 2
∂ 2 δs
∇
δP
+
∇ δT.
=
∂t2
ρ0
ρ0 ρn0
(2.23)
Using (2.22) in (2.21), we find
21
2. Two-fluid hydrodynamics: a review
This equation gives the local entropy fluctuations in terms of the local mass density δρ and
temperature δT fluctuations. It can also be re-written in terms of the local entropy per unit
mass s̄(r, t) defined earlier. Using
δs̄ = −
1
s0
δρ + δs,
2
ρ0
ρ0
(2.24)
(2.23) then takes the simpler form
∂ 2 δs̄
= s̄20
2
∂t
ρs0
ρn0
∇2 δT .
(2.25)
Expanding δP and δT in terms of δρ and δs̄ fluctuations
δP =
δT =
∂P
∂ρ
∂T
∂ρ
δρ +
s̄
δρ +
s̄
∂P
∂s̄
∂T
∂s̄
δs̄,
ρ
δs̄ ,
(2.26)
ρ
we see that (2.20) and (2.25) reduce to two coupled scalar equations for the time-dependent
fluctuations δρ and δs̄. In a uniform Bose superfluid, the hydrodynamic normal modes are
plane-waves and hence,
δρ, δs̄ ∝ ei(q·r−ωt) .
(2.27)
One finds these coupled algebraic equations have two solutions ω 2 = u2i q 2 , where u2i is a
solution of the quadratic equation
4
u −u
2
∂P
∂ρ
ρs0 T s̄20
ρs0 T s̄20
∂P
+
+
= 0.
ρn0 c̄v
ρn0
c̄v
∂ρ T
s̄
(2.28)
Here c̄v = T (∂s̄/∂T )ρ is the equilibrium specific heat per unit mass. In writing the coefficients
in (2.28), we have used some standard thermodynamic identities. One also has the exact
thermodynamic identity (see Sec. 16 of Landau and Lifshitz [67])
∂P
∂ρ
=
s̄
∂P
∂ρ
T
T 1
+
c̄v ρ20
∂P
∂T
2
c̄p
=
c̄v
∂P
∂ρ
,
T
(2.29)
22
2. Two-fluid hydrodynamics: a review
where c̄p = T (∂s̄/∂T )P is the specific heat per unit mass at constant pressure.
A key feature about superfluid 4 He is that (typical of any liquid) one finds that the
temperature and pressure fluctuations are essentially uncoupled, which means that in (2.29),
one can set (∂P/∂T )ρ ≃ 0. In this case, the adiabatic and isothermal compressibilities are
equal,
∂P
∂ρ
=
s̄
∂P
∂ρ
.
(2.30)
T
In this limit, (2.28) reduces to
u4 − u2 (A + B) + AB = 0,
(2.31)
where the two solutions for u2i are given by
u21
u22
∂P
= A≡
∂ρ s̄
ρs0 T s̄20
= B≡
.
ρn0
c̄v
(2.32)
(2.33)
These give explicit expressions for the velocities of first and second sound. Working out the
associated motions when (2.30) is valid, one finds the first sound mode (ω = u1 q) involves
the in-phase motion of the superfluid and normal fluid components (vn = vs ). One can show
that first sound in this case is a pure “pressure” wave (δT = 0). In contrast, the second
sound mode (ω = u2q) involves an out-of-phase motion of the two components, with δj ≃ 0
or
ρn0 vn + ρs0 vs = 0.
(2.34)
This corresponds to a pure “temperature wave” (δP = 0). These features depend crucially
on the condition in (2.30), which does not hold in a quantum gas.
First and second sound in a dilute uniform Bose gas have been studied by Zaremba,
Nikuni, and Griffin [47]. In Fig. 2.1, the velocities of first and second sound are plotted as a
2. Two-fluid hydrodynamics: a review
23
Figurep
2.1: First and second sound velocities in a dilute uniform Bose gas normalized by
ucr ≡ kB TBEC /m, where TBEC is the superfluid transition temperature (from Zaremba et
al. [47]).
function of temperature. These results were obtained from (2.28) using the thermodynamic
functions for a dilute Bose-condensed gas [47].
In this chapter, we have written down the non-dissipative Landau two-fluid hydrodynamic
equations. In the absence of a trapping potential, the solutions of the linearized equations
are plane-waves. In trapped superfluid gases, in contrast, the linearized Landau two-fluid
equations are very difficult to solve at finite temperatures. This is because they involve
partial differential equations with coefficients involving thermodynamic quantities that vary
rapidly in space. To overcome this problem, we will develop a way of solving the Landau
equations using a variational approach (see Chapter 5).
Chapter 3
Thermodynamics through the
BCS-BEC crossover: a microscopic
model
In his original paper on two-fluid hydrodynamics, Landau [1] calculated all thermodynamic
functions in terms of a noninteracting gas of thermal excitations (or quasiparticles). From experimental data, Landau inferred that there were two types of thermal excitations: phonons
at low energies, and rotons at higher energies [2]. By taking the normal fluid to be a gas
of these excitations, Landau was able to calculate all thermodynamic quantities required in
his two-fluid equations, such as entropy, the specific heat, and the superfluid density. Using
experimental data for these thermodynamic quantities, the energy spectrum of the postulated phonons and rotons could be determined. When the velocity of second sound was
later measured by Peshkov in 1946 [68], Landau [69] introduced a corrected single excitation
spectrum describing both the phonons (at low momenta) and rotons (high momenta) which
gave a better fit to the experimental data.
In this thesis, we are faced with the same task of calculating thermodynamic quantities
in order to determine the coefficients that enter the two-fluid equations for trapped Fermi
gases. Unlike superfluid 4 He, however, the spectrum of thermal excitations in a trapped
Fermi gas can be calculated to a very good approximation using the microscopic model
24
3. Thermodynamics through the BCS-BEC crossover: a microscopic model
25
given in this chapter. These trapped superfluid Fermi gases allow one to test a microscopic
model against experimental predictions, since we can carry out ab initio calculations of the
thermodynamic functions. However, the thermal excitations of Fermi gases in the BCSBEC crossover are more complicated than those in superfluid 4 He. A central theme of this
chapter and Chapter 4 is that the thermal excitations in the strongly-interacting region close
to unitarity cannot be described as a noninteracting gas. Because the superfluid is made
up of bosons (Cooper pairs) that are only weakly bound together, any excitation involving
these bosons can be damped by processes involving the break up of the Cooper pairs. Deep
into the BEC region, the dimer molecules are very tightly bound and in this limit Landau
damping vanishes. As a result, in the BEC limit, the thermodynamic quantities can be
calculated in terms of a noninteracting gas of Bogoliubov excitations of a BEC of dimer
molecules. In Chapter 4, we show that in the BEC limit, the normal fluid density reduces
to the well-known Landau formula.
The analysis in the present chapter is based on a functional integral formulation of the
theory of Nozières and Schmitt-Rink (NSR) [11], extended to the superfluid phase. Functional integration is a powerful technique that has become increasingly popular in theoretical
condensed matter physics. We do not review the details of this technique, referring the reader
to the excellent discussion given in the recent book by Altland and Simons [70]. The classic
account is by Popov [71] which deals with many of the topics discussed below, including
the theory of the bosonic excitations in a Fermi superfluid. However, Popov gives little
explanatory material. The functional integration method is also developed in a pedagogical
manner in the text by Negele and Orland [72]. The material described in this chapter, while
deriving new results, is based on a formalism first used for the BCS-BEC crossover in two
classic papers by Engelbrecht, Randeria, and Sá de Melo [54, 55].
3. Thermodynamics through the BCS-BEC crossover: a microscopic model
3.1
26
Thermodynamic potential for a uniform Fermi superfluid
Our first task is to calculate the grand canonical thermodynamic potential Ω, since this can
be used to generate all thermodynamic functions. For example, the pressure P is given by
Ω = −P V,
(3.1)
where V is the volume. The thermodynamic potential in turn is given by the partition
function Z,
Ω = −β −1 ln Z,
(3.2)
where β ≡ (kB T )−1 . All thermodynamic quantities of interest can be calculated once we have
a suitable approximation for the partition function Z. Functional integral techniques provide
a very efficient way of calculating the partition function, by expressing it as a functional
integral over fermionic Grassmann fields ψ and ψ̄ [71],
Z=
Z
D[ψ, ψ̄]e−S[ψ,ψ̄] .
(3.3)
ψ and ψ̄ are independent Grassmann fields (e.g., they are not complex or Hermitian conjugates; see Altland and Simons [70]). Any two Grassmann fields anti-commute so that, for
instance,
ψ̄ψ = −ψ ψ̄.
(3.4)
The imaginary-time action S[ψ, ψ̄] in (3.3) for a superfluid two-component Fermi gas with a
Feshbach resonance is given by [see (A.17)]1
S[ψ, ψ̄] =
1
Z
β
dτ
0
"Z
dr
X
#
ψ̄σ (x)∂τ ψσ (x) + H ,
σ
In what follows, we set ~ = 1 and the volume, V = 1.
(3.5)
3. Thermodynamics through the BCS-BEC crossover: a microscopic model
27
where the effective single-channel Hamiltonian H in (3.5) is
H =
Z
dr
X
σ
Z
∇2
− µ ψσ (x) − U0 dr ψ̄↑ (x)ψ̄↓ (x)ψ↓ (x)ψ↑ (x).
ψ̄σ (x) −
2m
(3.6)
Here we use the notation x = (r, τ ) where r denotes spatial coordinates and τ = it is the
imaginary time variable [70]. The effective interaction U0 between the two species of fermions
(↑, ↓) is related to the s-wave scattering length by [see (A.12)]
X 1
1
m
=−
+
.
U0
4πas
2ǫk
k
(3.7)
The derivation of (3.6) and (3.7) from a microscopic model that describes the Feshbach
resonance in a two-component gas of fermions is given in Appendix A. The important
feature is that the value of as can be “tuned” using the Feshbach resonance (see Fig. 1.1).
In this way, the entire range of the BCS-BEC crossover can be probed. We recall that when
as is small and negative, one is in the BCS region. A small but positive value of the s-wave
scattering length corresponds to being deep into the BEC region of dimer molecules. When
|as | = ∞, the system is at unitarity.
The functional integral in (3.3) can be evaluated analytically if the action is at most
quadratic in the field ψ. We refer to any action that is quadratic in its fields as Gaussian.
The interaction term in (A.19) is quartic in ψ, which requires that we introduce some approximation. A standard strategy is to treat U0 perturbatively and expand the partition
function in (3.3) in powers of U0 ,
Z =
Z
D[ψ, ψ̄]
"
= Z0 1 +
X U0
n=0
X h(U0
n=1
R
R
d4 x ψ̄↑ ψ̄↓ ψ↓ ψ↑
n!
n
e−S0 [ψ,ψ̄]
#
d4 x ψ̄↑ ψ̄↓ ψ↓ ψ↑ )n i0
.
n!
(3.8)
Here, we have introduced
1
h· · ·i0 ≡
Z0
Z
D[ψ, ψ̄](· · ·)e−S0 [ψ,ψ̄] ,
(3.9)
3. Thermodynamics through the BCS-BEC crossover: a microscopic model
28
the expectation value of a quantity with respect to the action for noninteracting fermions,
S0 ≡
Z0 =
R
Z
4
d x
X
σ
∇2
ψ̄σ (x) ∂τ −
− µσ ψσ (x).
2m
(3.10)
D[ψ, ψ̄] exp(−S0 ) is the partition function for the noninteracting Fermi gas.
Each term in the expansion in (3.8) can be evaluated analytically (up to quadrature)
since Wick’s theorem allows us to write the expectation value h(ψ̄ ψ̄ψψ)n i0 over the free-field
action S0 as the sum of all possible products of the two-particle expectation values hψ ψ̄i0 .
This defines the noninteracting single-particle Green’s functions,2
G0,σ (x, x′ ) ≡ −hψσ (x)ψ̄σ′ (x′ )i0 δσ,σ′ .
(3.11)
Using standard methods [70], the Gaussian integration involved in (3.11) can be carried out
to give the inverse Green’s function
′
G−1
0,σ (x, x )
∇2
= − ∂τ −
− µσ δ(x − x′ ).
2m
(3.12)
Owing to the number of permutations involved in applying Wick’s theorem, one usually
resorts to the techniques of diagrammatic perturbation theory. Generally, the most important contributions to the partition function involve a sum over an infinite number of terms
in (3.8) so that one has to decide in advance which are the most relevant contributions (i.e.,
the important classes of diagrams) for the problem of interest.
A powerful technique of evaluating the partition function involves the application of a
Hubbard-Stratonovich transformation to the functional integral in (3.3) (see, for instance,
Altland and Simons [70]). In this approach, one introduces an “auxilary” Bose field ∆(r, τ )
that decouples the interaction term in the action, reducing it to Gaussian in the Fermi field
ψ. The resulting functional integral over the Fermi fields can then be evaluated analytically,
2
Note that the time-ordering involved in the usual definition of the single-particle Green’s function is
implicit in this functional integral. See, for instance, the discussion by Stoof [73].
3. Thermodynamics through the BCS-BEC crossover: a microscopic model
29
leaving an expression for the partition function Z in terms of the new Bose field ∆(r, τ ). The
trade-off is that in order to evaluate the partition function, one must still calculate ∆(r, τ ).
However, if a suitable physically-motivated mean-field ansatz ∆0 can be made for this Bose
field, then one can simply expand the partition function in powers of fluctuations about this
mean-field. In superfluid Fermi systems, the natural ansatz is to work with ∆0 = U0 hψ↓ ψ↑ i,
describing the Bose condensate of BCS (Cooper) pairs. We now describe the details of this
approach.
Two useful identities for Gaussian integration that we will make extensive use of in this
chapter are [70]
Z
D[Ψ̄, Ψ]e−
P
ij
Ψ̄i Aij Ψj +
P
P ¯
i Ji Ψi + i Ψ̄i Ji
P
= DetA e
ij
J¯i A−1
ij Jj
,
(3.13)
for Gaussian integration over the Fermi Grassmann fields Ψ̄, Ψ, and
Z
D[Φ, Φ∗ ]e−
P
i,j
Φ∗i Mij Φj +
P
i
Ji∗ Φi +
P
i
Φ∗i Ji
P
∝ Det M−1 e
ij
−1
Jj
Ji∗ Mij
,
(3.14)
for Gaussian integration over the bosonic c-number fields Φ∗ , Φ. Here J is some arbitrary
field and can be either a c-number or a Grassmann field. In general, the sum over the indices
i, j in these expressions will involve a space-time integral over x = (r, τ ) (or, in Fourier space,
over momentum and the Matsubara frequency), as well as a sum over discrete indices such as
spin. We use “Det” with a capital D to denote a determinant over these continuous variables
in addition to spin indices [as in (3.13) and (3.14)]. We use “det” with a lowercase d to denote
a determinant only over the spin indices. The constant factor multiplying the right-hand
side of (3.14) can be ignored since it appears in both the numerator and denominator of any
expectation value.
In our problem, the Hubbard-Stratonovich transformation amounts to the following in-
30
3. Thermodynamics through the BCS-BEC crossover: a microscopic model
tegral identity:
eU0
R
dτ
R
dr ψ̄↑ ψ̄↓ ψ↓ ψ↑
=
Z
Z
∗
D[∆, ∆ ] exp −
β
Z
dτ
0
dr
|∆|2
− ∆∗ ψ↓ ψ↑ + ∆ψ̄↑ ψ̄↓
U0
.
(3.15)
This identity can be proven by integrating out the Bose field ∆ in the right-hand-side of this
expression using the result in (3.14), to reproduce the left-hand side. Inserting (3.15) into
the partition function in (3.3), one obtains
Z =
Z
∗
D[ψ, ψ̄]D[∆, ∆ ] exp
−
2
Z
β
dτ
0
|∆|
−∆∗ ψ↓ ψ↑ − ∆ψ̄↑ ψ̄↓ +
U0
Z
Z
∗
=
D[ψ, ψ̄]D[∆, ∆ ] exp −
β
dτ
0
X
∇2
− µ ψσ (x)
ψ̄σ (x) ∂τ −
2m
Z
dr
Z
|∆|2
dr Ψ −G−1 Ψ +
U0
σ
†
.
(3.16)
Here we have introduced the Nambu spinors
†
Ψ = ψ̄↑ , ψ↓ ,
Ψ=
ψ↑
ψ̄↓
,
(3.17)
and the inverse BCS Nambu-Gorkov Green’s function G−1 is given by
−1
′
G (x, x ) =
′
G−1
∆(x)δ(x − x′ )
0,↑ (x, x )
′
∆∗ (x)δ(x − x′ ) −G−1
0,↓ (x , x)
,
(3.18)
where the noninteracting Fermi Green’s function G−1
0,σ has been defined in (3.12). As we will
see shortly, the different time-ordering between the diagonal elements of (3.18) is critical to
−1 ′
ensuring correct results. The factor of −1 appearing in the (G−1 )22 = −G0,↓
(x , x) matrix
element is a consequence of the anticommutativity of the Grassmann fields.
Using (3.13), the integration over the Grassmann fields ψ in (3.16) can be performed in
straightforward fashion to give
Z=
Z
¯ −Seff ,
D[∆, ∆]e
(3.19)
3. Thermodynamics through the BCS-BEC crossover: a microscopic model
31
where [54]
∗
Seff [∆, ∆ ] =
Z
0
β
dτ
Z
dr
|∆(x)|2
− Tr ln[−G−1 ].
U0
(3.20)
The trace in (3.20) is performed over space and imaginary time variables, in addition to the
Nambu indices. We have used the standard identity ln DetA = Tr ln A [70]. We emphasize
that a theory based on (3.19) and (3.20) is still exact since no approximations have been
made yet.
Once some approximation is introduced in (3.20), the functional integration in (3.19)
can be carried out giving Z and hence the thermodynamic potential Ω for the uniform twocomponent Fermi superfluid. Following the standard procedure [54, 55, 71], we expand the
action in (3.20) in powers of fluctuations Λ(x) about the static and uniform mean-field BCS
pairing field ∆0 : ∆(x) = ∆0 + Λ(x). Thus, we write the BCS Green’s function in (3.18) as
−1
−1
G−1 = G−1
0 + Σ, where G0 ≡ G |∆(x)=∆0 , and
Σ≡
0
Λ(x)
∗
Λ (x)
0
δ(x − x′ ).
(3.21)
Physically, Λ(x) corresponds to the fermionic self-energies due to coupling to Bose collective
modes involving pair fluctuations in the Cooper pair channel.
Using the expansion
−1
Tr ln[−G−1
0 (1 + G0 Σ)] = Tr ln[−G0 ] + Tr ln[1 + G0 Σ]
= Tr ln[−G−1
0 ]+
X1
Tr[(G0 Σ)n ](−1)n+1 ,
n
n=1
(3.22)
we can expand (3.20) up to quadratic order in the Bose fluctuation field Λ to obtain the
Gaussian action,
SGauss ≡ S (0) + S (1) + S (2) .
(3.23)
3. Thermodynamics through the BCS-BEC crossover: a microscopic model
32
If ∆0 is chosen to satisfy the stationarity condition of the action,
∂Seff
∂∆
= 0,
(3.24)
∆=∆0
then the linear term in fluctuations S (1) in (3.23) automatically vanishes.3 As we discuss in
Section 3.2, however, there is an alternative definition for ∆0 given by the condition
∂Ω
∂∆0
= 0,
(3.25)
where Ω is the thermodynamic potential which goes past the BCS mean-field approximation.
In general, the solution of this equation gives a different value of ∆0 which will not satisfy
the stationarity condition in (3.24). Nonetheless, for a uniform system, or one for which
the action has no explicit time-dependence [i.e., where G0,σ (τ, τ ′ ) = G0,σ (τ − τ ′ )], it is
straightforward to show that the linear term S (1) vanishes anyway. Thus, our Gaussian
expression for the action only includes the mean-field contribution S (0) and the contribution
S (2) which is quadratic in fluctuations.
Fourier-transforming (3.20), the mean-field S (0) and fluctuation S (2) contributions are
given by
S (0) = β
∆20 X
−
tr ln[−G−1
0 (k)]
U0
k
(3.26)
and
S (2) = β
X |Λk |2
k
≡
U0
+
1X †
Λ MΛ.
2 q
1X
tr[G0 (k)Σ(−q)G0 (k + q)Σ(q)]
2 k,q
(3.27)
In (3.27), q ≡ (q, iνm ) is a 4-vector denoting the momentum q and boson Matsubara frequency νm = 2πm/β, where m is an integer. k ≡ (k, iωm ) is a 4-vector denoting the
3
This follows from the fact that the linear term in fluctuations about ∆0 is given by the second term in
the Taylor-series expansion S = S[∆ = ∆0 ] + (∂S/∂∆)∆=∆0 Λ(x) + · · ·. This term vanishes as a result of the
stationarity condition in (3.24).
33
3. Thermodynamics through the BCS-BEC crossover: a microscopic model
momentum k and fermion Matsubara frequency ωn = (2n + 1)π/β, where n is an integer. In
momentum-frequency space, the mean-field (denoted by the subscript “0”) Nambu-Gorkov
BCS Green’s function G0 (k) for the BCS superfluid is given by
G0 (k) =
1
(iωn )2 − Ek2
+
iωn eiωn 0 + ξk
−∆0
−iωn 0+
−∆0
iωn e
− ξk
.
(3.28)
Here we have introduced the well-known BCS quasiparticle dispersion relation
Ek =
q
ξk2 + ∆20 ,
(3.29)
with ξk ≡ k2 /2m − µ. The fluctuation self-energies in (3.27) are defined by
Σ(q) =
0 Λq
Λ∗−q 0
.
(3.30)
In the last line of (3.27), we have introduced the two-component spinor Λ† ≡ (Λ∗q , Λ−q ).
The exponential terms appearing in (3.28) involving the positive infinitesimal 0+ are
convergence factors. Although these factors are often left implicit in the literature, their
inclusion is critical in obtaining correct expressions for frequency sums over integrands that
go to zero slower than (iωn )−2 at large frequencies. These factors reflect the underlying
time-ordering involved in the definition of the Green’s functions [recall from (3.18) that the
diagonal elements have the opposite time-ordering]. We emphasize that had we handled the
functional integration leading to (3.20) more rigorously, these convergence factors would have
appeared naturally since the time-ordering of the Grassmann fields is implicit in the construction of the functional integral in (3.3) (for further discussion of this point, see Stoof [73]
as well as pg. 170 in Altland and Simons [70]).
The matrix elements of the inverse 2 × 2 matrix pair fluctuation propagator M which we
have introduced in (3.27) are given by [55]
M11 (q)
1X
M22 (−q)
1
G0,11 (k + q)G0,22 (k)
+
=
=
β
β
U0 β k
(3.31)
3. Thermodynamics through the BCS-BEC crossover: a microscopic model
34
and
M12 (q)
M21 (q)
1X
G0,12 (k + q)G0,12 (k).
=
=
β
β
β k
(3.32)
Here, G0,ij denotes the ij-th element of the matrix mean-field BCS Green’s function given
in (3.28). Summing over the fermion Matsubara frequencies in (3.31) and (3.32), the following explicit expressions for the matrix elements of the inverse matrix propagator for pair
fluctuations are obtained (Engelbrecht et al. [55]):
M22 (−q)
1
M11 (q)
=
=
β
β
U0
X
iνm 0+
+
[1 − f (Ek ) − f (Ek+q )] e
2
u2k u2k+q
vk2 vk+q
−
iν
iνm + Ek + Ek+q
m − Ek − Ek+q
k
2
X
u2k vk+q
vk2 u2k+q
iνm 0+
(3.33)
−
+
[f (Ek ) − f (Ek+q )] e
iνm + Ek − Ek+q iνm − Ek + Ek+q
k
and
M12 (q)
M21 (q)
=
=
β
β
X
uk vk uk+q vk+q
uk vk uk+q vk+q
[1 − f (Ek ) − f (Ek+q )]
−
iν
iνm − Ek − Ek+q
m + Ek + Ek+q
k
X
uk vk uk+q vk+q
uk vk uk+q vk+q
[f (Ek ) − f (Ek+q )]
+
−
.
iν
iν
m + Ek − Ek+q
m − Ek + Ek+q
k
(3.34)
Here
f (Ek ) =
1
eβEk
is the Fermi distribution function, while uk =
+1
p
(1 + ξk /Ek )/2 and vk =
(3.35)
p
(1 − ξk /Ek )/2
are the usual BCS-Bogoliubov coherence factors. We call attention to the convergence factors
+
eiνm 0 in the diagonal elements of M in (3.33), which arise from the convergence factors in
(3.28) and the definition in (3.31).
We now turn to the evaluation of the thermodynamic potential given in (3.2), using
the Gaussian action given by (3.23), (3.26), and (3.27). As defined, the thermodynamic
3. Thermodynamics through the BCS-BEC crossover: a microscopic model
35
potential is the sum of a mean-field component coming from the contribution of Fermi BCS
quasiparticles in S (0) , and a contribution from bosonic pair fluctuations in S (2) ,
Ω ≃ ΩF + ΩB .
(3.36)
The Fermi contribution ΩF to the thermodynamic potential due to BCS quasiparticles is
given by S (0) /β:
ΩF =
∆20
1X
tr ln[−G−1
−
0 (k)].
U0
β k
(3.37)
The Bose contribution ΩB from pair fluctuations to the thermodynamic potential is given
by
1
ΩB = − ln
β
Z
(2)
D[Λ, Λ̄]e−S .
(3.38)
With S (2) given by the quadratic expression in (3.27), the integration in (3.38) is easily done
using (3.14). We find
ΩB =
=
=
=
=
Z
1
(2)
− ln D[Λ, Λ̄]e−S
β
1 − ln Det M−1/2
β
1
− Tr ln M−1/2
β
1
Tr ln [M]
2β
1 X
ln detM(q).
2β q
(3.39)
The last step follows from the definition of the trace: it is a trace over both momentum and
frequency as well as the Nambu (spin) indices. The determinant appearing in the last line
is only for Nambu indices and hence, we denote it as “det”. In terms of the components in
(3.33) and (3.34), we have
2
detM(q) = M11 (q)M11 (−q) − M12
(q).
(3.40)
3. Thermodynamics through the BCS-BEC crossover: a microscopic model
36
Adding the Fermi contribution in (3.37) to the Bose contribution in (3.39), the full
Gaussian expression for the thermodynamic potential is given by
Ω=
∆20
1 X
1X
tr ln[−G−1
(k)]
+
ln detM(q).
−
0
U0
β
2β q
(3.41)
k
This result for the thermodynamic potential Ω will form the basis of our evaluation of the
thermodynamic functions in the BCS-BEC crossover in the rest of this chapter.
3.2
The gap and number equations
In this section, we discuss the gap and number equations that are solved self-consistently to
give the values of the BCS pairing gap ∆0 and the chemical potential µ from the Gaussian
thermodynamic potential in (3.41). These microscopic quantities are needed to calculate all
thermodynamic functions through the crossover, including the superfluid density.
Up to Gaussian order in fluctuations, the standard gap and number equations that arise
in the literature (see, for instance, Engelbrecht, Sá de Melo, and Randria [55] and Ohashi
and Griffin [48]) are given by
∂ΩF
∂∆0
= 0.
(3.42)
µ
and
n=−
∂Ω
∂µ
∆0
∂ΩF
≡−
∂µ
∆0
−
∂ΩB
∂µ
,
(3.43)
∆0
where the Gaussian thermodynamic potential Ω = ΩF + ΩB is given by (3.41). The number equation in (3.43) includes fluctuation contributions from ΩB , whereas the gap ∆0 is
determined from the mean-field thermodynamic potential.4 While it might seem desirable
4
Note that the mean field gap equation is equivalent to the stationarity condition of the action [see (3.24)],
(∂Seff /∂∆)∆=∆0 = 0, which defines the classical solution of the field equations, i.e., with no fluctuations in
space and time.
3. Thermodynamics through the BCS-BEC crossover: a microscopic model
37
to include fluctuations in the gap equation, this must be done carefully as we explain in
Appendix B.
We emphasize that the number equation in (3.43) leaves ∆0 fixed in evaluating the
derivative of Ω with respect to the chemical potential µ. That is, it does not include a
contribution from the derivative ∂∆0 /∂µ. An alternative number equation that has been used
recently (Keeling, Eastham, Szymanska, and Littlewood [74]; Hu, Liu, and Drummond [49];
Diener, Sensarma, and Randeria [75]) includes such a contribution:
n = −
≡ −
∂ΩF
∂µ
∂ΩF
∂µ
∆0
∆0
−
∂ΩB
∂µ
−
∂ΩB
∂∆0
∂ΩB
∂µ
∆0
∆0
−
∂ΩB
∂∆0
µ
∂∆0
∂µ
− δn∆ ,
(3.44)
(3.45)
where
δn∆ ≡ −
µ
∂∆0
∂µ
.
Calculations using this number equation together with the mean-field gap equation in (3.42)
improve on the earlier NSR calculations (Engelbrecht et al. [55]) described by (3.42) and
(3.43) by including the extra term δn∆ in the number equation. It was Keeling et al. [74]
who pointed out that the correction δn∆ in (3.45) can be of the same magnitude as (∂Ω/∂µ)∆0
in (B.3), and thus should not be ignored. In the context of the BCS-BEC crossover problem,
the work by Hu et al. [49, 50] based on (3.42) and (3.44) has shown the importance of
this correction. These papers obtained results that are in excellent agreement with Monte
Carlo simulations at both T = 0 (Astrakharchik, Boronat, Casulleras, and Giorgini [76])
and finite T (Bulgac et al. [51]). The work reported by in Refs. [50, 75] indicates that the
correction δn∆ in (3.45) brings in the effect of cubic and quartic fluctuations (Pieri and
Strinati [77]; Ohashi [78]) which have the effect of renormalizing the strength of the effective
interaction between stable Cooper pairs. In the BEC limit, it can be shown that (3.42) and
38
3. Thermodynamics through the BCS-BEC crossover: a microscopic model
(3.44) give rise to a chemical potential for dimer molecules that is consistent with a dimer
scattering length aM ≃ 0.6as . In contrast, solving (3.42) and (3.43) give results consistent
with aM = 2as in the BEC limit (Sá de Melo et al. [54]; Engelbrecht et al.). The value
aM ≃ 0.6as for the molecular scattering length was first obtained by Petrov, Salomon, and
Shlyapnikov [79] based on the Schrödinger equation for 4-body scattering involving fermions.
In Appendix B we show that the inclusion of the extra term in the number equation
given by (3.45) is equivalent to introducing fluctuations into the gap equation in (3.42).
For simplicity, the numerical results for the superfluid density to be discussed in Chapter 4 will be based on the approximate gap and number equations given by (3.42) and (3.43).
In contrast, the results presented in Chapter 6 for the two-fluid mode frequencies at unitarity, will be based on (3.42) and (3.44). While the results of solving (3.42) and (3.43) are
sufficiently accurate for many calculations, it fails to accurately describe the correct temperature dependence of the chemical potential close to unitarity. The chemical potential is the
key thermodynamic quantity appearing in our expressions for the two-fluid mode frequencies
at unitarity that we derive in Chapter 6. Consequently, we must use (3.42) and (3.44) to
calculate the chemical potential near unitarity.
We now discuss the solution of the gap and number equations given by (3.42) and (3.43)
in detail and give an important analytic result for the chemical potential in the BEC limit.
Using (3.37) in (3.42), we obtain
∆0
1X
G12 (k).
=
U0
β k
(3.46)
Performing the frequency summation and using (3.7) to replace U0 in terms of the s-wave
scattering length as , the mean-field gap equation in (3.46) becomes
X 1
m
1
=
−
tanh(βEk /2) .
4πas
2ǫk 2Ek
k
(3.47)
3. Thermodynamics through the BCS-BEC crossover: a microscopic model
39
Deep in the BEC region, we expect the chemical potential to become negative, equal to
one-half the binding energy of a dimer molecule since the energy cost of adding two fermions
(in different hyperfine states) should be roughly equal to the binding energy. We can show
this explicitly using the gap equation in (3.47). Assuming that the chemical potential is
large and negative, such that |µ| ≫ ∆0 , kB T , (3.47) reduces to
X 1
m
1
≃
−
.
4πas
2ǫ
2ǫ
k
k + 2|µ|
k
(3.48)
This integral can be evaluated straightforwardly to give [54]
|µ| =
Eb
1
≡− ,
2
2mas
2
(3.49)
where Eb = −1/ma2s is the dimer binding energy of the s-wave dimer molecule formed
between two fermions with scattering length as .
Using the Gaussian expression for the thermodynamic potential Ω in (3.41) in the number
equation, (3.43), we obtain
n =
∂
1 X ∂
1X
tr[G0 (k) G−1
(k)]
−
ln detM(q)
β k
∂µ 0
2β q ∂µ
+
+
1 X ∂
1 X iωn (eiωn 0 − e−iωn 0 ) + 2ξk
−
ln detM(q)
=
β k,iω
(iωn + Ek )(iωn − Ek )
2β q ∂µ
n
+
1 X
2iωn eiωn 0 + 2ξk
1 X ∂
=
−
ln detM(q)
β k,iω (iωn + Ek )(iωn − Ek ) 2β q ∂µ
n
X
1 X ∂
ξk
tanh(βEk /2) −
ln detM(q).
