Flow equations for
dissipative quantum systems
under non-equilibrium conditions
Andreas Hackl
Augsburg 2006
Flow equations for
dissipative quantum systems
under non-equilibrium conditions
Diplomarbeit
vorgelegt von Andreas Hackl
Universität Augsburg
Lehrstuhl für Theoretische Physik III
Juni 2006
Erstgutachter: Prof. Dr. D. Vollhardt
Zweitgutachter: Prof. Dr. T. Kopp
Flow equations for dissipative quantum systems under
non-equilibrium conditions
This work investigates dissipative quantum systems by means of flow
equations for Hamiltonians. We consider the time evolution of two models out of non-equilibrium initial states. As a first application the dissipative harmonic oscillator will be solved exactly. The insights gained
will be used in the second and nontrivial model, the spin-boson model.
A set of coupled differential equations will be derived that describes the
time-evolution of the spin operators within approximations. Numerical implementations of these coupled differential equations are used to
compare them with results from the non-interacting blip approximation
(NIBA) and to calculate correlation functions in non-equilibrium. The
thesis closes with a comparative discussion of the treated models and
the closely related Kondo model.
Contents
Contents
IV
1 Introduction
1
2 Dissipative quantum systems
2.1 The Caldeira-Leggett model . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 The Hamilton function and the classical equation of motion . . .
2.1.2 The spectral density of the bath . . . . . . . . . . . . . . . . . .
5
5
5
7
3 The flow equation method
3.1 History and applications . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Transformation of the Hamiltonian . . . . . . . . . . . . . . . . . . . . .
3.2.1 Infinitesimal unitary transformations . . . . . . . . . . . . . . . .
3.2.2 Multiparticle expansion of observables . . . . . . . . . . . . . . .
3.2.3 Flow equations for a general multiparticle expansion of an observable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.4 Approximations to the flow equations . . . . . . . . . . . . . . .
3.3 Continuos sequency of unitary transformations . . . . . . . . . . . . . .
3.4 Time evolution of observables . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Formulation of the Heisenberg equation in diagonal basis . . . .
3.4.2 Approximations to the Heisenberg equation in diagonal basis . .
3.4.3 Application of a time-evolved observable in diagonal basis representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
9
10
10
13
4 The dissipative harmonic oscillator I: preparations
4.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Applications, previous work and technical approaches
4.1.2 Discussion of the model Hamiltonian and initial states
4.2 Diagonalization of the Hamiltonian . . . . . . . . . . . . . . .
4.2.1 Definition of operator basises . . . . . . . . . . . . . .
4.2.2 Flow equations for the Hamiltonian . . . . . . . . . .
4.3 Flow equations for observables . . . . . . . . . . . . . . . . .
4.3.1 Transformation into diagonal basis . . . . . . . . . . .
4.3.2 Time evolution . . . . . . . . . . . . . . . . . . . . . .
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15
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II
CONTENTS
4.4
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 The dissipative harmonic oscillator II: applications
5.1 Non-equilibrium in a technical context . . . . . . . . .
5.1.1 Preparation of the initial state . . . . . . . . .
5.2 Entangled initial state . . . . . . . . . . . . . . . . . .
5.2.1 The position expectation value . . . . . . . . .
5.2.2 Correlation functions . . . . . . . . . . . . . . .
5.2.3 The fluctuation dissipation theorem . . . . . .
5.3 Product initial state . . . . . . . . . . . . . . . . . . .
5.3.1 Position evolution . . . . . . . . . . . . . . . .
5.3.2 Correlation functions . . . . . . . . . . . . . . .
5.3.3 Numerical test of the FDT . . . . . . . . . . .
5.4 Mode population in non-equilibrium . . . . . . . . . .
5.4.1 Mode occupation for a coherent state . . . . .
5.4.2 Mode occupation for an entangled initial state
5.5 General time-dependent fields . . . . . . . . . . . . . .
5.5.1 Exact solution for the operator x̃H (t) . . . . .
5.5.2 Discussion of the observable hx(t)i . . . . . . .
5.5.3 Discussion of correlation functions . . . . . . .
5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . .
44
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45
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6 The spin-boson model I: preparations
6.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.1 Formulation of the model Hamiltonian . . . . . . . . . . . . .
6.1.2 Previous work . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Diagonalization of the Hamiltonian . . . . . . . . . . . . . . . . . . .
6.2.1 The role of normal ordering . . . . . . . . . . . . . . . . . . .
6.2.2 Flow equations in O(λ2k ) . . . . . . . . . . . . . . . . . . . . .
6.2.3 The ground state approximation . . . . . . . . . . . . . . . .
6.3 Transformation of observables . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Transformation of spin operators in O(λ2k ) . . . . . . . . . . .
6.3.2 Approximate analytical solution of the flow equations . . . .
6.4 Preparation of the initial state . . . . . . . . . . . . . . . . . . . . .
6.4.1 Product initial states . . . . . . . . . . . . . . . . . . . . . . .
6.4.2 Non-factorizing initial state . . . . . . . . . . . . . . . . . . .
6.4.3 Normal ordering in the context of initial states . . . . . . . .
6.5 Discussion of transformations . . . . . . . . . . . . . . . . . . . . . .
6.5.1 Test of sum rules . . . . . . . . . . . . . . . . . . . . . . . . .
6.5.2 The ground state expectation values hσi iGS . . . . . . . . . .
6.5.3 Remarks on the SU(2) algebra for transformed spin operators
6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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83
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7 The spin-boson model II: spin dynamics
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119
CONTENTS
7.1
7.2
7.3
7.4
III
Time evolution of observables . . . . . . . . . . . . . . . . . . . . .
7.1.1 Time evolution of transformed observables . . . . . . . . . .
7.1.2 Time evolution using time-dependent flow equations . . . .
7.1.3 Discussion of transformations . . . . . . . . . . . . . . . . .
Application to a product initial state . . . . . . . . . . . . . . . . .
7.2.1 Formulation of the problem . . . . . . . . . . . . . . . . . .
7.2.2 The expectation values hσi (t)i . . . . . . . . . . . . . . . . .
7.2.3 How does a spin decay? . . . . . . . . . . . . . . . . . . . .
7.2.4 The correlation function Szz (t, tw ) . . . . . . . . . . . . . .
Application to an entangled initial state . . . . . . . . . . . . . . .
7.3.1 Analytical results in the limit of small field strength h . . .
7.3.2 Numerical results for an improved normal ordering scheme
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8 Comparative view on different models
8.1 Results for the time-dependent Kondo model
8.2 The expectation values hx(t)i and hσi (t)i . .
8.3 Correlation functions . . . . . . . . . . . . . .
8.4 The fluctuation dissipation theorem . . . . .
8.5 Conclusions . . . . . . . . . . . . . . . . . . .
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119
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9 Conclusions and outlook
165
A Properties of flow equations for spin operators
167
A.1 Asymptotic decay into bath operators . . . . . . . . . . . . . . . . . . . 167
A.2 Flow equations for Pauli matrices and the SU(2)-symmetry . . . . . . . 168
A.3 Analytical treatment of time-dependent flow equations . . . . . . . . . . 170
B Conditions for an equivalence of two initial preparations
C Used numerical approaches
C.1 Numerical approaches to solve flow equations . . . .
C.1.1 Time-independent flow equations . . . . . . .
C.1.2 Time-dependent flow equations . . . . . . . .
C.2 Normal mode transformation of the DHO . . . . . .
C.2.1 Definition of the normal mode transformation
C.2.2 Solution of an eigenvalue problem . . . . . .
C.2.3 Calculation of physical quantities . . . . . . .
C.2.4 Checks of numerical accuracy . . . . . . . . .
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175
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177
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188
D Evaluation of a time ordered commutator
191
List of Tables
193
List of Figures
196
IV
Bibliography
CONTENTS
197
Chapter 1
Introduction
Experimental motivation
Many examples in nature exhibit damped behaviour that is adequatily described by
equations of motion of the form
mẍ + η ẋ + v 0 (x) = F (t)
(1.1)
where the position x of the particle is influenced by the potential v(x) and a timedependent fluctuating force F(t). One immediately might think of the example of
Brownian motion where a collodial particle is immersed in a viscous fluid. Typically,
the fluctuating force F(t) would then be a stochastical noise stemming from randomly
occuring collisions between the Brownian particle and the molecules of the surrounding liquid. In fact, equation (1.1) has been extensively used to formulate a theory of
Brownian motion during the first half of the 20th century [1]. Due to the macroscopic
properties of a Brownian particle one does not need to bother about quantum effects.
If one thinks of the transport of rather microscopic particles where quantum mechanics
will get important for low temperatures, one can consider the example of a LCR circuit
that also obeys equation (1.1) if the position of a particle is replaced with the charge
on a capacitor. However, one has to introduce a quantum mechanical formulation for
the damped behaviour of charge transport in the low temperature regime.
Nowadays, much effort is spent on the topic of mesoscopic quantum systems that contain only few spin or charge degrees of freedom that are coupled to a larger macroscopic
environment. The still fast emerging industry of information technology continues to
be the driving force behind this branch of solid state physics. Moore’s law, that predicts a doubling of processing power every 18 months, was satisfied since the 1970’s by
scaling devices to smaller and smaller dimensions. Limits in fabrication of electronic
circuits and also the increasing importance of quantum mechanics at smaller length
scales will become of fundamental importance in the next decades. Already in 1985,
even transistors working with a single electron degree of freedom (Single electron transistors, SET’s) were realized.
2
Introduction
Leading behind the possibilities of semiconductor physics, mesoscopic quantum systems promise to give rise to a new concept of processing information, namely quantum
computing. Single spins are promising candidates for the realization of qubits because
they represent an ideal two state system. In quantum dots, it is possible to read out
a single spin state by a so called spin to charge conversion [2]. In such experiments,
the environmental coupling plays an important role since it tends to destroy the spin
state, especially during the measurement process. Weak currents in nano-scaled leads
help to decrease the decoherence effects of the environment during the time a readout
bias is applied.
Other promising realizations of mesoscopic two state systems are Josephson junctions
that can be reduced to effective two state systems at low temperatures. SET’s are proposed as suitable readout devices for these qubits. Again, the motivation is to reduce
decoherence effects during the measurement process by minimizing the readout device.
All these examples showed that a technical usage of mesoscopic systems with few degrees of freedom struggles with disturbing effects of environmental coupling. In all
technical applications, they will be open systems that interact with an environment,
most important through measurement processes and in general through limitations in
isolating them from different constituences of a device. Of central interest in theoretical
investigations is therefore to understand how the decoherence time can be improved.
This is the time scale in which pure quantum mechanical states of a small system
are destroyed by the randomly acting environment . However, theoretical descriptions
of mesoscopic quantum systems are inavoidable to understand their properties because
their behaviour is essentially governed by the very abstract rules of quantum mechanics.
These needs have lead to the formulation of dissipative quantum systems. These are
models that try to describe irreversible energy transfer processes from a small quantum
system to a macroscopic environment. This property is briefly adressed by the term
”dissipation”.
Theoretical motivation
Many models have been proposed to describe dissipation phenomena in quantum mechanical systems. The methods and applications of dissipative quantum mechanics are
described in great detail in the book of Weiss [9]. In the next chapter, we will introduce the common theoretical description of dissipative quantum systems that is often
referred to as the Caldeira-Leggett model [6-8]. Eg, this model has found a widespread
application to problems of quantum computing [69], quantum Brownian motion [8] and
dissipative quantum tunneling [7].
During the last decade, it became of interest to treat this model for out-of-equilibrium
preparations. Eg. the decohering influence of the bath that destroys pure quantum mechanical states is of interest. Such non-equilibrium situations lead to time-dependent
3
physical observables, eg a qubit prepared initially in a pure state will finally lose its
information due to decoherence. Several techniques have been developed to calculate
time-dependent observables in dissipative quantum systems and help to describe such
time-dependent non-equilibrium phenomena theoretically.
Analytical methods like the Keldysh technique [71] or Born and Markov approximations to quantum master equations mostly rely on perturbative approximations that
fail to describe important physical aspects in such problems. Eg these methods usually
fail to describe algebraic long-time tails in zero temperature correlation functions that
are important features of dissipative quantum systems [9].
More accurate are non-perturbative numerical techniques like the time-dependent numerical renormalization group (TD-NRG) [3] and the time-dependent density matrix
renormalization group (TD-DMRG). However, these numerical techniques are not suitable to identify exact analytical laws.
The main intention of this thesis is to apply the flow equation method as a new analytical method to treat time-dependent dissipative quantum systems under non-equilibrium
conditions. This method will be used to treat two different models:
1. Firstly, a trivial model for quantum dissipation is treated, the dissipative harmonic oscillator (DHO). It allows for exact solutions.
2. The second and non-trivial model will be the spin-boson model. It is necessary to
use approximations to treat this model and thus, this model treatment shall indicate that the flow equation method provides a controlled approximation scheme
for non-trivial dissipative quantum systems.
Both models will be treated in a similar way. Firstly, the model Hamiltonian is diagonalized with the flow equation method. Then, the Heisenberg equation for relevant operators is solved by exploiting the diagonalized Hamiltonian and these time-dependent
operators are finally evaluated in different non-equilibrium initial states.
These calculations aim at:
1. Calculating the time-dependent position expectation value hx(t)i of the DHO and
the spin expectation values hσi (t)i, i = x, y, z for the spin-boson model.
2. Calculating the two-time spin-spin correlation function hσz (t + tw )σz (tw )i of the
spin-boson model and the displacement correlation function hx(t + tw )x(tw )i of
the DHO.
3. Checking the fulfillment of the fluctuation-dissipation theorem for the real and
imaginary part of these correlation functions in the quantum limit.
4. Analyzing these results and comparing them in order to identify possible generic
behaviour.
4
Introduction
All results are compared against known results in order to show that the flow equation
method is able to give a proper description of non-equilibrium dynamics in dissipative
quantum systems.
Outline of this thesis
In chapter 2 the modern theoretical description of dissipative quantum systems is described. This chapter is based on the first chapter of reference [9]. Chapter 3 introduces
into the flow equation method. In chapter 4 the DHO and its applications are presented and the Hamiltonian is diagonalized by means of flow equations. In the end, the
Heisenberg equation for bosonic operators is solved. In chapter 5, these time-dependent
bosonic operators are used to calculate the position expectation value hx(t)i and the
displacement correlation function hx(t + tw )x(tw )i for two different non-equilibrium
states. In chapter 6, the spin-boson model is introduced and its Hamiltonian is diagonalized with the flow equation method. The time-dependent spin-operators are
calculated afterwards in chapter 7 and used to calculate the spin expectation values
hσi (t)i, i = x, y, z and the spin-spin correlation function hσz (t + tw )σz (tw )i. A comparison of all physical results is given in chapter 8 that also includes a comparison to
results for the time-dependent Kondo model [39]. The thesis closes with a summary in
chapter 9.
Chapter 2
Dissipative quantum systems
Several approaches exist that include effects of dissipation into quantum mechanics.
The most successful technique has been the Hamiltonian formalism that includes both
the environment and the system degrees into one conservative system. In this approach
dissipation is an effect that occurs in form of energy transfer from the system to the
environment. This energy transfer becomes ireversible if the environment consists of an
infinite number of degrees of freedom. In this chapter we treat the phenomenological
Caldeira-Leggett model that describes dissipative quantum systems. This model makes
an elimination of bath degrees of freedom from equations of motion of system observables possible. In the classical limit this elimination leads to a Langevin equation. A
formulation of the quantum mechanical dynamics is usually given with path integral
techniques that integrate out environmental degrees of freedom via path integrals. For
factorized initial density matrices this formulation was given first by Feynman and Vernon. It turned out that initial correlations between system and bath have important
influences onto dynamics of the system. Correlated initial density matrices can be
treated within the method of Grabert et al [4] that gives an extension of the approach
of Feynman and Vernon.
2.1
2.1.1
The Caldeira-Leggett model
The Hamilton function and the classical equation of motion
The model of a harmonic oscillator that is coupled linearly to a heat bath goes back
already to Ullersma [5]. Lateron, Caldeira and Leggett [6,7,8] used this model for a
description of dissipative effects for tunneling processes out of a metastable state.
Let us consider a small system of one or few degrees of freedom that is coupled to a heat
bath. This heat bath shall be described through an ensemble of oscillators. In order
to describe a macroscopic system, the limit of an infinite number of oscillators has to
be taken. Only in this case it is ensured that dissipated energy does not return to the
system, leading to an infinte Poincaré recurrence time. The fundamental assumption
6
Dissipative quantum systems
of the Caldeira-Leggett model is that every bath degree of freedom is only weakly
perturbed by the system. For macroscopic baths this is reasonable, since interactions
of the system with each bath degree of freedom are proportional to the inverse bath
volume. The coupling between system and bath can then be considered as linear in
the bath coordinates. This property makes it possible to effectively eliminate the bath
degrees of freedom from the dynamics of the system. In contrast, it is not necessary to
assume that the coupling of system and bath is linear in the system coordinate. Eg it
is possible to perturb the system degree of freedom strongly by external fields and it is
then necessary to describe nonlinear effects in the coupling to the system coordinate.
The Hamiltonian of the whole system then has the form
H = HS + HB + HSB
(2.1)
For a subsystem with a generalized coordinate q, the term HS will be
HS (q, p) =
p2
+ V (q, t)
2M
(2.2)
Time-dependent contributions to the potential V(q,t) are typically given in form of
time-dependent external fields that contribute the term -F(t)q to the potential. The
heat bath is described through
HB ({xα }, {pα }) =
X p2
1
α
+ mα ωα2 x2α
2mα 2
α
(2.3)
and the sum over α contains all N bath oscillators that are characterized by their masses
mα and frequencies ωα . The interaction part HSB has the general form
HSB (q, {xα }) = −
X
Fα (q)xα + ∆V (q)
(2.4)
α
The counter term ∆V (q) compensates renormalization effects of the potential introduced by the coupling to the heat bath. It is determined as
HSB (q, {xα }) = −
X F 2 (q)
α
2
2m
α ωα
α
(2.5)
In this way, the reservoir only introduces dissipation and not a renormalization of the
potential V(q). In all cases in which the coupling to the environment is also linear in
the systen coordinate, Fα (q) = cα q, we obtain the classical Hamilton function
N
H=
X h p2 1
p2
cα 2 i
α
+ V (q) +
mα ωα2 xα −
q
2M
1mα 2
mα ωα2
(2.6)
α=1
By using the canonical equations of motion it is possible to derive [9] an integrodifferential equation for the system coordinate q(t):
2.1 The Caldeira-Leggett model
t
Z
7
dt0 γ(t − t0 )q(t) +
M q̈(t) + M
−∞
∂V
= ζ 0 (t) − M γ(t)q(0)
∂q
(2.7)
This Langevin-equation describes a classical particle with linear damping under influence of a stochastic force. The damping kernel is given as
γ(t − t0 ) = Θ(t − t0 )
1 X c2α
cos[ωα (t − t0 )]
M α mα ωα
(2.8)
and the stochastic force as
ζ 0 (t) =
X
h
cα xα (0) cos(ωα t) +
α
i
pα (0)
sin(ωα t)
m − αωα
(2.9)
The stochastic equation of motion (2.9) still contrasts with the usual form of a Langevinequation by the spurious term −M γ(t)q(0). To eliminate this term one defines the
stochastic force
ζ(t) = ζ 0 (t) − M γ(t)q(0)
(2.10)
The modified random force ζ(t) does not vanish does not vanish on average when the
average is taken with respect to the equilibrium density matrix of the bath. In the
second step, ζ(t) is reconciled with the usual properties of a Gaussian random force
by performing the thermal average in the initil state of the reservoir with the shifted
canonical equilibrium distribution
n
2 io
Xh p(0)2
mα ωα2 n (0)
cα
α
ρR = Z −1 exp −β
+
xα −
q(0)
2mα
2
mα ωα2
α
(0)
(2.11)
(0)
Upon averaging xα and pα with the weight function (2.11), the random force ζ(t)
exhibits the proper statistical behaviour of a Gaussian coloured noise,
hζ(t)i = 0
0
(2.12)
0
0
hζ(t)ζ(t )i = M kB T γ(t − t ), t > t .
2.1.2
(2.13)
The spectral density of the bath
In order to describe the influence of the bath on the dynamics of the system, it is
not necessary to know the bath parameters mα , ωα and cα in particular. This is even
not possible for a macroscopic bath. Instead, the influence of the bath is completely
described through the spectral density
J(ω) =
π X c2α
δ(ω − ωα ).
2 α mα ωα
(2.14)
8
Dissipative quantum systems
For a heat bath with an infinite number of bath modes the spectral density can be
considered as a smooth function of ω. With this definition eq. (2.8) can be written in
the form
Z ∞
J(ω 0 )
2
dω 0
γ(t) = Θ(t)
cos(ω 0 t).
(2.15)
πM 0
ω0
The damping kernel γ(t) is therefore completely determined by the spectral density.
Any form of linear damping can be modeled by a suitable choice of the spectral density J(ω). The inversion of the Fourier integral (2.15) yields the frequency-dependent
damping constant 1
Z
2iω ∞ 0 J(ω 0 )
1
γ̃(ω) = −
dω
.
(2.16)
πM 0
ω 0 ω 02 − ω 2 − isgn(ω)0+
The real part of eq. (2.16) is given as
1 J(ω
.
(2.17)
M ω
Since the bath enters also into quantum mechanical equations of motion solely via the
spectral density J(ω), eq. (2.17) plays a major role 2 in the phenomenological modelling
of dissipative quantum systems. It is sufficient to know the classical damping constant
γ̃ 0 (ω) that can eg be obtained from molecluar dynamics simulations. Via eq. (2.17)
the phenomenological spectral density is then determined. A frequency independent
damping constant is known as Ohmic or Markovian damping.
γ̃ 0 (ω) =
γ̃ 0 (ω) = γ
(2.18)
In this case the spectral density is strictly Ohmic
J(ω) = M γω.
(2.19)
In the time-domain Ohmic damping corresponds to the ideal case of time-local damping
γ(t) = Θ(t)2γδ(t). Every real physical system is damped non-local in time and has a
frequency dependent damping. The underlying reason is that dissipation is based on
microscopic processes that occur on a finite time scale. Every realistic spectral density
vanishes in consequence in the limit ω → ∞. In many cases the spectral density can for
small frequencies below a cut off frequency ωc be described by a power law J(ω) ∝ ω s .
The frequency ωc is chosen well below a characteristic cut off frequency, eg the Fermi
frequency or the Debye frequency. Different parameter values s correpond to certain
types of baths. The case s=1 is known as Ohmic bath that is eg. important in Josephson
junctions. In addition, the value s < 1 is called sub-Ohmic bath and in analogy, the
case s > 1 is called a Super-ohmic bath. Eg. the cases s=3 and s=5 are characteristic
for a defect interacting with a bath of phonons at low temperature.
1
In the time domain, causality is ensured by the Theta-function in eq. (2.15). The poles of γ̃(ω) lie
in the lower half plain and lead also to causality.
2
The imaginary part of γ̃(ω) can be extracted from the real part via a Kramers-Kronig relation.
Chapter 3
The flow equation method
In the first two chapters of this thesis it became clear that we will have to deal with
interacting many particle systems. The flow equation method is a modern method
that was developed during the last few years eg. for an application to interacting many
particle systems. We will give an introduction into this method and will in the end
explain how it can be used to obtain-time dependent observables in dissipative quantum
systems.
3.1
History and applications
Interestingly, the flow equation method has two different roots. It was developed independently by Wegner 1994 [10] in the context of a problem of condensed matter
theory and by Glazek and Wilson in the field of high energy physics [11,12]. The latter
christened this method ’Similarity renormalization scheme’. Several successful ideas of
theoretical physics were combined in the flow equation method. In the famous Numerical Renormalization Group (NRG) method of Wilson [51], the energy spectrum of a
Hamiltonian is discretized logarithmically in order to transform a Hamiltonian iteratively into an effective low-energy Hamiltonian. This aspect of energy scale separation
is one of the main ingredients of the flow equation method. It applies a sequency of
infinitesimal unitary transformations that successively treat different energy scales of
the Hamiltonian and thus opens a controlled way to construct effective Hamiltonians.
An application to very different problems of condensed matter theory is possible since it
establishes a unitary transformation that preserves information, e.g. about high energy
scales, of the original Hamiltonian.
Eg. flow equations have been successfully applied to technical challenging strong coupling problems like the Kondo model [16,70]. In the field of dissipative quantum
systems, several publications [15,26] treated the spin-boson model that also shows
strong coupling behaviour. More recent applications aim at out-of-equilibrium situations where either the Hamiltonian contains an external bias [70] or the system is
prepared in a non-equilibrium initial state [16,17]. Eg., such situations were analyzed
10
The flow equation method
in the Kondo model with promising results. A comprehensive reference containing all
these works is the book on flow equations written by Stefan Kehrein [27].
3.2
Transformation of the Hamiltonian
Any application of flow equations starts with a transformation of a model Hamiltonian.
The aim is always to decouple interaction terms in order to make the Hamiltonian more
diagonal in its energy representation. This important property of the transformed
Hamiltonian can eg be exploited to calculate time-dependent observables, as we will
show in section 3.4.
3.2.1
Infinitesimal unitary transformations
To treat energy scales separately, flow equations provide a constructive approach to
formulate infinitesimal unitary transformations acting only on an infinitesimal large
energy interval at the same time. To achieve an effective transformation that makes
a Hamiltonian more energy diagonal, in a second step one has to formulate a whole
sequence of such differential transformations. A natural way to construct a transformation that establishes both unitarity and the desired infinitesimal transformation effect
is to use a parameter dependent unitary operator. The parameter of this operator will
be related to a particular energy scale and will lead to a unitary transformation that
decouples interaction matrix elements related to this energy scale.
Definitions
A unitary transformation U(B), dependent on a real parameter B is defined to transform the conventional operator space O of quantum mechanics that is related to a
Hamiltonian H. All operators of the space O will be referred to as observables.