1−
=
E
2β
∂µ
k
q
k
(3.50)
Here we have made use of the identities f (−Ek ) = 1 − f (Ek ) and 1 − 2f (Ek ) = tanh(βEk /2)
+
in the last line. Note that without the convergence factors e±iωn 0 , (3.50) would not include
the “1” inside the square brackets in the first term on the right-hand side. This shows
how the convergence factor plays a critical role in arriving at the correct expression for the
number equation. Similarly, the convergence factors in (3.33) are also needed to ensure the
correct contribution to the number equation from bosonic fluctuations.
3. Thermodynamics through the BCS-BEC crossover: a microscopic model
(a) ∆ /εF
∆(T)
∆(T=0)
Tc
2
1.5
1
0.5
0
0.1
T /εF
0.2
0.3
-2
(b) µ /εF
1
0
-1
-2
-3
0
0.1
T /εF
40
-1
0
1
2
-1
(kFas)
µ(T)
µ(T=0)
Tc
0.2
0.3
-2
-1
0
1
2
-1
(kFas)
Figure 3.1: (a) The order parameter ∆0 , and (b) the Fermi chemical potential µ in the BCSBEC crossover. The pairing interaction is measured in terms of the inverse of the two-body
scattering length as , normalized by the Fermi momentum kF . In these panels, the dotted line
shows Tc as a function of (kF as )−1 . In the strong-coupling regime, the apparent first-order
transition is unphysical, as discussed in the text.
3. Thermodynamics through the BCS-BEC crossover: a microscopic model
41
Solving (3.47) and (3.50) self-consistently through the crossover gives the results plotted
in Figs. 3.1 and 3.2. These expressions were evaluated numerically by Naoki Fukushima
and Yoji Ohashi. Figure 3.1 shows ∆0 and µ through the BCS-BEC crossover and as a
function of temperature. One immediately sees a “bend-over” in ∆0 close to unitarity and
on the BEC side of resonance near the transition temperature Tc . We emphasize that this
apparent first-order phase transition is an artifact of the NSR Gaussian treatment of pairing
fluctuations used to calculate ∆0 and µ self-consistently. This problem is equivalent to one
that arises in the self-consistent calculation of the condensate density and chemical potential
close to Tc in dilute Bose gases. An extensive discussion with references is given on pg. 34
of Shi and Griffin [80].
Figure 3.2 shows the superfluid transition temperature Tc across the crossover region. Tc
is determined by the temperature where ∆0 vanishes. We also show the chemical potential
at T = 0. We note how quickly the chemical potential approaches the BEC-limiting value
given by (3.49), namely one-half the dimer binding energy.
3.3
Thermal excitations
In this section, we discuss the two types of excitations that arise in our Gaussian approximation for the superfluid Fermi gas: the BCS quasiparticles and the Bogoliubov-Anderson
modes.
3.3.1
BCS quasiparticles
The BCS quasiparticles give the fermionic mean-field contribution to the thermodynamic
potential in (3.37). These excitations, with the BCS-type energy spectrum given by (3.29),
describe the breakup of a Cooper pair into two fermions. Evaluating the frequency sum in
3. Thermodynamics through the BCS-BEC crossover: a microscopic model
0.3
42
(a)
Tc / ε F
0.2
0.1
0
(b)
µ /εF
0
-1
-2
T=Tc
T=0
BEC
-3
-4
-2 -1.5 -1 -0.5
0
0.5
1
1.5
2
-1
(kF as)
Figure 3.2: (a) Superfluid phase transition temperature Tc , and (b) chemical potential µ(T =
Tc ) in the BCS-BEC crossover. In panel (b), µ at T = 0 and T = Tc is also shown.
The curve “BEC” gives the strong-coupling BEC limit, where one finds µ = −1/2ma2s .
In the dimensionless units used in the plot, this result is µ/ǫF = −(1/2ma2s )/(kF2 /2m) =
−1/(kF as )2 .
3. Thermodynamics through the BCS-BEC crossover: a microscopic model
43
(3.37), we find
1X
∆20
tr ln[−G−1
−
0 (k)]
U0
β
k
∆20 X
2X =
(ξk − Ek ) −
ln 1 + e−βEk .
+
U0
β
ΩF =
k
(3.51)
k
Using this expression, it is straightforward to calculate the contribution of the BCS quasiparticles to all thermodynamic quantities (such as entropy) that appear in the two-fluid
hydrodynamic equations.
3.3.2
Collective modes
The contribution ΩB to the thermodynamic potential from the bosonic collective modes
is given by the second term in (3.41). These modes play an increasingly important role
going from the BCS limit to the BEC limit (including unitarity). In general, there is no
simple expression for ΩB analogous to the BCS fermion contribution ΩF given by (3.51)
because the bosonic collective modes are damped by the BCS quasiparticles. The Gaussian
action we have used does not describe damping processes that involve interactions between
the bosonic collective modes. However, it does account for interactions between collective
modes and the BCS quasiparticles. In particular, Landau damping of the bosonic collective
modes arises due to scattering processes between collective modes and thermally excited BCS
Fermi quasiparticles:5 ωq + Ek = Ek+q . When such damping occurs, the collective modes
strongly hybridize with BCS Fermi quasiparticles, leading to the result that one cannot write
down a simple expression for ΩB in terms of well-defined excitations.
We note that Landau damping can vanish at T = 0 where there are no thermally-excited
BCS quasiparticles and also in the BEC limit, where the binding energy of the dimers is
sufficiently large that the BCS quasipartcles are “frozen out” (|µ| ≫ kB T ). Specifically, the
5
These (Landau) damping processes are described by the third lines in (3.33) and (3.34), involving
iνm ± (Ek − Ek+q ).
3. Thermodynamics through the BCS-BEC crossover: a microscopic model
44
spectrum ωq of Bose excitations is undamped at T = 0 if ωq ≪ ∆0 in the BCS region and
close to unitarity, and ωq ≪ |µ| in the BEC region. Close to unitarity and on the BCS
side of the resonance, if the collective modes have a higher energy than twice the excitation
gap, 2∆0 , they can spontaneously decay into BCS quasiparticle excitations.6 On the BEC
side of resonance, the collective modes of the molecular BEC will likewise decay when they
have energy greater than the molecular binding energy. Thus, at T = 0 when ωq ≪ ∆0 (in
the BCS region and at unitarity) and ωq ≪ |µ| (in the BEC region), the collective modes
are well-defined and expected to have a spectrum of the form
p
(c q)2 + (q2 /2m∗ )2 . To find
the coefficients (c and m∗ ) in this expression, we expand the inverse fluctuation propagator
matrix elements in (3.33) and (3.34) in powers of q (Engelbrecht et al. [55]),
M11 (q)
≃ A + B|q|2 + C(iνm )2 + D(iνm ),
β
(3.52)
M12 (q)
≃ A + F |q|2 + G(iνm )2 .
β
(3.53)
and
The energy ωq of the bosonic collective modes is determined in the standard way7 from the
solution of detM(q, iνm → ωq + i0+ ), where the determinant of M is defined in (3.40) and
iνm → ωq + i0+ denotes the usual analytic continuation from imaginary Bose frequencies.
Explicitly, ωq is the solution of
2
M11 (q, ωq )M11 (q, −ωq ) − M12
(q, ωq ) = 0.
(3.54)
Using the expressions in (3.52)-(3.54), one finds ωq = cq at T = 0, where the sound speed is
given by
c=
6
r
D2
2A(B − F )
.
+ 2AG − 2AC
(3.55)
The energy must be greater than twice the excitation gap since BCS quasiparticles can only be created
in pairs due to momentum conservation.
7
See, for instance, Popov [71].
45
3. Thermodynamics through the BCS-BEC crossover: a microscopic model
Outside the BEC region, or at finite temperatures throughout the crossover, it is well
known that one cannot carry out an expansion in powers of q and iνm [as in (3.52) and
(3.53)] because of Landau damping. These matrix elements of the fluctuation propagator M
are singular in the long wavelength, zero frequency limit [81, 82]. We can however expand
the matrix elements in powers of q and ω at T = 0, where there is no Landau damping. This
gives the following coefficients (Engelbrecht et al. [55]; see also Taylor, Griffin, Fukushima,
and Ohashi [56]):
A=
B=
X k
∆2
2 − 3 20
Ek
X ∆2
0
,
3
4E
k
k
ξk |k|2 cos2 φ
+
m
m2
C=
X ∆2
0
Ek2
k
D=−
F =
X
k
(3.56)
∆20
∆40
1
−2 + 13 2 − 10 4
,
Ek
Ek
16Ek3
−2
1
,
16Ek3
X ξk
,
3
4E
k
k
2
∆20 ξk |k|2 cos2 φ
∆0
∆40
1
−3 2 +
7 2 − 10 4
,
2
Ek m
m
Ek
Ek
16Ek3
(3.57)
(3.58)
(3.59)
(3.60)
and
G=
X ∆2 k
0
Ek2
1
.
16Ek3
(3.61)
We have made use of the gap equation given by (3.47) to eliminate 1/U0 from M11 (q).
In the strong-coupling BEC limit where ∆0 ≪ |µ|, we can expand the integrands above
in powers of ∆0 /|µ|. To leading order, using |µ| = (2ma2s )−1 , we find
A ≈ ∆20
X 1
∆20 a3s m3
,
=
4ξk3
16π
k
(3.62)
3. Thermodynamics through the BCS-BEC crossover: a microscopic model
46
X 1
|k|2 cos2 φ
mas
,
−
=
B≈
2
3
2
8mξ
4m
ξ
32π
k
k
k
(3.63)
X 1
m3 a3s
C≈−
,
=−
8ξk3
16π
k
(3.64)
X 1
m2 as
.
=
−
4ξk2
8π
(3.65)
and
D≈−
k
To leading order, we also find F ∼ ∆20 a5s and G ∼ ∆20 a7s , which are vanishingly small in the
BEC limit where as → 0. Similarly, since C ∝ (mas )3 , we can set this coefficient equal to
2
zero as well. However, since ∆20 ∝ a−1
s , one finds that A ∝ as , and thus we retain A in (3.52)
and (3.53).
The coefficients given by (3.56)-(3.61) are valid throughout the BCS-BEC crossover at
T = 0. In the BEC limit, these reduce to the expressions given by (3.62)-(3.65). Although
these BEC limit results were derived for T = 0, it is straightforward to show that (3.62)(3.65) are also valid at finite temperatures (when T ≪ |µ|/kB ), since the Fermi thermal
factors f (E) entering (3.33) and (3.34) are zero in the BEC limit. The only temperature
dependence enters through the gap ∆0 (T ) (see Fig. 3.1).
Using the coefficients given by (3.62), (3.63), and (3.65), and setting C = F = G = 0,
we obtain
detM(q, iν̃m ) = 2ABq2 + B 2 q4 − D 2 (iνm )2 .
Since the bosonic fluctuation spectrum is given by the zeros of detM(q, ωq ),
s
r
2 2
2AB 2 B 2 4
∆20 a2s 2
q
ωq =
.
q
+
q
=
q
+
D2
D2
4
2M
(3.66)
(3.67)
We recall that this expression for the fluctuation spectrum in the BEC limit is also valid at
finite temperatures, where ∆0 = ∆0 (T ) is temperature dependent.
3. Thermodynamics through the BCS-BEC crossover: a microscopic model
47
The fact that ∆0 (T ) is nonzero is associated with the existence of a molecular Bose
condensate in the BEC region. Thus the dispersion relation in (3.67) can also be written
in terms of the molecular condensate density nc (T ) instead of ∆0 (T ). One can show (see
Fukushima, Ohashi, Taylor, and Griffin [57]) that the corrections δnc to the mean-field
expression for the condensate density,
nc0 (T ) =
X ∆2 (T )
0
k
4Ek2
tanh2 (βEk /2),
(3.68)
are negligible throughout the BCS-BEC crossover (within our NSR Gaussian approximation).
Thus we can use (3.68) to determine the condensate density in the BEC limit (where |µ| ≫
kB T ), with the result
∆20 (T )M 2 as
nc (T ) =
.
32π
(3.69)
It is important to emphasize that in obtaining this expression, we have only assumed that
|µ| ≫ kB T . The gap ∆0 (T ) still has a strong temperature dependence (see Fig. 3.1) arising
from the thermally excited pairing fluctuations, which are not frozen out.
Using the relation between nc and ∆0 in (3.69), the phonon velocity in (3.67) reduces to
c2 ≡
UM nc (T )
∆20 (T )a2s
=
4
M
(3.70)
for an interacting gas of bosons of mass M = 2m. This is precisely the expected BogoliubovPopov phonon velocity with UM = 4πaM /M, with the molecular s-wave scattering length
given by the Born-approximation result aM = 2as (Sá de Melo et al. [54]). In order to get
the correct value of the molecular s-wave scattering length, namely aM ≃ 0.6as (Petrov et
al. [79]) in the BEC limit, one would have to include the effects of 4-body correlations which
are beyond the 2-body physics contained in our Gaussian NSR theory. That is, we would need
to expand the action to quartic order in fluctuations (Pieri and Strinati [77]; Ohashi [78]).
3. Thermodynamics through the BCS-BEC crossover: a microscopic model
48
As discussed in Section 3.2, Hu et al. [49] have shown that the correct renormalized value of
aM emerges when one calculates µ using the number equation given in (3.44) which includes
the contribution from (∂ΩB /∂∆0 )µ .
The results from this section can also be used to obtain an explicit expression for the
thermodynamic potential ΩB in the BEC limit, where the Bose excitations dominate. Recall
that the contribution ΩF from BCS quasiparticles vanishes in the BEC limit since these are
frozen out (i.e., |µ| ≫ kB T ). Using the expansion in (3.52) and (3.53) together with the
coefficient values given by (3.62)-(3.65), we obtain
M11 (q)
m2 as −iνm 0+
iνm e
− ǫq − nc UM
≃−
β
8π
(3.71)
m2 as
M12 (q)
≃
(nc UM ) ,
β
8π
(3.72)
and
where ǫq = q2 /2M. The frequency sum in this expression for the thermodynamic potential
ΩB in (3.39) can be evaluated via contour integration, to give
ΩB =
1X
1X
(ωq − ǫq − nc UM ) +
ln 1 − e−βωq .
2 q
β q
(3.73)
We again emphasize the crucial role played by the convergence factor in (3.71). The correct
expression for the first integrand in (3.73) is only obtained because of this convergence factor.
Equation (3.73) is precisely the thermodynamic potential one would write down for a Bosecondensate of molecules with noninteracting excitations given by the Bogoliubov spectrum
in (3.67).
We now turn to the BCS limit. Making use of the fact that ∆0 ≪ µ and µ ≃ ǫF , the
integrals in (3.56)-(3.61) can also be evaluated analytically with the result that [55]
vF
c= √ .
3
(3.74)
3. Thermodynamics through the BCS-BEC crossover: a microscopic model
49
0.6
c/vF
BEC
BCS
c/vF
0.5
0.4
0.3
0.2
2
1
0
(kFas)-1
-1
-2
Figure 3.3: The T = 0 phonon velocity through the crossover. “BEC” denotes the Bogoliubov expression for the phonon velocity p
of a condensate of dimer molecules, given by
(3.70). “BCS” denotes the phonon velocity 2µ/3m in (3.78), which is valid at unitarity
as well as the BCS limit. In the dimensionless units used in this plot, the BEC velocity corresponds
to c/vFp= ∆0 as /2vF = (∆0 /2ǫF )(kF as ). The BCS result corresponds to
p
c/vF = 2µ/3mvF2 = µ/3ǫF . The solutions for ∆0 /ǫF and µ/ǫF shown in Fig. 3.1 are
used.
This well-known result was first derived by Bogoliubov [83] and Anderson [84] in their pioneering discussions of the collective modes in a weak-coupling BCS superfluid. At T = 0, we
note that this collisionless phonon velocity coincides with the velocity of first sound, given
by (2.32):
2
c =
∂P
∂ρ
n
=
m
∂µ
∂n
.
(3.75)
Here, we have made use of the (T = 0) identity (∂P/∂ρ) = ρ(∂µ/∂ρ)/m [see (5.6)]. A
polytropic equation of state (see, for example, Giorgini et. al [13]) is defined by µ(n) = Anγ ,
where A and γ are constants. Using this in (3.75) gives
n
c =
m
2
∂µ
∂n
=γ
µ
.
m
(3.76)
In the weak coupling BCS limit, the chemical potential is given by the Fermi energy,
µ ≃ ǫF ≡
(3π 2 n)2/3
,
2m
(3.77)
3. Thermodynamics through the BCS-BEC crossover: a microscopic model
50
which corresponds to a polytropic equation of state with the exponent γ = 2/3. Using this
in (3.76), we recover the Bogoliubov-Anderson result in (3.75). As we discuss in Section 6.1,
at unitarity, the T = 0 chemical potential is also given by a polytropic equation of state with
γ = 2/3 due to universality. As a result, the T = 0 phonon velocity in the BCS limit as well
as at unitarity is given by
c2 =
2µ
.
3m
(3.78)
We emphasize, however, that the chemical potential µ ≃ 0.4ǫF at unitarity is different than
the chemical potential µ ≃ ǫF in the BCS limit [see Fig 3.2(b)].
In Fig. 3.3, the phonon velocity (in units of the Fermi velocity) given by (3.55) is plotted
through the crossover. We also plot the BEC limit given by (3.70) as well as the BCS limit
given by (3.78). Equation (3.78) is found to give the correct phonon velocity at unitarity as
well as in the BCS limit, in agreement with the predictions of universality (see Chapter 6).
Chapter 4
The superfluid density in the
BCS-BEC crossover
In this chapter, we extend the Gaussian Nozières-Schmitt-Rink theory for the pairing fluctuations used in Chapter 3 and consider the case of a current-carrying Fermi superfluid.
Using a standard definition of the superfluid density ρs (first given by Fisher, Barber, and
Jasnow [85]) in terms of a “phase twist” applied to the superfluid order parameter, we
show how to obtain an explicit expression for ρs from the thermodynamic potential for a
current-carrying uniform superfluid.
The superfluid density ρs was first introduced by Landau in connection with the two-fluid
hydrodynamic theory of superfluid 4 He discussed in Chapter 2. As we discuss in Chapters 5
and 6, the frequencies of the two-fluid hydrodynamic modes in our variational theory are
given completely in terms of the equilibrium thermodynamic functions, including the superfluid density. As we discussed in Chapter 3, the precise values of these equilibrium quantities
depend on the nature of the dominant thermal excitations, which are specific to different
superfluid systems. In superfluid 4 He, these thermal excitations are the phonon-rotons [2].
In a weakly interacting BCS superfluid, the thermal excitations are the BCS quasiparticles.
In a dilute Bose-condensed gas, the excitations are the Bogoliubov quasiparticles.
The relevant excitations in the BCS-BEC crossover are more complex, since they change
51
4. The superfluid density in the BCS-BEC crossover
52
as one goes from the BCS limit (where Fermi BCS quasiparticles dominate the thermodynamics) to the BEC limit (where the Bose collective modes–the Bogoliubov-Popov excitations–
dominate the thermodynamics). Clearly, an obvious requirement for any calculation of ρs
in the BCS-BEC crossover is that it correctly reproduces the well-known expressions for the
normal fluid density ρn = ρ − ρs in both the BCS and BEC limits (as noted in Section 4.1,
the usual definitions in the literature involve the normal fluid density rather than the superfluid density). As discussed by Lifshitz and Pitaevskii [88], in a weakly interacting superfluid
Fermi gas (i.e., the BCS limit of the BCS-BEC crossover), ρn is given by
2
ρn =
3m
Z
∂f (Ek )
d3 k 2
.
k −
(2π)3
∂Ek
(4.1)
Here Ek is the spectrum of the BCS quasiparticles, given in (3.29), and f (Ek ) is the Fermi
thermal distribution function. In contrast, for a superfluid dominated by Bose excitations
(such as 4 He and the BEC limit of the BCS-BEC crossover), the normal fluid density is given
by
1
ρn =
3M
Z
∂nB (ωq )
d3 q 2
.
q −
(2π)3
∂ωq
(4.2)
Here ωq is the spectrum of Bose excitations and nB (ω) = (eβω − 1)−1 is the Bose distribution function. In these equations, m and M are the masses of the fermions and bosons,
respectively.
4.1
Definition of the superfluid density
The defining feature of superfluid flow is that it is irrotational, ∇ × vs (r, t) = 0. This
naturally leads to the classic “rotating-bucket” definition of the superfluid density: if a
bucket of superfluid is slowly rotated, the superfluid component must remain stationary
since it cannot support rotational flow (neglecting the possibility of vortex nucleation). On
4. The superfluid density in the BCS-BEC crossover
53
the other hand, the interaction of the viscous normal component with the bucket walls will
quickly lead to an equilibrium state in which the normal component rotates with the same
angular velocity as the bucket. These simple considerations formally lead to the definition of
ρs and ρn in terms of the response of the superfluid to a small transverse probe (Nozières and
Pines [66]). In particular, the current response function (that describes how a small force
induces a change in the current) can naturally be divided into transverse and longitudinal
components. The walls in the rotating bucket experiment exert a transverse (Coriolis) force
on the fluid. Since it is the normal fluid that gets dragged along by the rotating walls,
linear response theory gives us the result that the normal fluid density ρn is given by the
long-wavelength limit1 of the transverse current-current correlation function (see Sec. 4.4 in
Nozières and Pines [66]; Griffin [86]).
While we could calculate the transverse current-current correlation function directly using some approximation,2 it is simpler within our functional-integral approach to impose a
velocity field (phase) on the condensate order parameter and calculate the resulting change
in the free energy (see Secs. 5.1 and 5.2 in Nozières and Pines [66]; Fisher et al. [85]), which
can then be related to the superfluid density ρs . We emphasize that it can be shown that
the definition of the superfluid density obtained in this way is equivalent to the standard
definition involving the current-current correlation function (see, for instance, Appendix A
of Taylor, Griffin, Fukushima, and Ohashi [56]).
To impose a current in our superfluid, one can apply a “phase twist” (Fisher et al. [85])
to the order parameter ∆(x):
∆(x) → ∆(x)eiQ·r .
1
(4.3)
Such that the wavelength is much larger than the coherence length (Nozières and Pines [66]).
In fact, it is not obvious which terms should be included in a direct diagrammatic evaluation of the
current-current correlation function in the BCS-BEC crossover. See Andrenacci, Pieri, and Strinati [87].
2
54
4. The superfluid density in the BCS-BEC crossover
The superfluid velocity vs associated with this imposed phase twist is
vs =
Q
,
M
(4.4)
where M = 2m is the bound state (Cooper-pair) mass. Treating Q as small, the superfluid
density can be obtained from the lowest-order change in the free energy of the system (F =
Ω + µN) due to the added kinetic energy of the imposed superfluid flow (Fisher et al. [85]).
This extra kinetic energy is
Q2
∆F = F (Q) − F (0) ≈
2
∂ 2 F (Q)
∂Q2
Q→0
1
≡ ρs mvs2 .
2
(4.5)
This expression gives the change in kinetic energy as a result of imposing a phase twist
(i.e., a velocity) on the broken symmetry order parameter. Physically, the superfluid density
ρs represents the density of atoms “dragged” along with the condensate (described by ∆0 )
when the condensate is given a finite velocity vs .3 Rearranging (4.5), we obtain the following
expression for the superfluid density:4
ρs ≡ 4m
∂ 2 F (Q)
∂Q2
.
(4.6)
Q→0
For a fixed total number of fermions n, we have
∂2F
∂Q2
=
Q→0
∂2Ω
∂2µ
+
n
.
∂Q2
∂Q2
(4.7)
Microscopically, the thermodynamic potential Ω of the current-carrying superfluid can be
expressed as a functional of the mean-field gap ∆0 , the chemical potential µ, and the phase
twist Q. In addition to an explicit Q-dependence, Ω also depends on the phase twist implicitly
through the gap ∆0 (Q) and the chemical potential µ(Q). Using this, we obtain
3
∂2Ω
∂Q2
Q→0
=
∂2Ω
∂Q2
∆0 ,µ
+
∂Ω
∂∆0
µ
∂ 2 ∆0
+
∂Q2
∂Ω
∂µ
∆0
∂2µ
.
∂Q2
(4.8)
For further discussion of this point, see Secs. 5.1 and 5.2 in Nozières and Pines [66].
Note that the superfluid density defined here is the superfluid number density and not the superfluid
mass density used in our discussion of two-fluid hydrodynamics in Chapter 2. As can be seen from (4.5),
ρs m is the total mass involved in the superfluid flow, with m being the Fermi atom mass.
4
55
4. The superfluid density in the BCS-BEC crossover
Here we have made use of the fact that the corrections to µ and ∆0 which are linear in Q
vanish when Q = 0: (∂µ/∂Q)Q→0 = (∂∆0 /∂Q)Q→0 = 0. The evaluation of the derivatives
at Q = 0 is left implicit on the right-hand side of (4.8). Using (4.8) in (4.7), we find
#
" 2 2
2 ∂ F
∂ Ω
∂Ω
∂2µ
∂ ∆0
∂Ω
+n
=
+
+
.
∂Q2 Q→0
∂Q2 ∆0 ,µ
∂∆0 µ ∂Q2
∂µ ∆0
∂Q2
(4.9)
Using the gap and number equations given by5 (∂Ω/∂∆0 )µ = 0 and n = −(∂Ω/∂µ)∆0 , we
see that all but the first term on the right-hand side of (4.9) vanish. Consequently, we find
2 2 ∂ F
∂ Ω
=
.
(4.10)
2
∂Q Q→0
∂Q2 ∆0 ,µ
Our final expression for the superfluid density in terms of Ω(Q) is
2
1 ∂ 2 Ω(vs )
∂ Ω(Q)
.
=
ρs = 4m
∂Q2
m
∂vs2
vs =0
Q→0
(4.11)
In (4.11) and elsewhere, the constancy of ∆0 and µ in taking derivatives with respect to Q
is left implicit. This formula is the basis for our discussion of ρs in the rest of this chapter.
4.2
The thermodynamic potential for a current-carrying
superfluid
The calculation of the thermodynamic potential Ω(Q) for a current-carrying superfluid is
identical to the calculation in Section 3.1, except that now the order-parameter includes the
phase-twist given by (4.3). The effective action for the current-carrying superfluid is given by
(3.20), with the phase twist applied to the order parameter that enters the Green’s function
G−1 . To remove the phase from the order parameter, we apply the unitary transformation
G̃−1 ≡ U−1 G−1 U (see, for instance, Eckern, Schön, and Ambegaokar [89] and Stone [90]),
where we take
U≡
5
eiQ·r/2
0
0
e−iQ·r/2
.
(4.12)
Note that these are the exact gap and number equations (i.e., no approximation has been made yet for
Ω), in contrast to the approximate gap and number equations used in our discussion of the Gaussian theory
in Sec. 3.2. As discussed in Appendix B, the exact number equation leaves ∆0 fixed [see (B.4) and (B.5)].
56
4. The superfluid density in the BCS-BEC crossover
Owing to the invariance of Tr ln[−G−1 ] under the action of this unitary transformation of
G−1 , the effective action with a phase-twisted order parameter is given by [compare with
(3.20)]
∗
Seff [∆, ∆ , Q] =
Z
β
dτ
0
Z
dr
|∆(x)|2
− Tr ln[−G̃−1 ],
U
(4.13)
where (p̂ ≡ −i∇)
−1
′
G̃ (x, x ) =
−∂τ −
(p̂−Q/2)2
2m
∗
+µ
∆(x)
−∂τ +
∆ (x)
(p̂+Q/2)2
2m
−µ
!
δ(x − x′ ).
(4.14)
In momentum-frequency space, the mean-field (denoted by the subscript “0”) NambuGorkov BCS Green’s function G̃0 (k) for the current-carrying BCS superfluid is defined by
its inverse,
G̃−1
0 (k) =
(iωn −
k·Q −iωn 0+
)e
m
∆0
− (ξk +
Q2
)
2m
(iωn −
∆0
k·Q iωn 0+
)e
m
+ (ξk +
Q2
)
2m
!
. (4.15)
Inverting this, we find [compare with (3.28)]
G̃0 (k) =
1
+
iω̃n eiωn 0 + ξ˜k
−∆0
−iωn 0+
−∆0
iω̃n e
− ξ˜k
(iω̃n )2 − Ẽk2
q
Here we have defined Ẽk ≡ ξ˜k2 + ∆20 , where
Q2
ξ˜k ≡ ξk +
,
2m
.
(4.16)
(4.17)
and the Doppler-shifted Matsubara frequency as
iω̃ ≡ iω +
k·Q
.
m
(4.18)
An expression for the thermodynamic potential Ω(Q) of a current-carrying superfluid
can now be obtained by expanding the action in (4.13) up to quadratic order in the pairing
fluctuations and doing the Gaussian integration as in Chapter 3. Following the same steps
4. The superfluid density in the BCS-BEC crossover
57
leading to (3.41), the thermodynamic potential Ω(Q)) for a current-carrying superfluid is
given by
Ω(Q) =
∆2
1 X
1X
tr ln[−G̃−1
(k)]
+
ln detM̃(q)
−
0
U0
β
2β q
k
≡ ΩF (Q) + ΩB (Q).
(4.19)
Here the matrix elements of the inverse 2 × 2 matrix pair fluctuation propagator M̃ for a
current-carrying superfluid are now given by
1X
1
M̃22 (−q)
M̃11 (q)
G̃0,11 (k + q)G̃0,22 (k)
+
=
=
β
β
U0 β k
(4.20)
and
M̃12 (q)
1X
M̃21 (q)
G̃0,12 (k + q)G̃0,12 (k).
=
=
β
β
β
(4.21)
k
Equations (4.19)-(4.21) will now be used to calculate ρs in (4.11).
We note that the values of the microscopic parameters ∆0 (Q) and µ(Q) for a currentcarrying superfluid should be obtained by self-consistently solving the gap equation and
number equations derived from the thermodynamic potential in (4.19). These are
∂Ω(Q)
∂∆0
=0
(4.22)
µ
and
n=−
Ω(Q)
∂µ
,
(4.23)
∆0
where Ω(Q) is given by (4.19). Recall that in our expression for the superfluid density
in (4.11), the derivative with respect to Q leaves ∆0 and µ fixed. Thus, we only require
the values of these quantities in the current-free state, as given by (∂Ω(0)/∂∆0 )µ = 0 and
n = −(Ω(0)/∂µ)∆0 .
58
4. The superfluid density in the BCS-BEC crossover
4.3
Superfluid density in the BCS-BEC crossover
In this section, we evaluate ρs using the NSR expression in (4.19). From (4.16) and (4.19),
one sees that the thermodynamic potential for a superfluid with a finite superfluid velocity
vs = Q/M is equivalent to the thermodynamic potential for a current-free superfluid [given
by (3.28) and (3.41)], but where the chemical potential and Matsubara frequencies are now
Doppler-shifted (see Stone [90]):
µ → µ − Q2 /8m ≡ µ̃,
(4.24)
iωn → iωn − k · Q/2m ≡ iω̃n ,
(4.25)
iνm → iνm − q · Q/2m ≡ iν̃m .
(4.26)
The physics of (4.24) is more transparent when it is written as an expression for the chemical
potential of pairs in the presence of a supercurrent: µB = 2µ̃ = 2µ − Q2 /2M. The shift in
this chemical potential is simply the kinetic energy per Cooper pair, Q2 /2M.
Considering separately the effects of the shifts to the chemical potential and the Matsubara frequencies, we can write the second-order derivative of Ω with respect to Q (keeping ∆0
and µ fixed) as
∂2Ω
=
∂Q2
2 ∂2Ω
∂ Ω
+
∂ µ̃∂Q
∂Q2 µ̃
2 Q ∂2Ω
∂ Ω
1 ∂Ω
.
−
+
= −
4m ∂ µ̃ 2m ∂ µ̃∂Q
∂Q2 µ̃
∂ 2 µ̃
∂Q2
∂Ω
+2
∂ µ̃
∂ µ̃
∂Q
(4.27)
Evaluated at Q = 0, the middle term in (4.27) vanishes. Using this in (4.11), we obtain
ρs
∂Ω
= −
∂ µ̃
= n + 4m
+ 4m
Q→0
2
∂ Ω
∂Q2
µ̃,Q→0
∂2Ω
∂Q2
.
µ̃,Q→0
(4.28)
59
4. The superfluid density in the BCS-BEC crossover
In the last line, we have made use of the number equation
∂Ω
n=−
∂ µ̃
∂Ω
=−
∂µ
µ,∆0 ,Q→0
.
(4.29)
∆0
Since n ≡ ρs + ρn , (4.28) gives us an explicit expression for the normal fluid density:
ρn = −4m
∂2Ω
∂Q2
.
(4.30)
µ̃,Q→0
Carrying out the summation over Fermi Matsubara frequencies in (4.19), the mean-field
BCS quasiparticle contribution to the thermodynamic potential in the presence of a current
is
2X h
i
∆20 X ˜
−β(k·Q/2m+Ẽk )
ξk − Ẽk −
.
ln 1 + e
+
ΩF (Q) =
U0
β k
k
(4.31)
Recall that the single-particle quasiparticle energies in the presence of a current are Ẽk =
q
ξ˜k2 + ∆20 , with ξ˜k ≡ k2 /2m − µ̃ and µ̃ is the Doppler-shifted chemical potential in (4.24).