In the usual way, the unitary operator U(B) can be representated as an exponential of an
antihermitean operator η(B), called its generator. We label any transformed operator
by the flow parameter B that correponds for B=0 to the untransformed operator. For
instance, the transformed Hamiltonian H(B) is defined as
H(B) = U (B)HU † (B)
(3.1)
In practice, equation (3.1) is not suitable to formulate a systematic transformation of
the Hamiltonian since in most cases a unitary transformation U (B) with the desired
property of energy scale separation cannot be obtained explicetely in a systematic way.
Eg via its generator η it has to expressed via a B-ordered exponential
†
Z
U (B) = T> exp
0
B
η(B 0 )dB 0
(3.2)
3.2 Transformation of the Hamiltonian
11
if the generator η(B) does not commute with itself for different values of B. In (3.2) we
used the operator T> that denotes B-ordering.
Equivalently to (3.2), the transformed Hamiltonian H can be derived from a differential
equation that is formulated using the generator η(B):
dH
= [η(B), H(B)]
(3.3)
dB
Equation (3.3) provides a differential formulation of the transformation of the Hamiltonian that we will refer to as flow equation of the Hamiltonian. While it is in most
cases not feasible to solve equation (3.3) exactly, it is possible to apply perturbative
approximations to it. For this purpose, we rearrange the formal representation (3.1) of
the transformed Hamiltonian H(B) into a more suggestive form 1 .
H(B) = H + U † (B)[H, U (B)]
Z B
h
i
†
= H + U (B)
dB 0 H, η(B 0 )
0
+
+
1
2!
Z
1
3!
Z
B
Z
B0
dB
0
h
i
dB 0 H, η(B)η(B 0 )
0
B
Z
B0
dB
0
dB 0
0
Z
B 00
h
i
dB 00 H, η(B)η(B 0 )η(B 00 )
!
0
+ ...
(3.4)
In this form, it is easily possible to apply perturbative approximations to the flow
equation (3.3) if the generator η(B) is proportional to a small expansion parameter λ.
Our aim is to split up the transformed Hamiltonian (3.1) into a perturbation series in
an expansion parameter λ with various operators H (n) (B) contributing to it.
H(B) = H
(0)
(B) + λH
(1)
2
(B) + λ H
(2)
(B) + ... = H
(0)
(B) +
∞
X
λn H (n) (B)
(3.5)
n=1
Physical interpretation of the perturbation series (3.5)
In order to relate the perturbation series (3.5) to the initial Hamiltonian H, in many
cases the expansion parameter λ is chosen as the prefactor of the dominating interaction
term in the model Hamiltonian. In a physical interpretation, the operators H (n) (B)
1
We use the series representation of the B-ordered exponential U(B) (cf. (3.2)).
12
The flow equation method
represent interaction terms while the operator H (0) (B) is the interaction-free part of
the Hamiltonian H(B). In the original Hamiltonian H usually only a small number
of interaction terms were included and the remaining terms in (3.5) are generated
during the flow of the Hamiltonian. Eg in dissipative quantum systems it is common
to treat the system-bath couplings λk as an expansion parameter [15,26,27]. In this
way, the operator λH (1) (B = 0) will correspond to the coupling term HSB in the
model Hamiltonian of a dissipative quantum system. Thus, the usual strategy is to
formulate a continuos sequence of transformations that eliminates the term H (1) (B)
from the perturbation series (3.5) in the limit B → ∞ of accomplished
The final
P∞ n flow.
(0)
(n)
Hamiltonian H(B = ∞) is then of the form H (B = ∞) + n=2 λ H (B = ∞).
The free Hamiltonian H (0) (B = ∞) will have the form
H (0) (B = ∞) = HS∞ + HB
(3.6)
and all interaction terms H (n) (B = ∞) have been eliminated at least up to the second
order in the expansion parameter λ.
In order to ensure a proper expansion, it has to be ensured that the expansion parameter is sufficiently small. Eg it is important to note that the perturbative series in λ
is conceptionally already formulated at the level of the infinitesimal transformation.
Thus, it is important to ensure that the series expansion retains its convergence properties during the integration of a continous sequence of differential transformations. Eg
in quantum impurity systems couplings with small energy differences to the impurity
levels decay very slow during the flow and higher orders in the perturbative series expansion can remain large [15,26,27].
Expansion of the flow equation
Our aim is to formulate a perturbative expansion of the flow equation (3.3). Already
from the series representation (3.4) of the transformed Hamiltonian H(B) it became
clear that it is useful to chose a generator η that is at least proportional to the expansion
parameter λ. In this way, the complicated higher commutators in the representation
(3.4) will also be of higher order in the expansion parameter. In an approximative
treatment of the flow equation it becomes then possible to neglect them by a perturbative approximation.
Due to the above motivation, the most general form of a generator we want to use is
given as a series expansion in λ that vanishes for λ = 0.
η = λη (1) + λ2 η (2) + ... =
∞
X
λn η (n)
(3.7)
n=1
Inserting the expanded Hamiltonian (3.5) and the generator (3.7) into the flow equation
3.2 Transformation of the Hamiltonian
13
(3.3), we arrive at a perturbative expansion of this flow equation.
dH
= λ[η (1) , H (0) ] + λ2 [η (1) , H (1) ] + [η (2) , H (0) ] + O(λ3 )
(3.8)
dB
We have finally shown the usefulness of a perturbative expansion of the Hamiltonian
and the generator η. From equation (3.8) it becomes obvious that it is conceptionally
possible to truncate all contributions from higher order interactions H (n) , n ≥ 2 in the
transformed Hamiltonian in a controlled way to approximate the flow of the Hamiltonian by neglecting terms in O(λ3 ) in the flow equation 3.3. However, this approach has
only a conceptional character and depends largely on the identification of a suitable
expansion parameter. In addition it is important for practical purposes to specify the
various terms H (n) in the expansion (3.5). It is of great practical importance to relate
these terms to operators of a canonic multiparticle representation of the Hamiltonian
that we will discuss in the next subsection.
We will see that the practical implementation of an infinitesimal unitary transformation emerges naturally from this multiparticle expansion of the Hamiltonian. In a more
abstract formulation, such infinitesimal unitary transformations can be applied to any
desired observable. Our main intention is to develop a formalism that can be applied
to calculate time-dependent observables. Based also on infinitesimal unitary transformations, this formalism also uses a multiparticle representation of observables.
From now on, we will discuss every technical aspect of the flow equation formalism
in full generality for any observable of the operator space O. All formal definitions of
differential transformations described above for the Hamiltonian apply in general also
for an arbitrary observable O, eg it will transform according to the flow equation
dO
= [η(B), O(B)]
dB
Eg in section 3.4 we will develop this general formalism further in order to calculate
time-dependent observables.
3.2.2
Multiparticle expansion of observables
In dissipative quantum systems a formulation of environmental degrees of freedom is
usually given in form of bosonic normal modes. The small subsystem that interacts
with this environment consists usually of one or few particles and its correponding
operator space is spanned by only few elementary operators that describe excitations
of this small system. We define a set of basis operators that can construct all these
elementary operators and denote its elements formally by the symbol Ol where l is an
index that counts the basis elements. Eg in the spin-boson model the basis operators
Ol act on a two state system and thus, these operators will be the Pauli matrices σx ,
14
The flow equation method
σy and σz combined with the two-dimensional unity matrix. In this way, it is possible
to expand every operator of a dissipative quantum system into a multiparticle basis of
the operator space O 2 .
Choice of a multiparticle basis
We define a multiparticle basis in the bosonic Fock space of the bath modes by chosing
the union of all subsets containing one creator or annihilator, two creators or annihilators and so on. Each of these subsets defines a subset of the full operator basis in O if
we build the tensor product of each of its elements with each of the operators Ol . We
l
denote these subsets of tensor products with the symbols Bnm
that define the following
subsets:
l
B0,0
= {Ol ⊗ 1}
(3.9)
b†k0 }
1
(3.10)
l
n
= {Ol ⊗ b†k1 bk10 } B0,2
= {Ol ⊗ bk1 bk2 } . . .
B1,1
(3.11)
l
B1,0
= {Ol ⊗ bk1 }
l
B0,1
= {Ol ⊗
l
The indices n,m of the symbol Bnm
define the number of bosonic annihilators (n) and
creators (m) that the elements of the subset contain. For brevitiy we chose a basis
with bosonic operators that are arranged according to the rule ’creators to the left of
annihilators’ 3 .
l
Each of the subsets Bn,m
can then be characterized by the number n of creation and
m of annihilation operators it contains and for completeness by the index l of the basis
operator Ol .
Multiparticle expansion of an operator
Any operator O that is element of the operator space O can be represented in this
multiparticle basis:
O=
XX X
l
m,n
X
† †
†
0
λlk1 k2 ...kn k10 k20 ...km
0 Ol ⊗ b
k1 bk2 ...bkn bk10 bk20 ...bkm
(3.12)
0
k1 k2 ...kn k10 k20 ...km
where we call the coefficients λlk1 k2 ...kn k10 k20 ...km
”generalized coupling constants”. A
0
differential transformation of the observable O will lead to a differential transformation
of the generalized coupling constants if the transformed observable is represented in
the same multiparticle basis. Thus, our aim is to describe the transformation of the
generalized coupling constants under a differential transformation of the observable O
2
Some of the notations in this subsection were used in close analogy in [72]
We will formulate this rule as part of the more abstract formalism of normal ordering in subsection
(3.2.4).
3
3.2 Transformation of the Hamiltonian
15
defined by the flow equation (3.3). For this purpose, we also represent the generator η in
the multiparticle basis, denoting its generalized coupling constants by ηlj1 j2 ...jn j10 j20 ...jm
0 .
3.2.3
Flow equations for a general multiparticle expansion of an observable
We want to develop the essential formalism that is characteristic for the flow equation
framework. This formalism determines the change of the generalized coupling constants
of an observable under infinitesimal unitary transformations. For this purpose, we insert
the generator η and the observable (3.12) in form of their basis expansions into the flow
equation (3.3):
dO(B)
= [η(B), O(B)] =
dB
XXX X
X
X
X
ηkj1 j2 ...jn j10 j20 ...jm
0 λlk k ...k k 0 k 0 ...k 0
o 1 2
1 2
p
0 k1 k2 ...kn k 0 k 0 ...k 0
k,l m,n o,p j1 j2 ...jn j10 j20 ...jm
m
1 2
† †
†
0 , Ol ⊗ b
×[Ok ⊗ b†j1 b†j2 ...b†jn bj10 bj20 ...bjm
k1 bk2 . . . bko bk10 bk20 . . . bkp0 ]
(3.13)
Even during a differential transformation of O, many new contributions to the observable O can occur in (3.13). The reason is that the multiparticle-commutators in (3.13)
may lead to a non-zero change of the generalized coupling constant of a basis operator
that did not contribute to the original representation of the observable O. After the
transformation the observable may contain new contributions that can be even of a
higher order in bosonic operators than overall were present in the initial observable .
At this development stage of the formalism, the reader might already guess that in
practice, approximations to the transformation (3.13) are often necessary. Inspite of
this problem, we can give a formal exact description that determines the change of the
generalized coupling constants during the flow of the observable O. For this purpose we
have to compare the multiparticle basis representation of the right hand side of equation (3.13) with the multiparticle basis representation of the observable O. In order
to rewrite the right hand side of eq. (3.13) into a multiparticle basis representation,
we have to rearrange all coefficients and define thereby the new generalized coupling
0
constants Λl(m0 ,n0 ) (η(k,n,m) (B), λl,o,p (B)) 4 . Through the various commutators in (3.13)
they will be functions of several generalized coupling constants λ(l,o,p) (B) of the observable O and also of those of the generator η (η(k,n,m) (B)). In this way the functions
0
Λl(m0 ,n0 ) (η(k,n,m) (B), λl,o,p (B)) contain the full information of the infinitesimal unitary
transformation. We rewrite the right hand side of eq. (3.13) by formally inserting the
0
functions Λl(m0 ,n0 ) (η(k,n,m) (B), λl,o,p (B)).
4
The symbols Λ(l0 ,m0 ,n0 ) ,η(k,n,m) and λ(l,o,p) denote the union of all generalized coupling constants
l0
k
l
of the operator subsets Bm
0 ,n0 , Bn,m and Bo,p , respectively.
16
The flow equation method
dO(B) X X
=
dB
0 0
l
m ,n
X
X
Λlk1 k2 ...kn0 k10 k20 ...k0 0 (η(k,n,m) (B), λ(l,o,p) )
m
k1 k2 ...kn0 k10 k20 ...k0 0
m
0
×Ol ⊗ b†k1 b†k2 ...b†k0 bk10 bk20 ...bkm
n
(3.14)
After inserting the multiparticle representation (3.12) of the observable O into the left
hand side of equation (3.14) a comparison of the generalized couping constants on both
sides is possible. It yields a set of coupled differential equations for the generalized
coupling constants that are commonly referred to as the flow equations.
dλlk1 k2 ...kn0 k10 k20 ...k0
0
m
= Λlk1 k2 ...kn0 k10 k20 ...k0 0 (η(k,n,m) (B), λ(l,o,p) )
(3.15)
m
dB
The coupled differential equations (3.15) determine the flow of the observable O, eg in
the limit of accomplished flow they determine the observable O(B = ∞). An integration
of the flow equations (3.15) is determined by the initial values of all involved generalized
coupling constants at the parameter value B=0. It is usual to approximate the flow
equations (3.15) since the flow of the observable O(B) may in principle contain terms
with an unlimited number of bosonic operators.
3.2.4
Approximations to the flow equations
The number and the complexity of the flow equations is reducable by a truncation of
the basis expansion (3.12) of the observable. A motivation for such a truncation procedure has to be carefully analyzed and it is not useful to truncate the untransformed
observable O from the beginning. Instead, it has to be carefully examined which terms
are newly generated during the flow due to a nonzero derivative of their generalized
coupling constants. Then it has to be decided if their generalized coupling constants
remain sufficiently small during the flow and if this is not the case, it has to be analyzed
if they contribute relevant physical aspects to the considered problem.
As we mentioned above, an approximated solution of flow equations is based on a truncation of the basis expansion (3.12) of the observable O and will restrict the flow of
this observable in the operator space O. Usually, the flow equation method leads to
a proliferation of such higher order terms during the flow. It is therefore necessary to
introduce a way of organizing these higher order terms into such a form that truncating
this series at some point leeds to as small an error as possible. A proper choice of a
truncation scheme is usually not clear at the stage of the formulation of differential unitary transformations since the generalized coupling constants can change appreciable
during the flow. Concerning the physical importance of the operators present in a basis
representation of an observable, a systematic formalism exists that helps to highlight
their physical properties. This is the formalism of ’normal ordering of operators’ that
is often used in quantum field theory to get rid of delicate vacuum energies.
3.2 Transformation of the Hamiltonian
17
The formalism of normal ordering
Normal ordering has been formally defined by Wick [13]. It represents a sophisticated
tool that can be used to relate creation and annihilation operators to a specified state
or density matrix. In its most trivial application, normal ordering amounts to the subtraction of ground state expecation values from an operator. Throughout this thesis,
normal ordering will be used as a crucial tool to approximate time-dependent observables (cf. chapter 7).
We will define this formalism for bosonic operators since we are interested only in
bosonic baths. From the beginning, we define both creators and annihilators by the
collective symbol Bk . Any normal ordering procedure is defined completely by its
contractions Ckl . These are real numbers that satisfy the property
[Bk , Bl ] = Ckl − Clk
(3.16)
In any application of normal ordering, we will define these contraction either as expectation values with respect to a given reference state | ψi
def
Ckl = hψ | Bk Bl | ψi
(3.17)
or as expectation values with respect to a mixed state described by a density matrix ρ
def
Ckl = T r(ρBk Bl )
(3.18)
A normal ordered operator O composed of bosons is denoted by :O: and defined by the
following three rules due to Wick:
1. C-numbers remain unchanged: :1:=1
2. Normal ordering is linear: : α1 O1 + α2 O2 := α1 : O1 : +α2 : O2 :
where α1 and α2 are complex numbers.
P
∂O
3. The recurrence relation Bk : O :=: Bk O : + l Ckl : ∂B
:, where O is considered
l
as an analytic function of the creation and annihilation operators Bl
If normal ordering is defined with respect to the bosonic vacuum state, these rules
amount to the trivial rule that creators are placed to the left of annihilators: : bk b†l :=
b†l bk .
The role of normal ordering
In order to explain the usefulness of normal ordering for an accurate truncation of higher
order operators in a multiparticle representation of an observable, we explain the most
important property of a normal ordered operator. Any normal ordered operator :O:
18
The flow equation method
that contains at least one creation or annihilation operator has a vanishing expectation
value h: O :i = 0 5 with respect to the state or density matrix that has been used
to define the contractions of the normal ordering prescription. It becomes obvious to
the reader that this property can be exploited if truncated operator expansions shall
be used to calculate expectation values of observables. If all truncated operators are
normal ordered with respect to the desired state or density matrix that is used to
calculate the expectation value of the observable, all truncated operators will have a
vanishing contribution to this expectation value.
3.3
Continuos sequency of unitary transformations
Up to now, we only discussed differntial transformations that describe the change of a
unitarily transformed observable O(B) under a infinitesimal variation of the parameter B. Moreover, it is important to understand the final outcome of a transformation
that used a continous sequence of such differential transformations described by the
flow equation (3.3). The effect of a sequence of differential unitary transformations is
controlled by the generator η.
Choice of a generator
A crucial motivation for the choice of a generator η is the goal to successively eliminate
the interaction terms of a Hamiltonian that couple environmental degrees of freedom
with those of a small subsystem. Of great importance is the aspect of energy scale
separation mentioned in the introduction of this chapter. Couplings between interaction matrix elements with large energy differences shall be decoupled first while nearly
resonant couplings are decoupled later. This approach is motivated by the successful
example of NRG [51] that uses a logarithmic discretization of energy scales in order to
separate physical irrelevant energy scales from relevant low energy scales that are discretized denser. A canonical generator that eliminates all interaction matrix elements
while respecting energy-scale separation has been constructed by Wegner [10]. We formulate its definition in a way suitable for an application to dissipative quantum systems.
Formally, the canonical generator is given by commuting the interacting part of a
Hamiltonian with the non-interacting part. In addition, the unitary dependence of this
parts on the flow parameter B has to be obeyed. For a dissipative quantum system,
the canonical generator is thus equivalent to:
η(B) = [HS (B) + HB (B), HSB (B)]
(3.19)
This generator is antihermitean as required. In addition, it leads to the consequence
that the parameter B has the dimension (Energy)−2 as a consequence of the flow equa5
This property was eg proven in [27].
3.4 Time evolution of observables
19
tion (3.3). More subtle, it can be shown that the canonical generator (3.19) eliminates
interaction matrix elements of HSB (B) with an energy difference according to the cor1
respondence ∆E = B − 2 [10,27]. In this way, the canonical generator makes it possible
that the interaction term HSB (B) vanishes in the limit B → ∞.
Other choices of generators are possible and eg were used in ref. [15] to decouple Hamiltonians of dissipative quantum systems.
Further formal definitions
The final outcome of a continous sequence of infinitesimal unitary transformations leads
to an unitarily transformed operator that depends on the continous parameter B that
denotes the end point of the continous sequence of transformations. Formally, the
effective transformation can be representated by the unitary operator U(B) introduced
in (3.2). It is useful to define a notation that makes the dependence of transformed
operators on the parameter B obvious. We will describe the effect of this transformation
by denoting any operator O that has been transformed by the unitary transformation
U(B) with the symbol O(B). The aim of the unitary transformations U(B) is to make
a Hamiltonian more energy diagonal. In the limit B → ∞, a final net transformation
has been applied that decoupled the Hamiltonian or made it block diagonal. Of all
transformations U(B) this net transformation is singled out in the limit B → ∞.
Eg. the B-dependent generalized coupling constants λlk1 k2 ...kn k10 k20 ...km
0 (B) (cf. (3.12))of
a multiparticle expansion will have converged in the limit B → ∞ and we denote
the converged generalized coupling constants by the symbol λlk1 k2 ...kn k10 k20 ...km
0 (∞). In
addition, we denote the transformed operators in this limit with the symbol O(∞)
and call the operator O(∞) the ’diagonal basis representation of the operator O’. This
definition of a diagonal basis allures to the diagonal form of the Hamiltonian H(∞).
3.4
Time evolution of observables
Our aim is to extend the formalism of flow equations in such a way that it allows to
calculate the expectation value of a time-dependent observable hO(t)i with respect to
a given initial state or an initial density matrix. While time-dependent quantum mechanical problems can be either discussed in the Schrödinger picture or the Heisenberg
picture, we argue that the flow equation method is useful only to formulate transformations in operator space. For this reason we restrict us to the Heisenberg picture with the
aim to employ the flow equation technique in order to obtain the time-dependent observable O(t). Any time-dependent operator has to satisfy the fundamental Heisenberg
equation of motion: 6
6
We restrict us to a Hamiltonian H that is time-independent in the Schrödinger picture.
20
The flow equation method
dO(t)
= i[H, O(t)]
(3.20)
dt
From many examples in standard quantum mechanics we know that it is usually not
possible to give a solution of this equation besides cases where the Hamiltonian is
trivial. In our case, we want to treat dissipative quantum systems with a Hamiltonian
that contains a coupling HSB between system and environment. The reader might
already imagine that the corresponding Heisenberg equation will be highly nonlinear
and a solution in closed form is not possible. Similarly to eq. (3.4), we can rearrange
the Heisenberg equation (3.20) by employing the Baker-Hausdorff identity: 7
O(t) =
X (it)n
n
n!
[H, O]n
(3.21)
In contrast to the perturbative expansion for the Hamiltonian H that has been inserted
into the flow equation (3.8), it is in general not possible to treat equation (3.20) in a
similar manner since the Hamiltonian is -unlike the generator η in eq. (3.4)- usually
not proportional to a small expansion parameter.
Our aim is to reformulate the Heisenberg equation (3.20) in the diagonal basis representation and try to formulate a perturbative approximation to this equation. The
underlying motivation is that in the diagonal basis, the Hamiltonian has a formal representation that eliminiated the coupling term H (1) (B = 0). Due to this aspect, we will
see that the representation of the Heisenberg equation in the diagonal basis is much
more suitable to perform a perturbative approximation of this differential equation.
Note that for an evaluation of time dependent observables in a given initial state we
cannot employ a solution of the Heisenberg equation in the diagonal basis and evaluate
it in a state that is not representated in diagonal basis. Either the state has to be
transformed into diagonal basis or the time-dependent operator is transformed back
into the initial basis. We will discuss both possibilities in subsection 3.4.3.
3.4.1
Formulation of the Heisenberg equation in diagonal basis
It is possible to formulate the Heisenberg equation (3.20) in any unitarily transformed
basis. Eg. we can use the basis that corresponds to the unitary transformation U (B)
defined in (3.4). The corresponding Heisenberg equation is formulated as
d U † (B)OU (B) (t)
= i[U † (B)HU (B), U † (B)OU (B) (t)]
(3.22)
dt
Our aim is to formulate eq. (3.22) in the limit B → ∞ using a representation
in the
†
multiparticle basis. In the limit B → ∞, the observable U (B)OU (B) (t) will be
7
As usual in the context of the Baker-Hausdorff identity, [H, O(t)]n denotes the n-fold commutator
with H.
3.4 Time evolution of observables
21
denoted by O(∞, t) and in analogy, we denote the transformed Hamiltonian by H(∞).
Due to the additional time-dependence, the observable O(∞, t) will have a multiparticle
basis representation with time-dependent generalized coupling constants that we denote
by λlk1 k2 ...kn k10 k20 ...km
0 (∞, t). We obtain the following representation for O(∞, t):
O(∞, t) =
XX X
l
X
† †
†
0
λlk1 k2 ...kn k10 k20 ...km
0 (∞, t)Ol ⊗ b
k1 bk2 ...bkn bk10 bk20 ...bkm
0
m,n k1 k2 ...kn k10 k20 ...km
(3.23)
In addition, we define the multiparticle expansion of the Hamiltonian H(∞) and denote
its generalized coupling constants by glj1 j2 ...jn j10 j20 ...jm
0 (∞). Now, we insert the multiparticle representations of the Hamiltonian H(∞) and the observable O(∞, t) into the
Heisenberg equation (3.22) and reformulate it in its multiparticle representation.
dO(∞, t)
= i[H(∞), O(∞, t)] =
dt
=
XXX X
k,l
m0 ,n0
X
X
X
gkj1 j2 ...jn0 j10 j20 ...jm
0 (∞)λlk k ...k k 0 k 0 ...k 0 (∞, t)
n 1 2
1 2
m
0 k1 k2 ...kn k 0 k 0 ...k 0
m,n j1 j2 ...jn j10 j20 ...jm
m
1 2
† †
†
0 ]
0 , Ol ⊗ b
×[Ok ⊗ b†j1 b†j2 ...b†j 0 bj10 bj20 ...bjm
k1 bk2 ...bkm bk10 bk20 ...bkm
n
(3.24)
It is obvious that the Heisenberg equation (3.24) is in general very difficult to solve and
is as complicated as the flow equation (3.3). However, we can try to find a perturbative
approximation of this equation.
3.4.2
Approximations to the Heisenberg equation in diagonal basis
In section 3.2 we worked out a truncation scheme that approximates the transformed
observables O(∞) and H(∞) by a truncation of contributions in higher order of an
expansion parameter. Eg the approximatively transformed Hamiltonian H(∞) is completely decoupled and has the general form HS (∞) + HB for a dissipative quantum
system. A tremendous simplification of the Heisenberg equation (3.24) results if we
can justify it to employ these approximated operators to approximate the Heisenberg
equation in the diagonal basis representation.
Expansion of the Heisenberg equation in the diagonal basis
Our aim is to formulate a perturbative expansion of the Heisenberg equation (3.24).
For this purpose, we use the series expansion of the Hamiltonian in the diagonal basis
representation that corresponds to the unitarily transformed expansion (3.5).
22
The flow equation method
H(∞) = H
(0)
2
(∞) + λ H
(2)
(∞) + ... = H
(0)
(∞) +
∞
X
λn H (n) (∞)
(3.25)
n=2
Note that in (), we omitted the first order contribution λH (1) (∞). As described above,
in dissipative quantum systems this term corresponds to the coupling term HSB of the
initial Hamiltonian and will be eliminated exactly in the limit of accomplished flow.
However, it has to be carefully analyzed that the series expansion (3.27) retained its
convergency properties during the unitary transformation of the operators H (n) (B). In
analogy, we write down the series expansion of the observable O(∞):
O(∞) = O
(0)
(∞) + λO
(1)
2
(∞) + λ O
(2)
(∞) + ... =
∞
X
λn O(n) (∞)
(3.26)
n=0
We assume that this perturbation series is known up to the first order term. In order
to derive an expansion of the Heisenberg equation (3.24) in the expansion parameter
λ, we insert the expanded Hamiltonian (3.25) and the expanded observable (3.26) into
it and arrive at:
O(∞, t) =
∞
X
(it)n
n=0
n!
[O(0) (∞) + λO(1) (∞), H (0) (∞)]n + O(λ2 )
(3.27)
Discussion of the expanded Heisenberg equation
Through the formulation of equation (3.28), we achieved a formally consistent approximation of the Heisenberg equation (3.24). We showed that it makes sense to employ the
truncated series expansions O(∞) = 0(0) (∞) + λO(1) (∞) and H(∞) = H (0) (∞) that
are both approximated in O(λ2 ) in order to approximate the Heisenberg equation in
O(λ2 ). As shown eg for the spin-boson model in [15], for dissipative quantum systems it
is usually trivial to solve equation (3.27) in closed form. The underlying reason is that
the term H (0) (∞) in eq. (3.27) is equal to the diagonal Hamiltonian HS (∞) + HB . Eg
the operator HB of the uncoupled bosonic bath leads to the trivial time- dependence
bk (t) = e−iωk t b†k of bosonic bath operators.
As already discussed for the truncation scheme of transformed operators, it has to
be ensured that tuncated higher orders in the approxmiated observable O(∞) lead
to negligible contributions in the expanded Heisenberg equation (3.27). However, the
same analysis has to be performed for the approximately diagonalized Hamiltonian.
In parts, this analysis has to be done already at the stage of the approximated diagonalization of the Hamiltonian and the corresponding transformation of the observable.
We stress the important difference that it is not possible to make use of a normal
ordering prescription since eq. (3.27) commutes these normal ordered operators and
yields usually as solution for O(∞, t) that is not normal ordered. Eg, the operators
3.4 Time evolution of observables
23
Figure 3.1: Algorithm to solve Heisenberg equation by means of flow equations
that are truncated from the exact solution will not be normal ordered and it has to be
carefully analyzed if they are of physical importance.
3.4.3
Application of a time-evolved observable in diagonal basis representation
In its diagonal basis representation, the approximated solution of the observable O(∞, t)
cannot be directly employed to calculate physical quantities since an initial state | ψi or
a density matrix ρ are usually given not in this basis representation. Two conceptional
approaches exist that can solve this problem:
1. If the problem allows, the state | ψi or the density matrix ρ can be representated
in diagonal basis. Eg. this is usually possible for the ground state of a dissipative quantum system since in the diagonal basis, the Hamiltonian is diagonal
up to normal ordered truncated terms. Eg. this approach has been used to calculate zero temperature equilibrium correlation functions of dissipative quantum
systems in [15].
2. The remaining second possibility is to transform the time-dependent operator
O(∞, t) back into the initial basis representation and evaluate it in the given
initial state or density matrix (cf. figure 3.1). Conceptionally, this transformation
uses the same formalism that was developed to transform observables from the
initial to the diagonal basis.
We postpone a discussion of the second approach to the first model we want to treat with
the flow equation method, the dissipative harmonic oscillator (DHO). This application
24
The flow equation method
will show in detail that it is possible to use flow equations also to transform the timedependent observable O(∞, t) back from the diagonal basis to the initial basis. Eg. the
latter transformation will be performed for the DHO without approximations and this
example will help the reader to understand this new method that has not been used in
previous work. In another model, the spin-boson model, we will also make use of this
approach and show that it can be used in principle for different dissipative quantum
systems without major modifications.
Chapter 4
The dissipative harmonic
oscillator I: preparations
In chapters 3 and 4, we will study the simplest case of a system reservoir model of
the type (2.1). Namely, this is a quantum mechanical oscillator subject to linear dissipation. Two underlying motivations can be mentioned: First of all, this model is
quadratic and can be solved exactly. Thus, it provides a good test bed for any approximation scheme in dissipative quantum physics. In especially, the flow equation method
works very similar for different dissipative quantum systems and therefore, in a model
treatment of the spin-boson model (chapers 6 and 7) many analogies to the dissipative
harmonic oscillator will be exploited.
This chapter serves to prepare some technical steps. It illuminates how flow equations
can be employed to treat time-dependent problems. Firstly, we derive flow equations
for a diagonalization of the Hamiltonian. Our aim is to transform the Hamiltonian
H = HS + HSB + HB into the form HS∞ + HB , where HS∞ will be the renormalized harmonic oscillator HS∞ = ∆∞ b† b. This work has been done by Kehrein and
Mielke [15]. In addition, important observables are transformed into the so defined
basis. Finally the time-dependence of these operators will be calculated by exploiting
the diagonal form of the Hamiltonian.
In chapter 5, these transformed operators will be employed in time-dependent nonequilibrium problems:
• Two different initial states will be considered. Our aim is to study situations
where the model is driven far from its equilibrium ground state. Closely related
to this problem is the time-dependent decay of dynamical correlation functions.
• Starting from the transformed representation of observables, the displacement
correlation function
26
The dissipative harmonic oscillator I: preparations
1
Sqq (t) = h{x(t), x}+ i
2
(4.1)
of the central oscillator and the position expectation value hx(t)i will be calculated.
• Not at least, these calculations are motivated by physical interests. Especial
attention will be paid to the fluctuation dissipation theorem which will be checked
under non-equilibrium conditions. In non-equilibrium the FDT is not generically
fulfilled, as it is in contrast the case for its equilibrium formulation within linear
response theory.
In recent works by Kehrein and Lobaskin [16,17] the time-dependent Kondo model
was treated using two different initial states that will motivate the non-equilibrium
situations for the DHO. In [16], it was shown that both situations lead to exactly the
same impurity spin dynamics. 1
With this example in mind, of especial interest will be to interpret physical results with
a focus on an identification of generic dynamical behaviour of observables.
4.1
The model
Classically, the relaxation of a damped osd2
cillator is typically described by m dt
2x +
d
2γ dt x + ω0 x where a mass m in a harmonic
potenial with curvature ω20 is damped proportional to its velocity. At the critical
damping strength γ 2 − ω02 = 0, the relaxation process changes from underdamped
oscillatory motion to overdamped exponential decay without oscillations. Also in the
quantum regime, one finds a transition between underdamped and overdamped motion of a damped quantum oscillator [18].
This transition can be induced by increasing the temperature up to a critical value.
The problem of a damped harmonic oscillator has been studied extensively since the Figure 4.1: Sketch of a dissipative harmonic oscillator exposed to an external
early sixties of the 20th century.
field
1
The interested reader is not necessarily deferred to the literature as chapter 6 of this thesis is
especially devoted to cite these results and compare them with the DHO and the spin-boson model.
4.1 The model
27
4.1.1 Applications, previous work
and technical approaches
Applications and original work
The problem of a quantum oscillator in contact with a heat bath has found widespread
applications. It can be used eg as a phenomenological model for flucuation effects in
Josephson junctions, low temperature quantum transport and quantum-optical systems. The original work of Schwinger and Senitzky [19] concerned itself with a theory
applicable to dissipative effects on the electromagnetic field in a resonant cavity. Lateron, Ullersma gave a formulation of the model Hamiltonian we will use and discussed
its properties in a series of four papers [5]. His first intention was to describe an electron
that is bound elastically to an electromagnetic field. These seminal works were mainly
concerned with a phenomenological description of the radiation field of lasers and described the radiation field as a fluctuating bosonic environment that couples to a small
quantum system. A comprehensive review on the theoretical description of lasers can be
found in [20]. Lax [21] in a series of classic papers on quantum noise also discussed the
problem of quantum Brownian motion. All important developments concerning models
of a dissipative harmonic oscillator have been summarized in great detail by Dekker [22].
Important results and technical approaches
Most of the early work on the dissipative harmonic oscillator was concerned with weakly
damped systems and relied on the Born and Markov approximations [22]. In this conventional approach the dynamics of the dissipative quantum system is described in
terms of quantum master or Langevin equations. It is known from many examples
that this techniques fail at low temperature and strong damping and do not reproduce
important quantum effects. Important progress beyond the limit of the weak coupling
approach has been made in the 1980s. In particular path integral techniques were
shown to be powerful means to describe such quantum dissipative systems [9]. This
has led to important results, such as, for instance, the nonexponential decay of correlation functions in the low temperature range [9], not directly available within the
conventional master equation approach.
In subsequent works [18], attention was put on the asymptotic low temperature characteristics which shows anormalous behavior beyond simple exponentially damped oscillatory relaxation ∝ cos(Ωt)e−Γt of the position expectation value hx(t) where Ω ≈ ∆
is approximately the unperturbed frequency of the central harmonic oscillator and the
real number Γ determines the rate of decay.
A whole variety of techniques has been employed to treat the model Hamiltonian formulated by Ullersma [5] since this model is trivial and can be used to test theoretical
28
The dissipative harmonic oscillator I: preparations
means against exact results. Eg. it is possible to exactly diagonalize the Hamiltonian
[5, 23] since it is quadratic (cf. equation (4.5)). An exact solution of the model has
been discussed in this way by Haake and Reibold [18].
Within the path-integral approach, one ends up with Gaussian integrals that can also
be evaluated exactly [9].
Much of the previous work focussed on the time-dependent expectation values of the
first and second moments of position and momentum of the central oscillator. Talkner
et al [24,25] introduced a phenomenological description that describes the bath influence
onto the central oscillator by a Gaussian stochstic treament. This approach is justified
if the bath is coupled linearly to the central oscillator. For Gaussian statistics, higher
moments of any observable can be expressed by its first two moments. We will therefore
concentrate on the observables hx(t)i and hx(t)xi that contain the complete information
of the dynamics of the central oscillator.
4.1.2
Discussion of the model Hamiltonian and initial states
The model Hamiltonian
We specify the general form of the potential V(q,t) from (2.2) as
1
V (q, t) = M ω02 q 2 − qh(t)
2
(4.2)
The inclusion of a time-dependent external force h(t) still leads to linear equations of
motion for the position operator that can be solved in analytic form. This calculation
will be done as an example of a time-dependent non-equilibrium situation in section
5.5. The physical interpretation of this force could be an electric field coupling to a
charged particle.
The dissipative harmonic oscillator in its general form is then given as 2
!
2
X p̂2
p̂2
1
1
λ
α
α
+ M ω02 x̂2 +
+ mα ωα2 xˆα −
x̂
+ h(t)x̂ + E0
2m 2
2m
2
m
ωα2
α
α
α
(4.3)
An interaction induced renormalization of the potential was introduced so that the
Hamiltonian (4.3) is bounded from below (cf. chapter 2). All operators are denoted
by a hat which shall be dropped from now on. They obey the canonical commutation
relations which read
[x, p] = i , [xα , p0α ] = iδα,α0
2
(4.4)
We replace the classical coordinates q and p with the quantum mechanical operators x̂ and p̂.
4.1 The model
29
For a full specification of a physical problem, the model Hamiltonian (4.3) has to be
supplemented with the initial state of the system-bath complex.
Preparation of the system
As explained above, we consider two different initial states that are motivated by reference [16]. They represent analogies of factorized initial states of the Kondo model that
are described by the state of an impurity spin and of an conduction band of electrons.
We give a short discussion of these analogies and discuss both initial states in more
detail in section 5.1. In order to clarify all analogies, we describe in a first sentence the
preparation of the impurity spin used in [16]. Completing the analogy, we describe the
corresponding preparation of the DHO in the next sentence.
Considered preparations of initial states:
1. An initial state that is prepared by exposing the system-reservoir complex to a large magnetic field that acts on the impurity spin until it
has equilibrated with the bath. In our case, an arbitrary electrical field
that displaces the initial position of a charged oscillator is chosen. The
bath is also allowed to come into equilibrium with the displaced central
oscillator.
2. A prepared state where the decoupled spin is projected onto an eigenstate and the bath is left in its decoupled ground state. Then, the
system-bath coupling is switched on. In our case, the state of the central oscillator is prepared in a coherent state and the bath is left in its
equilibrium as well.
We distinguish the initial states described under 1) and 2) with the ket symbols | Ii
and | IIi.
Motivation of the diagonalization approach
While it might appear exaggerated to use our sophisticated machinery for the simple
model in this chapter, we will later use the same machinery with very little change in
a much more complicated problem. For the reader, this chapter is also useful to get
familiar with the methods that will be used furtheron. For convenience, we will proceed
with a bosonic representation of the Hamiltonian (4.3) that lowers the calculational
effort.
30
The dissipative harmonic oscillator I: preparations
∆ 0 b† b +
X
λk (b + b† )(bk + b†k ) +
k
X
ωk b†k bk + E0
(4.5)
k
For a diagonalization of this Hamiltonian by means of flow equations, any type of bath
can be treated that obeys
J(ω) ∝ ω s , s > 0 for small ω
(4.6)
and goes to zero beyond some cut off frequency ωc . The simple structure of the Hamiltonian 4.5 is not only easy to treat within other methods, but also by our flow equation
approach. Due to the similar structure of the spin-boson Hamiltonian, this trivial
model provides a suitable case to develop the formalism that P
will be used throughout this thesis. Firstly, we eliminate the linear coupling term k λk (b + b† )(bk + b†k )
from (4.5) by means of flow equations. This essential step opens the way to obtain
closed solutions for time-dependent observables, since free bosons have the trivial timedependence bk (t) = e−iωk t bk . In section 4.3, the diagonalized Hamiltonian will be the
starting point to derive exact representations for time-dependent operators.
4.2
Diagonalization of the Hamiltonian
From the beginning, one might pose the question whether it is really necessary to use
infinitesimal unitary transformations. In the limit B → ∞ of accomplished flow a continous sequence of infinitesimal unitary transformations has been applied to transform
the Hamiltonian, defining a unitary transformation denoted by U (∞). In principle,
the effective transformation U (B = ∞) can be obtained by a solution of the equa(B)
tion dUdB
U = η(B) and written as an B-ordered product of factors exp(η(B)dB). In
general a generator η(B) does not commute with itself for different values of B and
it is not possible then to evaluate the infinite product (3.2) in closed form. Therefore
all transformations formulated in this chapter will be given in their differential formulation, leading to sets of coupled differential equations for the coupling constants in
transformed operators.
Before the Hamiltonian (4.5) is actually diagonalized, some conventions for bosonic
operators are introduced. Flow equations are related to a unitary transformation in
the underlying Hilbert space. Therefore two different basises of this Hilbert space have
to be considered that the unitary transformation U (∞) maps on each other. Since flow
equations transform exclusively operators, it is more suitable to discuss these basises
in operator space.
4.2.1
Definition of operator basises
In the Hamiltonian (4.5), both system and environmental degrees of freedom consist of
bosonic operators. Therefore we introduce a special notation for bosonic operators that
leads to a more suggestive distinction between environmental operators and operators
4.2 Diagonalization of the Hamiltonian
31
of the small subsystem consisting of a harmonic oscillator. We will refer to the operators b and b† and all linear compositions and products of them as system operators. In
analogy, the analogous combinations of operators in the bosonic Fock space spanned
by all bosonic bath modes will be called bath operators.
All bosonic creators and annihilators together represent the basis in the operator space
of system and bath where the model Hamiltonian is formulated in order to make its
physical properties obvious. E.g. the bath influence is described as a sum of bath
modes and the harmonic oscillator as one single mode.
Definition of physical and diagonal basis
A diagonalization of the Hamiltonian (4.5) by means of flow equations defines the
unitary transformation U (∞). This transformation will transform the system operators
b and b† into a combination of bath operators and system operators. 3 In order to adress
this complicated superposition of operators by a shortcut notation, a second basis in
operator space is introduced. We define the unitary transformation U now explicetely
and thus determine the additional basis in Hilbert space we want to introduce.
Z ∞
def
U = TB exp
dB 0 η(B 0 )
(4.7)
0
Eg. the bosonic operators b and bk will transform into the operators U bU † and U bk U † .
For these transformed operators, we introduce the notation
def
a0 = U bU †
def
ak = U bk U †
(4.8)
It is easy to verify that these operators are still bosonic operators with the additional
property that the Hamiltonian H is diagonal in this operator basis:
[ak , a†k0 ] = δkk0
X
H=
ω̄k a†k ak
kN0
(4.9)
Therefore, the operators ak represent normal modes -with corresponding normal frequencies ω̄k - that act in the full bosonic Fock space of system and bath operators. Due
3
In dissipative quantum systems or impurity models, it is generic that system operators decay into
a superposition of bath operators during the flow [15].
32
The dissipative harmonic oscillator I: preparations
physical basis
diagonal basis
b
a0
b1
a1
...
...
bN
aN
Table 4.1: Definition of the physical and diagonal basis in Fock space. Vertically aligned
operators are mapped onto each other by the transformation U that diagonalizes the
Hamiltonian.
to this property, a physical interpretation of the operators ak as one-particle excitations
is not possible. It is therefore useful to call the initial basis in Fock space the physical basis since it has a physical interpretation of system and bath modes. To make
a distinction between the second basis in state space, we will use the term diagonal
basis for the operators ak , kN0 , motivated by (4.9). Table 3.1 sums up all introduced
definitions. Note that also for a finite number of bath modes, the above definition of
a diagonal basis in operator space makes sense. This basis can also be defined by a
diagonalization of the Hamiltonian via a linear mapping. 4
Convention for operators in diagonal basis
It is often necessary to transform an operator O from its physical to the diagonal basis. An operator representated in diagonal basis will be often denoted by the symbol Õ
where an explicit representation is not convenient. This is mainly the case in section 5.5.
A representation of the Hamiltonian in normal modes may be seen as a passive diagonalization procedure. An active diagonalization that decouples the Hamiltonian
(4.5) by means of flow equations has been discussed very detailed in reference [15]. We
now follow strictly this previous work and refer for some of the subtle details of the
diagonalization procedure to the work of Kehrein and Mielke. While their focus was
mainly put on low temperature equilibrium correlation functions, we intend to give an
extention to non-equilibrium situations at zero temperature.
4.2.2
Flow equations for the Hamiltonian
Since the underlying transformed Hamiltonian is exactly diagonal, it can be used as
well as a starting point in non-equilibrium problems.
Extention to non-equilibrium problems
4
This diagonalization will be performed in section 5.3 because it results in less numerical effort than
a solution of flow equations will.
4.2 Diagonalization of the Hamiltonian
33
An exactly diagonalized Hamiltonian gives way to exact solutions for the time-dependent
operators b(t) and b† (t) both in their physical and diagonal basis representation (cf.
section 4.3). Then, in order to evaluate physical observables, only the correct nonequilibrium initial state has to be known for a specific non-equilibrium problem. In
this sense, flow equations for the dissipative harmonic oscillator are general and can be
used unchanged to solve any kind of physical problem that is given by a known initial
state of the system-reservoir complex.
This aspect of generality of flow equations gets in parts lost if one proceeds to other,
non-trivial models, where the ansatz for transformed operators has to be truncated.
We will especially have to deal with this problem in the case of the spin-boson model,
where it is necessary to introduce a normal ordering scheme for truncated operators.
Normal ordering with respect to non-equilibrium inital states is a delicate problem
that needs to be treated case by case. Nevertheless, the choice of a generator is always
motivated by a desired diagonalization of a Hamiltonian and depends not on a given
initial state.
Choice of a generator
To motivate a flow equation transformation of a Hamiltonian, the canonical choice
(3.19) of the generator is in most cases the first step. In general, there is no other
recipe at ones disposal to incorporate the aspect of decoupling into a generator. The
canonical generator reads:
η(B) =
X
λk (B)∆(B)(b† − b)(bk + b†k ) +
X
(b + b† )ωk (B)λk (B)(−bk + b†k )
(4.10)
k
k
At this stage, one crucial problem of any flow equation transformation gets obvious:
if we calculate the commutator [η(B), H] to motivate an ansatz for the transformed
Hamiltonian H(B), an additional term is generated that has not been included in the
initial form (4.5) of the Hamiltonian. It is always a goal to restrict the Hamiltonian
flow in operator space to an ansatz that preserves the initial form of the untransformed
Hamiltonian. In this way, one can avoid additional coupling terms in the ansatz that
need to be decoupled as well. For this purpose, one adds additional terms to the
(1)
(2)
canonical generator and choses general coupling parameters ηk , ηk , ηk,q and ηb that
are up to now not determined by necessarity.
η=
X
ηk (b − b† )(bk + b†k ) +
(1)
k
+
X
ηk,q (bk +
X
k
†
bk )(bq
ηk (b + b† )(bk − b†k )
(2)
− b†q ) + ηb (b2 − b†2 )
k,q
(4.11)
34
The dissipative harmonic oscillator I: preparations
The actual calculation of the commutator [η(B), H] imposes then restrictions onto all
coupling parameters in order to preserve the form of the initial Hamiltonian. To fulfil
the above mentioned restrictions, the generator is finally chosen as
(1)
ηk = −λk ∆f˜(ωk , B)
(2)
η = λk ωk f˜(ωk , B)
k
ηk,q
2λk λq ∆ωq ˜
(f (ωk , B) + f˜(ωq , B))
=− 2
ωk − ωq2
1 d∆
ηb = −
4∆ dB
(4.12)
The function f˜(ωk , B) is not known yet and can be chosen by convenience. Its choice
should at least achieve a final elimination of the couplings λk (B) in the limit B → ∞.
A suitable choice for all couplings of the type (4.6) is 5 :
ωk − ∆
f˜(ωk , B) = −
ωk + ∆
(4.13)
Derivation of flow equations
The ansatz for the transformed Hamiltonian H(B) can now be chosen as
∆(B)b† b +
X
λk (B)(b + b† )(bk + b†k ) +
X
ωk (B)b†k bk + E0 (B)
(4.14)
k
k
The ansatz (4.14) remains exactly closed during the flow. All operators present in this
ansatz will also be present in the commutator [η(B), H(B)] up to modified coefficients.
By comparing these coefficients in the differential equation dH(B)
dB = [η(B), H(B)] with
those in the ansatz (4.14), we finally can read off the flow equations
d∆
dB
dE0
dB
dωk
dB
dλk
dB
= 4
X
= 2
X
(2)
ηk λk
k
(2)
ηk λk + 2
k
X
(1)
ηk λk
k
1
= O( )
N
(1)
(2)
= ηk ∆ + ωk ηk +
X
+2
ηk,q λq + 2ηb λk
q
5
Profound arguments that justify this choice were given in [15].
(4.15)
4.2 Diagonalization of the Hamiltonian
35
In the thermodynamic limit of an infinite number of bath modes, the bath frequencies
turn out to be unrenormalized by a flow equation transformation. This is a consequence
of the property λk ∝ √1N . We finally obtain the diagonalized Hamiltonian in the limit
B → ∞. Then, all couplings λk (B) will have decayed to zero, and the diagonalized
Hamiltonian reads:
H = ∆∞ b† b +
X
ωk b†k bk + E∞
(4.16)
k
Discussion of the flow equations
The flow equations (4.15) are to complicated to allow for an analytical solution. Fortunately, the flow equations (4.15) affect the transformed Hamiltonian (4.16) only through
the renormalized frequency ∆∞ 6 . In physical results at zero temperature, it will enter
as a characteristic frequency scale for system observables like the position expectation
value hx(t)i. Therefore, qualitative behaviour of the transformed Hamiltonian can always be analyzed without solving any of the flow equations (4.15). It is only important
to ensure that ∆∞ has a finite value.
In dissipative quantum mechanics, it is generic for tunneling amplitudes ∆0 to get more
and more renormalized during renormalization procedures that integrate out couplings
to high energy degrees of freedom. E.g. in the spin-boson model (cf. chapter 5) it is
known that for Ohmic baths the tunneling matrix element vanishes at a critical damping
strength. Also the flow equation approach renormalizes such tunneling matrix elements
usually downwards to smaller values during the flow. 7 For a harmonic oscillator a
negative excitation energy ∆ leads to a Hamiltonian that is not bounded from below.
One may therefore ask for a boundary condition on the spectral density J(ω) that has
to be obeyed when the renormalized Hamiltonian shall be bounded from below. Indeed,
this leeds to the inequality [15]
Z ∞
dω
∆0 ≥ 4
J(ω, B)
(4.17)
ω
0
where we introduced the shorthand notation
J(ω, B) =
X
λ2k (B)δ(ω − ωk )
(4.18)
k
The case
Z
∆0 = 4
0
6
∞
dω
J(ω, B)
ω
(4.19)
Of course a renormalization of the energy E0 has no influence on any physical observable.
This has been confirmed for the spin-boson model and the dissipative harmonic oscillator for general
spectral densities of the bath in references [15,26,27].
7
36
The dissipative harmonic oscillator I: preparations
2
p
resembles the Hamiltonian of a free particle, since ∆∞ = 0 and HS (∞) reads 2m
.
The case ∆∞ = 0 will reappear again in chapter 5 in the very opposite context of a
well-known localisation phenomenon.