Summing over the fermion Matsubara frequencies in (4.20) and (4.21), the matrix ele-
ments of the inverse matrix propagator for pair fluctuations in the current-carrying superfluid
are given by
X
M̃11 (q)
1
M̃22 (−q)
+
=
=
+
eiνm 0
β
β
U0
k
+
+ fk− − fk+q
+
+ fk+ − fk+q
−
+ fk− − fk+q
−
fk+ − fk+q
u2k u2k+q
2
vk2 vk+q
iνm − q · Q/2m + Ẽk + Ẽk+q
iνm − q · Q/2m − Ẽk − Ẽk+q
vk2 u2k+q
iνm − q · Q/2m + Ẽk − Ẽk+q
2
u2k vk+q
iνm − q · Q/2m − Ẽk + Ẽk+q
and
M̃21 (q) X
uk vk uk+q vk+q
M̃12 (q)
−
=
=
fk+q
− fk+
β
β
iνm − q · Q/2m + Ẽk + Ẽk+q
k
u
v
u
k k k+q vk+q
+
+ fk+q
− fk−
iνm − q · Q/2m − Ẽk − Ẽk+q
(4.32)
60
4. The superfluid density in the BCS-BEC crossover
uk vk uk+q vk+q
iνm − q · Q/2m + Ẽk − Ẽk+q
uk vk uk+q vk+q
−
−
+ fk − fk+q
.
iνm − q · Q/2m − Ẽk + Ẽk+q
+
+ fk+ − fk+q
(4.33)
Here
fp± ≡ f p · Q/M ± Ẽp
(4.34)
q
q
˜
are the Fermi distribution functions, while up = (1 + ξp /Ẽp )/2 and vp = (1 − ξ˜p /Ẽp )/2
play the role of Bogoliubov coherence factors. Recall that the normal fluid density is evaluated at fixed µ̃ and consequently, the dependence of Ẽp on Q can be ignored in calculating ρn
in (4.30). Equations (4.32) and (4.33) reduce to (3.33) and (3.34) when vs = 0, as expected.
The Fermi distribution functions appearing in (4.32) and (4.33) involve Doppler-shifted
Fermi quasiparticle energies: p · Q/M ± Ẽp . The shift p · Q/M reflects the fact that
additional Fermi quasiparticles will be excited when the superfluid velocity is finite since
thermal equilibrium is defined with respect to the stationary lab frame (see Stone [90]).
Using these results in the thermodynamic potential in (4.19), the normal fluid density ρn
is given by the sum of Fermi quasiparticle and Bose collective mode contributions:
ρn ≡ ρFn + ρB
n,
(4.35)
where (Q̂ ≡ Q/|Q|)
ρFn
mX
≡ −
β k
k · Q̂
m
!2
tr[G0 (k)G0 (k)]
(4.36)
and
ρB
n ≡ −
2m X
β
q
1
detM̃

2 detM̃
2
∂ detM̃
∂Q2
!
µ̃
−
∂ detM̃
∂Q
!2 
µ̃

.
(4.37)
Q→0
We see that the Bose contribution ρB
n can be given in terms of the determinant of the inverse
fluctuation propagator detM̃, the zeros of which give the spectrum of the Bose collective
4. The superfluid density in the BCS-BEC crossover
61
modes. The simplicity of this expression for ρB
n is lost when expanded in terms of products
of current-free BCS Green’s function (as shown in Appendix B of Taylor, Griffin, Fukushima,
and Ohashi [56]).
The normal fluid density ρFn due to Fermi BCS quasiparticles given in (4.36) is readily
identified as the long-wavelength, static limit (q → 0) of the BCS current-current correlation
function (multiplied by −m). Carrying out the Matsubara frequency sum, (4.36) reduces to
ρFn
Z
d3 k
∂f (Ek )
2
(k · Q̂)2
= −
3
m
(2π)
∂Ek
Z
3
2
dk 2
∂f (Ek )
=
.
k −
3m
(2π)3
∂Ek
(4.38)
This is the well-known Landau formula for the normal fluid density of a uniform weakcoupling BCS superfluid [see (4.1)], arising from thermally-excited Fermi BCS quasiparticles
(Lifshitz and Pitaevskii [88]). In our case, it is valid for the entire BCS-BEC crossover, taking
into account that the quasiparticle spectrum depends on ∆0 and µ which are renormalized
from their mean-field BCS values by the inclusion of the effects of Bose fluctuations (see
Section 3.2). Of course, the Landau formula in (4.38) is also obtained by using the expression
for ΩF (Q) given by (4.31) in (4.30).
In the BCS limit, it is well known (see pg. 163 in Lifshitz and Pitaevskii [88]) that (4.38)
can be evaluated to give the normal fluid density
ρFn
=n
2π∆0
kB T
1/2
e−∆0 /kB T
(4.39)
in the limit of low temperatures (kB T ≪ ∆0 ). In contrast, in the strong-coupling BEC limit
given by (3.49), the fermions form bound pairs with a large binding energy: Eb = −1/ma2s .
As a result of this large binding energy, Fermi quasiparticle excitations arising from the
breakup of pairs are completely frozen out over the experimentally relevant temperature
4. The superfluid density in the BCS-BEC crossover
62
scale kB T ∼ kB Tc ≪ |Eb |. Using this fact, in this BEC region, (4.38) gives
ρFn
1
=
3
mkB T
2π 3
3/2
e−|Eb |/2kB T .
(4.40)
This result shows (as expected) that the Fermi contribution ρFn to the total normal fluid
density vanishes in the BEC limit of tightly-bound pairs (|Eb | ≫ kB T ).
Equation (4.37) describes the contributions to the normal fluid density ρB
n from fluctuations Λ(x) of the Bose pairing field. In general, the Bose pair excitations are damped at finite
temperatures due to coupling to the continuum of BCS quasiparticle states. As a result, the
Bose fluctuations will have a finite lifetime and ρB
n cannot in general be reduced to the
simple Landau formula in (4.2) involving Bose excitations. In the BEC limit, however, the
pair binding energy becomes very large and hence the contribution of BCS quasiparticles are
strongly suppressed. This is explicitly shown by the result in (4.40). As a result, damping of
the collective modes of the molecular Bose condensate by BCS quasiparticles will not occur.
In this limit, our expression for ρB
n in (4.37) does reduce to Landau’s formula in (4.2) for a
Bose superfluid. In the next section, we show this explicitly.
4.4
The normal fluid density in the BEC limit
Deep in the BEC region, the only contribution to the normal fluid density comes from the
contribution ρB
n from Bose collective modes given in (4.37). Based on the simplifications
discussed in Section 3.3.2 that arise in the BEC limit, we can show how the normal fluid
density reduces to the Landau equation in (4.2).
In the low-energy regime ωq ≪ |µ̃|, the spectrum of Bose pairing excitations in the BEC
limit is given by the Bogoliubov expression in (3.67). To extract the contribution of these
6 +
modes to the normal fluid density ρB
= 0 and f − = 1 in (4.20) and (4.21) and
n , we set f
6
In the BEC limit, ∆0 , Q2 /8m ≪ |µ| and, in the superfluid phase (T < Tc ), kB T ≪ |µ|, allowing us to
set f + = 0 and f − = 1 even at finite temperatures.
4. The superfluid density in the BCS-BEC crossover
63
then expand the inverse fluctuation propagator matrix elements in powers of q analogous to
the case of the current-free inverse propagator in (3.52) and (3.53). For the current-carrying
superfluid, this gives
M̃11 (q)
≃ A + B|q|2 + C(iνm − q · Q/2m)2 + D(iνm − q · Q/2m),
β
(4.41)
M̃12 (q)
≃ A + F |q|2 + G(iνm − q · Q/2m)2 .
β
(4.42)
and
Apart from the shift to the chemical potential given by (4.24), the expansion coefficients in
(4.41) and (4.42) are the same as in the Q = 0 case given by (3.56)-(3.61), but with the
replacements Ek → Ẽk and ξk → ξ˜k . As we did in arriving at (3.56)-(3.61), the mean field
gap equation is used to eliminate 1/U0 from M̃11 (q). For the current-carrying superfluid, the
gap equation (∂ΩF (Q)/∂∆0 ) = 0 becomes
1X
∆0
G̃0,12 (k).
=
U0
β k
(4.43)
In the strong-coupling BEC limit, (4.43) can be solved analytically, and we obtain µ̃ ≡
µ − Q2 /8m = −1/(2ma2s ). Thus, in the BEC limit where ∆0 , Q2 /2M ≪ |µ̃|, we can further
expand the integrands in the coefficients A, B, C, D, F, and G in (4.41) and (4.42) in powers
of ∆0 /|µ̃|. To leading order, using |µ̃| ≃ (2ma2s )−1 , these coefficients reduce to the values
for the current-free case given by (3.62)-(3.65). C, F , and G vanish as before. With these
coefficients, we find
2
q·Q
detM̃(q, iνm ) = 2ABq + B q − D iνm −
.
M
2
2 4
2
The Bose excitations are given by the solution of detM̃(q, ωq ) = 0, namely
s
r
2 2
2
q
2AB 2 B 4
UM nc (T ) 2
q + 2 q = q · vs +
q +
.
ωq (vs ) = q · vs +
2
D
D
M
2M
(4.44)
(4.45)
64
4. The superfluid density in the BCS-BEC crossover
This is identical to the Bogoliubov-Popov expression for the collective mode spectrum given
by (3.67), except that it is now Doppler-shifted by q · vs due to the presence of the superfluid
flow velocity.
Using the expression for detM̃(q, iν̃m ) in (4.44), it is straightforward to evaluate ρB
n in
(4.37). Making use of (3.67), (4.44) reduces to
detM̃(q, iν̃m ) = −D 2 (iνm − q · Q/M)2 − ωq2 (vs = 0) .
Using this, we find

1
2 detM̃
detM̃
2
∂ detM̃
∂Q2
!
µ̃
!2 
∂ detM̃ 
∂Q
−
µ̃


1
4
2
2
2 2D (iνm ) − ωq
D 4 (iνm )2 − ωq2 
(4.46)
=
Q→0
q · Q̂
M
!2
− 4D 4(iνm )2
!2 
q · Q̂ 
M
,(4.47)

where ωq = ωq (Q = 0) is the usual Bogoliubov-Popov excitation energy with vs = 0. Using
(4.47) in (4.37), we obtain
M X
ρB
=
n
β q,iν
m
q · Q̂
M
!2
2(iνm )2 + 2ωq2
.
(iνm − ωq )2 (iνm + ωq )2
(4.48)
To bring out the physics, (4.48) can also be written in terms of the transverse current
correlation function for a dilute Bose gas of interacting molecules (following Fetter [91]),
!2
X q · Q̂
M
ρB
=
tr [D(q, iνm )D(q, iνm )] ,
(4.49)
n
β q,iν
M
m
where
1
D(q, iνm ) =
(iνm )2 − ωq2
iνm + ωq
0
0
iνm − ωq
(4.50)
is the 2 × 2 Bose propagator describing the Bogoliubov excitations.
Carrying out the Bose frequency sum in (4.49), we find the expected Landau formula in
terms of Bogoliubov excitations,
ρB
n
2
= −
M
Z
d3 q
∂nB (ωq )
(q · Q̂)2
3
(2π)
∂ωq
65
4. The superfluid density in the BCS-BEC crossover
2
=
3M
Z
∂nB (ωq )
d3 q 2
.
q −
(2π)3
∂ωq
(4.51)
We recall from the definition in (4.11) that ρs (and hence ρn ) always refers to the number
of fermions. Thus, (4.51) is twice the usual expression [see (4.2)], reflecting the fact that
it is counting the number of fermions (not the number of bosons) associated with a normal
fluid composed of Bogoliubov excitations of a molecular BEC. The result in (4.51), valid
in the BEC limit, has also been recently derived by Andrenacci, Pieri, and Strinati [87]
using a diagrammatic approach (for further discussion, see Appendix B of Taylor, Griffin,
Fukushima, and Ohashi [56]).
As we discuss Appendix B, since we have used the mean field gap equation in arriving at
the expression for ρs in the BEC limit given by (4.51), we can also retain the δn∆ correction
term in (3.44) that leads to the important renormalization of the molecular s-wave scattering
length, from aM = 2as to aM ≃ 0.6as (Petrov et al. [79]). To be consistent, one must include
the analogous terms in (4.10) and additional terms will be generated in our definition of the
superfluid density given by (4.11):
ρs → ρs + 4m
∂ΩB
∂∆0
µ
∂ 2 ∆0
−
∂Q2
∂∆0
∂µ
∂2µ
.
∂Q2
(4.52)
In the extreme BEC limit, however, ∂ 2 ∆0 /∂Q2 → 0 as the BCS quasiparticles become frozen
out, and ∂ 2 µ/∂Q2 → 1/4m [as shown below (4.43)]. As a result, we obtain ρs → ρs + δn∆ ,
where δn∆ = −(∂ΩB /∂∆0 )µ (∂∆0 /∂µ) is the correction to the number equation given by
(3.45). Using this new expression in (4.28), δn∆ just adds another contribution to the total
density n and (4.30) remains unchanged in the BEC limit. Thus, even if we retain the extra
terms in (4.10) and (3.44) that lead to the renormalization of the molecular scattering length
aM , in the BEC limit ρn is still given by (4.30). Consequently, our result in (4.51) still holds
in the BEC limit when we include the higher-order corrections, except that aM is now the
66
4. The superfluid density in the BCS-BEC crossover
correct renormalized value aM ≃ 0.6as .
A similar analysis of the bosonic contribution ρB
n to the normal fluid in the BCS limit
can be carried out. If we limit ourselves to the low temperature region where kB T ≪ ∆0 , a
small q expansion analogous to (3.52) and (3.53) gives (Engelbrecht et al. [55])
v q 2 F
detM̃(q, iν̃m ) ∝ (iνm − q · Q/M) −
3
2
(4.53)
in place of (4.46). We emphasize, however, that the expansion leading to (4.53) leaves out
terms responsible for Landau damping and consequently, unlike its analogue in the BEC limit
given by (4.46), this expression is only valid at very low temperatures. Equation (4.53) is
just the propagator for undamped Bogoliubov-Anderson (BA) modes with a Doppler-shifted
√
dispersion ωq (vs ) = q · vs + (vF / 3)q. As a first approximation, we can neglect the effect of
Landau damping on ωq at finite temperatures, but introduce a sharp frequency cutoff at 2∆0 ,
where the BA collective mode enters the two-particle continuum. This gives an expression
√
for ρB
n in the BCS limit for BA phonons which is identical to (4.51) with ωq = (vF / 3)q, but
√
now the q-integration is limited to the wavevector region such that ωq = (vF / 3)q < 2∆0 .
However, as we noted at the beginning of this section, the finite lifetime of Bose collective
modes at finite temperature means that a Landau expression like (4.51) is never really valid
outside the extreme BEC limit, except in the limit of very low temperatures (where Landau
damping can be ignored). For the Bogoliubov-Anderson phonons, (4.51) gives (see pg. 92
in Lifshitz and Pitaevskii [88])
√ 4
4
3π
kB T
3
B
.
ρn = n
40
ǫF
(4.54)
In the BCS region, one has (kB Tc /ǫF ) ≪ 1, and the normal fluid contribution ρB
n from the
BA phonons is seen to be negligible compared to ρFn given by (4.39).
4. The superfluid density in the BCS-BEC crossover
67
Figure 4.1: Calculated superfluid density ρs in the BCS-BEC crossover. The self-consistent
solutions for ∆0 and µ shown in Fig. 3.1 are used. The dashed line shows Tc [see Fig. 3.2(a)].
4.5
Numerical results
Having discussed the BCS and BEC limits of our expressions for the normal fluid density
and shown analytically that they reproduce the well-known Landau formulae in those limits,
we now give the results of a full numerical calculation of ρs through the BCS-BEC crossover.
The expression for the superfluid density obtained from (4.36) and (4.37) has been evaluated
by Naoki Fukushima and Yoji Ohashi. The results are shown in Figs. 4.1 and 4.2 (see also
Fukushima, Ohashi, Taylor, and Griffin [57]).
Figure 4.1 shows the superfluid fraction ρs /n as a function of temperature through the
BCS-BEC crossover, from the BCS region [(kF aS )−1 = −2] into the BEC region [(kF aS )−1 =
2]. The “bend-over” in ρs close to unitarity and on the BEC side of resonance near Tc is due
to the discontinuity in ∆0 (T → Tc ) discussed in Section 3.2 (see Fig. 3.1). The superfluid
fraction on the BCS side of resonance is well described by the Landau formula in (4.38) until
68
4. The superfluid density in the BCS-BEC crossover
(kFas)-1-1= -2
(kFas)-1 = 0
(kFas) = 2
BCS
BEC
1
ρs / n
0.8
0.6
0.4
0.2
0
0
0.2 0.4 0.6 0.8
1
1.2 1.4
T / Tc
Figure 4.2: Superfluid density ρs as a function of temperature in the BCS region (solid
circles), unitarity limit (solid triangles) and BEC regime (open circles). “BCS” labels the
mean-field BCS result, given by ρs = n − ρFn with µ = ǫF and ρFn given by (4.38). “BEC”
B
gives ρs = n − ρB
n for a dilute Bose gas where ρn is given by (4.51), evaluated using the
excitation spectrum in (3.67).
fairly close to unitarity. It is only for (kF as )−1 & −0.5 that the fluctuation contribution from
(4.37) becomes significant.
Figure 4.2 shows the superfluid density as a function of temperature for three values
of the interaction parameter (kF aS )−1 corresponding to the BCS limit (kF aS )−1 = −2,
unitarity (kF aS )−1 = 0, and the BEC limit (kF aS )−1 = 2. These results emphasize that
the origin of the unphysical bend-over lies with the discontinuity in ∆0 (T ). In the BEC
region (kF as )−1 = 2, the superfluid density calculated from (4.36) and (4.37) is seen to agree
perfectly with the Landau expression in (4.51), as expected from our analysis in Section 4.4.
Landau’s formula is a straightforward integral expression and is not ill-behaved. However,
(4.51) is evaluated using the excitation spectrum in (3.67) which depends on ∆0 (T ). This
makes it clear that the discontinuity in ∆0 (T ) is responsible for the bend-over seen in the
superfluid density.
4. The superfluid density in the BCS-BEC crossover
69
Figure 4.3: Plot of the different superfluid density fractions at unitarity used in our calculations of the two-fluid hydrodynamic modes. The “bare” NSR data from Fig. 4.2 is shown by
the circles (The open circles denote the NSR data used in the curve fit). The squares give
the results of a path integral Monte-Carlo calculation of the superfluid density for N = 20
fermions (Akkineni, Ceperley, and Trivedi [92]), and the dashed line is the mean-field BCS
superfluid density scaled so that it vanishes at Tc = 0.225TF .
Even though the discontinuity in ρs (T ) for a uniform superfluid is only problematic close
to the transition temperature, in a trapped gas there will always be a region towards the
edge of the gas where the density is sufficiently small that the gas there will be in the normal
phase. In other words, the effective local temperature at the edge of the trap is above the
superfluid transition temperature. Consequently, even for relatively low temperatures, a
calculation of the superfluid density profile ρs (r) that will be used in our solution of the
Landau two-fluid equations requires us to have accurate knowledge of ρs close to Tc .
An improved calculation of the superfluid density close to Tc is an extremely difficult
problem, well beyond the scope of this thesis and our solution will be to use trial expressions
for the superfluid density at unitarity, where we carry out our calculations of the two-fluid
modes in Chapter 6. In particular, we use a) the low temperature fluctuation data in Fig. 4.2
4. The superfluid density in the BCS-BEC crossover
70
combined with a (Tc − T )2/3 curve fit7 for the high temperature data (T > 0.18TF ) and b)
the mean-field BCS result obtained from (4.36). In both cases, the data is scaled so that
the superfluid density vanishes at the transition temperature Tc = 0.225TF for a uniform
superfluid at unitarity obtained from the NSR fluctuation theory (based on (3.42) and (3.44);
see also Hu, Liu, and Drummond [49]). The results are shown in Fig. 4.3.
For comparison, we also plot the data from a restricted path-integral Monte-Carlo (RPIMC) calculation of Akkineni, Ceperley, and Trivedi [92] in Fig. 4.3. Using this data, they
predicted a superfluid transition temperature of Tc ≃ 0.25TF . Note that the R-PIMC data
does not give the correct behaviour of ρs close to Tc since Fermi Monte-Carlo calculations
are restricted to small particle numbers (the data plotted is for N = 20 particles). The fact
that their ρs does not go to zero at this temperature is due to finite size effects.
We emphasize that by scaling the mean-field BCS data for ρs , we are incorporating much
of the physics from the fluctuation theory since the scaled ρs (T ) vanishes at the fluctuation
theory result for Tc , which is much smaller than the mean-field result Tc ≃ 0.5TF .
In our calculation of the Landau two-fluid mode frequencies at unitarity reported in
Chapter 6, we use the fitted NSR fluctuation and scaled mean-field BCS superfluid densities
plotted in Fig. 4.3.
7
It is well known that for a superfluid with a two-component order parameter, close to the transition
temperature, ρs ∝ (Tc − T )2/3 . See, for instance, Sec. 28 in Lifshitz and Pitaevskii [88] and pg. 91 in
Griffin [86].
Chapter 5
Variational formulation of the Landau
two-fluid equations
In the collisional hydrodynamic region at finite temperatures, the collective modes of Bose
superfluids are described by the Landau two-fluid hydrodynamic equations reviewed in Chapter 2. In uniform systems, it is straightforward to solve the linearized Landau two-fluid equations since the solutions are just plane waves. In contrast, for trapped gases, it is not easy to
solve these differential equations since the coefficients are position-dependent thermodynamic
functions. Building on the approach initiated by Zaremba, Nikuni, and Griffin in 1999 [47]
for trapped atomic Bose gases, in this chapter we present a new variational formulation of
two-fluid hydrodynamic modes, based on the work of Zilsel [46] developed for superfluid
helium in 1950. Assuming a variational ansatz for the superfluid and normal fluid velocities
based on exact solutions at T = 0 and T > Tc , the frequencies of the hydrodynamic modes
are given by solutions of a small number of coupled algebraic equations, with constants only
involving spatial integrals over various equilibrium thermodynamic derivatives.
As emphasized by Shenoy and Ho [44], the primary difficulty in solving the two-fluid
differential equations for a trapped gas lies with fact that the superfluid and normal fluid
densities vary rapidly close to their interface. This makes a reliable brute-force calculation
extremely tricky and some approximation was required to deal with this problem [44] (for
71
5. Variational formulation of the Landau two-fluid equations
72
further discussion see Sec. 7B of Zaremba et al. [47]). We recall that the two-fluid equations
have also been solved for a two-component Fermi gas at unitarity by He et al. [45], although
few details of their numerical work are given. Since the numerical solutions of the two-fluid
equations are sensitive to rapid spatial changes in the superfluid density profile, one might
expect that the use of a local density approximation (LDA) for thermodynamic quantities
(as done in Refs. [44, 45] as well as Chapter 6 in this thesis) as well as the calculation of the
superfluid density for a uniform system could represent significant sources of error. Indeed,
as we show in Chapter 6, even our variational results for the two-fluid hydrodynamic modes
at unitarity are sensitive to small changes in the superfluid density profile. However, the
variational approach that we introduce in Section 5.2 is based on a Rayleigh-Ritz expansion
and consequently, has the advantage that it provides a rigorous upper bound on the two-fluid
mode frequencies for a given set of thermodynamic data, including the superfluid density.
Thus, our variational approach constrains the errors introduced by using approximate density
profiles.
5.1
Zilsel’s variational formulation
Zilsel’s variational principle [46] makes use of a Lagrangian density of the form
L = T − U,
(5.1)
where T is the kinetic energy density of the fluid and U is the internal energy density. In this
approach, we need to formulate the thermodynamics in terms of the internal energy density
U, rather than the total energy E = T + U, as is normally done in two-fluid hydrodynamics
(see, for example, pg. 521 in Landau and Lifshitz [60]). Our approach will be to obtain the
internal energy density U from the total energy density used by Landau and Khalatnikov
(LK) to derive the Landau two-fluid equations introduced in Section 2.2. For a trapped two-
73
5. Variational formulation of the Landau two-fluid equations
fluid system, we define the total energy density E as the sum of the kinetic and potential
energy densities,
1
1
E = ρs vs2 + ρn vn2 + U + ρVext ,
2
2
(5.2)
where ρn and ρs are the normal fluid and superfluid densities, vn and vs are the normal fluid
and superfluid velocities, and ρ = ρs + ρn is the total density. Vext is the harmonic trapping
potential defined in (2.9). This is included in our generalized version of the LK results.
Since two-fluid hydrodynamics describes a system in local equilibrium, all thermodynamic
quantities we discuss will be functions of position and time. However, this dependence is
usually left implicit. Even in static equilibrium, most thermodynamic quantities will be
position dependent in the presence of a trapping potential.
Following LK, the total energy density E0 (r, t) as measured in a frame of reference moving
with the local superfluid velocity vs is related to E by
1
E0 = E − ρvs2 − vs · [ρn (vn − vs )].
2
(5.3)
Using the LK identity for local equilibrium1
dE0 = T ds + µL dρ + (vn − vs ) · d [ρn (vn − vs )] ,
(5.4)
in conjunction with (5.2) and (5.3), we obtain the following identity for the internal energy
density:2
1
dU = (µL − Vext )dρ + T ds + (vn − vs )2 dρn .
2
(5.5)
We recall that s is the local entropy density. In similar fashion, the LK expression for the
pressure P ≡ ∂(E0 V )/∂V including the effects of a trapping can be recast in terms of the
1
Recall that µL = µ/m is the chemical potential per unit mass.
We have not considered the possibility that U also depends on the gradient of the superfluid density
∇ρs . Its inclusion would lead to additional terms in the variational principle and hence, to a modified form
of the Landau two-fluid equations. This approximation is equivalent to ignoring the “quantum pressure”
contribution to the energy. Geurst [93] discusses such contributions in his extension of Zilsel’s work.
2
5. Variational formulation of the Landau two-fluid equations
74
internal energy density, giving
1
P = −U − ρVext + T s + µL ρ + ρn (vn − vs )2 .
2
(5.6)
The two definitions given by (5.5) and (5.6) can be combined to give
ρdµL = dP − sdT − ρn (vn − vs ) · d(vn − vs ).
(5.7)
The three thermodynamic identities given by (5.5), (5.6), and (5.7) define all the local
equilibrium thermodynamic properties that we will require in developing our variational
approach. We note that (5.5) implies that
µL =
∂U
∂ρ
+ Vext
(5.8)
s,ρn
and
∂U
T =−
∂s
.
(5.9)
ρ,ρn
Equation (5.5) also tells us that the internal energy density is a function of the total density
ρ, the normal fluid density ρn , and the entropy density s:
U ≡ U(ρ, ρn , s).
(5.10)
Following Zilsel [46], the two-fluid equations can be obtained by equating to zero the
variations of the hydrodynamic action (S ≡
S =
Z
dr
Z
dt
R
drdt L) given by
i
1
(ρ − ρn )vs2 + ρn vn2 − U(ρ, ρn , s) − ρVext .
2
2
h1
(5.11)
Explicitly, the two-fluid equations result from the Euler-Lagrange equations
δS
= 0,
δx
(5.12)
5. Variational formulation of the Landau two-fluid equations
75
where x is any of the variables appearing in the action (e.g., ρn or vs ). This functional
derivative is defined by3
δS
∂
≡−
δx
∂t
∂L
∂ ẋ
−∇·
∂L
∂∇x
+
∂L
,
∂x
(5.13)
where the Lagrangian density L corresponding to (5.11) is
1
1
L = (ρ − ρn )vs2 + ρn vn2 − U(ρ, ρn , s) − ρVext .
2
2
(5.14)
In taking the variation of the action in (5.11), the variables vn , vs , ρ, ρn , and s will be treated
as independent. As an example, taking the variation of the action with respect to the density
ρ, the last term in (5.13) involves
∂U
≡
∂ρ
∂U
∂ρ
.
(5.15)
s,ρn
Zilsel first used the action in (5.11) in 1950 to derive Landau’s two-fluid equations in a
uniform superfluid in the non-dissipative limit (i.e., no transport coefficients which lead to
damping are included).
Two important conservation laws which are not incorporated into the action in (5.11) are
the conservation of mass and entropy. The variations of the action in (5.11) must be taken
subject to the constraints imposed by these two conservation laws. Following the approach
pioneered by Eckart [95] for classical electrodynamics, the constraints given by the mass and
entropy continuity equations [see (2.1) and (2.11)] can be incorporated into the variational
principle by introducing Lagrange multipliers φ and α (both dependent on r and t). In place
of (5.11), the new action is
S =
3
1
1
dr dt (ρ − ρn )vs2 + ρn vn2 − U(ρ, ρn , s) − ρVext
2
2
∂ρ
∂s
+φ
+ ∇ · [(ρ − ρn )vs + ρn vn ] + α
+ ∇ · (svn ) .
∂t
∂t
Z
Z
(5.16)
As discussed in usual textbook treatments of variational calculus (see, for example, Ref. [94]), the time
and spatial integration in (5.16) is done between two fixed points with the fluctuations of the variables
appearing in the action vanishing at both points.
76
5. Variational formulation of the Landau two-fluid equations
Setting the variation of the action given by (5.16) with respect to ρ, s, vs , and vn equal to
zero [using (5.13) with L given by the contents of the square brackets in (5.16)], and making
use of the thermodynamic identities implied by (5.5), we obtain the following relations:
1 2 ∂φ
δS
=
v −
− vs · ∇φ − µL = 0,
δρ
2 s
∂t
(5.17)
δS
∂α
=−
− vn · ∇α − T = 0,
δs
∂t
(5.18)
δS
= ρs (vs − ∇φ) = 0,
δvs
(5.19)
δS
= ρn (vn − ∇φ) − s∇α = 0.
δvn
(5.20)
and
Taking the variation of the action with respect to ρn and using (5.19), one recovers a thermodynamic identity already known from (5.5), namely
∂U
∂ρn
s,ρ
1
= (vn − vs )2 .
2
(5.21)
Taking the variation of the action with respect to φ and α recovers the two conservation
laws of two-fluid hydrodynamics, given by (2.1) and (2.11).
The results in (5.17)-(5.20) can be rearranged to yield useful expressions. From (5.17),
we find the important result
vs = ∇φ,
(5.22)
which means that the superfluid velocity vs is irrotational (i.e., ∇ × vs = 0). Using this
result for vs , (5.20) can be written as
ρn
(vn − vs ) = ∇α.
s
(5.23)
77
5. Variational formulation of the Landau two-fluid equations
Combining (5.23) with (5.19) gives
∂α
ρn
= −T − vn · (vn − vs ) .
∂t
s
(5.24)
Finally, using (5.24) in (5.20), we obtain
1 2
∂φ
= − µL + vs .
∂t
2
(5.25)
Taking the time derivative of (5.22) and the gradient of (5.25), the superfluid velocity satisfies
the equation of motion
1 2
∂vs
= −∇ µL + vs .
∂t
2
(5.26)
Thus we recover Landau’s equation for the superfluid velocity vs given in (2.12), as well as
the fact that vs is irrotational.
We can also use our results to write the equation of motion for vs in another way.
Rearranging (5.6) to obtain an equation for µL , and making use of the result
∇U =
∂U
∂ρ
∇ρ +
ρn ,s
= (µL − Vext )∇ρ +
∂U
∂ρn
ρ,s
∇ρn +
∂U
∂s
∇s,
ρ,ρn
1
(vn − vs )2 ∇ρn + T ∇s,
2
(5.27)
we obtain the expression
∇µL =
1
s
1 ρn ∇P − ∇T + ∇Vext −
∇ (vn − vs )2 .
ρ
ρ
2 ρ
(5.28)
Using this result in (5.26), the equation for the superfluid velocity can be rewritten in the
form
∂
s
1
1 ρn + vs · ∇ vs =
∇T − ∇P − ∇Vext +
∇ (vn − vs )2 .
∂t
ρ
ρ
2 ρ
Here, we have made use of the vector identity v · ∇v = (1/2)∇(v2 ) − v × (∇ × v).
(5.29)
5. Variational formulation of the Landau two-fluid equations
78
To derive an analogous equation to (5.29) for the velocity vn of the normal fluid, we take
the time-derivative of (5.23) and the gradient of (5.24). Then, using (2.1), (2.11), and (5.23),
one can show (after some labourious algebra) that the normal fluid velocity satisfies
ρs s
1
1 ρs ∂
+ vn · ∇ vn = −
∇T − ∇P − ∇Vext −
∇ (vn − vs )2
∂t
ρn ρ
ρ
2ρ
Γ
− (vn − vs ) ,
ρn
(5.30)
where a new “source function” Γ has been defined:
Γ≡
∂ρn
+ ∇ · (ρn vn ).