Discussion of the diagonalized Hamiltonian (4.16)
The resulting form (4.16) of the transformed Hamiltonian seems to pose the question
how it is possible that the underlying system shows dissipative behaviour as there
is no coupling present in this representation. The answer to this question is that
in the basis the above Hamiltonian is representated, the operators b and b† are no
longer representing physical observables of the damped oscillator. Instead, the inital
physical observables b and b† decayed completely into bath operators. Speaking in
other terms, the flow equation transformation created a non-trivial entanglement of
oneparticle states in the bosonic Fock space. Inspite of the Hamiltonian (4.16) being
quadratic in bosonic operators, in the corresponding transformed Hilbert space these
operators lead to collective excitations of all bath modes. In this way, transformed
operators incorporate now all aspects of dissipative behaviour that were formulated in
the initial model Hamiltonian (4.5). As already suggested in [27], this might give a new
view on dissipative quantum systems.
4.3
Flow equations for observables
Dynamical observables of the central oscillator system can be expressed through the
operators b(t) and b† (t) if one is aware of the initial state of the system.
Why to transform observables ?
Dependent on the basis in which the initial state is given, it is necessary to representate the operators b(t) and b† (t) as well in this basis. Then, observables like
hx(t)i = √12 hb(t) + b† (t)i can be evaluated. Flow equations are only suitable for transformations in operator space, not in state space. As discussed in section 3.4, the
diagonal basis is very suitable to solve the Heisenberg equation for operators. The
reason is that the diagonal Hamiltonian is quadratic.
bath operators will have
P Bosonic
dbk (t)
†
†
to obey the Heisenberg equation dt = [∆∞ bb + k ωk bk bk , bk ]. The simple solution
is bk (t) = e−iωk t bk . If the Heisenberg equation for an operator in the physical basis is
non-trivial and cannot be solved, the operator has to be transformed into diagonal basis
to allow for a solution. The discussion of transforming operators therefore starts with
the transformation of operators into the diagonal basis. Lateron, the time evolution of
operators by means of flow equations will be discussed in more detail.
4.3 Flow equations for observables
4.3.1
37
Transformation into diagonal basis
The generator (4.11) that has been used for the diagonalization of the Hamiltonian (4.5)
has to be used also to transform other operators into diagonal basis. This generator is
quadratic in creators and annihilators. If one commutes it with a linear ansatz
def
b(B) = β(B)b + β̄(B)b† +
X
βk (B)bk + β̄k (B)b†k
(4.20)
k
the commutator [η(B), b(B)] preserves the form of the ansatz (4.20) for b(B). The
differential equation db(B)
dB = [η(B), b(B)] can then be solved in closed form, leading
to an exact transformation of the system operators b and b† . For future purpose,
we use a slightly different parametrized ansatz than suggested by (4.20). It contains
only hermitian and antihermitian operators. Its coefficients are then more suitable to
calculate the time evolution of these operators. 8
b(B) = β[b + b† ] + β̄[b − b† ] +
X
αk [bk + b†k ] + ᾱk [bk − b†k ]
(4.21)
k
where the initial condition
1
αk0 (B = 0) = ᾱk0 (B = 0) = 0
(4.22)
2
has to be obeyed in order to fulfil b(B = 0) = b and b† (B = 0) = b† . Note that the above
flow of the operator b(B) allows for any B to circumvent the analogous transformation
of the hermitian conjungated operator b† by chosing the hermitian conjugated form
of equation (4.21). In other words, the unitary transformation induced by the flow
equations for ansatz (4.21) commutes with the operation of hermitian conjugation. The
ansatz (4.21) together with the generator (4.11) yields the following flow equations:
β(B = 0) = β̄(B = 0) =
dβ(B)
dB
= 2ηb β(B) + 2
dβ̄(B)
dB
= −2ηb β̄(B) − 2
(2)
X
αk (B)ηk
k
(1)
X
ᾱk (B)ηk
k
dαk (B)
dB
= 2ηk β(B) + 2
dᾱk (B)
dB
= −2ηk β̄(B) − 2
(1)
X
ηk,q αq (B)
q
(2)
X
ηq,k ᾱq (B)
q
(4.23)
Note that exactly the same flow equations with modified initial conditions could be
chosen to transform the bath operators bk . Finally it is worth to mention that the coefficients β(B) and β̄(B) vanish to zero in the limit B → ∞ [15]. This property can be
8
In section 4.3, the advantage of the ansatz (4.21) will become clear.
38
The dissipative harmonic oscillator I: preparations
seen as another interesting property of dissipative quantum systems: an observable of
a small system decays completely into bath operators. This statement holds as generic
and will become especially apparent in chapter 5.
For analytical evaluations of expressions that involve the transformed operators (4.21),
it is desirable to incorporate all transformation coefficients into one spectral function
that has a continous frequency dependence in the thermodynamic limit N → ∞. With
a slight abuse of the notation used in [15], we define:
sk
s̄k
def
=
def
=
ωk 12
ᾱk
∆0
ωk − 12
2
αk
∆0
2
(4.24)
As a straightforward consequence of the flow equations (4.23) together with sk (B =
0) = s̄k (B = 0) = 0 the identity sk ≡ s̄k holds 9 . In later calculations it will become
obvious that the spectral function
def
K(ω) =
X
sk (∞)2 δ(ω 2 − ωk2 )
(4.25)
k
is a suitable function to evaluate correlation functions and expectation values of the
position operator x = 12 (b + b† ). For practical usage, a correspondence to the bath
spectrum J(ω, 0) has been established in the form:
K(ω) =
2∆0 J(ω, 0)
R
02
0 ,0)
(∆20 − ω 2 + 2∆0 P ( dωω2J(ω
))2 + 4π 2 ∆20 J 2 (ω, 0)
−ω 02
(4.26)
P(.) denotes the principal value of the integral. K(ω) shows a maximum for some
ω < ∆∞ . The behaviour of K(ω) for small ω is determined by the behaviour of J(ω)
2 ωα
for small ω. For the cases of a Drude-like behaviour J(ω, 0) = γγ2 +ω
2 or an Ohmic bath
J(ω, 0) = 2αωΘ(ω − ωc ) with cut off frequency ωc the function K(ω) can be evaluated.
A Drude bath introduce a smooth cut off frequency γ that corresponds to a memory
time τ = γ1 of microscopic damping processes [9]. The result reads for a Drude-type
spectrum:
K(ω) =
2αγ 2 ∆0 ω(γ 2 + ω 2 )
(∆20 (γ 2 + ω 2 ) − 2παγ 3 ∆0 − ω 2 (γ 2 + ω 2 ))2 + 4π 2 α2 ∆20 γ 4 ω 2
(4.27)
For a Ohmic bath with hard cut off, one obtains:
K(ω) =
9
4αω∆0
c
2
2 2 2 2
[∆20 − ω 2 + 8α∆0 (−ωc + ω2 ln ω+ω
ωc −ω )] + 16π α ∆0 ω
Explicit arguments were given in [15].
(4.28)
4.3 Flow equations for observables
39
Figure 4.2: The function K(ω) for an Ohmic and a Drude-type bath type respectively.
The two different baths show a different peak structure. The qualitative features are
identical. K(ω) shows the typical spectral properties that can be found for many
observables of the DHO (cf. chapter 5). It shows a peak at approximately the renormalized frequency ∆∞ . This peak broadens and is shifted to lower frequencies if the
damping strength α is increased.
In the remaining chapters we stick to the Ohmic type of bath since a hard cut off
frequency ωc suits better for numerical implementations of this function.
4.3.2
Time evolution
When a certain time point t0 is achieved, the system will be prepared in one of the
initial states | Ii and | IIi. From t = t0 on the system-reservoir complex will be exposed to the equilibrium Hamiltonian 4.5 . The initial state | ψi can be given either
in physical basis or diagonal basis. E.g. the equilibrium ground state in diagonal basis
is simply a state of bosonic vacuum. In comparison it is not possible to give an exact
representation of the ground state in the physical basis. The initial state | Ii has the
same property than the ground state in this aspect. In contrast, the initial state | IIi
can only be expressed in the physical basis and not in the diagonal basis (cf. section 5.3).
Physical and diagonal Heisenberg picture
It is useful to distinguish between different Heisenberg pictures. We fix here the language diagonal Heisenberg picture and physical Heisenberg picture. An operator Õ(t)
in diagonal Heisenberg picture is defined to be a solution of the Heisenberg equation
in diagonal basis.
40
The dissipative harmonic oscillator I: preparations
X
dÕ(t)
= i[∆∞ b† b +
ωk b†k bk , Õ(t)]
dt
(4.29)
k
The physical Heisenberg picture corresponds to the usual Heisenberg picture that is
obtained by a solution of the Heisenberg equation in the physical basis.
X
X
dO(t)
λk (b + b† )(bk + b†k ) +
ωk b†k bk , O(t)]
= i[∆0 b† b +
dt
(4.30)
k
k
In order to evaluate observables both pictures are useful.
Evaluation of observables
We discuss now the calculation of a time-dependent observable hOi(t). O is some
system operator and h.i denotes the expectation value with respect to one of the initial
states | Ii and | IIi. In principle we could try to solve the Schrödinger equation. In
physical basis, it is very difficult to solve the Schrödinger equation due to the systemenvironment coupling. For the same reason, it is too difficult to solve the Heisenberg
equation dO
dt = [H, O(t)] in the physical basis. To simplify this problem, we will first
transform the operator O into the diagonal basis. This has been demonstrated in
subsection 4.3.1.
In diagonal basis, the Heisenberg equation is easy to solve for all system operators
since they are composed of bosonic operators that have the trivial time-dependence
bk (t) = e−iωk t bk in the diagonal Heisenberg picture. The initial state | Ii can then be
exactly evaluated in diagonal basis where its representation is known.
˜
hI | U † U O(t)U † U | Ii = hI˜ | Õ(t) | Ii
(4.31)
In (4.31), the unitary transformation U was inserted. This transformation transforms
from physical basis into diagonal basis. U O(t)U † = Õ(t) is the solution of the Heisenberg equation in diagonal basis. However, the initial states | IIi can only be evaluated
in physical basis where its representation is known.
hII | O(t) | IIi
(4.32)
The strategy then is to transform U O(t)U † back into physical basis. This approach has
not been discussed in previous literature. Nevertheless, it leads to an exact solution
for the observable hII | O(t) | IIi. Both strategies are based on a solution of the
Heisenberg equation in diagonal basis.
Heisenberg equation
4.3 Flow equations for observables
41
Flow equations were previously only used to solve time-dependent problems in diagonal
basis. The system operators a(t) = U b(t)U † and a† (t) in diagonal basis have to satisfy
the equation
X
da(t)
= i[∆∞ b† b +
ωk b†k bk , a]
dt
(4.33)
k
The trivial solution reads
a(t) =
X
e−iωk t (αk + ᾱk )bk + eiωk t (αk − ᾱk )b†k
(4.34)
k
Quite obvious, for equilibrium systems this solution is already sufficient. The operator
4.34 can be evaluated just in the trivial diagonal ground state, (restricted to T=0) or
the finite temperature density matrix W = Z −1 exp(−β H̃). The diagonal Hamiltonian
P
H̃ = ∆∞ b† b + k ωk b†k bk consists of free bosons.
We now want to obtain the solution b(t) of the Heisenberg equation in the physical
basis. The obvious way is to transform a(t) back into the physical basis. The operator
a(t) is linear in bosonic operators and the flow equations (4.23) can be used to transform
it. An ansatz for the transformed operator b(B, t) needs time-dependent coefficients.
The resultant flow equations will be also time dependent.
Time-dependent flow equations
The term time-dependent flow equations needs some explanation. If we calculate the
time evolution of bosonic bath or system operators in diagonal basis, we obtain the
following expression:
ak (t) =
X
cos(εi t) (bi − b†i )ᾱi + (bi + b†i )αi −
i=0
− i sin(εi t) (bi + b†i )ᾱi + (bi − b†i )αi
(4.35)
Now it is possible to set up the same procedure as we chose for the change into diagonal
basis in the time-independent case. The expression (4.35) is linear in bosonic operators.
It can be transformed exactly just like in case of the ansatz (4.21).
To transform an operator from physical to diagonal basis, one integrates the flow equations from B=0 to the limit B → ∞. To invert this transformation, one has to integrate
the flow equations from the limit B → ∞ to B=0. An essential difference to the usual
formalism of flow equations arises in this procedure. The operators a(t) that are transformed are time-dependent. A proper parametrisation of an ansatz then needs to
introduce a time-dependence into the coupling constants. With this example in mind,
we will call flow equations that have time-dependent initial conditions time-dependent
42
The dissipative harmonic oscillator I: preparations
Figure 4.3: Algorithm to solve Heisenberg equation by means of flow equations
flow equations. In our above example, we can integrate back real and imaginary part
of the flow equations for a(t) separately by choosing the ansatz:
b(B, t) = β(B, t)[b + b† ] + β̄(B, t)[b − b† ] +
X
+
αk (B, t)[bk + b†k ]ᾱk (B, t)[bk − b†k ] +
k
+ i δ(B, t)[b + b† ] + δ̄(B, t)[b − b† ] +
X
+i
γk (B, t)[bk + b†k ] + γ̄k (B, t)[bk − b†k ]
k
(4.36)
This ansatz has the advantage that the initial conditions (4.38) are real. A numerical
integration is then restricted to real-valued differential equations. We use the generator (4.11) and commute it with the ansatz (4.36) to obtain the time-dependent flow
equations for this ansatz in the usual way. Finally, the time-dependent flow equations
for the real part of (4.36) read:
4.3 Flow equations for observables
43
dβ(B, t)
dB
= 2ηb β(B, t) + 2
dβ̄(B, t)
dB
= −2ηb β̄(B, t) − 2
(2)
X
αk (B, t)ηk
k
(1)
X
ᾱk (B, t)ηk
k
dαk (B, t)
dB
= 2ηk β(B, t) + 2
dᾱk (B, t)
dB
= −2ηk β̄(B, t) − 2
(1)
X
ηk,q αq (B, t)
q
(2)
X
ηq,k ᾱq (B, t)
q
(4.37)
For the imaginary part the flow equations only need to be modified by the proper
coefficients. The initial conditions in the limit B → ∞ are determined by (4.35):
αk (∞, t) = αk cos(εk t)
ᾱk (∞, t) = ᾱk cos(εk t)
γk (∞, t) = −ᾱk sin(εk t)
γ̄k (∞, t) = −αk sin(εk t)
β(∞, t) = βk cos(∆∞ t)
β̄(∞, t) = β̄k cos(∆∞ t)
δ(∞, t) = −β̄k sin(∆∞ t)
δ̄(∞, t) = −βk sin(∆∞ t)
(4.38)
Discussion of the time-dependent flow equations
An analytical solution of the flow equations (4.37) is not possible. The spectral function
K(ω) cannot be employed to describe the time-dependent behaviour of these equations.
Numerical solutions turn out to converge very slow since the couplings λk (B) decay
1
only algebraically ∝ B − 2 .
Nevertheless the time-dependent flow equations can be seen as an instructive example
for problems where time-dependent flow equations can be integrated numerically with
less effort. The spin-boson model will be such an example. In chapter 7, the formalism
of time-dependent flow equations will be applied to solve physical problems. For the
DHO, an exact diagonalization technique yields exactly the same transformations of
operators. Where necessary, we will employ this technique since it yields much faster
numerical implementations. We suggest the time-dependent flow equations (4.37) as
an exact test for numerical routines developed to solve time-dependent flow equations.
44
4.4
The dissipative harmonic oscillator I: preparations
Summary
This chapter introduced an exact formalism to calculate exact representations of all
time-dependent system operators of the DHO. These can be employed to any physical
problem that is given by an initial state in diagonal or physical basis. We defined
the diagonal basis by an exact diagonalization of the DHO via flow equations. The
diagonal Hamiltonian enables to solve the Heisenberg equation in diagonal basis. Via
time-dependent flow equations, this solution can be exactly transformed into the physical basis. In the next chapter, this method 10 will be applied to two different initial
states that are given in diagonal and physical basis, respectively.
We conclude that flow equations provide an exact method that yields exact results for
system observables of the DHO. In the non-trivial spin-boson model, we will have to
extend this method by additional approximation schemes.
10
We only replace the numerical integration of the time-dependent flow equations by an equivalent
linear diagonalization routine.
Chapter 5
The dissipative harmonic
oscillator II: applications
In this chapter, different physical problems will be solved. Firstly, we discuss the nonequilibrium preparations | Ii and | IIi. In addition, we extend the inital states | Ii to a
problem with additional time-dependent fields in section 5.5. Using the flow equations
for bosonic operators and the diagonalized Hamiltonian derived in section 4.2, we will
then calculate dynamical observables with respect to the given initial states.
The symmetrized displacement correlation function Sqq (t, tw ) = 12 h{x(t + tw ), x(tw )}+ i
is calculated in dependence of waiting time tw . Exact results for the position expectation value hx(t)i are derived. In addition, the fluctuation dissipation theorem is
discussed.
5.1
Non-equilibrium in a technical context
For the DHO, the flow equation method was previously only employed to calculate
equilibrium correlation functions [15]. In this thesis, the method is applied to nonequilibrium problems. The aspect of non-equilibrium is formulated in form of nonequilibrium initial states.
On the context of non-equilibrium
Another approach would be to include time-dependent perturbations into the Hamiltonian that act also after the initial preparation. Such perturbations cannot be excluded
from the Heisenberg equation of any observable for the time scales after the initial
experimental preparation. We illustrate these technical aspects with two important
examples.
46
The dissipative harmonic oscillator II: applications
The well-known example of Fermi liquid theory assumes an adiabatically switched on
interaction between particles in order to retain a one-particle picture for interacting
fermions. It would be very interesting to investigate time-dependent variations of interaction strength between fermions that go beyond the adiabatic limit. Probably, a very
different state of condensed matter could be the result that leeds beyond Fermi liquid
theory. From experimental side, very promising experiments can drive interacting bose
systems from a superfluid phase to a Mott insulating phase by time-dependent variation
of the coupling strength on an optical lattice [28]. While this type of non-equilibrium is
certainly very interesting, it can be considered as one of the most complicated problems
in theoretical physics.
The actual technical problem within the flow-equation approach is that usually one
can only diagonalize a Hamiltonian for one point in time. Even if a time-dependent
diagonal Hamiltonian could be obtained, solving the Heisenberg equation in diagonal
basis for a given observable O is not trivial anymore. This is due to the additional
time-dependence in the diagonal Hamiltonian. 1 We illustrate this problem in section
5.5 where the equilibrium Hamiltonian 4.5 will be extended by a time-dependent field
coupling term ∆H = h(t)x. We derive an exact solution for x(t) in diagonal basis. It
will become clear that in most time-dependent problems a similar calculation is not
possible.
We will avoid all difficulties that are related to a generalization of the flow equation
method to time-dependent perturbations. The method we want to develop in this thesis
is in general formulated only for non-equilibrium initial states that evolve under an
equilibrium Hamiltonian. Nevertheless, an extension to time-dependent perturbations
in the Hamiltonian is an exciting future prospect of the flow equation approach. In
principle, it is possible to treat such time-dependent perturbations. We leave this topic
open for future work.
5.1.1
Preparation of the initial state
The role of the inital state
The model defined by the Hamiltonian (4.5) has to be supplemented by information
concerning the initial state. In earlier work it was frequently assumed that the initial
density matrix Wi of the system consisting of Brownian particle and heat bath factorizes according to Wi = ρi WR where ρ0 is the density matrix of the particle, while
−1
WR = ZR
exp(−βHR ) is the canonical density matrix of the bath at inverse temperature β = kB1T . The factorization is based on the assumption that there are no
correlations between the particle and the reservoir at time t=0 which is the case if
1
To solve the Heisenberg equation, the Hamiltonian has to be transformed into the Heisenberg
picture. If the Hamiltonian is time-dependent this is usually non-trivial, also for diagonal Hamiltonians.
5.1 Non-equilibrium in a technical context
47
the interaction is switched on for t > 0 only. Unfortunately, in most applications of
Brownian motion theory the coordinate q and the environmental degrees of freedom
are integral parts of the same system and their interaction is not at the disposal of the
experimentalist. The factorization assumption was introduced by Feynman and Vernon
to allow for their influence functional representation of environmental effects [30]. In
[4], the class of initial states was extended to correlated states.
In experiment, such correlations are usually present since the heat bath cannot be
fully decoupled from the system. Such correlations may sensitively influence the low
temperature decay of correlation functions. In [29] it was shown that the influence of
entanglement may vanish only algebraically in the relaxation of the expectation value
hx(t)i. Eg. these correlation effects decay even slower than the classical exponential
relaxation to equilibrium. These results are motivation enough to discuss two classes
of initial states and study their different behaviour. Also in our work, an important
interest will be the influence of entanglement onto the dynamics of the system.
As we will see, the initial states | IIi cannot be transformed into diagonal basis, while
the states | Ii can. It will become clear that the states | Ii are entangled in physical basis but not in diagonal basis and for the states | IIi, this property is fulfilled vice versa.
Therefore, we will furtheron classify initial states into the cathegories of entangled and
product initial states. Another reason that provokes these cathegories is that in these
examples, a representation of the initial states can be given only in the basis where the
states are not entangled (cf. sections 5.2 and 5.3). In consequence, time evolution of
observables has to be calculated either in diagonal or physical basis of Hilbert space,
depending on the category of initial state.
Preparation of a non-factorizing initial state
We assume that the initial density matrix of the system-bath complex has come into
equilibrium at some time t0 . Because of the above mentioned correlations between
system and bath, this density matrix cannot be factorized. 2 In t0 we switch on
a homogenous external field of constant magnitude h. This field shall couple to the
position x of the Brownian particle. This corresponds e.g. to an electrical field coupled
def
to an electrical dipole. It introduces an additional field coupling term ∆H = hx̂ into
the Hamiltonian at a large negative time t0 . If we decrease the time t0 to the limit
t0 → −∞, we can assume that the system has come into equilibrium in t = 0. In t=0,
the external field is switched of. The time-dependent Hamiltonian then reads:
2
For coupled quantum systems, entanglement is generic. We regard the ground state of the Hamiltonian as entangled in this generic sense. However, this assumption is only rigorous as far as the
system-environment coupling in the model Hamiltonian (4.5) is realistic.
48
The dissipative harmonic oscillator II: applications
Def
H(t) = ∆0 b† b +
X
λk (b + b† )(bk + b†k ) +
X
k
k
h
ωk b†k bk + Θ(−t) √ (b + b† ) + E0 (5.1)
2
3
In diagonal basis, the Hamiltonian is rewritten as
H̃(t) = ∆∞ b† b +
X
ωk b†k bk + Θ(−t)
k
X√
2hαk (bk + b†k ) + E0
(5.2)
k
For negative times, this Hamiltonian can be transformed into its unperturbed form by
making use of a unitary transformation T.
√ X αk
def
T = exp(h 2
(bk − b†k ))
ωk
(5.3)
k
The transformation T translates the bath operators: T † bk T = bk + µk . where the
√
def
translation of each bath mode is defined by µk = −h 2 αωkk . An application of the
transformation T diagonalizes the Hamiltonian (5.2) for t < 0.
T † H̃(h, t < 0)T = ∆∞ b† b +
X
ωk b†k bk − 2h2
k
X α2
k
k
ωk
(5.4)
Pictorially, in a diagonal basis the operation T shifts the bath oscillators by just the
amount that the electrical field polarized them due to the system-bath coupling and
thus restores the unperturbed ground state of the system-bath complex. In diagonal
basis the initial density matrix is in t=0 prepared as:
W̃i = T † Z −1 exp(−β H̃)T
(5.5)
Z −1 exp(−β H̃) represents the density matrix of the diagonalized equilibrium Hamiltonian (4.5). Since T acts only in the reservoir Hilbert space, the density matrix Wi is
also of factorizing form. Note that in the calculations of section (5.2), we will restrict us
to the limit of zero temperature where the initial density matrix reduces to the ground
state of the perturbed Hamiltonian (5.1).
We now turn to a class of initial states, namely coherent states [32], which are of interest
in various fields including quantum optics, optical communications, and high precision
measurements near the quantum limit.
Coherent states
3
We regard the ground state of the perturbed Hamiltonian (5.1) (t < 0) as entangled. We argue
exactly as in case of the equilibrium ground state.
5.1 Non-equilibrium in a technical context
49
Coherent states have been widely used to describe the radiation field of lasers [20] and
the question how a dissipative environment affects their time evolution was discussed
1
2
eg in [32]. Coherent states are eigenstates of the annihilation operator b = M2~ω0 q +
1
i(2~M ω0 )− 2 p of the harmonic oscillator defined by
b | αi = α | αi
(5.6)
where α is the complex eigenvalue. A coherent state is obtained by letting the displacement operator
D(α) = exp(αb† − α∗ b)
(5.7)
act on the ground state of a harmonic oscillator, i.e.
| αi = D(α) | 0i
(5.8)
A coherent state has a characteristic representation in energy basis.
2
− α2
| αi = e
N
X
αn
√ | ni
n!
n=0
(5.9)
In dependence of the complex number α momentum and position of the vacuum state
get displaced.
Mω 1
0
1
hqi + i(2~M ω0 )− 2 hpi
(5.10)
2~
In a coherent state the variances of position and momentum are equal and fulfil the
2
minimum uncertainty relation σp σx = ~4 .
α=
2
Preparation of a coherent state
Experimental means make it possible nowadays to switch quantum systems from an
uncoupled to a coupled regime. Especially suitable for such preparations are quantum
dots. During persistence of the uncoupled regime, the system is prepared in a coherent
state | αi. The bath is left in its ground state. The system-bath coupling is switched
on when the state is prepared, say in t=0. If the bath is cooled to zero temperature, it
stays in a bosonic vacuum state. The initial state | IIi will be a product state.
| IIi =| αi⊗ | 0i
(5.11)
The Hamiltonian becomes time-dependent due to the preparation procedure.
H(t) = ∆0 b† b + Θ(−t)(b + b† )
X
k
λk (bk + b†k ) +
X
k
ωk b†k bk + E0
(5.12)
50
The dissipative harmonic oscillator II: applications
The initial states | IIi were treated in [29] and [33]. It was shown that for α 1, these
states evolve as dissipative coherent states with the property b | IIi(t) = α(t) | IIi(t).
After both initial states have been discussed, we calculate time-dependent observables
with respect to these initial states.
5.2
Entangled initial state
In this section physical observables are derived for the initial state | IIi. This state
will be discussed only at zero temperature and thus, it will be a pure state. We expect that the aspect of quantum statistical non-equilibrium is most clearly accessible in
the ground state of the perturbed system without any additional thermal fluctuations.
Thermal fluctuations destroy pure quantum states, leading for high temperatures to the
classical limit. Classical non-equilibrium is different from its quantum analogy. Simple
phase space arguments can explain this.
As mentioned above (cf section 4.1), the expectation value hx(t)i and its corresponding
pair correlation function hx(t)xi are fundamental physical observables and can be used
to derive many other physical observables. Momentum related observables are accessible via M dhx(t)i
= hp(t)i. In equilibrium the stochastic process is a stationary Gaussian
dt
process since it the DHO is a linear system [25]. Correlation functions with an even
number of variables can then be written as the sum of all combinations of factorized
pair correlation functions.
The initial state | IIi is known only in diagonal basis. It can be representated as the
ground state of the Hamiltonian (5.2) (t < 0) in diagonal basis. This Hamiltonian can
be diagonalized with the unitary transformation T for t < 0, and thus, the ground state
in diagonal basis reads:
˜ = T | 0i
| IIi
The position operator x̃(t) =
√1 (a(t)
2
x̃(t) =
(5.13)
+ a† (t)) in diagonal basis follows from (4.34).
√ X
2
αk [e−iωk t bk + eiωk t b†k ]
(5.14)
k
5.2.1
The position expectation value
The observable hx(t)i is evaluated by making use of the diagonal basis. This calculation
amounts to evaluating the operator (5.14) with respect to the state (5.13).
5.2 Entangled initial state
51
hx(t)i = hII | x(t) | IIi
= hII | U † U x(t)U † U | IIi
˜ | x̃(t) | IIi
˜
= hII
(5.15)
The spectral function K(ω) (cf. def.(4.25)) can be employed and the result reads:
Z ∞
cos(ωt)
hx(t)i = −2∆0 h
K(ω)dω
(5.16)
ω
0
From
ishes.
dhx(t)i
dt
∝ hpi together with (5.16) one concludes that the initial momentum van-
hpi = 0
(5.17)
Indeed, this function shows damped oscillatory behaviour that is well-known from a
classical damped oscillator in the underdamped case. Note that the relaxation of the
position expectation value is completely independent of the field strength h that only
influences the initial position of the particle. Increased damping leads to a faster decay
and decreases the frequency of coherent oscillations. We chose the highest damping
strength near the critical value where ∆0 = 0. For a further discussion of the relaxation process, it is useful to analyze the function hx(t)i in the frequency domain.
Fourier transformed result
For large negative times, the preparation of the system is not at the disposal of the
experimentalist. To restrict us to the underlying experimental relevant time scale t ≥ 0
we use a half-sided fourier transform
Z ∞
K(ω)
def
hx(ω)i =
e−iωt hx(|t|)i = −2πh∆0
(5.18)
ω
−∞
This function has a finite offset and a vanishing slope in ω = 0 leading to an exponential
long-time decay for t ∆−1
0 .
5.2.2
Correlation functions
Definition of a two-time correlation function
The usual full time-translational invariance of equilibrium correlation functions is broken by the application of a time-dependent perturbation. In consequence, a two-time
correlation function is necessary if correlations shall be described for all times tw after
the initial preparation of the system. After tw has passed, two successive measurements
52
The dissipative harmonic oscillator II: applications
Figure 5.1: The coherent decay of the first moment of the position for and the halfsided Fourier transform of x(t) for different damping strengths α. All curves were
normalized to hx(t = 0)i = 1 by a suitable field strength h. We chose an Ohmic bath
with cut off frequency ωc = 10 and a potential curvature of ∆0 = 1.0. From the spectral
decomposition of the relaxation process it gets obvious that the Brownian particle will
show damped oscillatory behaviour. The damping of the oscillations increases with
broadening of the spectral peak’s full width at half maximum, whereas the frequency of
the coherent oscillations is situated approximately at the peak position of the spectrum.
Increased damping leads to stronger renormalization of ∆0 corresponding to a shift of
the peak position to a lower frequency scale. In the weak damping limit α 1, a
delta peak emerges at ω = ∆0 , leading to the trivial case of free oscillations hx(t)i ∝
cos(∆0 t). Another interesting part of the spectrum is the low frequency range. With
ω → 0 : K(ω) ∝ ω one obtains an exponential longtime-decay of the envelope due to
the finite offset in ω = 0.
5.2 Entangled initial state
53
are performed and their outcome depends on the time tw . Eventually, the system relaxes towards an equilibrium like state with increasing tw and correlations show a cross
over to an equilibrium correlation function.
1
hII | x(t + tw ), x(tw ) | IIi
(5.19)
2
This function will be evaluated in the initial state | IIi. In the case of vanishing field
strength, it reduces to the equilibrium version of the correlator. The symmetrized
equilibrium displacement correlator Sqq (t) = h{x(t), x}+ iGS shows an algebraic t−2
long-time decay for Ohmic baths [29]. Dissipative quantum systems at zero temperature
show generically algebraic long-time decay of equilibrium pair correlators [34]. However,
very high field strengths | h |> 0 can be exactly treated within our method and thus
we are able now to study initial states far from the equilibrium ground state. In these
cases two properties of the zero temperature equilibrium case are not expected to be
generic:
def
Cqq (t, tw ) =
1. Algebraic long time tails of pair correlations functions
2. A fulfillment of the fluctuation dissipation theorem for the displacement correlator
Sqq (t, tw ) and its related response function.
In analogy to the position expectation value (5.15) the correlator Cqq (t, tw ) will be
evaluated by making use of the diagonal basis.
˜ | x̃(t + tw )x̃(tw ) | IIi
˜
Cqq (t, tw ) = hII
(5.20)
We insert (5.13),(5.14) and employ the spectral function K(ω) in the result:
Z
∞
e−iωt K(ω)dω
0
Z ∞
Z ∞
cos(ω 0 (t + tw ))
cos(ωt)
2 2
K(ω)dω
K(ω 0 )dω 0
+ 4∆0 h
ω
ω0
0
0
Cqq (t, tw ) = ∆0
(5.21)
Discussion of the result
As already mentioned, in the case of vanishing field strength, the above correlation
function reduces to its equilibrium version. From Cqq (t, tw ), the symmetrized function
Sqq (t, tw ) and the antisymmetrized function Aqq (t, tw ) can be extracted as real and
imaginary part of (5.21). The dynamical response function χqq (t, tw ) can be extracted
via the Kubo formula
χqq (t, tw ) = 2iΘ(t)Aqq (t, tw )
(5.22)
54
The dissipative harmonic oscillator II: applications
cum (t) for different damping strengths
Figure 5.2: The half-sided Fourier transform of Sqq
eq
(t). We chose
α. This function is equivalent to the equilibrium correlation function Sqq
an Ohmic bath with cut off frequency ωc = 10 and a tunneling frequency ∆0 = 1.0. The
decay of correlations between two successive measurements of the particles position is
a consequence of damping. In the weak damping limit α 1 the positions become
delta-correlated at the frequency ∆0 of free oscillatory motion. Increased damping
broadens the spectrum in the peak region. Many frequencys of the bath spectrum
contribute and destroy the phase relation between to successive position measurements.
This broadening of the spectral peak is related to an increased decoherence caused
by the environmental coupling. For low frequencies ω 1 this function shows a
proportionality ∝ ω, leading to an algebraic t−2 long-time decay.
5.2 Entangled initial state
55
The field dependent term in (5.21) that represents the deviations from the equilibrium
correlation function is equal to the product hx(t + tw )ihx(tw )i. We conclude that the
cumulant function
def
cum
Sqq
(t, tw ) =
1
h{x(t + tw ), x(tw )}+ i − hx(t + tw )ihx(tw )i
2
(5.23)
is equal to the equilibrium version of the correlator Sqq (t, tw ). We know that the
observable hx(t)i decays exponentially in the long-time limit t ∆−1
0 (cf. (5.18)). The
cum
non-equilibrium correlator Sqq (t, tw ) approaches therefore the equilibrium correlator
def
Sqq (t) = 21 h{x(t), x}+ i exponentially fast for large tw ∆−1
0 . Obviously the dynamical
response function χqq (t) is translationally invariant in the time tw and equal to the
equilibrium correlator Sqq (t).
Z
∞
sin(ωt)K(ω)dωΘ(t)
χqq (t) = 2∆0
(5.24)
0
Exact results for the response function (5.24) have been first obtained by Ullersma [5]
and later by Riseborough et al [35].
The half-sided Fourier transform of (5.23)
4
cum
Sqq
(ω) =
can be performed analytically.
π
∆0 K(ω)
2
(5.25)
cum (ω) is equal to the equilibrium correlation function S (t). Since
We know that Sqq
qq
K(ω) ∝ J(ω), ω → 0, (4.26) yields the correct algebraic long-time decay ∝ t−2 of the
function Sqq (t) for Ohmic baths [15].
5.2.3
The fluctuation dissipation theorem
Formulation of the fluctuation dissipation theorem
The fluctuation dissipation theorem (FDT) establishes a correspondence between energy dissipation and quantum fluctuations in thermodynamical
equilibrium. Energy
R∞
dissipation is described by the imaginary part χ00qq (ω) = = −∞ e−iωt χqq (t)dt of the frequency dependent reponse function. Quantum fluctuations are contained in the symmetrized displacement correlation function Sqq (t). We give the fluctuation dissipation
theorem in the formulation of Callen and Welton [36].
βω χ00qq (ω) = tanh
Sqq (ω)
2
4
For a definition of the half-sided Fourier transform, see (5.18).
56
The dissipative harmonic oscillator II: applications
In general, this theorem can only be proved for equilibrium correlation functions. In
cum (ω)
reference [37] it was argued that a formulation of the FDT using the cumulant Sqq
5 is the suitable generalization for non-equilibrium situations.
βω cum
χ00qq (ω) = tanh
Sqq
(ω)
(5.26)
2
R
cum (ω) is defined as ∞ e−iωt 1 h{x(t), x} i − hx(t)ihxi dt. In
Here the cumulant Sqq
+
2
−∞
cum (ω) is usually equal to the corresponding symmetrized
equilibrium, the cumulant Sqq
correlation function since the product hx(t)ihxi vanishes in most cases. In the limit
T → 0 the FDT reduces to
cum
χ00qq (ω) = sign(ω)Sqq
(ω)
(5.27)
However, the fluctuation dissipation theorem is not more than a criterion for correlation
functions. Eg, it gives no insights into the physical reasons that cause a possible deviation from a fulfillment of the FDT. Our aim is to introduce an effective temperature
that can give a measure for the violation of the FDT.
The concept of effective temperature
cum (ω)
In reference [17], it has been exploited that the spin-spin correlation function Szz
6
of the Kondo model has a finite offset in ω = 0 that is sensitively influenced by
a non-equilibrium preparation of the Kondo model. This is related to the definition
of an effective temperature that gives a measure of the deviation from the behavior
of correlations in equilibrium. This concept is frequently used and well-established
in the investigation of classical non-equilibrium systems [38]. Formally, the effective
temperature Teff is defined by a zero frequency limit of the fluctuation dissipation
theorem (5.27) and replacing the temperature in the FDT by its appropriate value that
restores the FDT for the considered non-equilibrium correlation functions. This leads
to the following expression:
χ00qq (ω)
1
cum
= Teff Sqq
(ω = 0)
ω→0
ω
2
lim
In reference [17] the concept of effective temperature was suggested as a useful measure
for heating effects in quantum non-equilibrium systems. We want to discuss this concept in various non-equilibrium situations where it is possible to calculate the quantity
Teff .
Discussion of the fluctuation dissipation theorem
5
6
cum
The cumulant Sqq
(ω) will be defined in the next sentence.
cum
For a precise definition of Szz
(ω), see [17] or cf. chapter 8.
5.3 Product initial state
57
The fluctuation dissipation theorem is always fulfilled for equilibrium correlation funccum (t, t )
tions. From (5.21) we concluded that both of the correlators χqq (t, tw ) and Sqq
w
are equal to their equilibrium versions. In consequence the fluctuation dissipation theorem is therefore exactly fulfilled for any time tw . Even for large field strengths h,
corresponding to highly excited non-equilibrium initial states no assumption has been
stated that will break down. Moreover, the fulfillment is valid for any tw and can be
easily proved also for finite temperatures. An interpretation of this result is non-trivial.
No obvious or trivial arguments exist for a fulfillment of the FDT in non-equilibrium.
A lack of examples makes a comparison to other cases difficult. At least, in [17] it was
discussed in detail why the FDT is in general not fulfilled in non-equilibrium.
We will see in the next section that product initial states can have quite different
properties concerning the FDT.
5.3
Product initial state
In the previous section an initial state was treated that can be expressed in diagonal
basis. Knowledge of operators was therefore sufficient in diagonal basis, where a solution
of the Heisenberg equation was trivial. However, this method cannot be applied in any
case of initial preparation. We will treat the coherent states
| Ii =| αi⊗ | 0i
(5.28)
that have been introduced in section 5.1. These states cannot be transformed into diagonal basis. To evaluate time-dependent observables, we have to determine their time
evolution in the physical basis of the operator space. A systematic approach to this
problem has been introduced in section (4.3) by the concept of time-dependent flow
equations. We postpone their usage to chapter 7 for an application to the spin-boson
model. An exact diagonalization technique is for the DHO numerically more efficient.
Firstly, we will again calculate the observable hx(t)i analytically. For the correlator
hx(t)xi and the fluctuation dissipation theorem formal expressions will be derived.
These expressions are complicated and will be evaluated numerically.
5.3.1
Position evolution
We want to calculate the observable hI | x(t) | Ii. The state | Ii is known only in physical basis. From t=0 on, the state | Ii is exposed to the equilibrium Hamiltonian (4.5).
Inspite of this, a solution of the Heisenberg equation dx
dt = [H, x(t)] is not necessary to
evaluate hI | x(t) | Ii exactly. The basic idea is to chose an ansatz for the operator b(t)
- since x(t) = √12 (b(t) + b† (t)) - and to evaluate its coefficients by employing bosonic
commutation relations. Let us explain this further.
58
The dissipative harmonic oscillator II: applications
In section 4.3 the correct ansatz for the operator b(t) was shown to be linear in bosonic
operators (cf. equation (4.36)).
b(t) = µ(t)b + µ̄(t)b† +
X
(γk (t)bk + γ̄k (t)b†k )
(5.29)
k
The coefficients µ(t), µ̄(t), γk (t) and γ̄k (t) are complex functions. In the result hI |
x(t) | Ii only the coefficients µ(t) and µ̄(t) contribute what can be seen if (5.29) is
evaluated in the state | Ii. For the observable hxi(t) one obtains in analogy:
hxi(t) =
√
2<α[<µ(t)] − 2=α[=µ̄(t) + =µ(t)]
(5.30)
From the ansatz (5.29), the coefficients µ(t) and µ̄(t) can be extracted via the commutators [b, b(t)] = µ̄(t) and [b(t), b† ] = µ(t). These commutators are C - numbers and
invariant under the unitary transformation U from (4.7). As a whole, this strategy is
most clearly described as a sequence of equations.
(4.8)
(4.34)
µ(t) = [b(t), b† ] = U [b(t), b† ]U † = [U b(t)U † , U b† U † ] = [a(t), a† ] =
X
X
[αk + ᾱk ]b†k + (αk − ᾱk )bk ]
= [
e−iωk t [αk + ᾱk ]bk + eiωk t (αk − ᾱk )b†k ,
k
k
=
X
2 −iωk t
[αk + ᾱk ] e
2 iωk t
+ [αk − ᾱk ] e
(5.31)
k
We used the definition (4.8) for the operator a(t) and inserted the expression (4.34) for
a(t). In a completely analogous way µ̄(t) is obtained:
µ̄(t) = 2i
X
(αk2 − ᾱk2 ) sin(ωk t)
(5.32)
k
Inserting (5.31) and (5.32) and the spectral function (4.26) into equation (5.30) yields
the result for hx(t)i:
Z
√
hx(t)i = 2 2<α
√
− 2=α
"Z
0
∞
ω2 ∆0 −
K(ω) sin(ωt)dω + 2
∆0
Z
∞
0
∞
ωK(ω) cos(ωt)dω
#
ωK(ω) sin(ωt)dω
0
(5.33)
Discussion of the result
The initial position and momentum of the central oscillator influence the decay.
5.3 Product initial state
59
r
2
hxi =
<α
m∆0
p
hpi = 2m∆0 =α
(5.34)
We conclude from (5.34) that only parts of expression (5.33) contribute to the decay
of hx(t)i, if the initial momentum or position vanishes. Eg. the long-time behaviour
shows interesting properties.
If the particle has a finite initial momentum, α has an imaginary part. The Fourier
transform of the term proportional =α in (5.31) has the low-frequency behaviour of
K(ω) which is proportional to J(ω). 7 We make this behaviour explicit and write:
Z
∞
−
−∞
"Z
√
e−iωt 2=α
0
∞
#
Z ∞
ω2 ωK(ω) sin(ωt)dω] dt =
∆0 −
K(ω) sin(ωt)dω + 2
∆0
0
"
#
√
ω2 iπ 2=α ∆0 −
K(ω) + 2ωK(ω) ∝ J(ω), ω ∆0
∆0
(5.35)
t−2
For Ohmic baths, this term shows a sluggish algebraic
long-time decay. In contrast,
the initial momentum hpi for the initial states | IIi vanishes (cf. 5.17). To allow for a
closer comparison between the states | Ii and | IIi, we set therefore =α = 0 in (5.33)
and obtain:
√ Z ∞
hx(t)i = 2 2α
ωK(ω) cos(ωt)dω
(5.36)
0
Qualitatively this solution behaves similar to the entangled state (cf. section 5.2).
Firstly, the relaxation process is independent of the excitation strength α that only
fixes the initial position. Eg. the decay of hx(t)i will also show coherent oscillations
with a frequency that is renormalized to smaller values if the damping strength is
increased (cf. figure 5.3). This renormalization effect becomes obvious from the already
defined half-sided Fourier transform of (5.36) that can be performed analytically, with
the result:
√
hx(ω)i = 2 2απωK(ω) ∝ ω 2 , ω ∆0
(5.37)
t−3
Interestingly, the low-frequency behaviour of (5.37) yields an algebraic
long-time
decay for Ohmic baths. From (5.35) it can be seen that for any finite initial momentum,
this long-time decay is slower and changes to a t−2 behaviour. We conclude that the
initial momentum can control the long-time behaviour of the decay.
7
The long-time behaviour follows from the low-frequency behaviour both of the half-sided or the
full Fourier transform. For convenience, we chose here the full Fourier transform.
60
The dissipative harmonic oscillator II: applications
Figure 5.3: A comparison of the decay of the position hx(t)i for different damping
strengths. All curves were normalized to hx(t = 0)i = 1. For increased damping, the
renormalization effect of the oscillation frequency becomes pronounced. A shift of the
spectral peak to lower frequencies is accompanied by a broadening of the peak.We chose
a Ohmic bath with cut off frequency ωc = 10 and a potential curvature of ∆0 = 5.
5.3.2
Correlation functions
Also for the DHO, it turns out that correlation functions are difficult to evaluate analytically for product initial states. Firstly, we have to chose a method that leads to
reliable numerical results.
Choice of a method
In the ansatz (5.29) the coefficients γk (t) and γ̄k (t) remain unknown. However, in an
cum (t, t ), they would contribute. In principle we could
evaluation of the correlator Sqq
w
employ the time-dependent flow equations (4.37) to obain a numerical solution of these
coefficients. This involves a lot of numerical effort since the couplings λk (B) decay with
an algebraic asymptotic behaviour √1B . Further aspects of numerical solutions of flow
equations are discussed in appendix C.1. Fortunately, in many cases the couplings in
a Hamiltonian decay exponentially in the flow parameter B and it is possible to solve
time-dependent flow equations with much less effort. We will especially make use of
them in case of the spin-boson model.
It is worth to reemphasize that this chapter has two aims: on the one hand, technical
methods shall be introduced and tested, on the other hand, also results of physical
interest shall be derived. To achieve the latter, it is more appropriate in terms of
computational effort to diagonalize the Hamiltonian (4.5) by a linear transformation.
As will become clear, this linear transformation is time-independent and its matrix
elements have to be calculated only once. In contrast, a transformation by means of
5.3 Product initial state
61
time-dependent flow equations has time-dependent coefficients (cf. (4.36)). For every
point in time, one has to solve a different set of flow equations. Thus, computational
effort scales additionally in the number of time points for an evaluation. Eg. in order
to perform a Fourier transformation of dynamical correlation functions, this takes too
much computational effort.
Diagonalization by a linear transformation
P
†
We want to transform the Hamiltonian (4.5) into the diagonal form Hdiag = N
k=0 ω̄k ak ak .
We will use a linear transformation and a finite number of bath modes. Essentially,
this amounts to a normal mode transformation that yields the normal frequencies ω̄k .
In analogy to the algorithm of time-dependent flow equations, this transformation can
then be used to calculate the position operator x(t) in the physical basis. Below, we
mention just briefly the essential steps and definitions that yield finally the correlation
function Cqq (t, tw ) = hx(t + tw )x(tw )i. In appendix C.2, the whole calculation is presented step by step and its accuracy is checked against exact results. We obtain the
correlation function Cqq (t, tw ) within 3 steps:
1. Within the first step a normal mode transformation of the Hamiltonian (4.?)
in the full bosonic Fock space of system and reservoir is obtained. In the normal
PN mode† representation the Hamiltonian will be of the diagonal form H =
k=0 ω̄k ak ak where the operators ak define the N+1 bosonic normal modes with
[ak , a†k0 ] = δkk0 and the N+1 normal frequencies ω̄k . 8
2. In the normal mode representation, the normal modes have the trivial time dependence an (t) = an e−iω̄n t . By an inversion of the normal mode transformation,
it is then possible to express the time evolution of the relevant operators b(t)
physical basis via the matrix elements of the normal mode transformation. Then,
it is possible to derive a formal expression for the correlation function Cqq (t, tw ).
3. The resultant representation of Cqq (t, tw ) will be complicated in structure, e.g.
it is expressed through the matrix elements of the normal mode transformation.
Thus, a numerical implementation will be given. In addition, the dynamical correlation function will be numerically transformed into the frequency domain for
a discussion of the FDT.
8
After this thesis was nearly completed, the author became aware of reference [33]. There, this
diagonalization was performed analytically in the continuum limit, where [aω , a†ω0 ] = δ(ω − ω 0 ). Such
a transformation is known as a Fano-type transformation [23].
62
The dissipative harmonic oscillator II: applications
physical modes
normal modes
b0
a0
b1
a1
...
...
bN
aN
Table 5.1: Definition of the physical and the normal modes Vertically aligned modes are
mapped onto each other by the transformation U that diagonalizes the Hamiltonian.
Before we give the final representation of the correlation function Cqq (t, tw ), we want
to clarify the usage of notation.
Formal definition of the normal mode transformation
The normal modes are linear dependent on the physical modes, namely the bath modes
bk and the system mode b. To allow for an efficient numerical implementation, we will
denote all physical modes by the symbol bi 0 ≤ i ≤ N , where the system mode b
corresponds to i=0 and the bath modes to the remaining range of the index i (cf. table
5.1). Normal modes are denoted by the operators ak 0 ≤ k ≤ N . In this way, normal
modes are exactly the diagonal basis in Fock space defined in table 4.1.
By definition, the normal mode transformation takes the form:
def
an =
N
X
Ank bk + Bnk b†k
(5.38)
k=0
with the real matrices A, B ∈ M[(N + 1) × (N + 1), R], and the inverse transformation
is given as
def
bk =
N
X
Ckn an + Dkn a†n
(5.39)
n=0
with the real matrices C, D ∈ M[(N + 1) × (N + 1), R].
Following the derivation given in appendix C.2, the correlation function Cqq (t, tw ) finally
is obtained as:
5.3 Product initial state
63
Cqq (t, tw ) =
N
1 X
[C0,n + D0,n ][C0,n0 + D0,n0 ] ×
2 0
n,n =0
N X
−Ck,n Dk,n0 e−i[(ω̄n0 +ω̄n )tw +ω̄n t] − Dk,n Ck,n0 ei[(ω̄n0 +ω̄n )tw +ω̄n t] +
k=0
Ck,n Ck,n0 ei[(ω̄n0 −ω̄n )tw ]−ω̄n t] + Dk,n Dk,n0 ei[(−ω̄n0 +ω̄n )tw +ω̄n t] +
!
4α2 [C0,n C0,n0 − C0,n D0,n0 − D0,n C0,n0 + D0,n D0,n0 ] cos(ω̄n (t + tw )) cos(ω̄n0 tw )
(5.40)
It should be noted that the imaginary part Aqq (t, tw ) of (5.40) is equal to its equilibrium
version Aqq (t). The reason is that x(t) is linear in bosonic operators (cf. (5.29)). In
consequence the commutator [x(t + tw ), x(tw )] is a C-number. Therefore, Aqq (t, tw ) is
independent of the initial state and translationary invariant in tw .
cum (ω, t )
Results for the correlator Sqq
w
cum (ω, t ) (cf. def. (5.41)) is independent of
Firstly, it is important to note that Sqq
w
the eigenvalue α of the coherent state. Any dependence of the real part of (5.40) on
this eigenvalue is contained in the product hx(t + tw )ihx(tw )i that is subtracted in the
cum (t, t ).
cumulant Sqq
w
cum (ω, t ) numerically with a dicretizaUsing the representation (5.40), we evaluated Sqq
w
tion of 200 bath modes. Futher details of this calculations are given in appendix C.2. It
turned out that higher discretizations cannot improve the numerical accuracy on a sigcum (ω, t )
nificant value above 0.5% relative error. In the peak region the correlator Sqq
w
differs only 5 to 10% from the equilibrium correlation function Sqq (ω) during variation
of tw . We depicted the important cases tw = 0 and tw = ∆−1
0 in figure 5.4.