∂t
(5.31)
From the continuity equation given by (2.1), (5.31) implies that
∂ρs
+ ∇ · (ρs vs ) = −Γ.
∂t
(5.32)
Combining (5.29)-(5.32) to obtain an equation of motion for the current, j = ρs vs + ρn vn ,
the source terms of the superfluid and normal fluid components in (5.31) and (5.32) cancel
and we are left with
∂j
= −∇P − ρ∇Vext − ρs vs · ∇vs − ρn vn · ∇vn
∂t
−vs ∇ · (ρs vs ) − vn ∇ · (ρn vn ).
(5.33)
More familiarly, in component form, this equation can be written as
∂
∂Vext
∂ji
= −
[P δij + ρs vsi vsj + ρn vni vnj ] − ρ
,
∂t
∂xj
∂xi
(5.34)
where the index j is summed over. This is precisely Landau’s equation of motion for the
current, given by (2.2) and (2.8).
One can use the above results to reproduce the linearized two-fluid equations given in Section 2.2, generalized to include the effects of the external trapping potential Vext . Assuming
5. Variational formulation of the Landau two-fluid equations
79
that vs0 = vn0 = 0, linearizing (5.29) for the superfluid velocity gives
∂vs
1
∇P0
s0
∇δT − ∇δP + 2 δρ
=
∂t
ρ0
ρ0
ρ0
= −∇δµL .
(5.35)
This is the linearized Landau equation given in (2.19) for a uniform system. The effect of
the trapping potential enters (5.35) implicitly through the gradient ∇P0 of the equilibrium
pressure. In equilibrium, the chemical potential must be independent of position and hence,
∇µL0 = 0. Assuming that vn0 = vs0 = 0, (5.28) yields the well-known relation in trapped
gases,
∇Vext = −
1
∇P0 .
ρ0
(5.36)
Using this, (5.35) becomes
∂vs
s0
1
∇Vext
= ∇δT − ∇δP −
δρ.
∂t
ρ0
ρ0
ρ0
(5.37)
This shows explicitly how the external trapping potential enters into the linearized two-fluid
equation for the superfluid velocity.
In a similar manner, linearizing (5.30) for the normal fluid velocity gives
ρs0 s0
1
∇P0
∂vn
= −
∇δT − ∇δP + 2 δρ
∂t
ρn0 ρ0
ρ0
ρ0
s0
= −∇δµL −
∇δT.
ρn0
(5.38)
Using (5.36), we can write the equation of motion for the normal fluid velocity as
∂vn
1
∇Vext
ρs0 s0
∇δT − ∇δP −
δρ.
= −
∂t
ρn0 ρ0
ρ0
ρ0
(5.39)
Here we have made use of the fact that the source term Γ defined in (5.31) must vanish in
thermal equilibrium (Zaremba et al. [47]). The last term in (5.30) is second order in the
5. Variational formulation of the Landau two-fluid equations
80
fluctuations and thus vanishes in a linearized theory. We have also made use of the fact that
in equilibrium, ∇T0 = 0. Combining (5.37) and (5.39), we obtain
∂j
= −∇δP − δρ∇Vext .
∂t
(5.40)
This generalizes (2.17) to include a trapping potential Vext .
To summarize the discussion so far, equating to zero the variations of the action defined
in (5.16) has been shown to give (2.1), (2.11), (5.26), and (5.34). These are precisely the
Landau two-fluid equations in the non-dissipative limit, generalized to include a static external potential. This formulation of the two-fluid equations will be used in Section 5.2 to
derive an action for the two-fluid hydrodynamic modes in a trapped superfluid gas.
5.2
Action for linearized Landau two-fluid hydrodynamics
We have shown in Section 5.1 that the variation of the action in (5.16) with respect to ρ,
s, vs , vn , φ and α leads to the non-dissipative Landau two-fluid hydrodynamic equations
(Zilsel [46]). To determine the low-energy collective modes given by the solutions of the
linearized hydrodynamic equations, we could expand the action about the equilibrium values
of these variables. In discussing the collective modes, however, it is more convenient to
introduce displacement fields for the two velocity fields vs and vn . This allows one to
incorporate the conservation laws given by (2.1) and (2.11) directly into expressions for δρ
and δs, eliminating the need for the Lagrange multipliers in (5.16). In terms of the two
displacement fields defined by
vs (r, t) ≡
∂us (r, t)
,
∂t
vn (r, t) ≡
∂un (r, t)
,
∂t
(5.41)
the linearized continuity and entropy conservation equations can be expressed as
δρ(r, t) = −∇ · [ρs0 (r)us (r, t) + ρn0 (r)un (r, t)]
(5.42)
5. Variational formulation of the Landau two-fluid equations
81
and
δs(r, t) = −∇ · [s0 (r)un (r, t)] .
(5.43)
We will use this simpler approach in the following analysis.
Assuming that vn0 = vs0 = 0 in equilibrium, the action in (5.11) to second order in
fluctuations δs and δρ is given by
S
(2)
1
1 ∂2U
1
2
2
(δρ)2
=
dr dt ρs0 vs + ρn0 vn −
2
2
2 ∂ρ2 s,ρn
2 1 ∂2U
∂ U
2
δsδρ −
−
(δs) ,
∂s∂ρ ρn
2 ∂s2 ρ,ρn
Z
Z
(5.44)
subject to the constraints in (5.42) and (5.43) which relate δρ and δs to un and us . Using
the thermodynamic identity in (5.5), the coefficients involving second derivatives in (5.44)
can be replaced by various equilibrium thermodynamic derivatives as follows:
∂2U
∂µL
,
=
∂ρ2 s,ρn
∂ρ s,ρn
2 ∂µL
∂T
∂ U
=
=
,
∂s∂ρ ρn
∂s ρ,ρn
∂ρ s,ρn
2 ∂ U
∂T
.
=
2
∂s ρ,ρn
∂s ρ,ρn
(5.45)
Note that there is an alternative for the last term in the second line, given by the Maxwell
relation
∂T
∂ρ
s
=
∂µL
∂s
.
(5.46)
ρ
There is no contribution from fluctuations in ρn in (5.44) since (∂U/∂ρn )s,ρ = 0, which
follows from (5.21) and the fact that we are considering vs0 = vn0 = 0.
Using (5.45) and the constraints given by (5.42) and (5.43), the action to second order
in the displacement fields un and us is given by
S
(2)
=
Z
dr
Z
1
1
1 ∂µL
2
2
dt ρs0 u̇s + ρn0 u̇n −
[∇ · (ρs0 us + ρn0 un )]2
2
2
2 ∂ρ s,ρn
82
5. Variational formulation of the Landau two-fluid equations
∂T
−
∂ρ
1
[∇ · (s0 un )] [∇ · (ρs0 us + ρn0 un )] −
2
s,ρn
∂T
∂s
ρ,ρn
2
[∇ · (s0 un )]
.
(5.47)
There is no term in the action which is linear in the fluctuations (i.e., S (1) = 0), since these
vanish in accordance with the stationarity condition (see the related comment in the footnote
at the bottom of pg. 32). In evaluating the various local thermodynamic derivatives, ρn is
always fixed. However, for simplicity, this condition is left implicit.
Our variational principle has now been reduced to taking the variation of the quadratic
action in (5.47) with respect to un and us . The linearized hydrodynamic equations and
hence, the low-energy hydrodynamic modes of the system are completely determined by the
variational equations
δS (2)
= 0,
δus (r, t)
δS (2)
= 0.
δun (r, t)
(5.48)
Using (5.47) as well as the identities given by (5.42) and (5.43), (5.48) gives
"
∂µL
∂ρ
∂µL
∂s
#
" #
s0
∂T
∂T
−
, (5.50)
+ δs
∇ δρ
ρn0
∂ρ s
∂s ρ
üs = −∇ δρ
+ δs
s
∂µL
∂s
#
(5.49)
ρ
and
"
ün = −∇ δρ
∂µL
∂ρ
s
+ δs
ρ
where we have used the Maxwell relation in (5.46). Taking δF = δs(∂F/∂s)ρ + δρ(∂F/∂ρ)s ,
where F denotes either one of µL or T , with the displacement fields defined by (5.41), it
is apparent that (5.49) and (5.50) are equivalent to the linearized two-fluid hydrodynamic
equations given by (5.35) and (5.38). This confirms that the variational principle given by
(5.47) and (5.48) which we have constructed is correct.
Normal mode solutions of (5.48) corresponding to hydrodynamic modes of frequency ω
5. Variational formulation of the Landau two-fluid equations
83
have the form
us (r, t) = us (r) cos(ωt), un (r, t) = un (r) cos(ωt).
(5.51)
Because these are exact solutions of the linearized variational equations, we can insert these
expressions into the action directly. Substituting (5.51) into (5.47) and performing the timeintegration, we obtain (apart from an irrelevant constant factor) the following Lagrangian:
L(2) = K[us , un ]ω 2 − U[us , un ],
(5.52)
where we have defined
1
K[us , un ] ≡
2
Z
dr
ρs0 u2s + ρn0 u2n
(5.53)
and
Z
1
∂µL
U[us , un ] ≡
dr
[∇ · (ρs0 us + ρn0 un )]2
2
∂ρ s
∂T
∂T
2
[∇ · (s0 un )] [∇ · (ρs0 us + ρn0 un )] +
[∇ · (s0 un )] .
+2
∂ρ s
∂s ρ
(5.54)
Since the displacement fields us (r) and un (r) are no longer time-dependent, it suffices to
consider variations of the Lagrangian given by (5.52), and the variational equations now
become
δL(2)
= 0,
δus (r)
δL(2)
= 0.
δun (r)
(5.55)
Solving (5.55) is still equivalent to solving the linearized two-fluid hydrodynamic equations, the only simplification being that we have assumed an harmonic time dependence for
the normal mode solutions. The motivation for the preceding analysis is that one can obtain
approximate expressions for the frequencies of the hydrodynamic modes within a variational
approach. Following Zaremba et al. [47], we use a simplified Rayleigh-Ritz method and make
5. Variational formulation of the Landau two-fluid equations
84
an ansatz for the displacement fields of the form
us (r) = [asx fx (r), asy fy (r), asz fz (r)]
un (r) = [anx gx (r), any gy (r), anz gz (r)] ,
(5.56)
where the six coefficients asi and ani will be our variational parameters. Substituting this
ansatz into (5.52), and equating to zero the variation of the resulting expression with respect
to these parameters, we have the six variational equations4
∂L(2)
= 0,
∂asi
∂L(2)
= 0.
∂ani
(5.57)
In practice, the symmetry of the problem (e.g., solving the two-fluid equations in an axisymmetric trap) usually allows us to reduce the number of equations. From these equations, one
obtains a rigorous upper bound for the collective mode frequencies ω [96]. Fortunately, there
exist simple trial functions for fi (r) and gi (r) which are expected to give good approximations
to the exact solutions of the two-fluid equations.
Our ansatz for the function fi (r) describing the superfluid displacement at finite temperatures are the exact solutions at T = 0 originally worked out by Stringari [97]. Similarly,
the ansatz gi (r) for the normal fluid in (5.56) is taken to have the same form as the exact
solution for hydrodynamic modes of a normal Bose fluid above Tc . These were worked out
by Griffin, Wu, and Stringari [98]. Using the ansatz given by (5.56), the Lagrangian given
by (5.52)-(5.54) reduces to one that describes the dynamics of a pair of coupled harmonic
oscillators, with asi and ani representing the displacements of the two oscillators from equilibrium. The effective spring constants are determined by the equilibrium thermodynamic
quantities of the system. This simple picture is very useful to have when envisioning the low
4
Note that by using an ansatz involving time and spatially-independent variational parameters, the variational principle given by (5.12) and (5.13) has been reduced to finding the minimum of the Lagrangian.
5. Variational formulation of the Landau two-fluid equations
85
energy dynamics of the two fluids in trapped gases. It immediately implies, for instance, the
existence of in-phase as well as out-of-phase oscillation modes of the two fluids.
5.3
Examples of hydrodynamic modes
5.3.1
Uniform gas
In order to illustrate the variational formalism developed in Section 5.2, we first consider the
two-fluid modes in a uniform gas. In this case, the normal mode solutions of the linearized
two-fluid equations are first and second sound, as reviewed in Section 2.2. For a uniform
gas, ρs0 , ρn0 , and the equilibrium thermodynamic derivatives appearing in (5.53) and (5.54)
are all independent of position. As a result, the obvious ansatz for the displacement fields is
us (r) = ẑ N as cos(qz), un (r) = ẑ N an cos(qz),
(5.58)
where as and an are the variational parameters. We restrict ourselves to motion along
the z-axis, and ẑ is the corresponding unit vector (of course, in a uniform isotropic fluid, all
directions are the same and this choice is completely arbitrary). The normalization constant5
N is chosen so that
R
dr u2s = a2s . Inserting (5.58) into (5.53) and (5.54) gives
1
1
K[as , an ] = ρs0 a2s + ρn0 a2n
2
2
(5.59)
and
∂µL
∂µL
∂T
ω 2 a2s
2
(ρs0 )
+ as an ρs0 ρn0
+ s0 ρs0
U[as , an ] =
2
u
2
∂ρ s
∂ρ s
∂ρ s
2
∂µL
∂T
∂T
a
+ 2s0 ρn0
+ (s0 )2
.
(5.60)
+ n (ρn0 )2
2
∂ρ s
∂ρ s
∂s ρ
Using these expressions for K and U, the variational equations in (5.57) give the following
5
We emphasize that we are not required to normalize the plane wave ansatz in (5.58). The normal mode
frequencies obtained by solving (5.57) are independent of such a factor. Here we find it convenient to use
this normalization factor in order to make the connection with well-known results for uniform superfluids as
transparent as possible.
86
5. Variational formulation of the Landau two-fluid equations
quadratic equation for the sound velocities u2i ≡ (ω/q)2 :
u4
"
#
2
∂µ
∂T
s
∂T
L
− u2 ρ0
+ 2s0
+ 0
∂ρ s
∂ρ s ρn0 ∂s ρ
" 2 #
∂T
∂µL
∂T
ρs0 2
= 0.
s0
−
+
ρn0
∂s ρ ∂ρ s
∂ρ s
(5.61)
In the usual textbook discussions of two-fluid hydrodynamics [2, 38, 39], one works with
the entropy density s̄ = s/ρ, rather than the entropy s. Using the standard transformation
properties of thermodynamic derivatives (see Sec. 16 in Landau and Lifshitz [67]), we have
the following relations:
∂µL
∂ρ
s
= −2s̄
∂T
∂ρ
∂µL
∂ρ
∂T
∂s
ρ
∂T
∂ρ
=
s
−
∂T
∂s
s
s̄2
+
ρ
s̄
∂T
∂ρ
∂T
∂ρ
2
ρ
s̄
−
ρ
s̄
s
∂T
∂s̄
=
T
.
c̄v
1
+
ρ
ρ
1
= 2
ρ
1
=
ρ
∂T
∂s̄
∂T
∂s̄
∂P
∂ρ
,
∂P
∂ρ
,
(5.62)
s̄
,
(5.63)
ρ
T
∂T
∂s̄
,
(5.64)
ρ
(5.65)
ρ
and
∂T
∂s̄
ρ
(5.66)
Using these, one can show that (5.61) is identical to Landau’s equation for the velocities u2i
of first and second sound given by (2.28). Of course, this result was to be expected since for
a uniform superfluid, the variational plane wave solutions in (5.58) are the exact solutions
of the Landau equations.
5. Variational formulation of the Landau two-fluid equations
5.3.2
87
Dipole mode
We now discuss two-fluid hydrodynamic modes in a trapped superfluid gas using our variational formalism, where we do not have exact solutions of the Landau two-fluid equations.
The first and simplest example is the dipole mode. This mode is characterized by displacements of the centre-of-masses of the condensate and thermal cloud in a harmonic trap. That
is, we take (Zaremba et al. [47]),
δρs (r, t) = ρs0 (r − us (t)); δρn (r, t) = ρn0 (r − un (t)),
(5.67)
where ρs0 and ρn0 are the static equilibrium superfluid and normal fluid density profiles.
The time dependence of the (position-independent) displacement fields us (t) and un (t) are
given in (5.51). It is straightforward to show (using results we derive below for equilibrium
thermodynamic quantities in a harmonic trap) that a uniform displacement us = un of both
components together (the in-phase dipole mode) is an exact solution of the linearized twofluid equations with a harmonic trap potential, given by (5.35) and (5.40). Of course, this is
expected on the grounds that this mode is the generalized Kohn mode (for further discussion,
see Ref. [47]). This mode is a rigid in-phase oscillation of the superfluid and normal fluid
static density profiles.6 As a result, the interactions have no effect on the normal mode
frequency.
We also expect an out-of-phase dipole oscillation similar to the one discussed by Zaremba et
al. [47] in a dilute Bose gas at finite temperatures. This mode also corresponds to a centreof-mass oscillation of the superfluid and normal fluid components described by (5.67), but
now the two fluids have opposite velocities. To describe both types of mode (in-phase and
out-of-phase dipole modes), we introduce the following ansatz for the displacement fields
6
This mode is often called the “sloshing mode” in the experimental literature.
88
5. Variational formulation of the Landau two-fluid equations
along the z axis (the same results are found by considering the x and y displacements):
us = ẑas ,
un = ẑan ,
(5.68)
where the variational parameters as and an are allowed to take on any value.
In accordance with the exact result (for the generalized Kohn mode) discussed above, we
expect there to be an in-phase solution corresponding to as = an . Substituting this ansatz
into (5.53) and (5.54), we find
1
1
K[as , an ] = Ms a2s + Mn a2n
2
2
(5.69)
1
1
U[as , an ] = ks a2s + kn a2n + ksn as an ,
2
2
(5.70)
and
where Ms and Mn are the the masses of the superfluid and normal components, respectively,
given by
Ms ≡
Z
Mn ≡
dr ρs0 ,
Z
dr ρn0 .
(5.71)
The effective spring constants ks , kn , and ksn in the above expressions are defined as
ks ≡
kn
Z
dr
∂µL
∂ρ
s
∂ρs0 ∂ρs0
,
∂z ∂z
(5.72)
∂ρn0
∂µL ∂s0 ∂ρn0
+
≡
dr
∂s ρ ∂z ∂z
s ∂z
∂T
∂T
∂ρn0
∂s0 ∂s0
+
,
+
∂ρ s ∂z
∂s ρ ∂z ∂z
Z
∂µL
∂ρ
(5.73)
and
ksn ≡
Z
dr
∂µL
∂ρ
s
∂ρn0 ∂ρs0
+
∂z ∂z
∂T
∂ρ
s
∂s0 ∂ρs0
∂z ∂z
.
(5.74)
5. Variational formulation of the Landau two-fluid equations
89
The variational equations in (5.57) give the following coupled equations for the two dipole
modes:
Ms ω 2 − ks
−ksn
−ksn
Mn ω 2 − kn
as
an
= 0.
(5.75)
The frequencies of these modes are given by the zeroes of the determinant of this matrix,
namely
2
(Ms ω 2 − ks )(Mn ω 2 − kn ) − ksn
= 0.
(5.76)
Applying some thermodynamic identities, the spring constants ks and kn can be simplified
considerably. Specifically, with vn0 = vs0 = 0, (5.5) can be used to find useful expressions for
the gradient of various equilibrium thermodynamic quantities which appear in (5.72)-(5.74).
From the definition T = (∂U/∂s)ρ , one finds
2 ∂2U
∂ U
=
∇ρ0 +
∇s0
∂s∂ρ
∂s2 ρ
∂T
∂T
∇ρ0 +
∇s0 .
=
∂ρ s
∂s ρ
∇T0
(5.77)
Since ∇T0 = 0, this gives us
∂T
∂ρ
s
∂ρ0
+
∂z
∂T
∂s
ρ
∂s0
= 0.
∂z
(5.78)
(5.79)
Similarly, using (5.8), we obtain
∇µL0 =
∂µL
∂ρ
∇ρ0 +
s
∂µL
∂s
∇s0 + ∇Vext .
ρ
This is equivalent to (5.28) evaluated at equilibrium (when vn0 = vs0 = 0). For a harmonic
trapping potential given by (2.9) and noting that ∇µL0 = 0, (5.79) provides us with the
useful relation
∂µL
∂ρ
s
∂ρ0
+
∂z
∂µL
∂s
ρ
∂
∂s0
= − Vext
∂z
∂z
= −ωz2 z,
(5.80)
5. Variational formulation of the Landau two-fluid equations
90
where ωz is the trap frequency along the z-axis.
Substituting (5.80) and (5.78) into (5.72) and (5.73) and integrating by parts, the expressions for ks and kn simplify to
ks = ωz2Ms − ksn , kn = ωz2Mn − ksn .
(5.81)
Using these values for ks and kn in (5.76), we find the dipole modes are given by
Ms Mn ω 2 − ωz2 + ksn (Ms + Mn ) ω 2 − ωz2 = 0.
(5.82)
We note that the solutions only depend on the values of Ms , Mn , and ksn , as given above.
As expected, (5.82) has two solutions. One solution is
ω = ωz .
(5.83)
Using this in (5.75), we see that this corresponds to as = an . This is the expected generalized
Kohn mode in a harmonic trap. The other solution of (5.82) is
ω 2 = ωz2 −
ksn
.
Mr
(5.84)
Here we have defined the reduced mass of the superfluid and normal fluid components as
Mr =
Ms Mn
.
Ms + Mn
(5.85)
The frequency of this second mode does depend on interactions as these determine the
thermodynamic functions appearing in the ksn spring constant. This mode can be shown to
correspond to the solution Ms as +Mn an = 0. The displacements of the superfluid and normal
fluid have opposite signs, producing an out-of-phase oscillation of the two components. It
can be viewed as an analogue of second sound in superfluid 4 He [2].
Using (5.81), we can write also write the out-of-phase mode frequency in (5.84) in terms
of the simpler spring constant ks defined in (5.72), namely
ω2 =
Ms 2
ks
−
ω .
Mr Mn z
(5.86)
5. Variational formulation of the Landau two-fluid equations
91
This alternate formula will be useful when evaluating the frequency at unitarity in Chapter 6.
5.3.3
Breathing modes
We next use our formalism to describe the breathing hydrodynamic modes. At T = 0, it
has been proven that the “scaling solution” (Castin and Dum [99]; see also Stringari and
Pitaevskii [39])
δρs (r, t) = ρs0
x
y
z
,
,
ηx (t) ηy (t) ηz (t)
(5.87)
is an exact solution of the T = 0 hydrodynamic equations (see discussion in Section 5.4).
Here ηi (t) are the scaling factors and have harmonic time-dependence. Inserting this solution
into the T = 0 continuity equation, we can relate these scaling factors to the superfluid
displacement field,
us (r, t) = (ηx x, ηy y, ηz z) cos(ωt).
(5.88)
Similarly, one can show that
δρn (r, t) = ρn0
x
y
z
,
,
ηx (t) ηy (t) ηz (t)
(5.89)
is an exact solution of the hydrodynamic equations in the normal phase T > Tc (Griffin et
al. [98]). These solutions for the superfluid and normal phases motivate us to consider the following trial solution for the velocity displacements of the superfluid and normal components
involved in a breathing mode (first used by Zaremba et al. [47]):
us (r) = (asx x, asy y, asz z), un (r) = (anx x, any y, anz z).
(5.90)
Substituting (5.90) into (5.52), we obtain
K[as , an ] =
i
1 Xh
M̃si a2si + M̃ni a2ni ,
2 i
(5.91)
5. Variational formulation of the Landau two-fluid equations
92
and
U[as , an ] =
i
1 Xh
ks,ij asi asj + kn,ij ani anj + 2ksn,ij asi anj .
2 ij
(5.92)
Here the “mass moments” M̃i are defined by [compare with (5.71)]
Z
M̃si ≡
dr
ρs0 x2i ,
M̃ni ≡
Z
dr ρn0 x2i ,
(5.93)
and the effective spring constants ks,ij , kn,ij , and ksn,ij for the breathing modes are now given
by
Z
ks,ij =
Z
kn,ij =
dr
∂µL
∂ρ
s
∂(ρs0 xi ) ∂(ρs0 xj )
,
∂xi
∂xj
(5.94)
∂µL ∂(ρn0 xi ) ∂(ρn0 xj )
∂T
∂(ρn0 xi ) ∂(s0 xj )
dr
+2
∂ρ
∂xi
∂xj
∂ρ s ∂xi
∂xj
s
∂(s0 xi ) ∂(s0 xj )
∂T
,
+
∂s ρ ∂xi
∂xj
(5.95)
and
ksn,ij =
Z
dr
∂µL
∂ρ
s
∂(ρs0 xi ) ∂(ρn0 xj )
+
∂xi
∂xj
∂T
∂ρ
s
∂(ρs0 xi ) ∂(s0 xj )
.
∂xi
∂xj
(5.96)
Since we generally have six variational parameters (one for each Cartesian component of
the two displacement fields), the collective mode frequencies are found from the six coupled
algebraic equations, ∂L(2) /∂asi = 0, ∂L(2) /∂ani = 0. We find
M̃si ω 2asi =
M̃ni ω 2 ani =
1X
[(ks,ij + ks,ji)asj + 2ksn,ij anj ] ,
2 j
1X
[(kn,ij + kn,ji)anj + 2ksn,jiasj ] .
2 j
(5.97)
(5.98)
In experiments performed on trapped superfluid gases, one usually has an axisymmetric trap
such that, for instance, ωx = ωy ≡ ω⊥ , and the modes of interest are the radial and axial
93
5. Variational formulation of the Landau two-fluid equations
breathing modes (see, for example, Bartenstein et al. [30] and Kinast et al. [32], which deal
with Fermi gases close to unitarity). In this case, we have asx = asy and anx = any .
In Chapter 6, we shall use our variational formalism to evaluate the frequencies of the
breathing modes for a superfluid Fermi gas at unitarity. For this, it will be useful to have
the equations for the breathing mode frequencies given above in a different form. We first
define the following new coefficients involving the spring constants:
Kijs ≡ 2ks,ij + 2ksn,ij
(5.99)
Kijn ≡ kn,ij + kn,ji + 2ksn,ji.
(5.100)
and
Adding together the two equations for the breathing modes in (5.97) and (5.98), we find
h
i
1X
ω M̃si asi + M̃ni ani =
[(kn,ij + kn,ji + 2ksn,ij ) anj + (2ks,ij + 2ksn,ji) asj ]
2 j
1 X n
s
Kji anj + Kji
asj .
(5.101)
=
2 j
2
Subtracting the two equations from each other, we obtain
1X
ω [asi − ani ] =
2 j
2
s
Kji
2ks,ij 2ks,ij
+
−
M̃si
M̃ni
M̃ni
[asj − anj ] +
Kijs
M̃si
−
Kijn
M̃ni
anj .
(5.102)
After some rearranging and integrating by parts, one can derive the following results for
these new spring constants:
Kijs
=
DµL ∂ (ρs0 xi )
DµL
xj −
xi ρs0
Dxj ∂xi
Dxi
"
#
∂µL
∂
∂µL
∂
+ s0
,
−ρs0 xi ρ0
∂xi ∂ρ s
∂xi ∂s ρ
Z
dr
(5.103)
94
5. Variational formulation of the Landau two-fluid equations
and
Kijn
DµL ∂ (ρn0 xj )
DµL
DT ∂ (s0 xj )
DT
xi −
xj ρn0 +
xi −
xj s0
Dxi
∂xj
Dxj
Dxi ∂xj
Dxj
"
#
∂µL
∂
∂µL
∂
+ s0
−ρn0 xj ρ0
∂xj
∂ρ s
∂xj
∂s ρ
"
#
∂T
∂
∂T
∂
+ s0
,
(5.104)
−s0 xj ρ0
∂xj ∂ρ s
∂xj ∂s ρ
= 2
Z
dr
where we define
DµL
≡
Dxi
DT
≡
Dxi
∂µL
∂ρ
s
∂ρ0
+
∂xi
s
∂ρ0
+
∂xi
∂µL
∂s
ρ
∂s0
,
∂xi
(5.105)
and
∂T
∂ρ
∂T
∂s
ρ
∂s0
.
∂xi
(5.106)
In deriving these results, we have also made use of the Maxwell relation given in (5.46).
Analogously to (5.78) and (5.80), one can show that
DµL
DT
= −ωi2 xi ,
= 0.
Dxi
Dxi
(5.107)
Making use of (5.46) as well as the equilibrium thermodynamic identities [see (5.6) and (5.5)]
P = −U − ρVext + T s + µL ρ,
T =
∂U
∂s
,
(5.108)
ρ
and assuming that P, T , and µL are functions of the independent variables ρ and s, one can
show that
∂
∂xi
∂P
∂ρ
s
∂
= s0
∂xi
∂T
∂ρ
∂
+ ρ0
∂xi
s
∂µL
∂ρ
− ωi2xi .
(5.109)
∂T
∂ρ
(5.110)
s
Furthermore, making use of the fact that ∇T0 = 0, we find
∂
∂xi
∂P
∂s
ρ
∂
= s0
∂xi
∂T
∂s
∂
+ ρ0
∂xi
ρ
s
.
95
5. Variational formulation of the Landau two-fluid equations
Using (5.107), (5.109), and (5.110) in (5.103) and (5.104), the effective spring constants
Kijs and Kijn defined in (5.99) and (5.100), reduce to
Kijs
Kijn = 2
Z
= 2
(
Z
∂
∂P
2
dr ρs0 xi 2δij ωi xi −
,
∂xi ∂ρ s
∂
dr ρn0 xi 2δij ωi2 xi −
∂xi
∂P
∂ρ
s
∂
− s0 xi
∂xi
(5.111)
∂P
∂s
)
,
(5.112)
ρ
In summary, we have derived two equivalent expressions for the equations which determine the breathing mode frequencies. The more compact formulas are given by (5.97)
and (5.98). However, (5.101) and (5.102) will be more convenient in our calculation of the
breathing modes at unitarity in Chapter 6.
5.4
Hydrodynamic Theory at T = 0
Since the variational principle given by (5.52), (5.56), and (5.57) involves some fairly complex
expressions, it is useful to consider the case of T = 0, where the two-fluid Lagrangian
simplifies tremendously. We shall prove that the Landau two-fluid equations reduce to a
single hydrodynamic differential equation, first derived by Pitaevskii and Stringari [100].
They used it to discuss corrections to the mean-field Gross-Pitaevskii results [97] for the
collective mode frequencies in a dilute Bose gas at T = 0.
A hydrodynamic description means that a few local variables are sufficient to describe
the dynamics. While such a description requires rapid collisions and local equilibrium to be
valid in the case of a normal fluid, a “hydrodynamic” description of a superfluid is always
correct. Thus it is no surprise that the T → 0 limit of two-fluid hydrodynamics gives the
correct quantum description of the pure superfluid (see also the discussion on pg. 170 of
Pethick and Smith [38]).
96
5. Variational formulation of the Landau two-fluid equations
At T = 0, only the superfluid component exists and thus we have s = 0, ρn0 = 0, and
ρs0 = ρ0 . (5.52)-(5.54) then reduce to
1
L [us ] =
2
(2)
Z
3
d r
ρ0 u2s ω 2
−
∂µL
∂ρ
2
[∇ · (ρ0 us )]
.
(5.113)
From δL(2) /δus = 0, one obtains the equations for us :
2
ω us = −∇
∂µL
∂ρ
∇ · (ρ0 us ) .
(5.114)
Using the linearized continuity equation, δρ = −∇ · (ρ0 us ), (5.114) can be rewritten as
2
ω us = ∇
∂µL
∂ρ
δρ .
(5.115)
Multiplying both sides of this expression by ρ0 and taking the divergence, we obtain a closed
equation for the density fluctuations,
2
ω δρ = −∇ · ρ0 ∇
∂µL
∂ρ
δρ .
(5.116)
This is the basis of the T = 0 quantum hydrodynamic theory derived by Pitaevskii and
Stringari [100]. We note that (5.116) describes the low-energy collective modes at T = 0 of
both atomic Bose and two-component Fermi superfluid gases. The only difference between
these two quantum gases lies in the choice of the equation of state µL (ρ) in (5.116).
Our variational approach should give expressions for the collective mode frequencies
at T = 0 that agree with results derived from solving (5.116) directly (see, for example,
Pitaevskii and Stringari [100] for Bose gases, and the review by Giorgini et al. [13] for Fermi
gases close to unitarity). However, it is still useful to show explicitly how our formalism
reproduces the T = 0 results obtained in the recent literature. As a specific application, we
consider the breathing modes which were discussed at finite T in the previous section. The
breathing modes of trapped Fermi gases close to unitarity have been studied extensively at
97
5. Variational formulation of the Landau two-fluid equations
T = 0 (Giorgini et al. [13]), where a simple ansatz for the density dependence of the chemical
potential µL (ρ) often allows one to obtain analytic expressions for the frequencies of these
modes. We now show that our general variational results for the breathing mode frequencies
at finite T given by (5.97) and (5.98) reduce to these well-known expressions.
Since the normal fluid vanishes at zero temperature, we only have equations for the
superfluid component, given by (5.101), which reduce to
ω 2 M̃i asi =
1X s
Kji asj .
2 j
(5.117)
at T = 0, where Kijn and M̃ni are zero and
M̃i ≡
Z
dr ρ0 x2i .