For low frequencies ω ∆0 and high frequencies ω ∆ the numerical error exceeds
5% relative error (cf. appendix C.2) and it is difficult to interpret the dependence of
cum (ω, t ) on the time t .
the function Sqq
w
w
cum (ω, t ) from equilibrium correIn order to interpret the deviation of the correlator Sqq
w
cum (ω, t ) for different
lations, it is more adequate to evaluate the difference χ00qq (ω) − Sqq
w
damping strengths and different times tw . However, this calculation is at the same time
a test of the FDT.
64
The dissipative harmonic oscillator II: applications
cum (ω, t ) for different damping strengths. We
Figure 5.4: The correlation function Sqq
w
chose an Ohmic bath with cut off frequency ωc = 10 and a potential curvature ∆0 = 5.
Expectations in the context of the FDT
Unlike to the initial state | IIi, we cannot conclude from the reprentation (5.40) that
the FDT is obviously fulfilled. Nevertheless, we use the above representation of the
correlator Cqq (t, tw ) for a test of the fluctuation dissipation theorem. For this purpose,
cum (ω, t ) and χ00 (ω, t ) is calculated. These Fourier
the difference of the functions Sq,q
w
w
q,q
transforms have to be defined with care. 9
def
cum
Sqq
(ω, tw ) = 2
∞
Z
cum
Sqq
(t, tw ) cos(ωt)dt
0
χ00qq (ω, tw )
def
Z
= −2
∞
Aqq (t, tw ) sin(ωt)dt
(5.41)
0
Since Aqq (t, tw ) is equal to its equilibrium version, we could use the result (5.24) for
the response function χqq (t). For computational reasons, we will extract it from the
imaginary part of (5.40) via the Kubo formula (5.22).
Before we proceed with this last part of the section, it is worth to note what can be
expected in the context of the FDT.
cum (t, t ) is therefore independent of the eigen• We know that the correlator Sqq
w
value of the coherent state. Since the response function χqq (t, tw ) is equal to its
equilibrium version, the difference of both functions is completely independent of
R∞
The correlator χ00q,q (ω, tw ) is equivalent to its full Fourier transform −∞ e−iωt χqq (t, tw )dt since
Aqq (t, tw ) is antisymmetric in t (cf. the above mentioned arguments concerning Aqq (t, tw )). Since
cum
cum
Sqq
(−t, tw ) = Sqq
(t, tw ) holds only in equilibrium, the half-sided transform (5.41) is chosen. For
the equilibrium correlator Sqq (t) it is equivalent to the usual full Fourier transform.
9
5.3 Product initial state
65
the excitation strength α of the coherent state. That is the case when system and
bath are prepared in their ground states and then coupled together. This situation gives an example that shows that the fluctuation dissipation theorem is not
at all necessarily the stronger violated, the stronger the non-equilibrium situation
is prepared. As we will see below, this statement holds even when the FDT actually is violated. Previously, we mentioned the concept of effective temperature
(cf. section 5.2). This effective temperature is intended to give a measure for a
violation of the FDT. It was even suggested in [17] to be a measure for heating
effects in the bath during the process of energy dissipation.
In our case, the definition of an effective temperature would not be a suitable
measure for heating effects in the bath. Higher excitations α of the coherent state
should lead to higher effective temperatures. However, the effective temperature
is here independent of α.
• The time tw denotes a time that passed until the measurements that describe
the correlations were performed. In the limit tw → ∞, it can be expected that
an excess energy from an excited coherent state dissipated into the bath. Dissipation means also that this energy does not return to the central oscillator. It
was distributed to an infinite number of bath modes and dissipation is therefore
irreversible in the sense of statistical mechanics. Examples for the Kondo model
[16,17] show that non-equilibrium correlators reach their equilibrium form in the
limit tw → ∞. However, the fluctuation dissipation relation then holds. We
expect this behaviour also from the DHO. From this expectation, we can draw
another conclusion. We expect that if the FDT is violated at a finite time tw , the
violation will show a dependence on the time tw . Interesting comparisons can be
drawn to the non-equilibrium Kondo model [16]. We will give a more detailed
comparison to results [39] for this model in chapter 8.
An analytical result at zero waiting time tw = 0
Fortunately, the case tw = 0 is simple enough to allow for interesting analytical insights.
cum (t, t = 0) and A (t, t = 0) are extracted from (5.40).
The functions Sqq
w
qq
w
cum
Sqq
(t, tw = 0) =
N
1 X
[C0,n + D0,n ][C0,n0 + D0,n0 ]
2 0
n,n =0
N X
[Ck,n − Dk,n ][Ck,n0 − Dk,n0 ] cos(ω̄n t)
k=0
(5.42)
66
The dissipative harmonic oscillator II: applications
Figure 5.5: A plot of the expression (5.45) for different damping strengths. We chose an
Ohmic bath with cut off frequency ωc = 10 and a potential curvature of ∆0 = 5. The
delta functions in (5.46) have been broadened by a Gaussian with a standard deviation
that equals the bath discretization frequency.
N
1X
[C0,n + D0,n ]2 sin(ω̄n t)
Aqq (t, tw = 0) = −
2
(5.43)
n=0
Their half-sided Fourier transforms (cf. definition (5.40)) can be written as a sum over
delta functions.
cum
Sqq
(ω, tw = 0) =
N
πX 2
2
[C0,n − D0,n
] δ(ω + ω̄n ) + δ(ω − ω̄n )
2
(5.44)
n=0
and
χ00qq (ω, tw
N
πX
= 0) = −
[C0,n + D0,n ]2 δ(ω + ω̄n ) − δ(ω − ω̄n )
2
(5.45)
n=0
Obviously, the difference of these functions is finite.
cum
χ00qq (ω, tw = 0) − Sqq
(ω, tw = 0) = π
X
D0n [C0n + D0n ]δ(ω − ω̄n )
(5.46)
n
cum (ω, t )
For finite waiting times tw 6= 0 the we will evaluate the difference χ00qq (ω, tw )−Sqq
w
purely numerically.
5.3 Product initial state
67
Figure 5.6: A visualization of the violation of the FDT for two different damping
strengths α. Each plot contains different curves for different waiting times tw . We chose
a Ohmic bath with cut off frequency ωc = 10 and a potential curvature of ∆0 = 5.
5.3.3
Numerical test of the FDT
We evaluate the difference
def
cum
∆F DT (ω)tw = χ00qq (ω, tw ) − Sqq
(ω, tw )
(5.47)
numerically for different damping strengths and different times tw . We used a finite
number of 200 bath modes to evaluate the expression. An increasing of the discretization up to 3000 bath modes showed that convergence was already excellent for 200 bath
modes with deviations of less then 1% error for larger mode numbers.
Violation of the FDT
In figure 5.6, the function ∆F DT (ω)tw is depicted. For not too large times tw ≤ 10∆−1
0
it has a magnitude between 5% and 10% of the equilibrium response function χ00qq (ω) in
the region of the resonant frequency ∆∞ . In appendix D we show that the numerical
routine yields less than 1% error in this region. We conclude that the FDT is violated
for finite tw . In the limit of large waiting times tw ∆−1
0 , ∆F DT (ω)tw vanishes. Our
expectation that the FDT is fulfilled in the limit tw → ∞ is therefore confirmed.
The largest absolute violation of the FDT is situated in the resonant frequency ∆∞ . In
relative numbers, things become different. In ω = 0, ∆F DT (ω)tw vanishes for tw = 0.
For finite tw , ∆F DT (ω = 0)tw 6= 0 and the deviation from 0 exceeds the numerical
error. This is qualitatively the most intense violation of the FDT. A non-zero value
cum (ω = 0, t ) is the reason for this behaviour. In equilibrium, S cum (ω, t = 0)
of Sqq
w
w
qq
would be equal to the response function χ00qq (ω) that vanishes like J(ω), ω → 0. In the
68
The dissipative harmonic oscillator II: applications
cum (ω = 0, t ) 6= 0 is very meaningful since
context of an effective temperature Teff , Sqq
w
cum
Teff ∝ Sqq (ω = 0, tw ) (cf. subsection 5.2.3).
Discussion of the effective temperature
The effective temperature Teff has been defined in subsection 5.2.3 via
χ00qq (ω)
1 cum
=
S
(ω = 0, tw )
ω→0
ω
2Teff qq
lim
(5.48)
The left hand side of (5.48) can be obtained from the exact result χ00qq (ω) = π2 ∆0 K(ω)
χ00 (ω)
that is extracted from (5.25) via the FDT (5.27). It leads to the behaviour limω→0 qqω ∝
limω→0 J(ω)
ω . This limit is only finite for Ohmic baths. We conclude that for superOhmic and sub-Ohmic baths, Teff 6= 0 is not possible.
cum (ω =
For an extraction of the effective temperature, we would have to evaluate Sqq
cum (ω = 0, t ) also large time scales t ∆−1 of S cum (t, t )
0, tw ) numerically. In Sqq
w
w
qq
0
contribute. Similar to the dynamical sign problem in QMC [52], in the expression (5.40)
sums over rapidly oscillating phase factors lead to large numerical errors if t ∆−1
0 .
cum (ω = 0, t ) and
It is difficult to ensure a low enough absolute numerical error for Sqq
w
we sketch just some qualitative aspects by interpreting figure 5.6.
−1
cum (ω = 0, t ) is negative for two times t
The offset Sqq
w
w ≤ ∆0 . It is positive for
cum
tw = 0.6. A negative value of Sqq (ω = 0, tw ) corresponds to a negative effective temcum (ω = 0, t ) gets larger, corresponding to
perature. For larger dampings, the offset Sqq
w
a larger magnitude of effective temperatures (figure 5.6). According to epression (5.44)
cum (ω = 0, t ) vanishes in t = 0 and leads to T
and figure 5.6 the offset Sqq
w
w
eff = 0.
Various aspects of a violation of the FDT can be mentioned. Even a hierarchy of violations could be formulated, based on several categories of violation. A violation in ω = 0
corresponds to Teff 6= 0. A fulfillment in the limit lim tw → ∞ could be another category.
As we have seen, a negative effective temperature occurs for the DHO for several waiting times tw . This should not be considered prima facie as an artefact. Eg we know
the textbook example of population inversion in lasers. This may be regarded as a
Boltzman distribution with negative temperature. Eg if the bath modes are occupied
with an inverted Bose-Einstein distribution, a negative effective temperature could be
physically meaningful. Also for another reason, it is useful to get insights into the mode
population of the DHO. In a non-equilibrium situation the excitation of the first level
above the ground state could be weak. In this limit the central oscillator behaves like
a damped two-state system. One main part of this thesis is dedicated to the damped
5.4 Mode population in non-equilibrium
69
two-state system. A mapping of the damped two-state system onto a DHO has been
exploited in [15]. This mapping was used to calculate zero-temperature spectral functions in equilibrium.
5.4
Mode population in non-equilibrium
Up to now only observables of the central oscillator were calculated. Such observables
are often calculated from the reduced density matrix. It is obtained by integrating out
all bath degrees of freedom from the full density matrix. Previously, we discussed the
fluctuation dissipation theorem. We tried to relate its violation to energy dissipation
and heating effects. Heating effects in the bath can probably be related to the effective
temperature Teff . This has been suggested in [17]. However, this kind of temperature
is only a phenomenological description. A microscopic investigation of energy transfer
processes can only be achieved if the dynamics of bath degrees of freedom are included.
The underlying microscopic processes can be concluded from the transitions that are
induced by the coupling term in the Hamiltonian (5.4). These transitions involve a
change in the state of the harmonic oscillator by one quantum number, and there is
a concomitant change in the quantum number of one normal mode of the heat bath.
Those transitions that do not conserve energy are only virtual transitions while those
that do conserve the energy persist as real processes that dissipate energy. The requirement that energy is conserved means that a single transition can only present a real
dissipative process if the normal modes of the reservoir have a finite spectral weight at
the frequency ∆0 . We therefore expect that the resonant normal modes of the bath are
most important for the dissipation process.
Transitions that change the quantum number of bath modes can be traced through the
bath mode population.
nk (t) = hb†k (t)bk (t)i
(5.49)
We will denote the mode population of the central oscillator by n0 (t).
5.4.1
Mode occupation for a coherent state
We evaluate the observable (5.49) in the initial state | IIi. In section 5.3 we showed
that the fluctuation dissipation theorem is always violated for the coherent states | IIi.
An effective temperature occured that can probably be related to heating effects in
the bath. In this sense, Teff has a relation to the time-dependent occupation of the
bath modes. An uncoupled bosonic bath in thermal equilibrium is described by a
1
Bose-Einstein distribution nk (t) ≡ exp(βω
. The time-dependent non-equilibrium
k )−1
situation will lead to a non-trivial time-dependent behaviour of this distribution.
70
The dissipative harmonic oscillator II: applications
Analytical expansion in the short-time limit
We expand the operator b†k (t)bk (t) = eiHt b†k bk e−iHt in powers of t by employing the
Baker-Haussdorf identity [40]. This asymptotic expansion is done for the Hamiltonian
(4.5) in O(t4 ).
hα | b†k (t)bk (t) | αi = (4α2 + 1)λ2k t2 + O(t4 )
(5.50)
We want to interpret this behaviour in the context of an effective temperature. We
argued above that it might give a phenomenological description of mode population
corresponding to a Bose-Einstein distribution at a temperature that is related to Teff .
A power law for the couplings λ2k ∝ ωks , s > 0 leads to a population inversion. For
t 1 the mode population will behave as nk (t) ∝ ωk t2 and thus, the mode population
increases proportional to the mode frequency. In contrast, a Bose-Einstein distribution populates bosonic modes the higher, the lower their frequency is. In addition, a
negative effective temperature cannot be related to a negative temperature in the BoseEinstein distribution since it would lead to unphysical negative occupation numbers.
For larger times, the mode population has to be evaluated numerically. Below, we give
formal expressions for the mode population nk (t).
Formal expression derived with time-dependent flow equations
In order to obtain the observables nk (t) we could solve the time-dependent flow equations (4.37) in B=0 and evaluate the operator b†k (t)bk (t) in the coherent state | IIi,
leading to the result
nk (t) =
N
X
[αi (t) − ᾱi (t)]2 + [αi (t) − γ̄i (t)]2 + 4α2 [γ02 (t) + γ02 (t)]
(5.51)
i=0
The formal expression (5.51) demands a lot of computational time for a numerical
evaluation and it is again preferable to use a linear transformation. Nevertheless we
take (4.?) as another illustration how time-dependent flow equations can express timedependent observables. It becomes obvious that the time-dependent coefficients αi (t)
and ᾱi (t) in (5.51) yield a fluctuation of nk (t) even if the coherent state is not excited
(α = 0). An excited coherent state yields an additional fluctuating contribution ∝ α2
to the mode population nk (t).
Numerical evaluation with linear transformation
In section 5.3 we obtained the operator b(t) by two successive linear transformations
for a finite number of bath modes. Exactly the same linear transformation can be em-
5.4 Mode population in non-equilibrium
71
ployed to obtain the operator bk (t). The result can be expressed through the matrices
A, B, C and D defined in (5.38) and (5.39).
nk (t) =
N
X
Ck,n Ck,n0 ei(ω̄n −ω̄n0 )t
nn0 =0
N
X
+
+
Dk,n Dk,n0 ei(ω̄n0 −ω̄n )t
nn0 =0
N
X
Dk,n Ck,n0 e−i(ω̄n +ω̄n0 )t
nn0 =0
+
N
X
nn0 =0
+α2
N
X
Ck,n Dk,n0 ei(ω̄n +ω̄n0 )t
N
X
k=0
N
X
k=0
N
X
k=0
N
X
Bnk Bn0 k
Ank An0 k
Ank Bn0 k
Bnk An0 k
k=0
[Ckn eiω̄n t + Dkn e−iω̄n t ][Ckn0 e−iω̄n0 t + Dkn0 eiω̄n0 t ][An0 + Bn0 ][An0 0 + Bn0 0 ]
nn0 =0
(5.52)
For a specified finite number N of bath modes, both of the expressions 5.51 and 5.52
are exact. Expression 5.52 involves much less numerical effort. Eg one can evaluate
the time-dependent population of a finite set of bath modes in a 3D contour plot. This
would yield interesting insights into the dissipation process. We leave this calculation
open for future work. Instead we calculate just the time-dependent population of the
central oscillator mode.
Discussion of results
In figure 5.7 we show numerical evaluations of the expression (5.52) for an Ohmic bath
with different damping strengths α and two excitation strengths of the coherent state.
In the left plot, the coherent state has the eigenvalue a=0 and in the right plot it is
significantly excited to a=0.5. We make the following observations:
• The mode population shows always a decay to a finite expectation value in the
limit t ∆−1
0 . This expectation value increases with damping strengths for both
excitation strengths of the coherent states. Eg for weak dampings the convergence
to the asymptotic population is slower if the coherent state is excited.
• As seen already for many other observables before, the mode population is an
observable that shows a coherent oscillation at approximately the frequency of
the renormalized frequency ∆∞ . Eg. the peak of the population on the right
hand side is reached approximately at the inverse frequency ∆2π∞ .
72
The dissipative harmonic oscillator II: applications
Figure 5.7: We depict the time-dependent population n0 (t) of the central mode. The
numerical implementation was given for an Ohmic bath discretized with 200 bath modes
up to the cut off frequency ωc = 10. The left plot refers to a coherent state with
eigenvalue a=0 and the right one to an excited coherent state with a=0.5. Each plot
contains various curves with different damping strengths α.
• Even a coherent state that is not excited leads to an excitation of the mode
population for all positive times. This behaviour makes the transitions of the
central oscillator induced by the coupling to the heat bath obvious. However,
even for large damping strengths α ≥ 0.01 near the critical damping strength
where renormalization effects lead to a vanishing effective potential curvature
∆∞ , the central mode is for all times only slightly excited above the ground
state. The observed values n0 (t) < 0.05 should lead to a physical behaviour that
is similar to that of a two state system since effectively only the first excited state
will be accessible at zero temperature. This analogy has been pronounced in [15].
5.4.2
Mode occupation for an entangled initial state
We want to evaluate the mode population nk (t) for the inital states | Ii. The calculation
is performed in analogy to the observable hx(t)i (cf. section 5.2). We evaluate the
operator a†k (t)ak (t) (cf. 4.34)) in the diagonal representation | Ii =| 0i⊗ | Oi of the
initial state.
nk (t) =
N
X
i=0
[αi − ᾱi ]2 + 4
N
X
λ2i [αi2 cos2 (i t) + ᾱi2 sin2 (i t)]
(5.53)
i=1
The time-independent part of (5.52) is the ground state expectation value while the
time-dependent part is proportional to the squared field strength h2 since the coefficient
λk obeys (cf. (5.3))
h αk
λk = − √ 2
2 ωk
(5.54)
5.5 General time-dependent fields
73
We leave a numerical implementation of the expression (5.53) open for future work.
First tests show that it can be implemented with high accuracy.
5.5
General time-dependent fields
In the previous treatment of the DHO it became clear that flow equations are a suitable
method to calculate observables for any given initial state. However, after the initial
preparation we restricted us to the equilibrium Hamiltonian 4.5. We developed no
general arguments why the time-dependent flow equations (4.37) could be suitable to
treat also non-equilibrium Hamiltonians with time-dependence. In the following, the
non-equilibrium problem formulated in section 5.1 is extended to a more general timedependent Hamiltonian. This example illustrates two aspects:
1. The physical influence of a class of time-dependent perturbations onto physical
observables can be analyzed. Eg. this influence can be compared to the effects
caused by the non-equilibrium initial states from sections 5.2 and 5.3.
2. Technical problems that occur when a Hamiltonian has a non-trivial time-dependence
can be discussed.
Time-dependent preparation of the system
For negative times a field of arbitrary but constant magnitude shall act and prepare the
system-reservoir complex in the initial state | Ii discussed in section 5.1. In addition,
the central oscillator is now perturbed by an arbitrary time-dependent field h(t) that
acts only for positive times. This field is treated semiclassically and couples linear to
the position of the oscillator:
H(t) = ∆0 b† b +
X
k
(b + b† )λk (bk + b†k ) +
X
k
1
ωk b†k bk + h(t) √ (b + b† )
2
(5.55)
Typically this Hamiltonian describes an atom that interacts with an electromagnetic
field. The resultant Hamiltonian is suited eg. for quantum optics. In [19] Hamiltonians
of the type (5.55) were proposed to describe the radiation field of a cavity. The field
coupling term in (5.55) can describe a cavity that absorbs external radiation and dissipates it afterwards. For the formal solutions we want to derive, it will not be necessary
to specify the field h(t) further. It will have the general time-dependence:
(
h, t ≤ 0
h(t) =
(5.56)
h(t), t ≥ 0
Reformulation of the problem in diagonal basis
Again, we reformulate the Hamiltonian in diagonal basis and try to solve the Heisenberg
equation for the position operator in diagonal basis. Formally, this equation reads:
74
The dissipative harmonic oscillator II: applications
dx̃H (t)
= i[x̃H (t), H̃H (t)]
dt
(5.57)
Now an additional problem becomes obvious. H̃H (t) is an operator represented in the
diagonal Heisenberg picture. It is not explicetely known and cannot be assumed to be
equal to H̃(t) that is given in the Schrödinger picture:
H̃(t) = ∆∞ b† b +
X
ωk b†k bk + h(t)x̃
(5.58)
k
Nevertheless, it is possible to solve the Heisenberg equation (5.57) exactly.
5.5.1
Exact solution for the operator x̃H (t)
We will solve equation (5.57) exactly. In the next subsection we use this solution to
calculate position correlation functions and the observable hx(t)i by evaluating the op˜ given in diagonal basis (cf. (5.13)).
erator x̃H (t) in the state | Ii
Obviously, the first step for a solution of equation is to determine the unknown contributions of H̃H (t) in the Heisenberg equation (5.57).
Eliminiation of H̃H (t) from the Heisenberg equation
The unknown operator H̃H (t) in (5.57) is itself the solution of a Heisenberg equation:
∂ H̃(t)
∂h(t) †
dH̃H (t)
= Ũ † (t)
Ũ (t) =
Ũ (t)x̃Ũ (t)
dt
∂t
∂t
(5.59)
It is easy to show that H̃(t) does not commute with itself if h(t0 ) 6= h(t). In this case
equation (5.59) has to be supplemented by a time-ordered propagator Ũ (t).
Ũ (t) = T> e−i
Rt
0
dt0 H̃(t0 )
(5.60)
We used the Dyson time ordering operator T> in (5.60). Seemingly, (5.59) is a complicated differential equation. Nevertheless it is possible to evaluate its right hand side
up to an unknown real-valued function (cf. appendix D). The operator H̃H (t) is then
determined up to an unknown real-valued function. This function is not important
since it will drop out in the Heisenberg equation (5.57).
In appendix D, a solution for the right hand side of (5.60) is obtained up to an unknown
real valued function f(t). We insert this solution into (5.59) and obtain:
dH̃H (t) √ ∂h(t) X
[αk ][e−iωk t bk + eiωk t b†k ] + f (t)
= 2
∂t
dt
k
(5.61)
5.5 General time-dependent fields
75
Integrating equation (5.61) yields a formal solution for HH (t).
†
H̃H (t) = ∆∞ b b +
X
k
ωk b†k bk
+
X
(τk (t)bk +
τ̄k (t)b†k )
k
Z
+
t
f (t0 )dt0 + E∞
(5.62)
0
The coefficients τk (t) and τ̄k (t) defined in (5.62) obey differential equations that are
determined by a comparison of the time-derivative of (5.62) with (5.61):
dτk (t) √ ∂h
= 2 αk e−iωk t
dt
∂t
dτ̄k (t) √ ∂h
= 2 αk eiωk t
dt
∂t
(5.63)
The initial conditions for the differential equations (5.63) are determined by a comparison of (5.62) with the Hamiltonian H̃(t = 0) = H̃H (t = 0):
τk (t = 0) =
τ̄k (t = 0) =
√
√
2hαk
2hαk
(5.64)
Exact solution of the Heisenberg equation (5.57)
It is necessary to chose an ansatz for the operator x̃H (t) in order to solve its Heisenberg
equation (5.57). The operator x̃H (t) will be linear in bosonic operators since H̃H (t) is
quadratic in bosons.
def
x̃H (t) = β(t)b + β̄(t)b† +
X
[βk (t)bk + β̄k (t)b†k ] + r(t)
(5.65)
k
In t=0 we use the operator identity x̃H (t = 0) = x̃ and determine the initial conditions
for all coefficients the ansatz (5.65).
r(0) = β(0) = β̄(0) = 0
√
βk (0) = β̄k (0) = 2αk
(5.66)
Now, we insert the formal ansätze (5.65) for xH (t) and (5.62) for HH (t) into the
Heisenberg equation (5.57). A comparison of coefficients in the resultant equation
determines the solution for the ansatz (5.65) since all coefficients in this ansatz can
be obtained from the resultant differential equations. Using the conditions (5.66), it is
76
The dissipative harmonic oscillator II: applications
trivial to integrate the coefficients β(t), β̄(t), βk (t) and β̄k (t). Since the coefficient r(t)
will depend on the field h(t), it is difficult to integrate it for a general field. All these
steps are summarized in the results:
β(t) ≡ β̄(t) ≡ 0
√
βk (t) =
2[α ]e−iωk t
√ k iω t
β̄k (t) =
2[αk ]e k
√ X
dr(t)
= i 2
αk [eiωk t τk (t) − e−iωk t τ̄k (t)]
dt
k
(5.67)
Thus, ansatz (5.65) is completely determined by the the coefficients given in (5.67) and
the final solution for x̃H (t) takes the form:
x̃H (t) =
√ X
2
[αk ]e−iωk t bk + eiωk t b†k + r(t)
(5.68)
k
Discussion of the solution (5.68)
In the case h(t) ≡ 0, t ≥ 0 the contribution r(t) to the solution (5.68) vanishes. The
observable x̃H (t) becomes then equivalent to the solution (5.14) from section 5.2. We
denote this solution in the following by x̃0H (t) and define
def
x̃H (t) = x̃0H (t) + r(t)
5.5.2
(5.69)
Discussion of the observable hx(t)i
Due to the decomposition (5.68) into r(t) and the old solution (5.14), the observable
hx(t)i can be evaluated by inserting the old solution (5.16) for hx(t)i.