(5.118)
The expression for Kijs , given by (5.111), becomes
Kijs
=2
Z
dr ρ0 xi
2δij ωi2xi
∂
−
∂xi
∂P
∂ρ
,
(5.119)
since ρs0 = ρ0 at T = 0.
We now assume a polytropic equation of state (see Section 6.1 for further discussion),
defined by
µL (ρ) ∝ ργ .
(5.120)
With this, it is straightforward to show that the pressure P obeys P (ρ) ∝ ργ+1 at T = 0 [see
(5.6)]. Using this in (5.36), we find the spatial derivative in the integral of (5.119) reduces
to
∂
ρ0
∂xi
∂P
∂ρ
= γ
∂P
∂ρ
∂ρ0
∂xi
= −γρ0 ωi2xi .
(5.121)
5. Variational formulation of the Landau two-fluid equations
98
Substituting this result into (5.119), and making use of the definition of the mass moment
M̃i given by (5.118), we obtain
Kijs = 2M̃i ωi2 [2δij + γ] .
(5.122)
Combining this identity with (5.117) we obtain a simple expression for Kijs (noting that
s
Kji
= Kijs at T = 0),
ω 2asi =
X
j
2ωi2δij + γωi2 asj
= 2ωi2 asi + γωi2
X
asj .
(5.123)
j
These equations are identical to (3) in Astrakharchik, Combescot, Leyronas, and Stringari [101],
assuming a polytropic equilibrium equation of state as given by (5.120).
For an axisymmetric trap (ωx = ωy = ω⊥ ), the axial and longitudinal breathing modes
are characterized by variational solutions with ax = ay ≡ a. In this case, the normal mode
solutions of (5.123) are (Cozzini and Stringari [102]),
ω
2
1h
2
=
2(γ + 1)ω⊥
+ (γ + 2)ωz2 ±
2
q
i
2
2
2
.
− (γ + 2)ωz2 ] + 8γ 2 ωz2 ω⊥
[2(γ + 1)ω⊥
(5.124)
Although we have reproduced the T = 0 result of Cozzini and Strigari [102] for the
breathing mode frequency, we emphasize that the general two-fluid hydrodynamic formalism
described in this chapter is valid at finite temperatures. In general, the frequencies of the
two-fluid modes will not be given by simple analytic expressions as in (5.124). An exception
is at unitarity where, as we show in Chapter 6, the in-phase breathing mode is independent
of temperature, with a frequency given by the zero-temperature result in (5.124). This is
one of the major results of this thesis.
Chapter 6
Hydrodynamic modes in trapped
gases at unitarity
Understanding the nature of Fermi gases at unitarity, where the s-wave scattering length
is infinite, is a challenging many-body problem [13, 103, 104]. There has been considerable
recent interest in this question in connection with ultracold Fermi gases near a Feshbach
resonance. We argue that measurement of the breathing mode frequencies will provide
a sensitive test of current microscopic theories of a Fermi gas at unitarity, including the
predictions of “universal thermodynamics” (Ho [42]). We recall that the Landau two-fluid
equations for a superfluid predict two types of modes: an in-phase mode in which the
normal and superfluid components oscillate together, and an out-of-phase mode in which
they move against each other. Expressions for the frequencies of the dipole and breathing
modes were derived in Chapter 5. In this chapter, we now use these results to calculate the
frequencies of the dipole and breathing modes at unitarity as a function of the temperature.
Making use of the universality properties of thermodynamics at unitarity, we show how
the expressions for the mode frequencies simplify tremendously at unitarity. However, we
emphasize that our general formalism is valid anywhere in the BCS-BEC crossover where
two-fluid hydrodynamics is involved.
Thomas and coworkers [34] have discussed the collisional hydrodynamics of Fermi gases
99
6. Hydrodynamic modes in trapped gases at unitarity
100
at unitarity assuming that the dynamics of a trapped Fermi gas can be described by a single
hydrodynamic equation (Euler’s equation) for an ideal fluid with a single velocity field v(r)
for the entire gas, even below the superfluid transition temperature Tc . This assumption
immediately implies an in-phase hydrodynamic mode, with vs (r) = vn (r) ≡ v(r). Our
analysis in this chapter of the Landau two-fluid equations at unitarity confirms that such an
in-phase mode does exist at unitarity. Within our variational approximation, we prove that
the frequency of the in-phase mode is independent of temperature, with the value at T = 0
the same as in the normal phase well above Tc .
In our variational calculation of the breathing and dipole modes given in Chapter 5, the
frequencies of the two-fluid modes involved spatial integrals over a number of thermodynamic
derivatives. Due to the simplifications of universal thermodynamics at unitarity, we show
that only the superfluid density ρs and the chemical potential µ are needed.
6.1
Thermodynamics at unitarity
We first review the special features of thermodynamics at unitarity (for discussion and recent
references, see Giorgini et al. [13]). In particular, we show how universal thermodynamics
allow us to obtain analytic results for the gradients of the key thermodynamic derivatives
(∂P/∂ρ)s and (∂P/∂s)ρ that are involved in the coefficients [see (5.111) and (5.112)] of the
variational equations which determine the breathing mode frequencies.
In a uniform system of interacting fermions, there are three microscopic lengths scales
and hence, three energy scales. The three length scales are the mean interparticle spacing
−1/3
nF
, the thermal wavelength1 λ2T ≡ 2π/mkB T , and a length scale r0 that is characteristic of
the interaction. Here, nF ≡ (2mǫF )3/2 /3π 2 is the total fermion density (i.e., nF = n↑ + n↓ ),
where ǫF is the Fermi energy. The corresponding energy scales are the kinetic energy ǫF ,
1
We set ~ = 1.
6. Hydrodynamic modes in trapped gases at unitarity
101
kB T , and the interaction energy which can be expressed as a functional of the density nF
and r0 . At the low energies of interest in dilute cold gases, the relevant length scale that
characterizes the interaction is provided by the s-wave scattering length as . At unitarity, the
scattering length diverges, meaning that the only remaining length scales are the interparticle
−1/3
spacing nF
and the thermal wavelength, as first argued by Ho [42]. This also implies that
at unitarity, the only remaining energy scales are the Fermi energy and kB T . Consequently,
the only dimensionless energy scale at unitarity is kB T /ǫF ≡ kB T /kB TF . This immediately
means that all thermodynamic functions at unitarity can be written in dimensionless form
as a function of the ratio T /TF . The universal expressions for the internal energy, entropy,
and chemical potential are given below.
Owing to the fact that there is only one dimensionless energy scale, kB T /ǫF (ρ) ≡
kB T /kB TF (ρ), the energy density U and entropy S take the form (Kinast et al. [34] and
Ho [42])
U=
ρǫF (ρ)
fE [T /TF (ρ)].
m
(6.1)
Also, the total entropy S ≡ s∆V 2 of a fluid element of small (infinitesimal) volume ∆V is
S = NkB fS [T /TF (ρ)].
(6.2)
Here fE and fS are dimensionless functions of the reduced temperature T /TF (ρ). ǫF (ρ) is
the local Fermi energy and is a function of the mass density ρ(r). N(r) = ρ(r)∆V /m is
the total number of fermions in the small volume ∆V centered at position r. We emphasize
that both the energy density U(r) and the entropy S(r) of a small fluid volume centered at
r depend on position through the Fermi energy ǫF (ρ) and the local mass density ρ(r).
2
Recall that we used the entropy density s in our discussion of two-fluid hydrodynamics in Chapters 2
and 5.
102
6. Hydrodynamic modes in trapped gases at unitarity
The total local energy density is given by E0 = U + ρVext . Using (6.1), we see that
E0 ∆V = NǫF (ρ)fE [T /TF (ρ)] + NmVext .
(6.3)
The pressure P is defined by (see Landau and Lifshitz [67])
(E∆V )
P =−
∂∆V
,
(6.4)
N,S
From (6.2), we see that holding N and S constant requires holding the reduced temperature
T /TF (ρ) constant in (6.4) as well [34]. Thus, we find
(E0 ∆V )
∂∆V
= N
N,S
∂ǫF (ρ)
∂∆V
fE [T /TF (ρ)]
N
ρ2 ∂ǫF (ρ)
= −
fE [T /TF (ρ)]
m ∂ρ
2 ρǫF (ρ)
fE [T /TF (ρ)],
= −
3 m
(6.5)
where we have used ∂ǫF (ρ)/∂ρ = 2ǫF /3ρ. Using this result in (6.4), we obtain the following
result for the pressure in a Fermi gas at unitarity:
P =
2 ρǫF (ρ)
2
fE [T /TF (ρ)] = U.
3 m
3
(6.6)
This relation is identical to that for an ideal Fermi gas.
Since T = (∂U/∂s)ρ (where s = S/∆V is the entropy density), we can use (6.6) to give
∂P
∂s
∂
∂xi
ρ
2
=
3
∂U
∂s
ρ
2
= T0
3
(6.7)
and hence
∂P
∂s
=
ρ
2 ∂T0
= 0.
3 ∂xi
(6.8)
Combining the identity [see (5.8)]
µL =
∂U
∂ρ
s
+ Vext
(6.9)
6. Hydrodynamic modes in trapped gases at unitarity
103
with (6.6), we obtain
∂P
∂ρ
=
s
=
2
[µL − Vext ] .
3
(6.10)
Using this, we find
∂
∂xi
∂P
∂ρ
s
2 ∂µL0 2 ∂Vext
2
−
= − ωi2 xi .
3 ∂xi
3 ∂xi
3
(6.11)
Here we have made use of the fact that the equilibrium chemical potential is independent of
position and hence ∇µL0 = 0.
In closing, we use the results of this section to prove that the chemical potential at
unitarity follows the polytropic equation of state µ(n) = An2/3 (equivalently, µ(ρ) ∝ ρ2/3 )
at T = 0. We used this result earlier in deriving (3.78) for the phonon velocity at T = 0.
We also assumed a polytropic equation of state to derive (5.123), giving the breathing mode
frequency at T = 0. Combining (6.1) and (6.9), at T = 0 we find
∂ ρǫF (ρ)
fE (0) + mVext
∂ρ m
5(3π 2)2/3 2/3
n + mVext .
= fE (0)
6m
µ ≡ mµL =
(6.12)
Since fE (0) is a constant, this proves that the T = 0 chemical potential is given by a
polytropic equation of state at unitarity, with γ = 2/3.3
6.2
Locally isentropic dynamics
Before discussing our variational breathing and dipole mode solutions of the two-fluid equations at unitarity in Sections 6.3 and 6.4, we use the results of Section 6.1 to discuss
some general features of the solutions of the Landau two-fluid hydrodynamic equations for
trapped superfluid gases. In particular, we discuss the observation by Kinast, Turlapov, and
3
Note that the presence of an external potential Vext in (6.12) does not change the result given in (5.123),
since it is the compressibility (∂µ/∂ρ) that enters that expression.
6. Hydrodynamic modes in trapped gases at unitarity
104
Thomas [34] that the (in-phase) breathing mode at unitarity obeys a single Euler equation
for the velocity v ≡ vs = vn on the grounds of locally isentropic hydrodynamics. It followed
from the analysis of this Euler equation that the frequency of the breathing mode would be
independent of temperature. This surprising result was consistent with their experimental
results for the breathing mode (see Fig. 6.1). We now examine this problem starting from
Landau’s two-fluid hydrodynamic equations.
In our discussion of Landau’s two-fluid equations in Section 2.1, we showed that any
flow where the local superfluid and normal fluid velocities are the same (vs = vn ) is a
locally isentropic flow [see (2.15)]. For locally isentropic fluid flow, Landau’s expression for
the current in (2.6) reduces to j = (ρs + ρn )v = ρv. Using this result in (5.33), Landau’s
equation of motion for the current reduces to
∂j
= −∇P − ρ∇Vext − ρv · ∇v − v∇ · j.
∂t
(6.13)
Combining this equation with the continuity equation given by (2.1), we obtain the following
equation of motion for the velocity v:
∂v
= −∇
∂t
v2
+ Vext
2
−
∇P
.
ρ
(6.14)
This is precisely Euler’s equation for an ideal irrotational (such that ∇v2 = 2v · ∇v) fluid,
generalizing (2.5) to include the effect of an external trapping potential. This result shows
that for the special case where vs (r, t) = vn (r, t), Landau’s two-fluid hydrodynamic equations
reduce to a single equation of motion for the velocity which is just Euler’s equation.
So far, we have shown that for the special case of locally isentropic dynamics, vs =
vn ≡ v, Landau’s two fluid equations reduce to a single Euler equation for the irrotational
local velocity field v. Our present discussion develops the original suggestion in Ref. [34]
by showing how it is a rigorous consequence of Landau’s two-fluid equations. We have not
105
6. Hydrodynamic modes in trapped gases at unitarity
shown, however, whether such a solution of the two-fluid equations exists. We now derive a
condition for a locally isentropic (vs = vn ) normal mode solution of the Landau two-fluid
equations to exist.
Since each mass element evolves at constant entropy in a locally isentropic flow, these
elements do not exchange heat with their surroundings and hence the temperature remains
unchanged throughout the fluid. Subtracting the linearized two-fluid equation for vn given
in (5.38) from the linearized two-fluid equation for vs given in (5.35), one has
∂(vs − vn )
s0
=
∇δT.
∂t
ρn0
(6.15)
This implies ∇δT = 0 for an in-phase mode with vs (r, t) = vn (r, t), meaning that the
temperature remains constant everywhere for such a mode, as one expects for a locally
isentropic oscillation. This means that a locally isentropic mode is also a locally isothermal
mode. Using δT = (∂T /∂s)ρ δs + (∂T /∂ρ)s δρ and (5.42) and (5.43), we can write the
condition ∇δT = 0 as
∇
"
∂T
∂ρ
s
∇ · (ρ0 u) +
∂T
∂s
ρ
#
∇ · (s0 u) = 0,
(6.16)
where the displacement fields satisfy us = un ≡ u. Using (5.78) and (5.110), (6.16) reduces
to the following very useful condition:
" #
∂T
∂T
∂P
∇ (∇ · u) ρ0
+ s0
+ (∇ · u) ∇
= 0.
∂ρ s
∂s ρ
∂s ρ
(6.17)
To make contact with the results of Section 6.1, we want to express (6.17) in terms of
derivatives of the pressure. Using (5.6), (5.8), and (5.9), we find
∂P
∂s
= ρ0
ρ
∂T
∂ρ
+ s0
s
∂T
∂s
.
(6.18)
ρ
With this result, the condition in Eq. (6.17) can be written in the useful form
∇ (∇ · u)
∂P
∂s
∂P
+ (∇ · u) ∇
∂s
ρ
ρ
= 0.
(6.19)
6. Hydrodynamic modes in trapped gases at unitarity
106
Equation (6.19) thus gives the condition for there to exist a locally isentropic (or isothermal)
normal mode solution of the Landau two-fluid equations. We note that this relation is
completely general for an oscillation described by u, and not restricted to the case of a
superfluid in a harmonic trap.
At unitarity, the second term in (6.19) vanishes in accordance with (6.8). Thus we
conclude that a locally isentropic mode (vs = vn ) exists at unitarity if either [using (6.7)]
∂P
∂s
ρ
2
= T = 0,
3
(6.20)
or if
∇(∇ · u) = 0
(6.21)
is satisfied. The first condition in (6.20) is trivially satisfied at T = 0, where the normal
fluid vanishes and hence all particles move with the same velocity vs = v, and of course
any oscillation will be locally isentropic. This means that in order for a locally isentropic
mode to exist at finite temperatures, (6.21) must be satisfied. This is satisfied by the scaling
solution (5.88) of the hydrodynamic equation in (6.14) that describes the breathing mode. It
is also satisfied by the generalized Kohn (in-phase dipole) mode given by (5.68). Thus both
the in-phase breathing and dipole modes at unitarity are locally isentropic. We emphasize
however that that the in-phase dipole mode (for which ∇ · u = 0) always corresponds to a
locally isentropic mode, even away from unitarity, as shown by (6.19).
In Section 6.3, where we discuss the breathing modes at unitarity, we confirm that our
variational solution of the two-fluid equations gives a locally isentropic in-phase breathing
mode and show that its frequency is independent of temperature. This is the mode studied
by Kinast et al. [34]. In addition, our solution also predicts an out-of-phase breathing mode
which is not locally isentropic. Its frequency depends very strongly on temperature.
107
6. Hydrodynamic modes in trapped gases at unitarity
In closing this section we recall from Section 2.2 that in superfluid 4 He, first sound
also describes a locally isentropic mode. We now show that (6.19) also accounts for this.
However, we emphasize that first sound in uniform superfluid 4 He is locally isentropic for
different reasons than the in-phase breathing and dipole modes in a trapped Fermi superfluid
at unitarity. Since the plane-wave solutions of the uniform two-fluid equations do not satisfy
∇ · u = 0, we see that first sound will only be locally isentropic if (∂P/∂s)ρ = 0. Using the
identity (see Sec. 16 in Landau and Lifshitz, [67])
∂P
∂s
ρ
T
=
ρc̄v
∂P
∂T
,
(6.22)
ρ
where c̄v = T (∂s̄/∂T )ρ is the equilibrium specific heat per unit mass [first discussed below
(2.28)], one sees that (∂P/∂T )ρ ≃ 0 implies (∂P/∂s)ρ ≃ 0. As shown in Section 2.2,
(∂P/∂T )ρ ≃ 0 in superfluid 4 He, leading to the result that first sound is a locally isentropic
mode (vs = vn ).
Thus, we see that the condition in (6.19) for a locally isentropic mode to exist correctly
accounts for the situation in uniform superfluid 4 He. However, there is no reason to believe
that (∂P/∂T )ρ is especially small in Fermi gases at unitarity and we are led to the conclusion that a locally isentropic mode exists at unitarity only when u satisfies the condition
(6.21). We conclude that the existence of a locally isentropic mode [vs (r, t) = vn (r, t)] is
not a universal feature of hydrodynamics at unitarity, but rather is a special feature in a
harmonically confined gas.
6.3
Breathing modes at unitarity
In Section 5.3.3 we described the variational theory of the two-fluid hydrodynamic breathing mode frequencies. Using the properties of thermodynamics at unitarity, we now show
that these expressions simplify greatly, providing us with some interesting analytical results.
6. Hydrodynamic modes in trapped gases at unitarity
108
Using the identities in (6.8) and (6.11), (5.111) and (5.112) reduce to
Kijs = 2M̃si [2δij + 2/3] ωi2
(6.23)
Kijn = 2M̃ni [2δij + 2/3] ωi2 .
(6.24)
and
Substituting these values into (5.101) and (5.102), one arrives at the two equations
ω
2
M̃si asi + M̃ni ani
1 X n
s
Kjianj + Kji
asj
2 j
X
=
[2δij + 2/3] ωj2 M̃sj asj + M̃nj anj ,
=
(6.25)
j
and
s
s Kji
Kijn
Kij
1X
2ks,ij 2ks,ij
ω (asi − ani ) =
+
−
(asj − anj ) +
−
anj .
2 j
M̃si
M̃ni
M̃ni
M̃si M̃ni
#
"
X ks,ij ks,ij
M̃sj
+
−
(2δij + 2/3) ωj2 (asj − anj ).
(6.26)
=
M̃
M̃
M̃
si
ni
ni
j
2
One immediately sees that there is an in-phase solution of (6.26),
asi = ani ≡ ai .
(6.27)
Substituting this in-phase solution into (6.25), we find
M̃i ω 2ai =
X
M̃j ωj2 [2δij + 2/3] aj ,
(6.28)
j
where M̃i ≡ M̃si + M̃ni . From the definition of the spring constants Kijs and Kijn in (5.99)
s
n
and (5.100), one sees that Kijs + Kijn = Kji
+ Kji
. Applying this result, we see that the
expressions in (6.23) and (6.24) at unitarity imply
M̃i ωi2 = M̃j ωj2
(6.29)
6. Hydrodynamic modes in trapped gases at unitarity
109
for all coordinates, i, j = x, y, z. Making use of this result in the right-hand side of (6.28), it
reduces to
2 X
2
ω1B
ai = 2ωi2 ai + ωi2
aj ,
3
j
(6.30)
where we use ω1B to denote that this is the frequency of the in-phase breathing mode.
Equations (6.27) and (6.30) describe an in-phase oscillation of the normal and superfluid
components with vs (r) = vn (r). This is the expected locally isentropic breathing mode
discussed in Section 6.2.
There is an additional out-of-phase solution of (6.25) and (6.26). From (6.25), we see
that this solution corresponds to
M̃si asi + M̃ni ani = 0.
(6.31)
Substituting this solution into (6.26), the frequency of the out-of-phase breathing mode at
unitarity is given by
ω 2 asi =
X M̃sj
j
M̃si
"
#
M̃ri M̃sj
ks,ij
−
(2δij + 2/3) ωj2 asj .
M̃rj
M̃rj M̃ni
(6.32)
The reduced mass moment M̃ri is defined as
1
1
1
≡
+
.
M̃ri
M̃si M̃ni
6.3.1
(6.33)
In-phase mode at unitarity
We now discuss the frequency of the locally isentropic in-phase mode at unitarity (asi = ani )
given by (6.30). This is the same as the breathing mode frequency at T = 0 [given by
(5.123)] using the polytropic exponent γ = 2/3. As discussed before, this value of the
polytropic exponent is an exact result at unitarity and at T = 0. Thus, the frequency of the
in-phase two-fluid hydrodynamic breathing mode at unitarity is independent of temperature
110
6. Hydrodynamic modes in trapped gases at unitarity
and equal to the zero temperature value. We emphasize that this fact is a consequence of
the thermodynamic properties at unitarity, and is not expected to hold away from unitarity.
For an axisymmetric trap, ωx = ωy ≡ ω⊥ , the axial and longitudinal breathing modes
are characterized by solutions of the form ax = ay . In this case, (6.30) gives
2
ω1B
4
1
5 2
+ ωz2 ±
= ω⊥
3
3
6
q
2
2
2
− 8ωz2) + 32ωz2ω⊥
.
(10ω⊥
(6.34)
We further note that for an isotropic trap (ωx = ωy = ωz ≡ ω0 ), we have ax = ay = az , and
the solution of (6.30) is
ω1B = 2ω0 .
(6.35)
Castin [105] proved by a very general analysis that this is an exact result at unitarity,
independent of temperature. It is reassuring that two-fluid hydrodynamics gives a result in
agreement with this prediction.4 The temperature independence of this in-phase breathing
mode was also found from the numerical solution of the Landau equations by He, Chen,
Chien, and Levin [45].
Our predicted temperature independence of the in-phase breathing mode frequency is
consistent with the recent experimental results of Kinast et al. [34], as well as their theoretical results based on locally isentropic dynamics and the Euler equation in (6.14). The trap
used in their experiments was highly anisotropic (ωz ≪ ω⊥ ). In this limit, the radial hydrodynamic breathing mode frequency [given by the upper branch of (6.34)] is well-approximated
by
ω1B ≃
4
p
10/3ω⊥ .
(6.36)
In addition to the assumption that two-fluid hydrodynamics is applicable, we have also made use of the
scaling ansatz in (5.97) and (5.98) in arriving at the result in (6.35). As shown by Castin [105], there is an
exact eigenstate of the isotropic trap Hamiltonian such that all atoms move with velocity v(r, t) = ar cos(ωt).
6. Hydrodynamic modes in trapped gases at unitarity
111
Figure 6.1: Experimental values (filled circles) for the radial breathing mode in a highly
anisotropic trap at unitarity. The dot-dash line gives the result in (6.36) (from Kinast et
al. [34]).
This should be contrasted with the result ω = 2ω⊥ for a trapped noninteracting gas. The
data of Thomas and coworkers showed very little difference between the measured radial
breathing mode frequency and (6.36) over the large temperature range, 0.12TF . T . 1.1TF
(see Fig. 6.1). One notices a very slight increase in the mode frequency [less that 3% of the
value given by (6.36)] over this range. This is possibly due to a systematic error (the data in
Fig. 6.1 is scaled from the “raw” data points given by the empty circles to account for nonharmonic corrections to the Gaussian trapping potential), or the breakdown of hydrodynamic
behaviour near the trap edges.
6.3.2
Out-of-phase breathing mode at unitarity
We next discuss the out-of-phase breathing mode at unitarity, with frequency given by (6.32).
In the special limit of an isotropic trap, (6.32) simplifies tremendously. In this case, asi → as
and M̃si → MsB /3, M̃ni → MnB /3, where we have defined
MsB
=
Z
2
dr ρs0 (r)r ,
MnB
=
Z
dr ρn0 (r)r 2 .
(6.37)
112
6. Hydrodynamic modes in trapped gases at unitarity
5
breathing
mode
3
B/
0
4
2
1
0
0.0
0.1
T/T
0.2
0.3
F
Figure 6.2: The frequency of the out-of-phase breathing mode as a function of temperature
for an isotropic trap for a number of superfluid density profiles, as discussed in the text.
The solid line gives the breathing mode frequency calculated using the fitted NSR data for
ρs (see Fig. 4.3). The curved dashed line is obtained using the BCS mean-field ρs (scaled to
the correct Tc ). The horizontal dashed line gives frequency of the in-phase mode, ω1B = 2ω0 .
The blue arrow denotes the superfluid transition temperature Tc ≃ 0.27TF (for a trapped
gas).
With these identities, (6.32) reduces to
2
ω2B
=
ksB
MsB 2
−
4
ω ,
MrB
MnB 0
(6.38)
where
ksB ≡ 3
=
Z
X
ks,ij =
j
dr
X
ks,ij
i,j
∂µL
∂ρ
s
{∇ · [rρs0 (r)]}2 .
(6.39)
MrB ≡ MsB MnB /(MnB + MnB ) is the breathing mode reduced mass moment. As follows from
(6.31), the out-of-phase breathing mode in an isotropic trap at unitarity corresponds to the
following eigenvector:
MsB as + MnB an = 0.
(6.40)
6. Hydrodynamic modes in trapped gases at unitarity
113
We note that in order to calculate the mass moments MsB and MnB , and the spring constant
ks , one needs to know the equilibrium density profiles ρs0 (r) and ρ0 (r), as well as the local
compressibility (∂µL /∂ρ)s [using a local density approximation (LDA)] as a function of temperature at unitarity. We recall from Section 4.5 that our original fluctuation calculation
of the superfluid density suffered from a bend-over problem approaching the transition temperature Tc . This makes a reliable calculation of the superfluid density profile ρs (r) within
LDA very difficult. Consequently, we use the results of two different superfluid density calculations, shown in Fig. 4.3, to calculate these profiles. This way, we obtain an estimate of
the error introduced by our approximate superfluid density profiles. The LDA calculation
of ρs (r) using the data shown in Fig. 4.3 has been carried out by Hui Hu and Xia-Ji Liu.
Details, and a plot of the density profiles, are given in Section 6.5. Using the two different superfluid density profiles, they evaluated (6.38) for the frequency of the out-of-phase
breathing mode. The results are plotted in Fig. 6.2 as a function of temperature for both of
the superfluid density profiles.
One immediately sees that the out-of-phase breathing mode frequency is quite sensitive
to the superfluid density profile. This underlines the importance of calculating this quantity
with better accuracy. However, the broad features of the temperature dependence of ω2B
are the same for both density profiles. Namely, the frequency diverges at low temperatures
and decreases with increasing temperature, before increasing again as Tc is approached. In
both cases, the frequency of the out-of-phase breathing mode is greater than the frequency
of the in-phase breathing mode ω1B = 2ω0 . We emphasize that these features are all very
different from the results of He et al., reported in Ref. [45]. They find the out-of-phase
breathing mode frequency starts below the in-phase mode frequency at low temperatures,
and increases monotonically to Tc .
6. Hydrodynamic modes in trapped gases at unitarity
114
The large “up-turn” in the frequency of the out-of-phase breathing mode plotted in
Fig. 6.2 as T → 0 can be understood within our variational formalism as follows. We recall
that the Lagrangian describing the two-fluid hydrodynamics [see (5.52)] can be thought of as
describing two coupled harmonic oscillators with effective masses given by the mass moments
for the mode in question [for the breathing mode, these are given by (6.37)]. As T → 0, the
mass of the normal fluid “oscillator” goes to zero. As with two coupled harmonic oscillators,
in this limit, the small (normal fluid) mass executes a high frequency (and large amplitude)
oscillation about the heavy (superfluid) mass, which is essentially static. We emphasize
that our variational ansatz is probably not very good in this very low-temperature region.
Moreover, Landau’s two-fluid hydrodynamics is not valid in the limit of low temperatures.
6.4
Dipole modes
We also consider the in-phase and out-of-phase dipole modes at unitarity for an isotropic
trap. Recall from Section 5.3.2 that the in-phase dipole mode is a generalized Kohn mode
with frequency equal to the trap frequency. For an isotropic trap with frequency ω0 , the
frequency ω1D of this in-phase mode is
ω1D = ω0 .
(6.41)
The frequency of the out-of-phase mode is given by (5.86). For an isotropic trap, we have
(denoting this out-of-phase dipole mode frequency by ω2D )
2
=
ω2D
MsD 2
ksD
−
ω .
MrD MnD 0
(6.42)
Here, the dipole mass moments MsD and MnD are defined by (5.71), where we now use the
superscript D to denote dipole. The spring constant ksD is likewise defined in (5.72):
ksD
=
Z
dr
∂µL
∂ρ
s
∂ρs0 ∂ρs0
.
∂z ∂z
(6.43)
115
6. Hydrodynamic modes in trapped gases at unitarity
5
dipole
mode
3
D/
0
4
2
1
0
0.0
0.1
T/T
0.2
0.3
F
Figure 6.3: The frequency of the out-of-phase dipole mode in an isotropic trap as a function
of temperature. The solid line gives the breathing mode frequency calculated using the fitted
NSR data for ρs (see Fig. 4.3). The curved dashed line is obtained using the BCS mean-field
ρs (scaled to the correct Tc ). The horizontal dashed line gives the frequency of the in-phase
dipole (generalized Kohn) mode, ω1D = ω0 .
Recall from the discussion in Section 5.3.2 that this mode, analogous to (6.40) for the
isotropic breathing mode, is described by
MsD as + MnD an = 0.
(6.44)
The frequency ω2D of the out-of-phase dipole mode is plotted in Fig. 6.3 as a function of
temperature and for a number of superfluid density profiles as explained in Section 6.3.2.
Note that the expression for this out-of-phase dipole frequency in (6.42) is very similar to the
analogous expression for the out-of-phase breathing mode frequency in (6.38). This explains
why the temperature dependence of the frequencies of these two modes (plotted in Figs. 6.2
and 6.3) are so similar.
As with the out-of-phase breathing mode, the calculation of the out-of-phase dipole
mode frequency requires us to calculate the superfluid and total density profiles as well as
6. Hydrodynamic modes in trapped gases at unitarity
116
the isentropic compressibility within LDA.
6.5
Calculation of the isentropic compressibility and
superfluid density at unitarity
This section briefly describes the numerical calculations of Hui Hu and Xia-Ji Liu for the
out-of-phase breathing and dipole modes. Further discussion can be found in Taylor, Hu,
Liu, and Griffin [58]. The local isentropic compressibility (∂µL /∂ρ)s = (∂µ/∂n)s is extracted
from the equation of state at unitarity as follows. For convenience, we assume that that the
volume V = 1. Our starting point is the LDA expression for the “local” (uniform gas)
chemical potential µL,unif (Hu, Liu, and Drummond [50]),
µL = µL,unif [ρ0 (r), T /TF (r)] + Vext (r),
(6.45)
where µL is the chemical potential at the centre of the trap (r = 0). µL,unif is the chemical
potential calculated for a uniform Fermi superfluid, as described is Section 3.2, but as a
function of the local density ρ0 (r), which is nonuniform. Comparison with (6.9) gives us the
following result:
µL,unif =
∂U
∂ρ
.
(6.46)
s
Once µL,unif is calculated as a function of ρ0 and T , (6.45) can then be inverted to give
the equilibrium total density profile ρ0 (r). Using (6.1), we can write (6.46) for µL,unif (at
unitarity) as
µL,unif =
5 ǫF (ρ)
fE (T ′ ) ≡ ǫF (ρ)fµ (T ′ )
3 m
(6.47)
The local chemical potential µL,unif and hence the constant fµ (T ′ ) is found by solving the
gap and number equations in (3.42) and (3.44) self-consistently. These calculations were
carried out by Hui Hu and Xia-Ji Liu. Within LDA, one also finds a superlfuid transition
6. Hydrodynamic modes in trapped gases at unitarity
117
temperature of Tc ≃ 0.27TF , as determined by the temperature at which the order parameter
∆0 [ρ(0)] at the centre of the trap goes to zero.
From (6.1) and (6.2), we see that the local chemical potential5 µL and the entropy
density s can be expressed as functions of the density ρ and the reduce temperature T ′ ≡
T /TF . Using (6.1), we obtain the following expression for the fluctuation dµL of the chemical
potential:
dµL
2 ∂2U
∂ U
=
dT ′
dρ +
2
∂ρ T ′
∂ρ∂T ′
2 ǫF
=
fµ dρ + ǫF fµ′ dT ′ ,
3 ρ
(6.48)
where fµ′ ≡ dfµ /dT ′ . Using (6.2), we similarly find (for the fluctuation in entropy)
ds = kB fs dρ + ρkB fs′ dT ′ ,
(6.49)
where we have absorbed a factor of 1/m into the constant fs and fs′ ≡ dfs /dT ′. Using this
expression, we see that at constant entropy (ds = 0),
dT ′ =
dρ fs
.
ρ fs′
(6.50)
Substituting this equation into (6.48), we find
∂µL
∂ρ
s
ǫF
=
ρ
fµ′ fs
2
fµ − ′ .
3
fs
(6.51)
The entropy and hence fs is calculated from the thermodynamic potential Ω in (3.41) using
s = − (∂Ω/∂T )µ . Equation 6.51 is calculated for a uniform superfluid as a function of the
density ρ0 and temperature T . Using the value of ρ0 (r) obtained from (6.45), one obtains
a spatially-dependent compressibility that is used in the evaluation of the various spring
constants.
5
We drop the subscript “unif”, but this is still the chemical potential for a uniform superfluid.
6. Hydrodynamic modes in trapped gases at unitarity
118
Figure 6.4: Superfluid density profiles within LDA using the data for a uniform superfluid
in Fig. 4.3 at T = 0.2TF .
The superfluid density profile ρs (r) is calculated using the superfluid fraction f ≡ ρs0 /ρ0
for a uniform superfluid, and then using the density profile ρ0 (r) calculated using LDA
(discussed above) to obtain
ρs0 (r) ≡ f ρ0 (r).
(6.52)
As discussed in Sections 4.5 and 6.3.2, we cannot use our NSR fluctuation theory data for
the superfluid fraction plotted in Fig. 4.2 due to the bend-over problem approaching Tc .
Instead, we use the data plotted in Fig. 4.3, namely, a fit to the fluctuation data in Fig. 4.2
that removes the bend-over and a scaled mean-field BCS superfluid fraction. The resulting
density profiles are plotted in Fig. 6.4.
Chapter 7
The two-fluid hydrodynamic density
response function
In previous chapters, we have discussed the Landau two-fluid modes in trapped superfluid
Fermi gases and evaluated the normal mode frequencies at unitarity for an isotropic trap.
In this chapter and Chapter 8, we address the problem of how to excite and measure the
frequencies of these two-fluid hydrodynamic normal modes. In experiments done on both
ultracold Bose and Fermi gases, a standard technique of exciting low-lying normal modes
is to modulate the harmonic trapping potential for a short period of time. For instance,
by displacing the trap and bringing it back to its equilibrium position, one excites a dipole
mode. Similarly, a single oscillation of the trap frequency between two values will excite
a breathing mode. Once the trap is modulated (and the relevant mode has been excited),
after waiting some time ∆t, the gas is released from the trap and the oscillating condensate
is imaged after a period of free expansion. Repeating this for a number of values of ∆t, the
oscillation of the condensate can be measured as a function of time, giving the excitation
frequency ω in a very direct manner.
In the case of probing two-fluid hydrodynamic normal modes in the BCS-BEC crossover,
the technique described above can only be used to study the in-phase modes. Probing the
out-of-phase modes requires a different approach. One technique that has already been used
119
7. The two-fluid hydrodynamic density response function
120
very successfully to study the excitation spectrum in Bose-condensed gases is two-photon
Bragg spectroscopy (for a review, see Ozeri, Katz, Steinhauer, and Davidson [4]; see also
Stamper-Kurn et al. [106] and Steinhauer et al. [107, 108]). As we discuss in more detail
in Chapter 8, the scattering cross-section for two-photon Bragg scattering can be shown to
be directly proportional to the imaginary part of the density response function, Imχρρ (q, ω)
(Stringari and Pitaveskii [39]; Zambelli, Pitaevskii, Stamper-Kurn, and Stringari [52]). In the
present chapter, we discuss the structure of the two-fluid density response function and derive
a variational expression that we can use to derive the contribution of low-lying hydrodynamic
modes in the two-fluid region to the density response. In Chapter 8 we give numerical results
for resonances corresponding to the dipole and breathing modes at unitarity (see Chapter 6)
and discuss the prospects for measuring these modes using Bragg scattering experiments.
7.1
Density response function for a uniform superfluid:
review
To make contact with the literature on the dynamic structure factor used in the context of
superfluid 4 He, we first review the derivation of the density response function for a uniform
superfluid in the hydrodynamic region. Our review follows closely the discussion in Chapter
14 of Griffin, Nikuni, and Zaremba [65]. In Sections 7.3 and 7.4, we show how to extend these
results to trapped superfluids by extending the variational formalism developed in Chapter 5.
The non-dissipative two-fluid equations for a superfluid discussed in Section 2.2 can be
used to calculate the density response function in the two-fluid hydrodynamic region. We
illustrate this procedure by solving the two-fluid equations in the presence of a small timedependent external potential Vpert (r, t). Equations (2.17) and (2.19) become
∂j
= −∇δP − ρ0 ∇Vpert ,
∂t
(7.1)
7. The two-fluid hydrodynamic density response function
121
∂vs
= −∇ (δµL + Vpert ) .
∂t
(7.2)
There is no static trapping potential. As in Chapter 5, Vpert is an external potential divided
by the atomic mass (i.e., Vpert ≡ Upert /m, where Upert is the actual perturbing potential).
We now repeat the calculations given in (2.20)–(2.25). In place of (2.20), we find
∂ 2 δρ
= ∇2 δP + ρ0 ∇2 Vpert .
2
∂t
(7.3)
Equation (2.25) is unchanged. Taking δP to be a function of s̄ and ρ, (7.3) becomes
∂ 2 δρ
=
∂t2
∂P
∂ρ
2
∇ δρ +
s̄
∂P
∂s̄
∇2 δs̄ + ρ0 ∇2 Vpert .
(7.4)
ρ
These reduce to the coupled equations of Chapter 2 if we set the small time-dependent
perturbation Vpert (r, t) equal to zero.
In order to model the potential induced by the Bragg laser beams (or neutron beams in
inelastic neutron scattering), we consider an external potential described by a single Fourier
component,
Vpert (r, t) = Vq,ω ei(q·r−ωt) .
(7.5)
The solutions of (7.3) and (7.4) will be plane-waves,
δρ(r, t) = δρq,ω ei(q·r−ωt) , δs̄(r, t) = δs̄q,ω ei(q·r−ωt) .
(7.6)
After substitution, we obtain two coupled algebraic equations for the fluctuations in density,
δρq,ω , and entropy density, δs̄q,ω ,
2
ω δρq,ω = q
2
∂P
∂ρ
s̄
δρq,ω + q
2
∂P
∂s̄
δs̄q,ω + q 2 ρ0 Vq,ω ,
(7.7)
ρ
and
ω 2 δs̄q,ω
" #
∂T
ρs0 2 2 ∂T
s̄ q
δρq,ω + q 2
δs̄q,ω .
=
ρn0 0
∂ρ s̄
∂s̄ ρ
(7.8)
122
7. The two-fluid hydrodynamic density response function
The solution of these equations is simple, giving
δρq,ω = ρ0 q 2
s0
ω 2 − s̄20 ρρn0
(ω 2 −
∂T
q2
∂ s̄ ρ
Vq,ω
u21 q 2 )(ω 2 − u22 q 2 )
(7.9)
and
δs̄q,ω = ρ0 q 4
s0
s̄20 ρρn0
∂T
∂ρ
s̄
(ω 2 − u21 q 2 )(ω 2 − u22 q 2 )
Vq,ω
(7.10)
where u1 and u2 are the first and second sound velocities given by the solutions of the
quadratic equation in (2.28).
Using standard linear response theory discussed in standard texts (Pines and Nozières [109];
Griffin [86]), the solution in (7.9) can be written in terms of the density response function.
One finds
δρq,ω =
χρρ (q, ω)
Vq,ω ,
V
(7.11)
where V is the sample volume. The density response function χρρ (q, ω) in (7.11) is the
Fourier transform of the retarded spatial density response function1
χρρ (r, r′ ; t) = −iθ(t)h[δ ρ̂(r, t), δ ρ̂† (r′ , 0)]i.
(7.12)
Here, the square brackets [A, B] = AB − BA denote the commutator of the density fluctuation operators δ ρ̂† and δ ρ̂ at different positions and times. θ(t) is the step function, equal
to zero for t < 0 and unity for t > 0. Performing a Fourier transform, χρρ (q, ω) in (7.11) is
given by (Griffin [86])
χρρ (q, ω) ≡
=
Z
dr e
Z
′
iq·r′
dr e
Z
∞
dt eiωt χρρ (r, r′ ; t)
−∞
1 − eβ(Em −En )
|hm|ρ̂†q |ni|2
.
Z
ω − (Em − En )
X e−βEn
m,n
1
−iq·r
(7.13)
For further details on linear response formalism, see Ch. 3 in Mahan [110] and also Ch. 7 in Altland and
Simons [70].
123
7. The two-fluid hydrodynamic density response function
Here |ni denotes the exact eigenstate of the grand-canonical Hamiltonian Ĥ, i.e., Ĥ|ni =
En |ni. ρ̂†q is the Fourier transform of the density fluctuation operator, ρ̂†q =
R
dr e−iq·r δ ρ̂† (r).
For later use, we note the linear response theory results in (7.11)-(7.13) are also valid in the
case of a static trapping potential, i.e., for a spatially nonuniform superfluid.
The imaginary part of the density response function in (7.13) is directly related to the
well-known dynamic structure factor S(q, ω) (see for example Ch. 2 of Griffin [86])2
Z ∞
Z
Z
1
′ iq·r′
−iq·r
S(q, ω) ≡
dr e
dt eiωt hδ ρ̂(r, t)δ ρ̂† (r′ , 0)i
dr e
2πNm2
−∞
1 X e−βEn
=
|hm|ρ̂†q |ni|2 δ[ω − (Em − En )].
2
Nm m,n Z
(7.14)
Comparison of (7.13) with (7.14) gives the well-known expression (0+ is a positive infinitesimal)
S(q, ω) = −
1
[nB (ω) + 1] Imχρρ (q, ω + i0+ ),
πNm2
(7.15)
where Imχρρ (q, ω + i0+ ) is the imaginary part of the density response function and nB (ω) =
(eβω − 1)−1 is the Bose distribution function. In arriving at (7.15), we have used the Dirac
identity,
1
1
= P − iπδ(x),
+
x + i0
x
(7.16)
where P is the principal part. One easily sees from the explicit expression in (7.13) that
Imχρρ (q, ω) is an odd function of ω [86],
Imχρρ (q, −ω + i0+ ) = −Imχρρ (q, ω + i0+ ).
(7.17)
Using this in (7.15), we see that
S(q, −ω) = −
2
1
nB (ω)Imχρρ (q, ω + i0+ ),
2
πNm
(7.18)
The factor N in this definition of S(q, ω) is omitted in some treatments, such as in the texts by Pines
and Nozières [109] as well as Pitaevskii and Stringari [39]. The additional prefactor factor 1/m2 is due to
our use of the mass density ρ = mn instead of the number density n.
7. The two-fluid hydrodynamic density response function
124
using the Bose identity nB (ω) = −[nB (−ω) + 1].
We can combine these results to write the imaginary part of χρρ in terms of S(q, ω) and
S(q, −ω),
Imχρρ (q, ω + i0+ ) = −πNm2 [S(q, ω) − S(q, −ω)] .
(7.19)
As we discuss in Chapter 8, two-photon Bragg scattering measures the imaginary part of
this density response function. This is in contrast to inelastic neutron scattering measurements which measure S(q, ω), which includes the “detailed balance factor” nB (ω) + 1. It is
convenient to split S(q, ω) = SS + SA into even SS and odd SA components (with respect
to ω), with
SA (q, ω) ≡
1
1
[S(q, ω) − S(q, −ω)] = −
Imχρρ (q, ω + i0+ ).
2
2
2πNm
(7.20)
SA (q, ω) is the correlation function which is measured by two-photon Bragg spectroscopy.
We now return to the results for two-fluid hydrodynamics given above. By a direct
comparison of (7.9) with (7.11), we see that the density response function for a uniform
superfluid described by the non-dissipative Landau two-fluid equations is given by
χLρρ (q, ω) = Nmq 2
ω2 − v2q2
.
(ω 2 − u21 q 2 )(ω 2 − u22 q 2 )
(7.21)
Here we have introduced a new velocity v defined by
2
v ≡
ρs0
s̄20
ρn0
∂T
∂s̄
ρ
=T
s20 ρs0
.
c̄v ρn0
(7.22)
We attach the superscript “L” to the density response function in (7.21) to emphasize that
this is the contribution to the density response given by Landau two-fluid hydrodynamics.
In the spatially uniform system we are considering, χLρρ has poles corresponding to first
and second sound. The two-fluid density response function in (7.21) was first discussed in
125
7. The two-fluid hydrodynamic density response function
detail for superfluid 4 He by Hohenberg and Martin [3, 111] and was first applied to weakly
interacting superfluid Bose gases by Gay and Griffin [40].
Expanding the two-fluid hydrodynamic response function in (7.21) in partial fractions
and using (7.16), the imaginary part of the density response function is found to be
ImχLρρ (q, ω + i0+ ) = −πNmq 2 [Z1 δ(ω 2 − u21 q 2 ) + (1 − Z1 )δ(ω 2 − u22 q 2 )].
(7.23)
Here, the weight Z1 of the first sound mode in the density response function is given by
Z1 =
u21 − v 2
.
u21 − u22
(7.24)
The relative weight of the second sound mode is found to be Z2 = 1 − Z1 . In arriving at
(7.23), we have used the identity
δ(ω 2 − ω12 ) =
1
1
δ(ω − ω1 ) −
δ(ω + ω1 ).
2ω1
2ω1
(7.25)
Using the two-fluid approximation for the density response function in (7.23), the contribution of first and second sound to the even part of S L (q, ω) for a uniform two-fluid superfluid
is given by
SAL (q, ω) =
7.2
q2
[Z1 δ(ω 2 − u21 q 2 ) + (1 − Z1 )δ(ω 2 − u22 q 2 )].
2m
(7.26)
The f -sum rule for the density response
The dynamic structure factor defined in (7.14) satisfies the well-known f -sum rule,
Z
∞
−∞
dω ωS(q, ω) =
q2
1 X e−βEn
†
2
(E
−
E
)|hm|ρ̂
|ni|
=
.
m
n
q
Nm2 m,n Z
2m
(7.27)
This gives an exact constraint on the energy spectrum of the dynamic structure factor. Since
this result is model-independent, being a consequence only of mass conservation (Pines and
Nozières [109]), this constraint is very useful as a check on theoretical and experimental
126
7. The two-fluid hydrodynamic density response function
results for S(q, ω). It has been particularly useful in inelastic neutron scattering studies of
superfluid 4 He (see Griffin [86]). Because of its importance, we review the derivation of this
f -sum rule.3
To prove (7.27), consider a microscopic Hamiltonian Ĥ of the form
Ĥ =
Z
∇2
dr ψ̂ (r) −
+ Uext (r) − µ ψ̂(r) + Ĥint ,
2m
†
(7.28)
where Ĥint is the interaction part of the Hamiltonian and Uext (r) is a static potential. The
spin index on these fermion operators has been suppressed. The exact commutator between
this Hamiltonian and ρ̂q is easily worked out, giving
∇2r′
′
′
†
† ′
+ Uext (r ) − µ ψ̂(r )
dr mψ̂ (r)ψ̂(r), ψ̂ (r ) −
[ρ̂q , Ĥ] =
dr e
2m
Z
h
i
1
−iq·r
†
†
dr e
q · ψ̂ (r)∇ψ̂(r) − ∇ψ̂ (r)ψ̂(r)
=
2i
Z
−iq·r
Z
′
≡ q · ĵq .
(7.29)
Here we have defined the Fourier transform ĵq =
operator ĵ(r),
R
dre−iq·r ĵ(r) of the (mass) current density
i
1 h †
†
ψ̂ (r)∇ψ̂(r) − ∇ψ̂ (r)ψ̂(r) .
ĵ(r) =
2i
(7.30)
We note that (7.29) is just the operator version of the continuity equation. The density
operator commutes with the interaction term in the Hamiltonian and consequently, it does
not contribute to the right-hand side of (7.29). Similarly, because [δ ρ̂(r), Uext (r′ )δ ρ̂(r′ )] = 0,
the external potential does not contribute to the commutator.
We now show how (7.29) leads to the f -sum rule. Taking the matrix element with respect
to eigenstates of Ĥ, we find
hm|[ρ̂q , Ĥ]|ni = (En − Em )hm|ρ̂q |ni = −q ·
3
X
m,n
hm|ĵq |ni.
(7.31)
Our derivation follows closely that given in Ch. 2.3 in Pines and Nozières [109], which also allows for an
external potential.
127
7. The two-fluid hydrodynamic density response function
Similarly, the double commutator [[ρq , H], ρ†q ] is easily worked out,
[[ρ̂q , Ĥ], ρ̂†q ] = [q · ĵq , ρ̂†q ]
Z
Z
h
i
m
′
−iq·r
dr′ eiq·r q · ψ̂ † (r)∇ψ̂(r) − ∇ψ̂ † (r)ψ̂(r) , ψ̂(r′ )ψ̂ † (r′ )
dr e
=
2i
= N̂mq 2 ,
where N̂ ≡
states
P
n
R
(7.32)
dr ψ̂ † (r)ψ̂(r) is the total number operator. Inserting a complete set of eigen-
|nihn| = 1 into the double commutator [[ρq , H], ρ†q ] and using (7.31), we find the
thermal average of (7.32) is
hN̂imq 2 =
X e−βEm
hm|[[ρ̂q , Ĥ], ρ̂†q ]|mi
X e−βEm
(En − Em ) hm|ρ̂q |nihn|ρ̂†q |mi + hm|ρ̂†q |nihn|ρ̂q |mi
Z
m
o
X e−βEm n
=
hm|[ρ̂q , Ĥ]|nihn|ρ̂†q |mi − hm|ρ̂†q |nihn|[ρ̂q , Ĥ]|mi
Z
m,n
=
m,n
= 2
Z
X e−βEn
m,n
Z
(Em − En )|hm|ρ̂†q |ni|2 .
(7.33)
Using (7.33) in (7.27), we have thus proven the f -sum rule.
Since a factor ω appears in the integrand, the frequency moment in (7.27) only picks up
the contribution from the odd part of the dynamic structure factor. Thus, the f -sum rule
in (7.27) can be written as
Z
∞
−∞
dω ω SA (q, ω) =
q2
.
2m
(7.34)
The two-fluid hydrodynamic dynamic structure factor in (7.26) for a uniform superfluid
exactly satisfies this f -sum rule, since by explicit calculation
Z
∞
−∞
dω ω SAL (q, ω) = [Z1 + (1 − Z1 )]
q2
q2
=
.
2m
2m
(7.35)
At first sight, this result seems surprising, since two-fluid hydrodynamics only describes the
low frequency dynamics. Thus one might have thought the high energy spectrum in SA (q, ω)
7. The two-fluid hydrodynamic density response function
128
might be important in (7.34). As noted above, the f -sum rule is a consequence of the exact
continuity equation for the quantum mechanical density operator given in (7.29). Two-fluid
hydrodynamic modes also involves a continuity equation, given by (2.1), but this is only
for coarse-grained macroscopic variables. The way to understand why SAL (q, ω) satisfies the
exact f -sum rule is to remember that our two-fluid expression in (7.26) is only valid for
small values of q, corresponding to long wavelength density fluctuations. One expects that
at these small (hydrodynamic) values of q, the exact SA (q, ω) is indeed well approximated
by SAL (q, ω), as given by (7.26) and which only has contributions at low frequencies ω = ui q.
The key point is that the two-fluid approximation for SA (q, ω) only satisfies the exact f -sum
rule as in (7.35) for small values of q. Another way of saying this is that at small q, the
macroscopic continuity equation of Landau two-fluid hydrodynamics is a good approximation
to the exact continuity equation on which the f -sum rule is based.
In Section 7.5, we discuss the f -sum rule in trapped superfluids using SAL (q, ω) as given
by our variational formalism. The f -sum rule provides a useful way of determining the
relative weights of the normal modes excited by two-photon Bragg spectroscopy as one
sweeps through the spectrum of values for q and ω. If a particular normal mode (e.g., the
out-of-phase dipole mode) exhausts the f -sum rule for a given value of q and ω, that tells
us that Bragg scattering at this wavevector and frequency will only excite this mode. This
is important for trapped gases since, unlike a uniform superfluid where the collective modes
are plane waves characterized by the wavevector q, there are many types of normal modes
that can be excited for a given momentum transfer from a Bragg pulse.
129
7. The two-fluid hydrodynamic density response function
7.3
Density response function for trapped superfluid
Fermi gases
In Section 7.1, we reviewed the density response function in uniform Bose superfluids. We
solved the linearized two-fluid hydrodynamic equations to find the induced density fluctuation δρ when a perturbing potential Vpert is applied to the system. For a uniform hydrodynamic system with a plane-wave perturbing potential, this is a straightforward procedure
since the coupled hydrodynamic equations admit plane-wave solutions. In contrast, for
trapped gases, such analytic solutions do not exist in general. This same difficulty provided
the original motivation for our variational formulation of Landau’s two-fluid hydrodynamics
given in Chapter 5. In this section, we extend the variational approach and show how it can
be used to also calculate the density response function in trapped superfluid gases. This is
a major result of this thesis.
For a perturbing potential Vpert (r, t) that is both time and position-dependent, the corresponding term in the fluctuation Hamiltonian is
Hpert =
Z
dr Vpert (r, t)δρ(r, t).
(7.36)
Treating the perturbation as small, there will be a density response of the form given by
(7.11) given in terms of the density response function χρρ . Our task is to determine the
structure of χρρ which is imposed by the Landau two-fluid equations in the presence of both
Vpert r, t) and the trap potential Vext (r).
As we showed in Chapter 5, the dynamics of the two-fluid hydrodynamic fluctuations
(2)
can be described using an action given by (5.47). We now label this action S0 , using the
subscript “0” to denote that this is the hydrodynamic action in the absence of the small timedependent perturbing potential Vpert . To this action, we add the perturbation contribution
7. The two-fluid hydrodynamic density response function
130
R
(2)
Spert = − dt Hpert from (7.36):
(2)
(2)
S (2) [us , un ] ≡ S0 [us , un ] + Spert [us , un , Vpert ],
(7.37)
Z
(7.38)
where
(2)
Spert
=
drdt Vpert (r, t)∇ · [ρs0 (r)us (r, t) + ρn0 (r)un (r, t)] .
Following (5.42), we have expressed δρ in terms of the displacement fields us and un . The
solutions of the two-fluid equations for hydrodynamic fluctuations are given by the EulerLagrange equations in (5.48). Using uLs and uLn to denote the solutions of the two-fluid
equations, we rewrite these equations more explicitly as
δS (2) [us , un ] = 0,
δus
L
us =uL
,u
=u
n
s
n
δS (2) [us , un ] = 0,
δun
L
L
(7.39)
us =us ,un =un
where S (2) is given by (7.37). This way, we can write the density fluctuation solution δρL of
the two-fluid equations as
δρL (r, t) = −∇ · ρs0 (r)uLs (r, t) + ρn0 (r)uLn (r, t) .
(7.40)
We again use the superscript “L” on δρ to denote the fact that this density fluctuation is
required to be a solution of the Landau two-fluid equations that include the effects of the
potential Vpert . Thus, solving (7.39) using the action in (7.37) which includes the contribution
from the perturbing potential Vpert (r, t) will give the two-fluid hydrodynamic density response
using (7.40). In Section 7.4, we introduce a variational ansatz for the displacement fields
us , un for breathing and dipole modes, to obtain an explicit expression for the two-fluid
density response function χLρρ .
131
7. The two-fluid hydrodynamic density response function
7.4
Variational formulation
We now discuss the variational solutions of (7.40) for the contribution of the dipole and
breathing modes to the two-fluid density response functions for a trapped Fermi superfluid.
To keep the discussion simple, we restrict our analysis to an isotropic trap, but the method
can easily be extended to anisotropic traps used in experiments. The breathing and dipole
modes in an isotropic trap have the simplifying feature that the ansatz for the displacement
fields, given in (5.56), can be expressed in terms of a single vector component. Each mode
can be described using two variational parameters (as , an ), instead of six.
We recall that the normal mode solutions of the linearized two-fluid equations have a
harmonic time-dependence [see (5.51)]. However, for our derivation of the density response
function χLρρ (q, ω) for the low-lying two-fluid normal modes, it is convenient to allow these
fields to have arbitrary time-dependence represented by a Fourier series,
us (r, t) =
X
e−iωt as (ω)f (r)û, un (r, t) =
ω
X
e−iωt an (ω)g(r)û.
(7.41)
ω
Here û is a unit vector giving the displacement direction. For the breathing modes in an
isotropic trap, we choose û = r̂. For the dipole and plane-wave collective modes (along
the z-direction), we choose û = ẑ. Since us and un must be real functions of t, we also
require that the variational parameters satisfy a(−ω) = a(ω). Substituting (7.41) into the
unperturbed action given by (5.47), we obtain
(2)
S0 =
1X T
Λ (ω)A(ω)Λ(ω),
2 ω
(7.42)
where the transpose of the spinor Λ is ΛT (ω) ≡ [as (ω), an (ω)]. The 2 × 2 matrix A is given
by
A(ω) ≡
M̃s ω 2 −ks
−ksn
.
−ksn
M̃n ω 2 −kn
(7.43)
132
7. The two-fluid hydrodynamic density response function
The mass moments M̃s , M̃n , and spring constants ks , ksn , and kn , are found to be given by
Z
M̃s ≡
2
dr ρs0 (r)f (r), M̃n ≡
Z
dr ρn0 (r)g 2(r),
(7.44)
and
ks =
kn =
ksn =
Z
Z
dr
∂µL
∂ρ
∂µL
∂ρ
∂µL
∂ρ
s
{∇ · [ûρs0 (r)f (r)]}2 ,
2
∂T
∂ρ
(7.45)
{∇ · [ûρn0 (r)g(r)]} + 2
{∇ · [ûρn0 (r)g(r)]}
s
∂T
2
× {∇ · [ûs0 (r)g(r)]} +
{∇ · [ûs0 (r)g(r)]} ,
(7.46)
∂s ρ
Z
dr
s
{∇ · [ûρs0 (r)f (r)]} {∇ · [ûρn0 (r)g(r)]}2
dr
s
∂T
+
{∇ · [ûρs0 (r)f (r)]} {∇ · [ûs0 (r)g(r)]} .
∂ρ s
(7.47)
The action given by (7.42), together with the mass moments M̃s , M̃n defined in (7.44) and
the spring constants defined in (7.45)-(7.47), describe the in-phase and out-of-phase dipole
(û = ẑ) and breathing modes (û = r̂) in an isotropic trap as well, as first and second sound
(û = ẑ) in a uniform superfluid. Our ansatz for the functions f (r) and g(r) is the same as
we used in Chapter 5. The variational equations of the Landau two-fluid equations given by
(7.39) reduce to
∂S (2) ∂S (2) =
= 0,
∂as as =aLs (ωL )
∂an an =aLn (ωL )
(7.48)
where aLs , aLn describe the normal mode solutions of the Landau two-fluid equations with
frequency ωL .
Using (7.41) in (7.38), the perturbation contribution to the action is
(2)
Spert
=
XZ
ω
drdt Vpert (r, t)e−iωt ∇ · [ûρs0 (r)f (r)as (ω) + ûρn0 (r)g(r)an (ω)] + H.c., (7.49)
133
7. The two-fluid hydrodynamic density response function
where “H.c.” denotes the Hermitian conjugate.4 To simulate the electric dipole potential
due to Bragg laser beams, we take this perturbing potential to be a plane-wave,
Vpert (r, t) = Vq,Ω ei(q·r−Ωt) ,
(7.50)
where Ω is the energy imparted by the Bragg pulse (see Chapter 8). Substituting this into
(7.49), we obtain
(2)
Spert
= Vq,Ω
Z
dreiq·r ∇ · [ûρs0 (r)f (r)as (Ω) + ûρn0 (r)g(r)an(Ω)] + H.c.
≡ Vq,Ω [Fs∗ (q)as (Ω) + Fn∗ (q)an (Ω)] + H.c.,
(7.51)
where we have introduced the useful functions
Fs (q) ≡
Z
−iq·r
dre
∇ · [ûρs0 (r)f (r)] , Fn (q) ≡
Z
dre−iq·r ∇ · [ûρn0 (r)g(r)] .
(7.52)
In terms of the spinor Λ, the perturbation action in (7.51) can be written as
(2)
Spert = Vq,Ω F† (q)Λ(Ω) + H.c.,
= Vq,Ω F† (q)Λ(Ω) + Λ† (Ω)F(q) ,
(7.53)
where
F† (q) ≡ [Fs∗ (q), Fn∗(q)] .
(7.54)
Note that the F factors defined in (7.52) have an imaginary component and consequently,
F is a column vector with Fs and Fn as elements.
(2)
(2)
Using (7.42) and (7.53), we can write the total action S (2) = S0 + Spert as
S (2) =
4
1X T
Λ (ω)A(ω)Λ(ω) + Vq,Ω F† (q)Λ(Ω) + Vq,Ω ΛT (Ω)F(q).
2 ω
(7.55)
We need to add the conjugate since our perturbing potential will be a complex-valued plane wave
function.
134
7. The two-fluid hydrodynamic density response function
Applying the stationarity condition in (7.48) to this action, we find this corresponds to
A(Ω)ΛL (Ω) + Vq,Ω F(q) = 0,
(7.56)
where ΛTL (Ω) ≡ (aLs , aLn ) is a solution of the two-fluid equations as given by (7.48). More
explicitly, (7.56) involves two coupled equations for aLs and aLn ,
A11 (Ω)aLs + A12 (Ω)aLn + Vq,Ω Fs∗ (q) = 0,
(7.57)
A12 (Ω)aLs + A22 (Ω)aLn + Vq,Ω Fn∗ (q) = 0,
(7.58)
where Aij denotes the ij matrix element of the 2 × 2 matrix A. Making use of the standard
matrix identity A−1 A = 1, we can write (7.56) as
ΛL (Ω) = −Vq,Ω A−1 (Ω)F(q).
(7.59)
We can now put things together to obtain an explicit expression for the hydrodynamic density
response function in a trapped gas. Using the definition of δρL (r, t) in Eq (7.40), one can
show that
δρLq,Ω
Z
drdt ei(q·r−Ωt) ρL (r, t)
Z
= − drdt ei(q·r−Ωt) ∇ · ρs0 (r)uLs (r, t) + ρn0 (r)uLn (r, t)
Z
XZ
iq·r
dt e−i(Ω−ω)t ∇ · ûρs0 f (r)aLs (ω) + ûρn0 (r)aLn (ω)
= − dr e
≡
ω
= −
Fs∗ (q)aLs (Ω)
+ Fn∗ (q)aLn (Ω) .
(7.60)
Thus we have shown that
δρLq,Ω = −F† (q)ΛL (Ω)
= F† (q)A−1 (Ω)F(q)Vq,Ω .
(7.61)
7. The two-fluid hydrodynamic density response function
135
Comparing (7.61) with (7.11), the two-fluid hydrodynamic density response function χLρρ (q, ω)
is given by5
χLρρ (q, ω) = F† (q)A−1 (ω)F(q).
(7.62)
More explicitly, performing the matrix multiplication, this result corresponds to
χLρρ (q, ω) =
h
1
|Fs (q)|2 A22 (ω) − Fs∗ (q)Fn (q)A12 (ω)
detA(ω)
i
−Fn∗ (q)Fs (q)A12 (ω) + |Fn (q)|2 A11 (ω) ,
(7.63)
where
detA(ω) = A11 (ω)A22 (ω) − A212 (ω).
(7.64)
Equation (7.63) gives us an explicit variational expression for the two-fluid density response function associated with the dipole and breathing modes in an isotropic trap. Since
we are dealing with linear response theory, the linear density response function defined in
(7.11) must be independent of Vq,Ω . That is, the matrix elements of A−1 (ω) should be expressed in terms of the frequencies of the two-fluid modes in the absence of the perturbing
potential. Setting Vq,Ω = 0 in (7.56), we obtain the following equation determining the
frequencies and variational parameters which describe these modes:
A(ω)ΛL(ω) = 0.
(7.65)
Using a standard result of linear algebra, the solutions of this set of coupled linear equations
for the two-fluid mode frequencies ωL are given by
detA(ω = ωL ) = 0.
5
We set V = 1.
(7.66)
136
7. The two-fluid hydrodynamic density response function
Since the density response function in (7.63) is proportional to [detA(ω)]−1, (7.66) shows
that the linear density response function will have poles at the unperturbed two-fluid hydrodynamic mode frequencies ωL , as required.
As a check of our formalism, it is useful to consider a uniform superfluid with a plane-wave
ansatz for the displacement fields, given by [see (5.58)]
f (r) = g(r) = cos(qz).