Z ∞
cos(ωt)
hx(t)i = −2∆0 h
K(ω)dω + r(t)
(5.70)
ω
0
From (5.14) we know that hx(t)i shows an exponential long-time decay if r(t) ≡ 0. Via
r(t) the field h(t) can therefore sensitively influence this long-time behaviour. Thus,
our aim is to discuss the function r(t). For a general field h(t), (5.67) is difficult to
integrate in the time domain. We prefer to discuss it in the frequency domain.
Solution for the function r(t) in the frequency domain
For negative times the field h(t) can have a constant non-zero value that prepares a
non-equilibrium initial state (cf. (5.56)). We define a half-sided Fourier transform of
r(t) to avoid divergencies.
5.5 General time-dependent fields
def
77
Z
∞
r(ω) =
e−iωt r(t)dt
(5.71)
0
We perform this half-sided Fourier transform also with equations (5.63), and (5.67, line
4) and make use of the condition limt→∞ r(t) = 0. After these steps, we arrive at:
(5.67) i X
iωr(ω) = √
[αk + ᾱk ][τk (ω − ωk ) − τ̄k (ω + ωk )]
2 k
(5.72)
ωh(ω)
(5.63) 1
τk (ω − ωk ) = √ [αk + ᾱk ]
(5.73)
ω
− ωk
2
Inserting the coefficients (5.73) into expression (5.72) and emploing the spectral function
K(ω) for the coefficients αk yields the solution for r(ω) in dependence of the field h(ω).
Z ∞
ω0
(5.74)
r(ω) = 2∆0 h(ω)
dω 0 K(ω 0 ) 2
ω − ω02
0
We conclude that the response of r(t) to incoming electromagnetical radiation is determined by the function
Z ∞
ω0
2∆0
dω 0 K(ω 0 ) 2
(5.75)
ω − ω02
0
We can reexpress K(ω) by the imaginary part of the dynamical response function via
χ00qq (ω) = π∆0 K(ω) (cf. (5.48 ff)). Finally, the principal value integral in (5.76) can be
R ∞ χ00qq (ω)
eliminated by employing the Kramers-Kronig relation χ0qq (ω 0 ) = π1 −∞ ω−ω
0 dω and
we arrive at
r(ω) = h(ω)χ0qq (ω)
(5.76)
Discussion of physical aspects
Equation (5.76) leads to two interesting observations. We will explain the existence of
an absorption gap at a certain frequency where electromagnetic radiation is fully transmitted and r(t ≡ 0). In addition, from (5.76) we can conclude that electromagnetic
radiation usually will stabilize the long-time decay of hx(t)i from an exponential to an
algebraic long-time decay.
Remarkably r(ω) is proportional to the field spectrum h(ω) and the real part χ0qq (ω)
of the dynamical susceptibility. Thus, the function χ0qq (ω) contains all information
on the dissipative influence of the bath modes onto the absorption of electromagnetic
radiation. For an Drude type spectral function 10
J(ω) =
2
M ΓωωB
1+
10
All notations were taken from [35].
2
ωB
ω2
78
The dissipative harmonic oscillator II: applications
0
the exact
2 low frequency
asymptotics from Riseborough et al [35] is given by χqq (ω) ∝
−1
2
M
(ω0 − ΓωB ) − ω , ω ∆0 . The long-time decay of r(t) is therefore given by the
low frequency behaviour
r(ω) ∝ h(ω)M −1 (ΓωB − ω 2 ), ω ∆0
(5.77)
It is important to remind that an instantaneous switch off of the external field was
shown to lead to an exponential long-time decay of hx(t)i (cf. section 5.2). In contrast,
we conclude from (5.77) that Ohmic 11 baths lead always to an algebraic long-time
decay of hx(t)i with an exponent that is determined by the low frequency spectrum
of the external field. We interpret this sluggish decay as a stabilisation of the decay
process due to absorption of external radiation. However, due to (5.76) a stabilisation
of hx(t)i is only possible at frequencies where χ0qq (ω) 6= 0.
It is remarkable that χ0qq (ω) has always a zero below the frequency ∆0 (compare figure
5.8). This represents a gap in the absorption spectrum of the damped oscillator. For a
Drude type bath the absorption gap is centered at
q
ω = ω02 − ΓωB
Increased damping Γ will shift the absorption gap more and more below the resonant
frequency ω0 in dependence of the cut off ωB of the bath spectrum.
5.5.3
Discussion of correlation functions
We want to calculate the dynamical response function χqq (t, tw ) and the correlator
cum (t, t ) for the initial states | Ii and an arbitrary field dependence h(t), t ≥ 0. It
Sqq
w
is easy to show that the additional component r(t) in the operator x̃H (t) (cf. (5.69))
will not appear in formal expressions of these correlators. In the commutator [x̃H (t +
tw ), x̃H (tw )], the C-number r(t) drops out. The dynamical response function χqq (t, tw ) is
therefore independent of h(t). As shown in section 5.2 it reduces then to the equilibrium
reponse function χqq (t).
χqq (t, tw ) ≡ χqq (t)
(5.78)
cum (t, t ). AccordA simple calculation shows that r(t) drops also out of the correlator Sqq
w
cum (t, t ) reduces then to the symmetrized
ing to the discussion given in section 5.2 Sqq
w
equilibrium correlator Sqq (t).
cum
Sqq
(t, tw ) ≡ Sqq (t)
(5.79)
This is the situation we know already from subsection 5.2.3 . We conclude that the
fluctuation dissipation theorem (5.27) is fulfilled for all times tw and as argued in 5.2.3,
also for finite temperatures.
11
This statement should not depend on the Drude type cut off we used since high energy modes will
not influence the long-time decay.
5.5 General time-dependent fields
79
Figure 5.8: A plot of the real (dispersive) part χ0qq (ω) of the dynamical response function for different damping strengths α. It describes the absorption of elecromagnetic
radiation by the central oscillator. The dispersion shows two peak regions of absorption
distributed around an absorption gap. With increasing damping the absorption peaks
and the gap broaden and spectral weigth is shifted to lower frequencies.
cum
Sqq
(ω, tw ) ≡ sign(ω)χ00qq (ω, tw )
(5.80)
Conclusions
cum (t, t ) and χ (t, t ) are indepenWe saw that the two-time correlation functions Sqq
w
qq
w
dent of influences of a time-dependent external field and also of the time variable tw that
determines the time for the first measurement. All results from the field-independent
case for these functions remained valid, eg the fulfillment of the FDT. In this sense, these
correlation functions are not sufficient to analyze non-equilibrium processes in general.
Arbitrary time-dependent fields can lead to complicated time-dependent emission and
absorption processes that lead to complicated non-equilibrium behaviour. It is desirable to investigate other observables that describe such processes more detailed. The
observable Sqq (t, tw ) is more suitable to discuss influences of the time-dependent fields
cum (t, t ), it contains
since it depends on the field. In comparison to the cumulant Sqq
w
eg the additional component r(t + tw )r(tw ) that showes an alebraic long-time decay in
the time tw derived above. Therefore the correlator Sqq (t, tw ) contains the algebraic
long-time crossover (tw ∆−1
0 ) to the equilibrium correlation function Sqq (t).
80
5.6
The dissipative harmonic oscillator II: applications
Summary
We analyzed the non-equibrium dynamics of two different initial states at zero temperature. In addition, we obtained an analytical solution that describes the time-dependent
influence of an external field onto the initial state | IIi. The observables Sqq (t, tw ) and
hx(t)i were used to discuss the time-dependent cross over behaviour of system observables to equilibrium expectation values. Equilibrium behaviour is indicated by a
fulfillment of the FDT that relates different correlation functions.
It turned out that the cross over regime to equilibrium like properties depends very
sensitively on the initial preparation of the system. Eg the observable hx(t)i decays
exponentially in the long-time limit t ∆−1
0 for the entangled initial states | IIi. In
presence of any finite external field h(t), t > 0 coupling to the position of the central
oscillator, the decay changes to an algebraic long-time behaviour with an exponent that
depends on the field. Coherent states show generically algebraic long-time decay if the
bath modes are initially not entangled with the coherent state. Again, the initial preparation influences the long-time decay. A non-vanishing initial momentum decreases the
exponent of the algebraic long-time behaviour from -2 to -3. A comparison of the observable hx(t)i for the two different initial states | Ii and | IIi is given in figure (5.9)
where we show results from sections 5.2 and 5.3.
In any considered case the correlation function Sqq (t, tw ) showed a cross over to the
equilibrium correlation function Sqq (t) in the limit tw → ∞. However, we showed that
the long-time approach depends sensitively on the initial preparation and can occur as
well exponentially or algebraically in the tw with different exponents.
Interestingly, the FDT was fulfilled for the initial states | IIi also in presence of an
external field. For coherent states, the FDT was violated for finite tw but showed a fast
cross over to a fulfillment for tw ≥ 10∆−1
0 . Further interpretations of the FDT in the
context of real physical processes are difficult. For coherent states the violation was
independent of the eigenvalue of the coherent state. Heating effects in the bosonic bath
surely depend on this value and an application of the concept of effective temperature
fails if it shall describe such heating effects properly.
We conclude this summary with the statement that the trivial DHO has a qualitatively diverse non-equilibrium behaviour. In contrast, for the technically much more
challenging Kondo model very uniform results for two different non-equilibrium preparations were derived in [39]. A detailed comparison of these contrasting results for the
two different models will be given in chapter 8 that also includes a comparison to the
spin-boson model.
5.6 Summary
81
Figure 5.9: We compare the normalized expectation value hx(t)i/hx(0)i in the time
domain (upper plot) and the frequence domain (lower plot) for the inital states | Ii
(red curves) and | IIi (blue curves). The damping strength increases from top to
bottom among the values α = 0.003, α = 0.006, α = 0.011 and α = 0.012. For large
dampings α ≥ 0.01, the spectral peak positions start to differ and the coherent state
| Ii shows faster coherent oscillations than the entangled state. It is not possible to
identify the difference between the exponential and algebraic long-time behaviour of
both states on the depicted time scale.
82
The dissipative harmonic oscillator II: applications
Chapter 6
The spin-boson model I:
preparations
This chapter intends to step to the second model treated in this thesis. The treatment
of this model is divided into two parts that are organized as one chapter, respectively.
The first chapter gives an introduction to the model and, most important, it provides the static part of the flow equation treatment. That is the diagonalization of
the Hamiltonian and additionally, the transformation of all spin operators into the so
defined diagonal basis. A thorough discussion of all performed approximations will be
given. All transformations are performed in order to calculate the time-dependent spin
expectation values hσi (t)i, i = x, y, z and the symmetrized spin correlation functions
1
2 h{σz (t + tw ), σz (tw )}+ i for different non-equilibrium initial states.
Chapter 6 can be considered as the dynamical part of the model treatment. Firstly, it
derives approximated representations for time-dependent spin operators both in diagonal and physical basis. Finally, this operators are evaluated in different non-equilibrium
initial states and the above mentioned physical observables are discussed. While all
technical steps are performed for general bath types of the form J(ω) ∝ ω s , s > 0,
observables will be evaluated only for Ohmic baths (s=1).
6.1
The model
Quantum mechanical systems are often restricted to an effectively two-dimensional
Hilbert space. Both in physics and chemistry two-state systems are ubiquitous. Many
systems are intrinsic two state systems that posess only two different quantum states,
like a nucleus with spin 12 or the polarization in case of a photon. Even if the system
posesses a formally continous degree of freedom in form of an effective potential energy
function dependent on a continous parameter, it can as well behave like a two state
system if it has two separated minima (See figure 6.1). Suppose the barrier height V0
is much larger compared to the excitation energies ~ω− and ~ω+ , and also the temper-
84
The spin-boson model I: preparations
Figure 6.1: A double-well system in the ”two-state” limit
ature obeys kB T ~ω+ , ~ω− . Then, the Hilbert space of the full system is effectively
spanned by the two ground states of the single wells. Of course the possibility of tunneling between the two wells has to be taken into account.
The continous degree of freedom q parametrizing the effective potential V (q) need not
be geometrical and can as well be the magnetic flux threading a SQUID. A detailed
description of this problem was given in [41]. It is related to the exciting topic of
macroscopic quantum tunneling [42] and a promising experimental realization of qubits
[43]. In fact, firstly the problem of macroscopic quantum tunneling motivated many
theorists to establish theoretical results of two-state system dynamics in dependence
of realistic system-environment couplings. If the system does not interact with the
environment, the system undergoes
p quantum-coherent tunneling between the two states
with a tunneling rate of order ∆20 + 2 , where is the level asymmetry. However, the
difficult task now is to grasp the influence of the environmental coupling onto the ideal
of undamped quantum coherent tunneling. In many real examples, these influences
become important. Already mentioned was the tunneling of flux states in SQUIDS.
Further examples are charge transport processes, where in its simplest form an electron
localized at a donor size is tunneling to an acceptor site. Interactions between charge
and the polarization cloud of the environment strongly influence the tunneling rate
[44,46]. Other experimental examples are tunneling processes of defects in crystalline
solids and electron tunneling between quantum dots [45].
6.1 The model
6.1.1
85
Formulation of the model Hamiltonian
The general problem of a double-well system coupled to a dissipative environment can
be reduced to the Hamiltonian 1
H=−
X
1 X
∆0
σx + σz
λk (b†k + bk ) +
ωk b†k bk + E0
2
2
k
(6.1)
k
where the two-state system is formulated as a pseudo-spin with corresponding eigenstates | +iz and | −iz of the spin projection operator σz 2 and a tunnel splitting ∆0 of
these two levels. Tunneling dynamics is strongly influenced by a coupling to bosonic
bath modes that obey the standard Bose commutation relations [bα , b†α0 ] = δαα0 . Therefore, the global model Hamiltonian has been dubbed spin-boson model. In general, the
bath couples to all spin components σi , i = x, y, z. For most applications of this model,
the system-environment coupling to σx and σy is negligible [41]. In the Hamiltonian
(6.1), high energy modes have already been eliminated by the introduction of a cut-off
frequency ωc . This is possible since these modes affect only the process of transition
through the potential barrier and thereby renormalize the effective tunneling matrix element. These modes can be eg integrated out by adiabatic renormalization
and yield in
α
∆ 1−α
for α < 1, disthe Ohmic case J(ω) = 2αω the well-known result [47] ∆r = ∆( ωc )
regarding an unspecified numerical constant. In a renormalization scheme, the physical
results shall not depend on the choice of an high-energy cut off and thus, the physical
behaviour depends on the frequency ωc only via ∆0 . In consequence, the whole model
is fully described by the three quantities ∆0 , J(ω) and . From these quantities, the
bath spectrum J(ω) turns out to influence the physics most sensitively. For sub-Ohmic
baths the spin is localized in one well at T=0 and relaxates exponentially at T 6= 0. In
the Ohmic case, a Kosterlitz-Thouless transition occurs at the critical damping α = 1
for T=0 and in the super-Ohmic case, a cross over from coherent to incoherent tunneling takes place at a finite temperature [41].
Typically, the spin-boson model is related to many other prominent models in theoretical physics - most strikingly to the anisotropic Kondo model (cf. chapter 8). The
Kondo model as well as the spin-boson model belong to the so called strong-coupling
problems which make the use of renormaliztion group techniques almost indispensable.
The Kondo model was first solved by Wilson employing the numerical renormalization
group [51]. These ideas were also applied to the spin-boson model [48,49].
6.1.2
Previous work
Many methods were employed to investigate the spin-boson model and in part, this
is due to the fact that it represents one of the simplest non-trivial models in dissi1
A derivation of this Hamiltonian has been given for very general system-environment couplings in
[41].
2
In the double-well problem these eigenstates represent states that are localized in one of the two
wells respectively.
86
The spin-boson model I: preparations
pative quantum mechanics. In 1987 Leggett, Chakravarty, Dorsey, Fisher, Garg and
Zwerger summarized the theoretical efforts on the spin-boson model in Ref. [41], propagating the Non-Interacting Blip Approximation (NIBA). This method is based on the
Feynman-Vernon influence functional approach and formulates the problem as one single path integral over four different states, two diagonal states and two off-diagonal
states. The NIBA is essentially a perturbative treatment of this path integral in the
tunneling matrix element ∆0 .
Except for some special parameter values, an exact solution of the spin-boson model
is not known. Only with numerical real time quantum Monte Carlo (QMC) methods
[52], it is possible to evaluate the exact path integral representation of the reduced
density matrix. For large times, real time QMC leads to large numerical errors due to
the well-known dynamical sign problem [50,52]. Recent efforts concerning this problem can be found in [50]. Other approaches try to use a mapping of the spin-boson
model on the anisotropic Kondo model and solve the latter using numerical renormalization group (NRG), Bethe Ansatz or conformal field theory (CFT). However, NRG
and Bethe Ansatz provide only spectral properties or dynamics at very short time
scales, and CFT has solved so far only the unbiased case = 0 for the diagonal elements of the reduced density matrix. In addition, the mapping on the Kondo model
cannot be proved rigorously, and the relation of the parameters is not precisely known.
Only recently, a direct application of NRG was possible by extending its applicability
to bosonic baths [53]. During the last years, renormalization-group techniques were
extended to go beyond spectral properties in order to describe time-dependent physics
in non-equilibrium situations. Among them are time-dependent NRG (TD-NRG) [3]
and the real-time renormalization group [54]. In this thesis, it will be shown that the
flow equation method is a renormalization group technique especially suitable for timedependent non-equilibrium situations.
Lateron, we will compare our results to the NIBA since it provides good results in large
parts of the parameter space of temperature and the spectral function J(ω), especially
for Ohmic baths J(ω) = 2αω with α ≤ 0.5.
After the model has been introduced, the first prerequisite of our model treatment
will be derived. We pursue the same strategy that we developed to calculate timedependent observables for the DHO. A diagonalized version of the Hamiltonian is the
starting point for all time-dependent problems that we want to treat:
1. The diagonalized Hamiltonian will be used to solve the Heisenberg equation for
spin operators in diagonal basis.
2. A solution of the Heisenberg equation for spin operators in diagonal basis will be
transformed back to the physical basis via time-dependent flow equations .
3. The time-dependent spin operators in diagonal and physical basis will be evalu-
6.2 Diagonalization of the Hamiltonian
87
ated in different non-equilibrium initial states.
6.2
Diagonalization of the Hamiltonian
As we shall see in the following section, the spin-boson Hamiltonian cannot be diagonalized exactly since an ansatz for a transformed Hamiltonian H(B) remains not closed
during the flow like in the case of the DHO. Therefore, it is necessary to truncate the
ansatz for transformed operators at a certain order of the expansion parameter. Additional attention should be paid to the first truncated terms in an ansatz. Here, the
importance of normal ordering becomes pronounced, since these terms will always be
normal ordered in order to increase accuracy.
6.2.1
The role of normal ordering
A truncation scheme for an operator expansion is performed more accurately if the
truncated operators are normal ordered. In consequence, the technical approach used
in case of the DHO (cf. chapter 4) has to be improved by an important additional
ingredient. We will see that normal ordering is especially important to describe the
influence of non-equilibrium initial states on a physical observable.
If one talks of normal ordering in the context of flow equations, it should be clarified
that the choice of a definite normal ordering prescription never leads to complications
in the setup of the flow equations. With specified ansatz and generator, even different
normal ordering procedures can be employed without any significant formal influence
on the flow equations and on the choice of a truncation scheme. This can be clearly
understood by the fact that normal ordering only modifies contractions that enter as
constants into the flow equations. In order to formulate all flow equations for a wide
variety of possible initial states we will use generalized contractions. These generalized
contractions are defined as a general formal representation of all contractions with respect to the initial states that are treated within this thesis. Only if flow equations are
applied to a definite initial state, the contractions will be specified with respect to the
given initial state.
In case of a diagonalization of the Hamiltonian, normal ordering will always be used
with respect to the equilibrium ground state, regardless of the correct non-equilibrium
initial state. This leads to the important consequence that one has only to carry
out one diagonalization procedure for one Hamiltonian and can then treat different
initial states without making any difference in the diagonalization procedure of the
Hamiltonian. Only flow equations of other observables will be modified by different
contractions. Following closely ref. [27], we now proceed with the diagonalization of
the Hamiltonian. We keep the pedagogical order of presentation used in this reference
because it is useful to understand the origin and motivation of the approximations that
will be introduced. Since the transformation used in our reference has not especially
88
The spin-boson model I: preparations
been prepared for an application to non-equilibrium problems, we attempt to identify
the crucial aspects of all approximations. In a later discussion, their consequences for
the transformation of observables will be considered.
6.2.2
Flow equations in O(λ2k )
Formal description of a truncation scheme
In a first step, the Hamiltonian is diagonalized by neglecting contributions to the flow
equations that are generated at least as a double sum over the couplings λk or coupling
constants that are proportional to the couplings λk . Such flow equations will be called
”flow equations in O(λ2k ) of the couplings” or briefly ”flow equations in O(λ2k )”. Note
that transformations of this type can for Ohmic baths with J(ω) = 2αω only yield
good results for dampings α ≤ 0.2 [15] 3 . This treatment is a necessary motivation
for an approximation that leads to improved flow equations formulated in O(λ3k ) of
the couplings. After introducing this approximation, the diagonalized version of the
Hamiltonian is derived by neglegting truncated terms in O(λ3k ) of the couplings.
Derivation of the flow equations
In order to motivate the derivation of flow equations, a bosonic representation of the
Hamiltonian is used
H=−
X
∆0
1 X
ωk b†k bk + E0
σx + σz
λk (b†k + bk ) +
2
2
(6.2)
k
k
where the bath modes are described by the bosonic creators b†k and annihilators bk
that obey the standard bose commutation relation [bk , b†k0 ] = δkk0 . Like in the case of
the DHO, this representation simplifies the flow equation treatment alot. Clearly, the
interaction term we want to eliminate is the coupling HSB of system and bath.
1 X
HSB = σz
λk (bk + b†k )
2
(6.3)
k
P
In addition we denote the remaining parts ∆20 σx + k ωk b†k bk + E0 of the Hamiltonian
by the symbol H0 . By definition, the canonical generator takes the structure
η = iσy
X
k
(y)
ηk (bk + b†k ) + σz
(y)
X
ηk (bk − b†k )
(z)
(6.4)
k
(z)
with ηk = λ2k ∆ ηk = − λ2k ωk . For our purpose to obtain a form invariant Hamiltonian flow, this generator is not the best choice. Working out the commutator
[η, H0 ], one immediately obtains a new coupling of system and bath with the structure
3
When we specify the damping strength α to concrete numbers, we will always refer to Ohmic baths.
6.2 Diagonalization of the Hamiltonian
89
P
iσy k µk (bk − b†k ) that was not contained in the initial Hamiltonian. Even in lowest
order of a flow equation expansion this term would have to be included in the ansatz,
since it is linear in the couplings λk . With the same form of generator, but the modified
parametrisation
λk ω k − ∆
(y)
ηk = − ∆
2 ωk + ∆
(z)
ηk = −
λk ωk − ∆
ωk
2
ωk + ∆
one avoids the undesired coupling. As another useful aspect of this choice the derivative
of the couplings λk will be quadratic in | ωk − ∆ | (cf. equations (6.7)). This ensures
the important property of energy scale separation since during the flow, the coupling
constants λk (B) will decay at a very different rate, dependent on the energy difference
| ωk − ∆ |.
The full commutator [η, H] now reads:
1 X
σz
λk (ωk − ∆)2 (bk + b†k )
2
k
X (y)
− σx
ηk λk (2n(k) + 1)
[η, H] = −
k
1 X (y)
(y)
−
(ηk λl + ηl λk ) : (bk + b†k )(bl + b†l ) :
2
k,l
X (z)
+
ηk λk
k
(6.5)
where we used normal ordering with respect to the bosonic vacuum or the finite temperature free density matrix. Such a normal ordering prescription results in the contractions
1
def
hb†k bk i = n(k) =
exp(βωk ) − 1
where n(k) is the standard bose factor of the bosonic mode with index k. The final
Hamiltonian then has the structure
H(B = ∞) = −
X
∆∞
σx +
ωk b†k bk
2
k
1 X
− σx
ωkl (B = ∞) : (bk + b†k )(bl + b†l ) : +E(∞)
2
k,l
(6.6)
90
The spin-boson model I: preparations
where the renormalized coefficients ∆∞ , E∞ and ωkl (∞) are determined by the flow
equations:
X
d∆
ωk − ∆(B)
= −∆(B)
λ2k (B)
dB
ωk + ∆(B)
k
dλk
= −(ωk − ∆(B))2 λk (B)
dB
h ω − ∆(B) ω − ∆(B) i
λk (B)λq (B)
dωkl (B)
q
k
=−
∆(B)
+
dB
2
ωk + ∆(B) ωq + ∆(B)
dE(B)
1 X 2 ωk − ∆(B)
=−
λk ωk
dB
2
ωk + ∆(B)
k
(6.7)
In the final Hamiltonian H(B = ∞) coupling terms were eliminated up to quadratic order in the couplings λk . The remaining problem are the resonant couplings ωkk (where
ωk ≈ ∆∞ ) that cannot be neglected because they have a large magnitude [14]. The
normal ordered term in (6.6) cannot yield any contribution to physical observables if
one evaluates it in the initial state that was used for normal ordering. Nevertheless, it
inavoidable starts contributing to the time evolution of σz on a time scale of order ∆2π∞
if it is evaluated in a time-dependent non-equilibrium initial state [14]. It is a known
result that the transformation in O(λ2k ) fails to describe
the line shape of the symR∞
metrized spin-spin correlation function Szz (ω) = −∞ e−iωt 12 h{σz (t), σz }+ iGS dt close
to the resonant modes ω ≈ ∆∞ [26]. There are two possibilities under discussion to
achieve a more accurate diagonalization of the Hamiltonian.