(7.67)
Using this in (7.44)-(7.47), the matrix elements of A defined in (7.43) reduce to
∂µL
2
2
A11 (ω) = ρs0 ω − q ρs0
,
∂ρ s
A12 (ω) = q
2
∂µL
∂ρ
ρs0 ρn0 +
s
∂T
∂ρ
s
ρs0 s0 ,
(7.68)
(7.69)
and
∂T
∂T
∂µL
2
2
ρn0 + 2
ρs0 +
s0 .
A22 (ω) = ρn0 ω − q
∂ρ s
∂ρ s
∂s ρ
(7.70)
Using these, the determinant of A becomes
detA(ω) ≡ A11 (ω)A22 (ω) − A212 (ω) = ρs0 ρn0 (ω 2 − u21 q 2 )(ω 2 − u22 q 2 ),
(7.71)
where we have introduced the first and second sound velocities as given by the solution of
(2.28). Substituting (7.68)-(7.70) into (7.63), we finally obtain
χLρρ (q, ω) = Nmq 2
ρs0
ω 2 − q 2 s20 ρn0
ρ0
(ω 2 −
∂T
∂s ρ
.
u21 q 2 )(ω 2 − u22 q 2 )
(7.72)
Using s̄ ≡ s/ρ and (∂T /∂s)ρ = ρ−1 (∂T /∂s̄)ρ , we see that this density response function
given by our variational method is identical to the result in (7.21) and (7.22), obtained by
direct solution of the linearized Landau two-fluid equations for a uniform superfluid.
7. The two-fluid hydrodynamic density response function
7.5
137
The contribution to the f -sum rule from two-fluid
normal modes
In this section, we use our variational expression for the two-fluid density response function
in (7.63) to derive the contribution to the f -sum rule in (7.34) from the two-fluid normal
modes we are considering. As discussed in Section 7.2, we expect other hydrodynamic
normal modes to contribute to the f -sum rule, in addition to the dipole and breathing
modes. This motivates us to define a quantity γq that represents the contribution of a
particular hydrodynamic mode to the f -sum rule for a given momentum transfer q.
It is convenient to define the frequency moment hωiq (see, for instance, Pitaevskii and
Stringari [39]),
hωiq ≡
Z
+∞
dω ωSA (q, ω) =
−∞
q2
.
2m
(7.73)
This moment hωiq can be expressed in terms of the high-frequency limit of the density
response function (see Ch. 2.2 of Griffin [86]). Using the definition of the density response
function in (7.13) and the Dirac identity in (7.16), it is straightforward to prove the spectral
relation
χρρ (q, ω) = −
∞
Z
dω ′
−∞
Imχρρ (q, ω)
.
ω − ω′
(7.74)
Inserting (7.15) into this relation and using (7.17), we obtain
2
χρρ (q, ω) = 2Nm
Z
∞
ω ′ SA (q, ω ′)
dω
.
ω 2 − ω ′2
−∞
′
(7.75)
In the high-frequency limit, this reduces to [using (7.73)]
lim χρρ (q, ω) =
ω→∞
Nmq 2
2Nm2
hωi
=
.
q
ω2
ω2
This result is just an alternative way of expressing the f -sum rule in (7.27).
(7.76)
7. The two-fluid hydrodynamic density response function
138
Using our expression for the matrix A(ω) [in (7.43)] which describes the two-fluid dipole
and breathing modes, we find
A11 (ω)
1
,
=
ω→∞ detA(ω)
M̃n ω 2
lim
A22 (ω)
1
,
=
ω→∞ detA(ω)
M̃s ω 2
lim
(7.77)
and A12 (ω)/detA(ω) ∼ ω −4. Combining these expressions with (7.63) and (7.76), we arrive
at the following result (valid for both the dipole and breathing modes in a trapped Fermi
superfluid):
1
|Fs (q)|2 |Fn (q)|2
hωiq =
+
≡ γq ,
2Nm2
M̃s
M̃n
(7.78)
where Fs , M̃s , Fn , and M̃n are defined in (7.44) and (7.52). Clearly, γq represents the contribution to the f -sum rule in (7.34) from the two-fluid hydrodynamic modes characterized
by the displacement fields f (r) and g(r) in (7.41). The value of γq tells us how much a particular mode is excited for a given momentum transfer q from a Bragg pulse. In Section 8.3
of Chapter 8, we give explicit numerical results for γq for the dipole and breathing two-fluid
modes in a trapped Fermi superfluid in an isotropic trap at unitarity.
Chapter 8
Bragg scattering from two-fluid
hydrodynamic modes at unitarity
In this chapter, we present and discuss the results of explicit calculations of the two-fluid
hydrodynamic density response function for the lowest energy dipole and breathing modes
discussed in Chapter 6. Using the variational formalism developed in Chapter 7, we compute
the dynamic structure factor for these modes. We also review two-photon Bragg scattering
and discuss the use of this well established technique to measure the two-fluid hydrodynamic
modes.
8.1
A brief review of two-photon Bragg scattering
In Bragg scattering experiments (see Sec. 12.9 of Pitaevskii and Stringari [39]), the superfluid
gas is subjected to two laser beams, the difference in their wavevector and frequency being
denoted by q ≡ k1 − k2 and Ω ≡ ω1 − ω2 respectively. The atoms in the trapped superfluid
can absorb a photon from one beam and coherently re-emit into the other. In this process,
energy Ω and momentum q is transferred to the superfluid gas.1 In the limit of an infinitely
long Bragg pulse duration, this two-photon scattering process only happens if the values of
Ω and q coincide with the spectrum of the (undamped) excitations of the superfluid. In
experiments, the resonance width is broadened since the Bragg “pulse” is only applied for a
1
Recall that we have set ~ = 1.
139
8. Bragg scattering from two-fluid hydrodynamic modes at unitarity
140
finite amount of time ∆t. To simulate the effect of such a pulse, we use
Vpert (r, t) =
Vq,Ω ei(q·r−iΩt) for 0 < t < ∆t
.
0
otherwise
(8.1)
This is identical to the potential used in (7.50) except that now it is “turned on” at t = 0
and “turned off” at t = ∆t. Using (8.1), it is straightforward to show that the rate of energy
transferred to the superfluid as a function of the pulse duration is given by (Pitaevskii and
Stringari [39])
dE(q)
2
= Vq,Ω
d(∆t)
Z
∞
−∞
dω SA (q, ω)
sin[(Ω − ω)∆t]
.
Ω−ω
(8.2)
This shows how the energy transferred to the superfluid is directly related to a frequency
integral over the dynamic structure factor SA (q, ω) introduced in Chapter 7 [see (7.20)]. In
experiments, it is usually the momentum transferred to the superfluid rather than the energy
that is measured. For further discussion, see Ozeri et al. [4] and Stringari and Pitaveskii [39].
In the limit of a very long Bragg pulse (Ω∆t ≫ 1), sin[(Ω−ω)∆t]/(Ω−ω) → πδ(Ω−ω) and
rate of energy transferred given in (8.2) is directly given by SA (q, Ω), with sharp resonances
at the hydrodynamic frequencies. For finite pulse durations, the resonance width is of the
order ∆ω ∼ 1/∆t. The frequency spacing between the normal modes of trapped superfluids
is of the order of the trap frequency ω0 (see Chapter 6). This means that to clearly resolve
the hydrodynamic normal mode frequencies, the pulse duration should be (ideally) much
larger than the inverse of the trap frequency, ω0 ∆t ≫ 1.
In trapped atomic gases, one can identify several regions of energy and momentum where
different kinds of excitations will dominate the response of the system (Zambelli et al. [52]).
For large enough values of q, the response will be dominated by “free atom” excitations,
ω = q 2 /2m, where m is the Fermi atom mass. At smaller values of q, but in the region where
qRT F ≫ 2π (RT F being the smallest Thomas-Fermi radius of the trapped gas), one expects
8. Bragg scattering from two-fluid hydrodynamic modes at unitarity
141
to excite phonon excitations ω = uq similar to those found in a uniform system since the
wavelength λ = 2π/q of the excitations is much smaller than the spatial size of the condensate. If this q region is described by collisional hydrodynamics (ωτ ≪ 1), these phonons
will correspond to first (ω = u1 q) and second (ω = u2 q) sound in a uniform superfluid.
Otherwise, this q region describes collisionless excitations (discussed in Section 3.3.2).
An additional region of q space relevant in highly anisotropic traps is where the wavelength of collective excitations is small compared to the axial length of the condensate
(qRT F,z ≫ 2π), but comparable to the radial width of the condensate (qRT F,⊥ ∼ 2π).
In this case, the axial sound modes couple into the radial normal modes (e.g., the radial
breathing and quadrupole modes) of the condensate, leading to a Bogoliubov spectrum with
several branches (Steinhauer et al. [108]).
At lower values of q, such that qRT F,z . 2π, Bragg scattering will mainly excite the low
energy normal modes characteristic of trapped gases. In this chapter, we reserve the term
“normal modes” to refer to these types of modes, and not the collective modes of a uniform
system with small wavelengths which can be excited in trapped gases via Bragg scattering
using large momentum transfers. Specifically, we are interested in the lowest energy modes
of trapped superfluid gases in the collisional hydrodynamic region described by the two-fluid
equations.
In experiments done so far on Bose-condensed gases, Bragg scattering has only been used
in the large q region (qRT F ≫ 2π) in order to probe the phonon excitations characteristic
of uniform (bulk) systems (Ozeri et al. [4]; Stamper-Kurn et al. [106]). Bragg scattering
has not been used to study the low-lying normal modes in the region qRT F,z . 2π, since
these modes can be excited by simply modulating the harmonic trap. However, as we show
in Section 8.4, this kind of modulation technique does not excite the interesting out-of-
8. Bragg scattering from two-fluid hydrodynamic modes at unitarity
142
phase two-fluid hydrodynamic modes in the region near unitarity. For these modes, Bragg
spectroscopy seems potentially very useful and we hope that our predictions (Taylor, Hu,
Liu, and Griffin [58]) will encourage experimental studies of this kind.
8.2
Dynamic structure factor for hydrodynamic normal modes at unitarity
We now use the results given in Section 7.4 and calculate the weights of the in-phase and
out-of-phase dipole and breathing two-fluid modes in SAL (q, ω) at unitarity in an isotropic
trap. The frequencies of these modes were calculated in Chapter 6.
8.2.1
Dipole modes
The dipole mode discussed in Sections 5.3.2 and 6.4 is characterized by displacements of the
centre-of-masses of the two fluids along one of the axes of the harmonic trap (we take this to
be the z-axis). In this case, we use the following ansatz for the displacement fields in (7.41)
[see also (5.68)]:
f (r) = g(r) = 1.
(8.3)
Using this in (7.44)-(7.47), and making use of (5.81), the matrix elements of A reduce to
A11 (ω) = MsD ω 2 − ksD , A12 (ω) = ksD − MsD ω02 ,
(8.4)
A22 (ω) = MnD ω 2 − (MnD − MsD )ω02 − ksD .
(8.5)
and
Thus, the determinant of A becomes
2
detA = MsD ω 2 − ksD MnD ω 2 − (MnD − MsD )ω02 − ksD − ksD − MsD ω02
MsD 2
ksD
ksD 2 MsD 4
2
D
D
4
2
= Ms Mn ω − ω ω0 − D ω0 + D + D ω0 − D ω0
Mn
Mr
Mr
Mn
2
2
2
= MsD MnD ω 2 − ω1D
ω − ω2D
.
(8.6)
8. Bragg scattering from two-fluid hydrodynamic modes at unitarity
143
Here ω1D and ω2D are the frequencies of the in-phase and out-of-phase dipole oscillations
given by (6.41) and (6.42).
Substituting these results into (7.63), the density response function for the dipole modes
is given by
χLρρ (q, ω) =
2
(ω 2 −ω02 ) |FsD (q)|2 MnD +|FnD (q)|2 MsD +(MsD ω02 −ksD ) FsD (q)+FnD (q)
, (8.7)
2
2
MsD MnD (ω 2 − ω1D
)(ω 2 − ω2D
)
where we have defined
FsD (q)
≡
Z
−iq·r ∂ρs0
dr e
∂z
,
FnD (q)
≡
Z
dr e−iq·r
∂ρn0
.
∂z
(8.8)
Following the same procedure leading to the result in (7.26) and using the identity [from
(5.81)]
2
2
MsD ω02 − ksD = MrD ω1D
− ω2D
,
(8.9)
we find the odd part of the two-fluid dynamic structure factor is
SAL (q, ω)
D
|Fs (q)|2 |FnD (q)|2 1
2
2
+
Z1 (q)δ(ω 2 − ω1D
) + [1 − Z1 (q)] δ(ω 2 − ω2D
)
=
2
D
D
2Nm
Ms
Mn
2
2
= γqD Z1 (q)δ(ω 2 − ω1D
) + [1 − Z1 (q)] δ(ω 2 − ω2D
) .
(8.10)
It is useful to compare this result with SAL (q, ω) for a uniform gas in the two-fluid region give
in (7.26). In the present case, the q dependence only enters into the weight of the dipole
modes, not the frequencies.
The weight Z1 (q) of the in-phase (generalized Kohn) mode in (8.10) is given by
D
F (q) + F D (q)2
s
n
,
(8.11)
Z1 (q) ≡ D
|Fs (q)|2 MnD /MrD + |FnD (q)|2 MsD /MrD
while the relative weight of the out-of-phase mode is found to be Z2 (q) ≡ 1 − Z1 (q). The
contribution γqD of the dipole modes to the f -sum rule in (7.34) [using the ansatz in (8.3)] is
" R
R
#
2
dr eiqz ns0 (r)2
dr eiqz nn0 (r)2
q
1
1
R
R
γqD ≡
+
.
(8.12)
2m N
N
dr ns0 (r)
dr nn0
8. Bragg scattering from two-fluid hydrodynamic modes at unitarity
144
In obtaining this, the functions Fn,s in (8.8) have been integrated by parts. We observe that
the weight factors Z1 (q) and γqD only involve spatial integrals over the superfluid and normal
fluid densities [ρs0 (r) and ρn0 (r)]. The numerical calculation of these factors is very similar
to that for the effective masses, as discussed in Section 6.5.
8.2.2
Breathing modes
In this section, we use the results in Section 6.3 to derive the contribution of the two-fluid
breathing modes in an isotropic trap at unitarity to the dynamic structure factor SAL (q, ω).
For the breathing modes in an isotropic trap, we use the ansatz discussed earlier,
f (r) = g(r) = r.
(8.13)
Substituting this ansatz into (7.44), MsB and MnB are given by (6.37). Using the identity in
(6.23), we can write
X
Kijs = 2
ij
X
M̃si [2δij + 2/3] ω02 = 8MsB ω02.
(8.14)
ij
From the definition in (5.99), we also have
X
Kijs = 2
ij
X
B
[ks,ij + ksn,ij ] = 2ksB + 2ksn
,
(8.15)
ij
B
where ksB and ksn
are given by (7.45) and (7.47) [with û = r̂ and f (r) = g(r) = r]. The
spring constant ksB is given explicitly by (6.39):
ksB
≡
Z
dr
∂µ
∂ρ
s
{∇ · [rρs0 (r)]}2 .
(8.16)
Combining (8.14) and (8.15), we obtain the following relation at unitarity:
B
ksn
= 4MsB ω02 − ksB .
(8.17)
Similarly, using (6.24), (5.100), and (8.17), it is straightforward to show that
B
knB = 4MnB ω02 − ksn
= 4(MnB − MsB )ω02 + ksB .
(8.18)
145
8. Bragg scattering from two-fluid hydrodynamic modes at unitarity
Substituting these results into (7.43), the matrix elements of A are easily calculated:
A11 (ω) = MsB ω 2 − ksB , A12 (ω) = ksB − 4MsB ω02,
(8.19)
A22 (ω) = MnB ω 2 − 4(MnB − MsB )ω02 − ksB .
(8.20)
and
The determinant of A is
2
detA = MsB ω 2 − ksB MnB ω 2 − 4(MnB − MsB )ω02 − ksB − ksB − 4MsB ω02
ksB
ksB 2
MsB 4
MsB 2
2
4
2
B
B
= Ms Mn ω − ω 4ω0 − 4 B ω0 + B + 4 B ω0 − 16 B ω0
Mn
Mr
Mr
Mn
2
2
2
= MsB MnB ω 2 − ω1B
ω − ω2B
.
(8.21)
Here ω1B = 2ω0 and ω2B (T ) are the frequencies of the in-phase and out-of-phase breathing
mode oscillations in an isotropic trap at unitarity, given by (6.35) and (6.38), respectively.
Substituting these results into (7.63), the density response function for the two-fluid
breathing modes is
2
B
B
2
2
2
B
B
2
B
B 2
B B
F
(q)+F
(q)
(ω
−ω
)
|F
(q)|
M
+|F
(q)|
M
+(4M
ω
−k
)
s
n
1B
s
n
n
s
s
0
s
χLρρ (q, ω) =
,
2
2
MsB MnB (ω 2 − ω1B
)(ω 2 − ω2B
)
(8.22)
where [see (7.52)],
FsB (q)
≡
Z
−iq·r
dr e
∇ · [rρs0 (r)] ,
FnB (q)
≡
Z
dr eiq·r ∇ · [rρn0 (r)] .
(8.23)
These results lead to the dynamic structure factor,
2
2
SAL (q, ω) = γqB Z1 (q)δ(ω 2 − ω1B
) + [1 − Z1 (q)] δ(ω 2 − ω2B
) ,
(8.24)
where we have made use of the identity
4MsB ω02
−
ksB
=
MrB
= MrB
MB
kB
− sB + 4 sB ω02
Mr
Mn
2
2
ω1B
− ω2B
.
4ω02
(8.25)
8. Bragg scattering from two-fluid hydrodynamic modes at unitarity
146
The result for SAL (q, ω) in (8.24) for the breathing modes has the same form as that for the
dipole modes given in (8.10).
The weight of the in-phase breathing mode is now given by [the equivalent of (8.11)]
B
F (q) + F B (q)2
s
n
.
Z1 (q) ≡ B
|Fs (q)|2 MnB /MrB + |FnB (q)|2 MsB /MrB
(8.26)
The relative weight of the out-of-phase mode is given by Z2 (q) ≡ 1−Z1 (q). The contribution
γ B (q) to the f -sum rule from the breathing modes is given by [compare with (8.12) for dipole
modes]
8.3
" R
#
R
2
dr eiqz zns0 (r)2
dr eiqz znn0 (r)2
q
1
1
R
R
γqB ≡
.
+
2m N
dr ns0 (r)
N
dr nn0
(8.27)
Two-fluid resonances as a function of q and T
In this section, we present numerical results for the two-fluid dynamic structure factor
SAL (q, Ω) given in (8.10) and (8.24). These expressions have been evaluated by H. Hu and
X.-J. Liu. As a representative example, all results in this section are based on the fitted
NSR data for ρs (see Fig. 4.3). For the frequencies of the breathing and dipole modes, this
corresponds to the solid lines in Figs. 6.2 and 6.3.
As discussed earlier in Section 8.1, the response SAL (q, Ω) is expected to depend sensitively
on the value of the momentum q = 2π/λ transferred by the Bragg laser beams. Our first goal
is to determine the optimal values of q at which the response of the out-of-phase modes at
unitarity is maximized. The maximum weight a mode can have in the f -sum rule in (7.34) is
γq ≃ q 2 /2m. This happens in a uniform superfluid where the Bragg laser beams excites only
in-phase and out-of-phase modes with well defined momentum q. For a trapped superfluid
gas, in contrast, the Bragg potential couples simultaneously to a number of modes for a
given value of q. Their f -sum weight γq in (8.10) and (8.24) can be significantly reduced
147
8. Bragg scattering from two-fluid hydrodynamic modes at unitarity
1.0
0.5
(a)
(b)
dipole
mode
breathing
Z2 ( q )
f-sum weight,
q
1.0
0.5
mode
0.0
0.0
0.0
0.5
1.0
qR
0.0
0.5
1.0
1.5
/(2 )
TF
Figure 8.1: Momentum dependence of (a) the f -sum weight γq (in units of q 2 /2m) and (b)
out-of-phase weight Z2 of the lowest-lying dipole and breathing modes at T = 0.18TF ≃
0.7Tc .
as a result. This is shown in Fig. 8.1(a), where we plot γq for both dipole and breathing
modes at T = 0.7Tc (in units of q 2 /2m). In this and other graphs, the wavevector q is given
in units of the 2π/RT F , where RT F =
p
~/mω0 (24N)1/6 is the Thomas-Fermi radius of an
ideal two-component Fermi gas (see Giorgini et al. [13]). Figure 8.1(a) shows that in the
q ≪ 2π/RT F limit, the dipole mode saturates the f -sum rule. However, it is only the inphase generalized Kohn mode that is appreciably excited. As q is increased, the breathing
modes begin to acquire a finite weight in the density response. When q ∼ 2π/RT F , the
combined f -sum weight γq of the dipole and breathing modes is much smaller than q 2 /2m.
This indicates that hydrodynamic modes other than the dipole and breathing modes have
appreciable weight at such values of q.
Figure 8.1(b) shows the dependence of the weight Z2 (q) of the out-of-phase modes as
a function of q, at T = 0.7Tc . The out-of-phase modes are seen to have negligible weight
(compared to the in-phase modes) when q ≪ 2π/RT F and have significant spectral weight
when q ∼ 2π/RT F . Figure 8.2 shows our results for the density response function SAL (q, Ω)
vs. Ω at T = 0.7Tc , for several different values of q. These results show that the weights of the
8. Bragg scattering from two-fluid hydrodynamic modes at unitarity
148
Figure 8.2: Evolution of the spectral function SA (q, Ω) with increasing Bragg momentum q
at T = 0.7Tc .
out-of-phase mode are very sensitive functions of the value of q. The optimal momentum to
observe the out-of-phase modes is about 2π/RT F , where the overall weight γq Z2 (q) reaches
a maximum.
The results in Figs 8.1 and 8.2 are for an intermediate temperature in the superfluid
phase (T = 0.7Tc ). At T = 0, only the in-phase modes have finite weight. This can be
seen by noting that as T → 0, Mr → Mn , and Fn (q) = 0. Moreover, Fn2 (q) goes to zero
faster than Mn as T → 0, and hence Z1 (q) → Fs2 (q)Mr /Fs2 (q)Mn = 1. This means the
weight of the out-of-phase mode (Z2 ) vanishes at T = 0, as expected since there is no
normal fluid. As the temperature is increased, however, the out-of-phase mode acquires a
finite weight (i.e., Z2 (q) 6= 0) and shows up in SAL (q, Ω) as an additional resonance. This
is shown in Fig. 8.3, which shows SAL (q, Ω) at a series of temperatures (shifted for clarity).
In the limit T → Tc , the weight of the out-of-phase modes again vanishes since Mr → Ms
and Z1 (q) → Fn2 (q)Mr /Fn2 (q)Ms = 1. This can be seen in Fig. 8.3 where the top curve
corresponds to SAL (q, Ω) at T = 0.28TF , just above the superfluid transition temperature of
Tc = 0.27TF . Thus we conclude that in the normal phase (T > Tc ) as well as at T = 0, only
the in-phase modes have spectral weight in SAL (q, Ω).
8. Bragg scattering from two-fluid hydrodynamic modes at unitarity
149
Figure 8.3: Temperature dependence of SAL (q, Ω) as a function of the frequency Ω at unitarity
due to (a) dipole and (b) breathing modes, for an isotropic trap. The momentum transferred
q = 2π/RT F is given in terms of the Thomas-Fermi radius of a ideal trapped Fermi gas. From
bottom to top, the temperature T increases from 0.04TF to 0.28TF , with a step 0.04TF . TF
is the Fermi temperature for a noninteracting trapped Fermi gas. The top curve is above
the superfluid transition temperature Tc ≃ 0.27TF . The dashed lines with arrows trace the
T -dependence of the out-of-phase mode frequencies. A spectral broadening of 0.02ω0 has
been used to simulate a Bragg pulse of duration ∆t ∼ 50(2π/ω0 ).
8. Bragg scattering from two-fluid hydrodynamic modes at unitarity
150
The results for SAL (q, Ω) are for q = 2π/RT F . At this value of q, one notes that the inphase mode weight Z1 (q) decreases with increasing T . For smaller values of q, the in-phase
modes continue to have weight even at higher temperatures.
The calculations shown in this chapter are just a few illustrations to show that Bragg
scattering looks like a useful probe of the out-of-phase hydrodynamic modes in trapped Fermi
gases. A much more extensive study as a function of q and T will be presented elsewhere,
including results for anisotropic traps of the kind used in experiments.
8.4
Direct excitation of normal modes
In addition to the plane-wave perturbation in (7.5), the formalism introduced in Chapter 7
can also be applied to consider the response produced by other perturbing potentials. Two
standard techniques of directly exciting normal modes are the use of a “dipole” potential
Vpert (r, t) = VΩ zeiΩt ,
(8.28)
and a “breathing” potential given by
Vpert (r, t) = VΩ x2i eiΩt .
(8.29)
The dipole potential describes a linear displacement of the harmonic trapping potential2 along
some arbitrary axis (here, the z-axis) and can be used to excite the generalized Kohn mode.
The breathing potential in (8.29) corresponds to a harmonic modulation of the harmonic
trap frequency and is used to excite the in-phase breathing mode. Using these potentials
in (7.38), one can generate different response functions and hence probe different aspects of
the two-fluid hydrodynamic normal mode spectrum. For further discussion, see Sec. 12.4 in
Stringari and Pitaevskii [39].
2
Displacing the trap from its equilibrium position by a small distance a, the trapped gas experiences a
linear restoring potential, ωz2 z 2 → ωz2 (z − a)2 ≃ ωz2 z 2 − 2aωz2 z, which can be treated as a perturbation.
8. Bragg scattering from two-fluid hydrodynamic modes at unitarity
151
It was noted earlier that modulation of the trap frequency does not excite the out-ofphase breathing mode at unitarity. We now prove this statement. We also show that the
dipole potential cannot excite the out-of-phase dipole mode. That is, a linear displacement
of the trap excites only the in-phase generalized Kohn mode, which has a frequency given
by the trap frequency in the direction in which the displacement is applied. This fact is
well known to experimentalists, since they use this result to measure the harmonic trap
frequencies with great accuracy.
For simplicity, we consider an isotropic trap. Substitituing (8.29) into (7.38), and using
the breathing mode ansatz for an isotropic trap in (8.13), we find
(2)
Spert
Z
= −VΩ dr r 2 ∇ · r [ρs0 (r)as + ρn0 (r)an ] .
Z
= 2VΩ dr r 2 [ρs0 (r)as + ρn0 (r)an ]
= 2VΩ MsB as + MnB an .
(8.30)
We recall that the out-of-phase breathing mode at unitarity corresponds to the solution
MsB as + MnB an = 0 [see (6.40)]. Thus, at unitarity, the breathing mode potential given by
(8.29) does not couple into the out-of-phase breathing mode. The modulation of the trap
frequency will only excite the pure in-phase mode as = an , as discussed in Section 6.3.1. We
emphasize that this result is only “exact” at unitarity.
We next consider the dipole potential in (8.28). Using the ansatz for the dipole mode in
(8.3), (7.38) becomes
(2)
Spert
∂
= −VΩ dr z [ρs0 (r)as + ρn0 (r)an ]
∂z
Z
= VΩ dr [ρs0 (r)as + ρn0 (r)an ]
Z
= VΩ MsD as + MnD an .
(8.31)
This expression clearly vanishes for the out-of-phase dipole mode, since this is characterized
8. Bragg scattering from two-fluid hydrodynamic modes at unitarity
152
by the solution MsD as + MnD an = 0. Unlike the breathing mode result, however, this solution
is valid everywhere and is independent of interactions or temperature. Thus, displacing the
trap will only excite the in-phase dipole mode, in all regions of the BCS-BEC crossover and
at all temperatures.
Chapter 9
Conclusions and future work
Since the discovery of BEC in trapped Bose gases, the field of ultracold atoms has become a
major research area. These new quantum gases have allowed us to study a rich new domain of
physics. However, so far two-fluid collisional hydrodynamics of ultracold gases has not been
studied very much. In this thesis, we have calculated the two-fluid hydrodynamic normal
modes of a Fermi superfluid close to unitarity, using the Landau two-fluid equations. Our
results are based on a new variational formalism that we have developed to solve the twofluid equations in nonuniform systems, such as trapped atomic Fermi gases. This formalism
is valid quite generally for any superfluid, whenever collisions are strong enough to produce
local equilibrium. We have concentrated on the BCS-BEC crossover, where a Feshbach
resonance can be used to produce short collision times. However, we emphasize that our
variational approach can also be used for any trapped Bose-condensed gas which is in the
collisional hydrodynamic region.
At finite temperatures, two-fluid hydrodynamics predicts a new type of mode in which
the superfluid and normal fluid components oscillate out-of-phase. In superfluid 4 He, this
is the famous second sound hydrodynamic oscillation. We have described the analogue
of second sound in trapped Fermi superfluid gases, namely the out-of-phase dipole and
breathing modes. In contrast to the in-phase modes that have been extensively studied in
153
9. Conclusions and future work
154
recent experiments, the out-of-phase modes await exploration. We have argued that twophoton Bragg spectroscopy can be used to excite and measure the frequencies of these modes,
giving explicit predictions at finite temperatures for the dynamic structure factor SA (q, ω),
a quantity that is directly related to the two-photon Bragg scattering cross-section.
In our numerical calculation of the lowest energy hydrodynamic normal modes and the
associated dynamic structure factor, we have concentrated on the strong interaction region
at unitarity. However, our method of calculating the thermodynamic functions (Chapters 3
and 4) needed in the two-fluid equations can also be used to give results on the BEC and BCS
sides as well. This is planned in the future. We also plan on extending our work on Bragg
spectroscopy of the two-fluid modes to deal with anisotropic traps used in experiments.
Our explicit calculations in this thesis of the hydrodynamic dipole and breathing modes
at finite temperature have been based on the simplest possible variational ansatz for the
motion of the superfluid and normal fluid components. In the future, we hope to improve
on this by working with an improved variational ansatz.
Our numerical results for the out-of-phase breathing and dipole modes at unitarity emphasize the importance of an accurate calculation of the superfluid density profile. This is
clearly a challenging problem that requires a more careful handling of higher-order fluctuation terms in the self-consistent solution of the gap and number equations close to Tc .
Finally, one would like to be able to calculate the size of the two-fluid hydrodynamic
region (see Fig. 1.2), in terms of (kF as )−1 and T . This requires one to define and calculate
the appropriate collisional relaxation time τ , which controls the time taken to reach local
equilibrium (see Griffin, Nikuni, and Zaremba [65]). Closely related to this problem is the
calculation of the damping of hydrodynamic modes arising from transport coefficients. Initial
studies of these quantities in the normal phase of a trapped Fermi gas have been discussed
9. Conclusions and future work
by Massignan, Bruun, and Smith [114], and by Bruun and Smith [115].
155
Appendix A
A single-channel model for a Feshbach
resonance
In this appendix, we give a brief review of the Feshbach resonance in two-component Fermi
gases. The physics of a many-body system with a Feshbach resonance is naturally described
in terms of a so-called “two-channel model” [17, 18]. Starting from the Hamiltonian for this
two-channel model, we give a simple derivation of an effective single-channel Hamiltonian
that was originally used in the literature to discuss the BCS-BEC crossover problem. This
single-channel model is used in our calculation of thermodynamic quantities in Chapter 3
and the superfluid density in Chapter 4, quantities which enter in the linearized Landau
two-fluid equations.
A.1
A brief review of Feshbach resonance physics
In atomic Fermi gases, the Feshbach resonance is a scattering resonance between two unbound
fermions in different hyperfine states (labelled by ↑, ↓) with a low-energy bound state of
fermions in the hyperfine states ↑′ , ↓′ . The fermions in the first set of hyperfine states (↑, ↓)
are said to be in the “open channel”. Fermions in the ↑′ , ↓′ hyperfine states are in the “closed
channel”. Since the magnetic moment (µo ) of the open channel fermions differs from the
magnetic moment (µc ) of the closed channel fermions, the energy-level splitting between
156
157
A. A single-channel model for a Feshbach resonance
dimers in the closed channel and the free fermions in the open channel can be altered by
tuning an externally applied magnetic field B, which couples to the magnetic moment of the
atoms. At a critical value B0 of the magnetic field (see Fig. 1.1), the energy of the open
channel fermions coincides with the energy of the dimer in the closed channel. This leads
to a diverging scattering length between the fermions in the open channel (see Sec. 5.4.2 in
Pethick and Smith [38]),
as = abg 1 −
∆B
B − B0
.
(A.1)
Here abg is the background (nonresonant) scattering length between open channel fermions,
and ∆B gives the “width” of the Feshbach resonance. As we discuss below, this width ∆B
plays an important role in determining whether a Fermi gas close to a Feshbach resonance
can be described in terms of a simple single-channel model. The latter corresponds to an
interacting Fermi gas with an adjustable interaction.
In many-body systems, the physics of a Feshbach resonance can be conveniently formulated in terms of a Hamiltonian for a two-channel model [17, 18] that describes the conversion
of fermions (a) in the open channel to bound dimers (b) of fermions in the closed channel
(we set ~ = V = 1):
Ĥ =
X
k,σ
(ǫk − µ) â†k,σ âk,σ +
X
q
† †
ǫM
q − 2µ + ν b̂q b̂q
4πabg X †
+
â
â†
âk′ +q/2,↓ â−k′ +q/2,↑
m q,k,k′ −k+q/2,↑ k+q/2,↓
X
b̂†q âk+q/2,↓ â−k+q/2,↑ + â†−k+q/2,↑ â†k+q/2,↓ b̂q .
+g
(A.2)
q,k
Here ǫk = k2 /2m is the kinetic energy of fermions of mass m, and µ is the chemical potential.
2
ǫM
q = q /2M is the energy of the dimer molecules of mass M = 2m. The parameter
g describes the coupling between a bound dimer in the closed channel and open channel
A. A single-channel model for a Feshbach resonance
158
fermions. The detuning parameter ν gives the energy difference between the closed channel
dimers and open channel fermions in terms of the applied magnetic field B,
ν = ∆µM (B − B0 ),
(A.3)
where ∆µM ≡ µc −µ0 is the difference between the magnetic moments of the closed and open
channel fermions. We now turn to a discussion of the conditions needed for the two-channel
model (A.2) to reduce to a single-channel model.
A.2
The single-channel model
The appropriate microscopic model of a Feshbach resonance in a two-component gas of
fermions is given by the two-channel Hamiltonian in (A.2). However, the essential physics
of this model (namely, a BCS limit in which the fermions are paired up and are attracted
to one another, and a BEC limit in which the fermions are paired up into tightly bound
dimer molecules that repel each other) is the same as the BCS-BEC crossover model studied
originally by Leggett [8]. This model involves only two fermion species and thus constitutes
a single-channel model. This similarity suggests that the physics of a many-body Fermi gas
with a Feshbach resonance can be understood in terms of a single-channel model. Strong
evidence for this was provided by Diener and Ho [22], who pointed out that in the limit of a
broad Feshbach resonance, the mean-field two-channel gap equation gave equivalent results
as an effective single-channel model with a renormalized effective interaction.
In this section, we give a new derivation of the equivalence of a single-channel model with
the two-channel model in (A.2), using the technique of functional integration (Popov [71]).
This approach gives a very clear picture of how the single-channel model arises from a twochannel one by “integrating out” the closed channel fermions. This leaves an effective theory
in terms of open channel fermions, with an effective interaction mediated by virtual closed
159
A. A single-channel model for a Feshbach resonance
channel molecules.
The defining feature of a broad Feshbach resonance is that there are essentially no Feshbach molecules over the range of magnetic fields used to probe the crossover region (see for
instance, Refs. [22, 113]). Of course, numerous experiments have studied the BEC region
of the crossover and reported observations of a molecular Bose-Einstein condensate. We
emphasize, however, that for a broad Feshbach resonance, these molecules are dimers of
open channel fermions, which are distinct from the closed channel Feshbach molecules. Even
though the two-channel Hamiltonian given in (A.2) only includes a small background interaction between the open-channel fermions, close to the Feshbach resonance, virtual Feshbach
molecules mediate a strong interaction between the open channel fermions. For this reason,
apart from their contribution to the effective interaction between open-channel fermions, the
closed-channel molecules can be ignored for a broad Feshbach resonance. The two-channel
problem thus reduces to a single-channel one with an effective interaction.
We now describe the detailed derivation of the above results. The most straightforward
way of discussing the physics of the open channel fermions is to start with the partition
function for the two-channel model, and then integrate out the closed-channel molecules.
Constructing the partition function Z =
R
∗
D[a, ā, b, b∗ ]e−S[a,ā,b,b ] from (A.2) and integrating
out the molecular Bose fields (b∗ , b), one obtains the following effective action for the openchannel fermions:
Seff [a, ā] =
X
k,σ
+
1 X
ā−k+q/2,↑ āk+q/2,↓ Veff (q)ak′ +q/2,↓ a−k′ +q/2,↑
β
q,k,k ′
− 2µ + ν .
(A.4)
āk,σ (−iωn + ξk ) ak,σ −
X
q
ln −iνm + ǫM
q
Here ωn and νm are Fermi and Bose Matsubara frequencies. The effective interaction Veff (q)
160
A. A single-channel model for a Feshbach resonance
in (A.4) is given by
g2
4πabg
−
m
iνm − ǫM
q − ν + 2µ
4πabg
= −
− g 2D0 (q, iνm ),
m
Veff (q) ≡ −
(A.5)
where ν is the detuning defined in (A.3). In the second line of (A.5), we have written the
effective interaction in terms of the bosonic Green’s function D0 (q) for the closed channel
molecules. Writing it this way underscores the fact that the virtual Feshbach molecules
mediate a new interaction between the open channel fermions, a point first stressed in papers
by Ohashi and Griffin [19, 48].
Carrying out the summation over the Bose Matsubara frequencies iνm , the last term in
(A.4) can be written as
X
q,νm
ln −iνm +
ǫM
q
− 2µ + ν =
i
h
−β(ǫM
q −2µ+ν)
.
ln 1 − e
X
q
(A.6)
Using the identity Ω = S/β relating the thermodynamic potential Ω for noninteracting
particles with the action S, we see that (A.6) represents the contribution ΩM to the thermodynamic potential of the Fermi gas from Feshbach molecules:
ΩM
i
1X h
−β(ǫM
q −2µ+ν)
=
.
ln 1 − e
β q
(A.7)
This allows us to obtain an expression for the number Nm of dimer Feshbach molecules (i.e.,
dimer molecules in the closed channel), namely
1
Nm ≡ −
2
∂ΩM
∂µ
T
=
X
q
1
β(ǫM
q −2µ+ν)
e
−1
.
(A.8)
This involves the Bose distribution function for noninteracting dimer molecules with energy1
ωq = ǫM
q − 2µ + ν. The factor of 1/2 in (A.8) reflects the fact that there are two fermions in
each dimer molecule; i.e., there are 2Nm fermions bound in Nm dimer molecules.
1
Ohashi and Griffin [48] calculate an improved expression for the molecular energy that includes selfenergy corrections due to coupling with the open channel fermions.
161
A. A single-channel model for a Feshbach resonance
Equation (A.4) gives us the effective action Seff for the fermions in the open channel.
The closed channel dimers (Feshbach molecules) enter explicitly in two places: through
the mediated interaction defined in (A.5) and in the term given in (A.6). When the fraction
2Nm /N of fermions that are bound in Feshbach molecules (N is the total number of fermions)
is very small, the thermodynamic potential ΩM in (A.7) vanishes [and hence, so does the
last term in (A.4)]. In this case, we see that the two-channel model reduces to a singlechannel one. A similar derivation of an effective action for open channel fermions was given
by Holland and coworkers [112], although they did not discuss the circumstances in which
the contribution in (A.6) from the closed channel Feshbach molecules vanishes.2
We now discuss the situation of a broad Feshbach resonance, where the last term in (A.4)
is small and can be neglected. In order to make contact with discussions in the literature
regarding the definition of a broad resonance, we first need to regularize the effective interaction potential given by (A.5) and define a scattering length as for fermions in the open
channel [see (A.1)]. The use of a coupling constant g in (A.2) which is independent of the
momenta of the scattering fermions and bosons leads to an unphysical ultraviolet divergence, and thus the effective interaction Veff (q) must be renormalized. This is easily done
by solving the Lippmann-Schwinger equation3 for two fermions in vacuum (µ = 0) using the
effective interaction Veff (q). The final result is that Veff (q) in (A.4) can be replaced by the
renormalized contact interaction U0 , defined by
U0 ≡
Veff (0)
P
1 + Veff (0) k
1
2ǫk
.
(A.9)
From (A.5), we have (recalling that µ = 0 in vacuum)
4πabg
Veff (0) = −
m
2
mg 2 /4πabg
1−
.
ν
(A.10)
Ref. [112] predated the later interest in broad vs. narrow Feshbach resonances, and was not appreciated
in the more recent debate surrounding the validity of a single-channel model.
3
See for instance, Sec. 5.1.2 in Pethick and Smith [38].
162
A. A single-channel model for a Feshbach resonance
The effective action for the open channel fermions in (A.4) thus reduces to
Seff [a, ā] =
X
k,σ
+
āk,σ (−iωn + ξk ) ak,σ −
X
q
U0 X
ā−k+q/2,↑ āk+q/2,↓ ak′ +q/2,↓ a−k′ +q/2,↑
β q,k,k′
i
h
−β(ǫM
q −2µ+ν)
,
ln 1 − e
(A.11)
where we have used (A.6) for the last term.
Inverting the expression in (A.9) for U0 gives the well-known expression (see Randeria [10])
X 1
m
1
=−
+
,
U0
4πas
2ǫ
k
k
(A.12)
where the effective atomic s-wave scattering length between fermions in the open channel is
defined by Veff ≡ −4πas /m, with
as
mg 2 /4πabg
≡ abg 1 −
ν
2
mg /4πabg ∆µM
= abg 1 −
B − B0
(A.13)
Comparing this expression with (A.1), the width ∆B of the Feshbach resonance is now given
in terms of the microscopic parameters of the original two-channel model,
∆B =
m
g2
.
4πabg ∆µM
(A.14)
For a broad Feschbach resonance, described by the condition [22, 113]
2m(∆µM )2 (∆Babg )2 =
g 4 m3
≫ ǫF ,
8π 2
(A.15)
it is straightforward to show that the fraction 2Nm /N of Feshbach molecules is small. In
this case, the last term in (A.11) vanishes, and we are left with a single-channel model,
Seff [a, ā] =
X
k,σ
āk,σ (−iωn + ξk ) ak,σ −
U0 X
ā−k+q/2,↑ āk+q/2,↓ ak′ +q/2,↓ a−k′ +q/2,↑ . (A.16)
β
′
q,k,k
163
A. A single-channel model for a Feshbach resonance
The two-channel model has thus been replaced by an effective single-channel model where
the (closed channel) molecular degrees of freedom only enter the theory through the effective
interaction U0 defined in (A.12). In space and imaginary time coordinates (r, τ ), we can write
(A.16) as
S[ψ, ψ̄] =
Z
β
dτ
0
"Z
dr
X
#
ψ̄σ (x)∂τ ψσ (x) + H ,
σ
(A.17)
where
1 X
ak,σ ei(k·r−ωn τ ) .
ψσ (x) = √
β k,ω
(A.18)
n
Here we use the notation x = (r, τ ) where r denotes spatial coordinates and τ = it is the
imaginary time variable [70]. The effective single-channel Hamiltonian H in (A.17) is given
by
H =
Z
dr
X
σ
Z
∇2
ψ̄σ (x) −
− µ ψσ (x) − U0 dr ψ̄↑ (x)ψ̄↓ (x)ψ↓ (x)ψ↑ (x). (A.19)
2m
This model Hamiltonian is the microscopic basis of our treatment of the BCS-BEC crossover
in Chapters 3 and 4.
Appendix B
Fluctuations in the gap and number
equations: a critical review
This appendix gives a careful discussion of the gap and number equations that arise in
Chapter 3, with the aim of clarifying different approximations used in the recent literature
on the BCS-BEC crossover. In particular, we show how to properly include the effects of
the bosonic fluctuations at the Gaussian level.
Our discussion is motivated by calculations that solve for ∆0 and µ using a number equation that includes Gaussian fluctuations and a mean-field gap equation that does not (Hu,
Liu, and Drummond [49]; Diener, Sensarma, and Randeria [75]). These improve on earlier
broken-symmetry NSR calculations (Engelbrecht et al. [55]) by including an extra term in
the number equation. This leads to a chemical potential in the BEC limit of dimer molecules
that is consistent with a molecular scattering length aM ≃ 0.6as , a result first obtained by
Petrov, Salomon, and Shlyapnikov [79] based on directly solving the Schrödinger equation
for 4-body scattering involving fermions. In contrast, calculations based on the original NSR
scheme (Engelbrecht et al. [55]) produce results consistent with the Born approximation for
the molecular scattering length, aM = 2as [54, 55] [see, for example, (3.70) in Chapter 3].
As discussed in Chapter 3, the results presented in this thesis make use of both the original
NSR scheme (to calculate the superfluid density in Section 4.5), and the “improved” NSR
164
B. Fluctuations in the gap and number equations: a critical review
165
scheme (Chapter 6). All results are based on the mean-field gap equation in (3.42).
Since the major numerical results of this thesis (in Chapters 6 and 8) make use the
improved NSR scheme [49, 75] that includes an extra term in the number equation, in this
appendix we give a careful justification of adding this extra term. We also discuss how
one can include Gaussian fluctuations into the gap equation self-consistently for improved
accuracy.
Before proceeding, it is useful to summarize what has been done in the literature in tabular form. In Table B.1, we show the different gap and number equations that we will be
discussing in this appendix. We follow the notation used in Chapter 3, F standing for the
mean-field BCS Fermi excitations. B stands for the contributions from Bose pairing fluctuations. The gap and number equations given by (I) are the broken-symmetry NSR equations
first solved by Engelbrecht, Sá de Melo, and Randeria [55] and subsequently by many authors
(see, for instance, Ohashi and Griffin [48] and Taylor, Ohashi, Fukushima, and Griffin [56]).
The second set of equations, (II), bring in the correction δn∆ ≡ −(∂ΩB /∂∆0 )(∂∆0 /∂µ), first
included by Keeling, Eastham, Szymanska, and Littlewood [74], and subsequently by Hu,
Liu, and Drummond [49], as well as Diener, Sensarma, and Randeria [75]. (III) incorporates
the effects of fluctuations in the gap equation, but neglects the δn∆ correction in the number
equation. We are unaware of any discussion of (III) in the literature. Finally, (IV) includes
both fluctuations in the gap equation and the δn∆ correction in the number equation. Diener et al. [75] give a detailed discussion of (IV), and concluded that these equations lead to
significant self-consistency problems. The primary result of this appendix is that we show
(II) and (III) are both self-consistent ways of including fluctuations beyond (I). We also find
that (IV) cannot by solved consistently, confirming the conclusions in Ref. [75].
We emphasize that the situation described above is a consequence of the approximations
166
B. Fluctuations in the gap and number equations: a critical review
gap equation
number equation
(I)
∂ΩF
∂∆0
=0
n=−
∂ΩF
∂µ
−
∂ΩB
∂µ
(II)
∂ΩF
∂∆0
=0
n=−
∂ΩF
∂µ
−
∂ΩB
∂µ
(III)
∂ΩF
∂∆0
+
∂ΩB
∂∆0
=0 n=−
∂ΩF
∂µ
−
∂ΩB
∂µ
(IV)
∂ΩF
∂∆0
+
∂ΩB
∂∆0
=0 n=−
∂ΩF
∂µ
−
∂ΩB
∂µ
µ
µ
µ
µ
µ
µ
∆0
∆0
∆0
∆0
∆0
∆0
−
∂ΩB
∂∆0
∂∆0
∂µ
−
∂ΩB
∂∆0
∂∆0
∂µ
µ
∆0
∆0
µ
Table B.1: The gap and number equations discussed in the text.
used to solve the gap and number equation (i.e., by using the Gaussian thermodynamic potential). The exact gap and number equations are given by standard results from equilibrium
statistical mechanics:
∂Ω
∂∆0
=0
(B.1)
µ
and
n=−
∂Ω
∂µ
.
(B.2)
T
Treating the equilibrium gap ∆0 as a function of the chemical potential µ, we can write the
number equation in (B.2) as (leaving the constancy of T implicit)
n = −
∂Ω
∂µ
∆0
−
∂Ω
∂∆0
µ
∂∆0
∂µ
.
(B.3)
Using (B.1), the second term on the right-hand side of (B.3) should be zero, in which
case the gap and number equations reduce to
∂Ω
∂∆0
µ
=0
(B.4)
167
B. Fluctuations in the gap and number equations: a critical review
and
∂Ω
n=−
∂µ
.
(B.5)
∆0
We note that the number equation (B.5) leaves ∆0 fixed. While this analysis suggests
that ∆0 should always be kept fixed when evaluating the number equation, we show in the
discussion below that this is not necessarily the case when one uses some approximation for
the thermodynamic potential Ω [e.g., the Gaussian approximation in (3.41)]. Depending on
how the approximation is handled, one can include a correction to the number equation that
involves (∂∆0 /∂µ), as with II in Table B.1.
We now go back to (B.1) and (B.2) and use the Gaussian approximation for the thermodynamic potential Ω = ΩF + ΩB , given by (3.41). Separating out the F and B contributions,
the gap and number equations are
∂ΩF
∂∆0
+
µ
∂ΩB
∂∆0
=0
(B.6)
µ
and
∂ΩF
n=−
∂µ
∆0
−
∂ΩB
∂µ
∆0
−
"
∂ΩF
∂∆0
µ
+
∂ΩB
∂∆0
#
µ
∂∆0
∂µ
.
(B.7)
As before, using the gap equation in (B.6), the last term in the number equation (B.7)
proportional to (∂∆0 /∂µ) should vanish. This gives the set of equations denoted as (III) in
Table B.1. On the other hand, if instead of (B.6) we use the mean-field gap equation
∂ΩF
∂∆0
= 0,
(B.8)
µ
then clearly the contents of the square brackets in (B.7) need not be set to zero. Only the
term involving (∂ΩF /∂∆0 ) vanishes, in accordance with (B.8). In this case, the number
equation (B.7) reduces to
∂ΩF
n=−
∂µ
∆0
−
∂ΩB
∂µ
∆0
−
∂ΩB
∂∆0
µ
∂∆0
∂µ
.
(B.9)
B. Fluctuations in the gap and number equations: a critical review
168
(B.8) and (B.9) are the gap and number equations given by (II) in Table B.1 and include
the correction δn∆ ≡ −(∂ΩB /∂∆0 )(∂∆0 /∂µ) in the number equation.
The preceding analysis shows that one can either solve the coupled equations for µ and
∆0 given in (III), or the coupled equations given by (II) and in both cases be consistent.
In contrast, solving the coupled equations given by (IV) in Table B.1, leads to an obvious
inconsistency. (IV) involves the “Gaussian gap equation” given in (B.6) and the number
equation in (B.9). As shown above, however, one only arrives at (B.9) as a consequence
of using the mean-field gap equation in (B.8). If one uses the Gaussian gap equation in
(B.6), the correction δn∆ to this number equation should vanish. A more detailed analysis
of (IV) in Ref. [75] gives an explicit discussion of the problems that arise with this approach.
Consequently, we conclude that at the Gaussian level, any attempt to include fluctuations
in the gap equation should be restricted to solving (III).
In closing this appendix, we write down an explicit expression for the Gaussian gap
equation in (B.6). Using (3.41) in (B.6), we find
∆0
∂
1X
1 ∂ΩB
−1
tr[G0 (k)
=
G (k)] −
U0
β k
∂∆0 0
2 ∂∆0 µ
1 X
1X
∂detM(q)
1
G12 (k) −
=
.
β k
4β q detM(q)
∂∆0
µ
(B.10)
Bibliography
[1] L. D. Landau, J. Phys. U.S.S.R. 5, 71 (1941).
[2] I. M. Khalatnikov, An Introduction to the Theory of Superfluidity (W. A. Benjamin, New
York, 1965).
[3] P. C. Hohenberg and P. C. Martin, Ann. Phys. (N.Y.) 34, 291 (1965).
[4] R. Ozeri, N. Katz, J. Steinhauer, and N. Davidson, Rev. Mod. Phys. 77, 187 (2005).
[5] K. Huang, Statistical Mechanics (Wiley, New York, 1987).
[6] B. De Marco and D. S. Jin, Science 285, 1703 (1999).
[7] J. Bardeen, L. N. Cooper, and, J. R. Schrieffer, Phys. Rev. 106, 162 (1957).
[8] A. J. Leggett, J. Phys. C 41, 7 (1980).
[9] D. M. Eagles, Phys. Rev. 186, 456 (1969).
[10] M. Randeria in Bose-Einstein Condensation, edited by A. Griffin, D. Snoke, and
S. Stringari (Cambridge University Press, Cambridge, 1985).
[11] P. Nozières and S. Schmitt-Rink, J. Low Temp. Phys. 59, 195 (1985).
[12] Q. Chen, J. Stajic, S. Tan, K. Levin, Physics Reports 412, 1 (2005).
[13] S. Giorgini, L. P. Pitaevskii, and S. Stringari, e-print: arXiv:0706.3360v1.
169
B. Fluctuations in the gap and number equations: a critical review
170
[14] R. Grimm, in Proceedings of the International School of Physics “Enrico Fermi” Course CLXIV “Ultra-Cold Fermi Gases”, Varenna, June 2006, edited by M. Inguscio,
W. Ketterle, and C. Salomon (to be published).
[15] I. Bloch, J. Dalibard, and W. Zwerger, e-print: arXiv:0704.3011.
[16] W. C. Stwalley. Phys. Rev. Lett. 37, 1628 (1976); E. Tiesinga, B. J. Verhaar, and
H. T. C. Stoof, Phys. Rev. A 47, 4114 (1993).
[17] E. Timmermans, K. Furuya, P. W. Milonni, and A. K. Kerman, Phys. Lett. A 285, 228
(2001).
[18] M. Holland, S. J. J. M. F. Kokkelmans, M. L. Chiofalo, and R. Walser, Phys. Rev. Lett.
87, 120406 (2001).
[19] Y. Ohashi and A. Griffin, Phys. Rev. Lett 89, 130402 (2002).
[20] C. A. Regal and D. S. Jin, Phys. Rev. Lett. 90, 230404 (2003).
[21] R. A. Duine and H. T. C. Stoof, Phys. Rep. 396, 115 (2004).
[22] R. Diener and T.-L. Ho, e-print: arXiv:cond-mat/0405174.
[23] M. W. J. Romans H. T. C. Stoof, Phys. Rev. Lett. 95, 260407 (2005).
[24] G. B. Partridge, K. E. Strecker, R. I. Kamar, M. W. Jack, R. G. Hulet, Phys. Rev. Lett.
95, 020404 (2005).
[25] E. Taylor and A. Griffin, Phys. Rev. A 72, 053630 (2005).
[26] M. J. Wright, S. Riedl, A. Altmeyer, C. Kohstall, E. R. Sánchez Guajardo, J. Hecker
Denschlag, and R. Grimm, Phys. Rev. Lett. 99, 150403 (2007).
B. Fluctuations in the gap and number equations: a critical review
171
[27] C. A. Regal, M. Greiner, and D. S. Jin, Phys. Rev. Lett. 92, 040403 (2004).
[28] T. Bourdel, L. Khaykovich, J. Cubizolles, J. Zhang, F. Chevy, M. Teichmann, L. Tarruell, S. J. J. M. F. Kokkelmans, and C. Salomon, Phys. Rev. Lett. 93, 050401 (2004).
[29] M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupach, A. J. Kerman, and
W. Ketterle, Phys. Rev. Lett. 92, 120403 (2004).
[30] M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, C. Chin, J. H. Denschlag, and
R. Grimm, Phys. Rev. Lett. 92, 120401 (2004).
[31] C. Chin, M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, J. Hecker Denschlag, and
R. Grimm, Science 305, 1128 (2004).
[32] J. Kinast, S. L. Hemmer, M. E. Gehm, A. Turlapov, and J. E. Thomas, Phys. Rev.
Lett. 92, 150402 (2004).
[33] M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, C. Chin, J. H. Denschlag, and
R. Grimm, Phys. Rev. Lett. 92, 203201 (2004).
[34] J. Kinast, A. Turlapov, and J. E. Thomas, Phys. Rev. Lett. 94, 170404 (2005);
J. E. Thomas, J. Kinast, and A. Turlapov, Phys. Rev. Lett. 95, 120402 (2005).
[35] S. Stringari, Europhys. Lett. 65, 749 (2004).
[36] H. Heiselberg, Phys. Rev. Lett. 93, 040402 (2004).
[37] H. Hu, A. Minguzzi, X.-J. Liu, and M. P. Tosi, Phys. Rev. Lett. 93, 190403 (2004).
[38] C. J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases (Cambridge
University Press, Cambridge, 2002).
B. Fluctuations in the gap and number equations: a critical review
172
[39] S. Stringari and L. Pitaevskii, Bose-Einstein Condensation (Oxford University Press,
Oxford, 2003).
[40] C. Gay and A. Griffin, Journ. Low Temp. Phys. 58, 479 (1985).
[41] A. Griffin and E. Zaremba, Phys. Rev. A 56, 4839 (1997).
[42] T.-L. Ho, Phys. Rev. Lett. 92, 090402 (2004).
[43] T. Schaefer, e-print: arXiv:cond-mat/0701251; G. Rupak and T. Schaefer, e-print:
arXiv:0707.1520.
[44] V. B. Shenoy and T.-L. Ho, Phys. Rev. Lett. 80, 3895 (1998).
[45] Y. He, Q. Chen, C.-C. Chien, and K. Levin, Phys. Rev. A 76, 051602(R) (2007).
[46] P. R. Zilsel, Phys. Rev. 79, 309 (1950).
[47] E. Zaremba, T. Nikuni, and A. Griffin, Journ. Low Temp. Phys. 116, 277 (1999).
[48] Y. Ohashi and A. Griffin, Phys. Rev. A 67, 063612 (2003).
[49] H. Hu, X.-J. Liu, and P. D. Drummond, Europhys. Lett. 74, 574 (2006).
[50] H. Hu, X.-J. Liu, and P. D. Drummond, Phys. Rev. A 73, 023617 (2006).
[51] A. Bulgac, J. E. Drut, and P. Magierski, Phys. Rev. Lett. 96, 090404 (2006).
[52] F. Zambelli, L. P. Pitaevskii, D. M. Stamper-Kurn, and S. Stringari, Phys. Rev. A 61,
063608 (2000).
[53] A. J. Leggett, Quantum Liquids (Oxford university Press, Oxford, 2006).
[54] C. A. R. Sá de Melo, M. Randeria, and J. R. Engelbrecht, Phys. Rev. Lett 71, 3202
(1993).
B. Fluctuations in the gap and number equations: a critical review
173
[55] J. R. Engelbrecht, M. Randeria, and C. A. R. Sá de Melo, Phys. Rev. B 55, 15153
(1997).
[56] E. Taylor, A. Griffin, N. Fukushima, and Y. Ohashi, Phys. Rev. A 74, 063626 (2006).
[57] N. Fukushima, Y. Ohashi, E. Taylor, and A. Griffin, Phys. Rev. A 75, 033609 (2007).
[58] E. Taylor, H. Hu, X.-J. Liu, and A. Griffin, To be published.
[59] E. Taylor, H. Hu, X.-J. Liu, and A. Griffin, e-print: arXiv:0709.0698.
[60] L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Elsevier, Oxford, 1987).
[61] B. Clancy, L. Luo, and J. E. Thomas, Phys. Rev. Lett. 99, 140401 (2007).
[62] S. J. Putterman, Superfluid Hydrodynamics (North-Holland, Amsterdam, 1974).
[63] N. N. Bogoliubov, Lectures on Quantum Statistics, Vol. 2 (Gordon and Breach, New
York, 1970).
[64] Z. M. Galasiewicz, Superconductivity and Quantum Fluids (Pergamon, London, 1970).
[65] A. Griffin, T. Nikuni, E. Zaremba, Bose-condensed Gases at Finite Temperatures (Cambridge University Press), to be published.
[66] P. Nozières and D. Pines, The Theory of Quantum Liquids, vol. II (Addison-wesley,
Redwood City, 1989).
[67] L. D. Landau and E. M. Lifshitz, Statistical Physics (Elsevier, Oxford, 2003).
[68] V. P. Peshkov, J. Phys. U.S.S.R. 10, 389 (1946); Nature 157, 300 (1946).
[69] L. D. Landau, J. Phys. U.S.S.R. 11, 91 (1947).
B. Fluctuations in the gap and number equations: a critical review
174
[70] A. Altland and B. Simons, Condensed Matter Field Theory (Cambridge University
Press, Cambridge, 2006).
[71] V. N. Popov, Functional Integrals and Collective Excitations (Cambridge University
Press, Cambridge, 1987).
[72] J. W. Negele and H. Orland, Quantum Many-Particle Systems (Westview Press, Boulder, Colorado, 1988).
[73] H. T. C. Stoof, e-print: cond-mat/9910441v1.
[74] J. Keeling, P. R. Eastham, M. H. Szymanska, and P. B. Littlewood, Phys. Rev. B 72,
115320 (2005).
[75] R. B. Diener, R. Sensarma, and M. Randeria, e-print: arXiv:0709.2653.
[76] G. E. Astrakharchik, J. Boronat, J. Casulleras, and S. Giorgini, Phys. Rev. Lett. 93,
200404 (2004).
[77] P. Pieri and G. C. Strinati, Phys. Rev. B 61, 15370 (2000).
[78] Y. Ohashi, J. Phys. Soc. Jpn. 74, 2659 (2005).
[79] D. S. Petrov, C. Salomon, and G. V. Shlyapnikov, Phys. Rev. Lett. 93, 090404 (2004).
[80] H. Shi and A. Griffin, Phys. Rep. 304, 1 (1998).
[81] E. Abrahams and T. Tsuneto, Phys. Rev. 152, 416 (1966).
[82] H. T. C. Stoof, Phys. Rev. B 47, 7979 (1998).
[83] N. N. Bogoliubov, V. V. Tolmachev, and, D. V. Shirkov, A New Method in Superconductivity Theory (U.S.S.R. Acad. Sci. Press, Moscow, 1958).
B. Fluctuations in the gap and number equations: a critical review
175
[84] P. W. Anderson, Phys. Rev. 112, 1900 (1958).
[85] M. E. Fisher, M. N. Barber, and D. Jasnow, Phys. Rev. A 8, 1111 (1973).
[86] A. Griffin, Excitations in a Bose-Condensed Liquid (Cambridge University Press, Cambridge, 1993).
[87] N. Andrenacci, P. Pieri, and G. C. Strinati, Phys. Rev. B 68, 144507 (2003).
[88] E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics, Part 2 (Butterworth-Heinemann,
Oxford, 2002).
[89] U. Eckern, G. Schön, and V. Ambegaokar, Phys. Rev. B 30, 6419 (1984).
[90] M. Stone, Int. J. Mod. Phys. B 9, 1359 (1995).
[91] A. L. Fetter, Ann. Phys. (N. Y.) 60, 464 (1970).
[92] V. K. Akkineni, D. M. Ceperley, and N. Trivedi, Phys. Rev. B 76, 165116 (2007).
[93] J. A. Geurst, Phys. Rev. B 22, 3207 (1980).
[94] I. M. Gelfand and S. V. Fomin, Calculus of Variations (Prentice Hall, Englewood Cliffs,
1963).
[95] C. Eckart, Phys. Rev. 54, 920 (1938).
[96] G. Arfken, Mathematical Methods for Physicists (Academic Press, Orlando, 1985), p.
957.
[97] S. Stringari, Phys. Rev. Lett. 77, 2360 (1996).
[98] A. Griffin, W.-C. Wu, and S. Stringari, Phys. Rev. Lett. 78, 1838 (1997).
B. Fluctuations in the gap and number equations: a critical review
176
[99] Y. Castin and R. Dum, Phys. Rev. Lett. 77, 5315 (1996).
[100] L. Pitaevskii and S. Stringari, Phys. Rev. Lett. 81, 4541 (1998).
[101] G. E. Astrakharchik, R. Combescot, X. Leyronas, and S. Stringari, Phys. Rev. Lett.
95, 030404 (2005).
[102] M. Cozzini and S. Stringari, Phys. Rev. Lett. 91, 070401 (2003).
[103] G. F. Bertsch, “Many body X challenge problem”, see R. A. Bishop, Int. J. Mod. Phys.
B 15, iii (2001).
[104] L. P. Pitaevskii and S. Stringari, Science 298, 2144 (2002).
[105] Y. Castin, Comptes Rendus Physique 5, 407 (2005); this paper can also be found on
the arXiv: cond-mat/0406020.
[106] D. M. Stamper-Kurn, A. P. Chikkatur, A. Görlitz, S. Inouye, S. Gupta, D. E. Pritchard,
and W. Ketterle, Phys. Rev. Lett 83, 2876 (1999).
[107] J. Steinhauer, R. Ozeri, N. Katz, and N. Davidson, Phys. Rev. Lett. 88, 120407 (2002).
[108] J. Steinhauer, N. Katz, R. Ozeri, N. Davidson, C. Tozzo, and F. Dalfovo, Phys. Rev.
Lett. 90, 060404 (2003).
[109] D. Pines and P. Nozières, The Theory of Quantum Liquids, vol. I (Benjamin, New
York, 1966).
[110] G. D. Mahan, Many-Particle Physics (Plenum Press, New York, 1993).
[111] P. C. Hohenberg and P. C. Martin, Phys. Rev. Lett. 12, 69 (1964).
[112] J. N. Milstein, S. J. Kokkelmans, and M. J. Holland, Phys. Rev. A 66, 043604 (2002).
B. Fluctuations in the gap and number equations: a critical review
[113] G. M. Bruun, Phys. Rev. A 70, 053602 (2004).
[114] P. Massignan, G. M. Bruun, and H. Smith, Phys. Rev. A 71, 033607 (2005).
[115] G. M. Bruun and H. Smith, Phys. Rev. A 75, 043612 (2007).
177