6.2.3
The ground state approximation
The actual method that will be used for an improved diagonalization of the spin-boson
Hamiltonian is based on an approximation that is performed in order to simplify the
canonic way of improving the accuracy of diagonalization. Therefore, we start with a
discussion of this canonic procedure.
Preliminary discussion of flow equations in quadratic order
The dissatisfying transformed Hamiltonian (6.6) can be improved in a canonic way if
we extend the ansatz for the transformed Hamiltonian H(B) to quadratic order in the
couplings λk in order to truncate normal ordered terms in O(λ3k ). This approach leads
to the consequence that the truncated terms in O(λ3k ) will have negligible contributions
to any time-dependent spin operator for not too large damping α ≤ 0.2 [15]. In order to
contain all terms of O(λ2k ) in the couplings in an improved ansatz for the Hamiltonian,
also to the generator η such terms -called η (2) - have to be added 4
4
This is eg motivated by the definition of the canonical generator (cf. section 3.3).
6.2 Diagonalization of the Hamiltonian
η (2) = σx
X
91
(ωk + ωl )tkl (b†k b†l − bk bl ) + 2(ωk − ωl )skl b†k bl
(6.8)
k,l
with the additional coupling parameters tkl (B) and skl (B). The idea to achieve this
improvement is straightforward but leads to complicated flow equations due to the additional couplings in (6.8). We can avoid much of the additional effort that is involved
in the above proposal if we can achieve an elimination of the undesired term (6.6) in
the Hamiltonian otherwise.
Improved flow equations with normal ordered observable σx
A property of the diagonalized Hamiltonian (6.6) is that the expectation value of σx in
its ground state is equal one. At this moment, it is important for the reader to remember
the essential advantage of a normal ordering procedure. Evaluating a normal ordered
bosonic operator in the state that has been used for normal ordering allows a save way
for neglecting the contribution of normal ordered operators to physical quantities. In
this spirit, one can also introduce a normal ordering procedure for the operator σx so
that one eliminates the ground state expectation value:
def
σx = hσx iGS + ∗σx ∗
(6.9)
The ground state approximation consists of neglegting the fluctuating part ∗σx ∗ and
just considers the expectation value hσx iGS . If normal ordering with respect to the
ground state
˜ =| +ix ⊗ | 0i
| GSi
in diagonal basis is used, the ground state approximation becomes exact in the sense
that hσx iGS = 1 and h∗σx ∗i = 0 is fulfilled. The ground state approximation has the
great advantage that one has to deal with a C-number instead of the operator σx and
it is now easy to avoid the coupling term in (6.6) by adding an additional part η (2) to
the generator:
η (2) =
X
ηkl (B) : (bk + b†k )(bl − b†l ) :
(6.10)
k,l
with
λk λl ∆ωl
ηkl =
2(ωk2 − ωl2 )
ωk − ∆ ωl − ∆
+
ωk + ∆ ωl + ∆
!
(6.11)
We formulate the full generator that makes use of the ground state approximation:
η = iσy
X
k
ηk (bk + b†k ) + σz
(y)
X
k
ηk (bk − b†k ) +
(z)
X
k,l
ηkl (B) : (bk + b†k )(bl − b†l ) : (6.12)
92
The spin-boson model I: preparations
This generator will be used in every remaining application if no other generator is
specified explicitely. In some cases, we will use explicetely the restricted generator
(6.4) where η (2) ≡ 0. If we now employ the generator (6.13) and make use of the
ground state approximation, the flow equations for the ansatz
H(B) = −
X
∆(B)
1 X
σx + σz
λk (B)(bk + b†k ) +
ωk b†k bk
2
2
k
(6.13)
k
then remain closed up to truncated normal ordered operators in O(λ3k ). These improved
flow equations then read:
!
X λ2p ωp
dλk
ωk − ∆ ωp − ∆
2
(6.14)
= −(ωk − ∆) λk + ∆λk
+
dB
ωk2 − ωp2 ωk + ∆ ωp + ∆
p
and the flow equation for ∆(B) from (6.7) remains formally unchanged, however the
flow of ∆(B) itself is slightly modified due to its implicit dependence on the modified
flow of the couplings λk (B). Like in the flow equation equation (6.14), the term (6.2) in
the generator (6.12) will modify flow equations for the observables H, σx , σy , σz only
via double sums over ηkl -like contributions. One can give the general rule of thumb
that for small dampings α < 0.2, contributions from these ηkl -like terms to the flow
of the coupling constants are small except near resonant couplings where ωk ≈ ∆∞ . In
several remaining applications of the generator (6.12), we will refer to this property.
On the validity of the ground state approximation
It should be emphasized that the ground state approximation can -if it does so- only
affect calculations of physical observables on the resonant time scale t ≈ ∆2π∞ . On other
time scales, it leads to very accurate results and a discussion of its validity is not necessary. Since we will choose different initial states for the normal ordering procedure,
most accurately the validity of the ground state approximation should be discussed case
by case. In the ground state in diagonal basis, the property h∗σx ∗iGS = 0 is exactly
fulfilled. However, in time-dependent problems with non-equilibrium initial states ∗σx ∗
cannot be fully neglected since h∗σx ∗ (t)i 6= 0 then. A quantitative discussion of the
possible errors can be based on the following idea:
We want to evaluate a time-dependent observable O(t) in diagonal basis.PWe can
assume that this observable is in diagonal basis of the form O(B = ∞) = k,l Ok σl
where Ok are operators solely composed of bosonic operators and σl might be either
Pauli matrices or the identy operator in the spin Hilbert space. For the purpose of
evaluating this observable O(B = ∞) in a given state | ψ̃i, a formal representation of
the expectation value can be written as
hψ̃ | eiH(∞)t ei∗σx ∗tU O(B = ∞)e−i∗σx ∗tU e−iH(∞)t | ψ̃i
(6.15)
6.3 Transformation of observables
93
where | ψ̃i is a non-equilibrium initial state given in diagonal basis, H(∞) = − ∆2∞ σx +
P
†
5 the truncated normal ordered part of the diagonalized Hamilk ωk bk bk and ∗σx ∗ U
6
tonian (6.7). As argued above, the component ei∗σx ∗tU of the full propagator can only
have significant influences on the resonant time scale t ≈ ∆2π∞ . By virtue of the BakerP
1
Haussdorf identity ei∗σx ∗tU O(B = ∞)e−i∗σx ∗tU = ∞
n=0 n! [i ∗ σx ∗ tU , O(B = ∞)]n the
undesired contribution of the operator U to (6.15) is readily rewritten as
2π
hψ̃ | eiH(∞) ∆∞
X
2π
µkl ∗ σl ∗ πk (U, Ok )e−iH(∞) ∆∞ | ψ̃i
(6.16)
kl
where πk (U, Ok ) is a permutation of a product of k operators U and the operator Ok ,
∗σl ∗ are the now normal ordered -with respect to the ground state in diagonal basisversions of the above operators σl and µkl are some C-numbers. In addition, the time
t was fixed to t = ∆2π∞ in (6.16) 7 .
The remaining question is to give a criterion under which conditions the expression
(6.16) will lead to a negligible contribution to the expectation value hO(t)i.
2π
To the extinct that the state eiH(∞) ∆∞ | ψ̃i has already reached a steady state similar to
˜ =| +ix ⊗ | 0i, the expectation values hψ̃ | ∗σi ∗ (t) | ψ̃i
the diagonal ground state | GSi
of the operators ∗σl ∗ are negligible. In this context, it must be put as a warning that
this criterion has at least to be carefully examined for every non-equilibrium initial
state that is considered.
6.3
Transformation of observables
In this section, we want to transform all three components σi , i = x, y, z of the spin- 21
operator separately into the diagonal basis with the above introduced generator (6.12)
that uses the ground state approximation. If we wanted to restrict us to the generator (6.4), we could formally set ηkl ≡ 0 in the resultant flow equations. For the
most important component σz that describes the population of the underlying two
state system, several results have already been obtained [15,26]. E.g. in [15] also the
generator (6.12) has been applied to transform σz and we can to stick to some results derived in this reference. It was shown that a transformation neglecting normal
ordered terms in quadratic order of the couplings in the ansatz for σz is already sufficient to describe
the full line shape of the symmetrized equilibrium correlation function
R
1 ∞ −iωt
h{σz (t), σz }+ idt. Remarkably the spin operator σz can seemingly
Szz (ω) = 2 −∞ e
be transformed with an accuracy of one order less in the expansion parameter than we
used for the diagonalization of the Hamiltonian.
P
def
We define U = − 12 kl ωkl (B = ∞) : (bk + b†k )(bl + b†l ) : as the normal ordered bosonic part of the
truncated operator in the diagonalized Hamiltonian (6.7).
6
The full propagatorPin diagonal basis can be decomposed into two parts like in (6.15) since the
components − ∆2∞ σx + k ωk b†k bk and ∗σx ∗ U of the full Hamiltonian (6.6) in diagonal basis commute
with each other.
7
Note that the operators ∗σl ∗ eg obey ∗σy ∗ = σy , ∗σz ∗ = σz and ∗1∗ ≡ 0.
5
94
The spin-boson model I: preparations
In addition we will discuss flow equations for the two operators σx and σy , called coherence operators. Motivated by the successful transformation of σz , these flow equations
are also derived in O(λ2k ). Note that in the up to now published literature no attempts
were made to transform these operators by means of flow equations. Therefore we
will discuss these transformations in more detail. Here it is worth to note a principle
we pursued in any case of transformations of spin operators. Before we apply any
transformation to physical problems, we try to derive transformations for all three spin
operators. Not in every case our demand of completeness leads to applications that
are of considerable interest to experts working on the topic of dissipative quantum systems. We justify this by the very methodological character of this thesis that is not
only aiming on physical results, but also has an important focus on technical aspects.
In order to check properties of the transformed coherence operators, the ground state
expectation value of σx is calculated in dependence of damping strength in subsection
6.5.2 and in section 7.1 also the symmetrized equilibrium correlation functions Syy (t)
and Sxx (t) 8 will be calculated. Important properties of these quantities will be checked
against results from the literature.
6.3.1
Transformation of spin operators in O(λ2k )
In order to motivate the transformation of all spin operators, the example of the population operator σz is discussed. However, the same procedure can be applied to transform
the remaining operators σx and σy .
The transformed spin operator σz can be formally represented by U † σz U , where U is
the unitary transformation induced by the generator η. For an exact transformation of
the spin operator in its representation σz one cannot construct a representation of U
but has to start from the exact equation
d
σz (B) = [η(B), σz (B)]
dB
(6.17)
Of course η(B) does not commute with the spin operators and generates instead complicated additional contributions to the transformed spin operators. Therefore, the
transformed observables will become complicated and highly nonlinear functions of the
bath operators in the limit of accomplished flow. Solving equation (6.17) is therefore
of comparable difficulty to a solution of the corresponding Heisenberg equation for the
operator σz . The starting point to obtain an approximate solution is to consider the
easiest possible operator expansion for σz that can be used for an ansatz. The generator η(B) generates only coupling terms linear in bosons if it is commuted with any
Pauli matrix. In this way, an ansatz in linear order of the couplings λk can be read
8
These correlation functions are defined in analogy to the equilibrium correlation function Szz (t) =
.
1
h{σz (t), σz }+ iGS
2
6.3 Transformation of observables
95
P (y)
off from the commutator [η, σz ] = −2σx k ηk (bk + b†k ). If then quadratic terms in
bosons generated by the operation [η(B), [η(B), σz ]] 9 are normal ordered and finally
truncated, the flow equations for the linear ansatz are closed in O(λ2k ) and can be
derived by comparing coefficients in equation (6.17). Indeed, this kind of ansatz has
been used in references [15,26] to transform σz into the diagonal basis. We adapt the
notation that was used in these references and use the ansatz
σz (B) = h(z) (B)σz + σx
X
χk (B)(bk + b†k )
(z)
(6.18)
k
In order to understand which operators should be truncated and also how normal
ordering influences this truncation, we write down the commutator [η(B), σz (B)]:
[η(B), σz (B)] = −2σx
X
ηk h(B)(bk + b†k )
(y)
k
+σz
X
+iσy
X
(y) (z)
ηk χk0
: [bk + b†k , bk0 + b†k0 ]+ :
kk0
ηk χk0 : [bk − b†k , bk0 + b†k0 ]+ :
(z) (z)
kk0
+σz
X
+iσy
X
ηk χk0 h[bk + b†k , bk0 + b†k0 ]+ i
(y) (z)
kk0
ηk χk0 h[bk − b†k , bk0 + b†k0 ]+ i
(z) (z)
kk0
+2σx
X
ηk,l χl [bk + b†k ]
(z)
k,l
(6.19)
Normal ordering of truncated operators
The two normal ordered terms in (6.19) are truncated and instead of bosonic operators,
only the expectation values 10 with respect to the desired density matrix or initial
state are kept. For the convenient case of normal ordering with respect to the finite
−βH
def
temperature free density matrix ρ = Tere−βH0 0 11 these contractions resemble the welldef
known Bose factor n(k) =
1
exp(βωk )−1 :
hbk b†k0 i = δkk0 n(k)
(6.20)
In the limit of zero temperature, one recovers the common normal ordering with respect
to the bosonic vacuum.
9
This commutator contains -besides from coefficients- essentially the same operators than the commutator [η(B), σz (B)] if the ansatz σz (B) is truncated after second order in the couplings.
10
These are denoted in (6.19) by the symbol h i.
P
11
H0 is the spin-boson Hamiltonian without coupling: H0 = − ∆20 σx + k ωk b†k bk
96
The spin-boson model I: preparations
T→0
hbk b†k0 i = δkk0
(6.21)
It becomes now obvious that the initial preparation influences also transformations in
operator space, eg the influence of temperature is contained fully in contractions that
result from the truncation scheme.
Introduction of generalized contractions
During the various applications of the flow equations for spin operators, different normal
ordering procedures will be used and it is useful to introduce generalized contractions.
In all applications the contribution of the term in the 2nd line from below in (6.19)
will always vanish, note that otherwise the ansatz (6.18) for σz (B) would have to be
modified to include also this term. As it will turn out in various applications in chapter
7, the contractions (6.22) will always be of the general form
hbk b†k0 i = δkk0 n(k) + Ckk0
(6.22)
where Ckk0 are elements of a symmetric matrix. The additional matrix elements Ckk0
will be non-zero only in non-equilibrium situations and therefore contain important information about the influences of a non-equilibrium preparation. Note that operators
in linear order of bosonic operators bk and b†k are not necessarily normal ordered if one
deviates from the usual normal ordering with respect to equilibrium initial states, eg in
the ground state of a biased spin-boson Hamiltonian (cf. section (7.3)) bath modes will
be shifted out of their equilibrium position due to the applied bias. One often uses a
completely normal ordered ansatz for observables in order to distinguish between oneparticle and two-particle properties of the Hamiltonian or an observable (cf. section
3.2), but it would be necessary to introduce a more complicated ansatz. Anyway, the
linear bosonic terms will be evaluated exactly and it leads to no difference in the result
of our calculations.
We will denote all flow equations that contain the generalized contractions (6.22) generalized flow equations.
Generalized flow equations for the spin operators
The above choice of a truncation scheme together with the normal ordering prescription
(6.22) leads to the generalized flow equations for σz .
6.3 Transformation of observables
97
dh(z) X (y) (z)
=
ηk χk0 (2δkk0 coth(βωk /2) + 8Ckk0 )
dB
0
k,k
(z)
X
dχk
(y)
(z)
= −2ηk h(z) (B) − 2
ηkl χl
dB
l
(6.23)
Motivated by the transformation of σz , we chose also in the case of the observables
σx and σy ansätze that are motivated by the commutators [η, σx ] and [η, σy ]. These
ansätze read
σy (B) = h(y) (B)σy + iσx
X
χk (B)(bk − b†k )
(y)
(6.24)
k
and
σx (B) = h(x) (B)σx + σz
X
χk (bk + b†k ) + iσy
(x)
X
µk (bk − b†k ) + α(B)
(6.25)
k
k
It is worth to mention the constant α(B) appearing in the ansatz (6.25) for σx (B). As
we will see below, its origin is the normal ordering procedure for truncated operators.
Finally, the commutators [η, σy (B)] and [η, σz (B)] yield the flow equations for the
respective observables and are helpful in order to identify truncated operators and the
approximations involved with that.
[η(B), σy (B)] = −2iσx h(y)
−σy
X
(z) (y)
ηk χk0
X
: [bk −
ηk (bk − b†k )
(z)
k
b†k , bk0
− b†k0 ]+ :
kk0
−σy
+iσz
X
X
(z) (y)
ηk χk0 2δkk0 coth(βωk /2)
kk0
(y) (y)
ηk χk0
: [bk + b†k , bk0 − b†k0 ]+ :
kk0
−2iσx
X
ηkl χk (bl − b†l )
(y)
kl
(6.26)
Note that [bk +b†k , bk0 −b†k0 ]+ will in all treated initial states have a vanishing expectation
value since positive and negative contributions cancel each other 12 . In (6.26), the two
normal ordered terms have been truncated in the ansatz for σy (B). The commutator
[η(B), σx (B)] reads:
12
This aspect becomes clear when these initial states are treated.
98
The spin-boson model I: preparations
[η(B), σx (B)] = 2h(x) (B)σz
X
ηk (bk + b†k )
(y)
k
+2σz
X
ηkl χl (bk + b†k )
(x)
kl
+2ih
(x)
(B)σy
X
ηk (bk − b†k )
(z)
k
−2iσy
+2
X
X
ηkl µk (bl − b†l )
kl
(y)
ηk µk +
2
X
k
−σx
X
(x) (y)
χk ηk0
: [(bk +
(x) (z)
χk ηk
k
b†k ), (bk0
+ b†k0 )]+ :
kk0
+σx
X
ηk µk0 : [bk − b†k , bk0 − b†k0 ]+ :
(z)
k,k0
−σx
X
kk0
1
(x) (y)
χk0 ηk (2δkk0 coth( βωk ) + 8Ckk0 )
2
X (z)
1
−σx
ηk µk0 2δkk0 coth( βωk )
2
0
kk
(6.27)
In (6.27), the two normal ordered terms have been truncated in the ansatz for σx (B).
Before we discuss the resultant flow equations for the spin operators, it is worth to
discuss again the peculiar case of a C-number appearing in the ansatz (6.25) for σx (B).
One should keep in mind that this C-number is completely a consequence of the normal
ordering prescription, e.g. if we had not normal ordered the truncated terms in (6.27),
the constant would not appear at all in the flow equations. Also, for a different kind
of normal ordering, it could be of a different number value. Since we chose normal
ordering with respect to an initial state | ψi or a density matrix ρ, one can imagine
that parts of the expectation value hψ | σx | ψi 13 get transformed into this C-number
by application of the flow equation transformation. 14 In this example it becomes
obvious once more that normal ordering has a strong influence on transformations of
operators. In any case, the flow equations for σx (B) determined by our truncation
scheme can be read off by comparing coefficients in (6.27).
13
Or for the density matrix ρ, the expectation value T r(ρσx )
In subsection 6.5.2 it will be shown that α(B) indeed has a significant magnitude of order the
ground state expectation value hσx iGS when normal ordering with respect to the ground state is used.
14
6.3 Transformation of observables
99
X (x) (y)
X
dh(x)
(z)
=−
χk0 ηk (2δkk0 coth(βωk /2) + 8Ckk0 ) −
2δkk0 ηk µk0 coth(βωk /2)
dB
0
0
kk
kk
(x)
dχk
dB
(y)
= 2h(x) (B)ηk + 2
X
(x)
ηkl χl
l
X
dµk
(z)
= 2h(x) (B)ηk + 2
ηlk µl
dB
l
X (y)
X (z) (x)
dα
=2
ηk µk + 2
ηk χk
dB
k
k
(6.28)
In the same way, one obtains all generalized flow equations for σy (B).
X
dh(y)
βωk
(y)
(z)
=−
2δkk0 χk0 (B)ηk (B) coth(
)
dB
2
0
kk
(y)
dχk
dB
(z)
= −2ηk (B)h(y) (B) − 2
X
(y)
ηlk χl
l
(6.29)
Discussion of the flow equations
In general, all flow equations derived above are coupled differential equations. Although
we already had to make approximations in order to derive flow equations for the spin
operators, the set of coupled flow equations itself is to complicated to allow for an
analytical solution. Two possibilities remain:
1. For a numerical solution, one needs to implement these systems of differential
equations on a computer. In practice, a versatile and fast algorithm is an adaptive
step-size 4th order Runge-Kutta algorithm as can be found in many compilations
of numerical algorithms (see, e.g. [55]).
2. In [14], several approximative analytical treatments of such systems of differential
equations are proposed, in principle based on simplifying parametrizations of the
coupling constants in the Hamiltonian.
We start with a discussion of an analytical treatment in order to get first qualitative
insights into the effective transformation of the spin operators with the above flow
equations. In chapter 7, we will in addition discuss also numerical solutions.
100
The spin-boson model I: preparations
The most controllable part of the flow equations are the coupling parameters λk of
the Hamiltonian since the choice of a generator is usually designed to optimize their
decay. Equation (6.14) suggests that the non-resonant couplings (ωk 6= ∆0 ) show
an exponentional decay for sufficiently large flow parameter B. Below, an analytical
parametrization of these couplings is used in order to approximate their flow. Then,
this approximate flow of the couplings is inserted into all flow equations of the spin
operators in order to obtain an approximate analytical solution of the corresponding
couplings in the limit of accomplished flow.
6.3.2
Approximate analytical solution of the flow equations
The flow equation for the couplings λk is solvable exactly if we neglect the additional
contribution from the η (2) -term in the generator (6.12) to eq. (6.14). We then have:
λk (B) = λk (0)e−
RB
0
(ωk −∆(B))2
(6.30)
A closer look on the flow equations (6.11) and (6.14) shows that we can neglect contributions from the η (2) -term if ωk is away from the resonant frequency ∆∞ . We
can then consider the solution (6.30) as a good approximation for the non-resonant
couplings λk (B). If we plug this approximate solution into the other flow equations
(6.23),(6.28),(6.29) under additional assumptions, we can finally derive approximate expressions for the coupling constants of the transformed spin operators. As a first step,
the flow equations (6.23),(6.28),(6.29) are formally integrated and (6.30) is inserted:
(y)
χk (B)
(x)
ωk − ∆(B)
2
λk (0)e−B(ωk −∆(B))
ωk + ∆(B)
∆(B)h(x) (B)
ωk − ∆(B)
2
λk (0)e−B(ωk −∆(B))
ωk + ∆(B)
=
0
Z
χk (B) = −
B
h(y) (B)ωk
Z
B
0
Z
µk (B) = −
B
ωk h(x) (B)
0
(z)
χk (B)
Z
=
B
ωk
0
ωk − ∆(B)
2
λk (0)e−B(ωk −∆(B))
ωk + ∆(B)
ωk − ∆(B) 2
2
λk (0)e−B(ωk −∆(B))
ωk + ∆(B)
(6.31)
The exponential decay of the integrands in (6.31) yields contributions to the integrals
1
approximatively only up to B = O( (ωk −∆)
2 ) and suggests to neglect the B-dependence
of the functions h(x) (B)),h(y) (B), h(z) (B) and ∆(B) in the integrand since these functions decay much slower 15 Therefore, we set h(x) (B), h(y) (B), h(z) (B) ≡ 1 respectively
def
and approximate ∆(B) with the constant value ∆r = ∆(B = (ωk − ∆0 )−2 ). In the
limit of accomplished flow, this yields the following approximative results:
15
cf. [15] for a discussion of this aspect
6.3 Transformation of observables
101
(y)
ωk λk (0)
ωk2 − ∆2r
(6.32)
(z)
∆r λk (0)
ωk2 − ∆2r
(6.33)
χk (∞) =
χk (∞) =
(x)
χk (∞) = −
µk (∞) = −
∆r λk (0)
ωk2 − ∆2r
ωk λk (0)
ωk2 − ∆2r
(6.34)
(6.35)
Obviously, the approximated coupling constants (6.32)-(6.35) diverge in the resonant
frequency ωk = ∆r since we could not justify the approximations in this frequency
range. An exact solution for the coupling constants smears out this divergency and
remains finite in the resonant frequency. Our approximate solution is especially useful
to describe the low-frequency behaviour of the concerning observables and thus, their
behaviour for large times t ∆−1
0 . As discussed in [15], the approximated coupling
(z)
constants χk (∞) yield the correct long-time behaviour of the zero temperature equilibrium correlation function Szz (t). In addition, the approximated coupling constants
(6.32) will be used furtheron to discuss the long-time behaviour of the equilibrium correlation function Syy (t).
Asymptotic behaviour of the operator flow for B → ∞
In the above derived expressions for the asymptotic values of the coupling constants, no
approximate solution for the constants h(x) (∞)),h(y) (∞) and h(z) (∞) has been given.
In a sense, the flow of these coefficients describes the decay of the initial observable
σi (B = 0), i = x, y, z into a complicated superposition of bath operators. With increasing flow parameter B, the magnitude of all coupling constants increases and one
expects to observe a decrease in magnitude of the coupling constants h(x) (B),h(y) (B)
and h(z) (B). The intuitive imagination that has been formed by the flow equations for
the DHO (cf. chapter 4) is that the thermodynamic largeness of the bath leeds finally
to a complete decay of the initial observable into operators that contain bosonic bath
operators. As already mentioned, this result is generic and also holds for the observables σy and σz . In appendix A.1, we will give a prove for the asymptotic behaviour
limB→∞ h(y) (B) = 0 and limB→∞ h(z) (B) = 0.
Up to now, only transformations of operators were discussed. In order to treat problems
of physical interest, these operators have to be supplemented with initial states.
102
6.4
The spin-boson model I: preparations
Preparation of the initial state
Like in the case of the harmonic oscillator, we want to treat non-equilibrium initial
states in order to describe non-equilibrium relaxation processes. Again, the choice of
initial states is motivated by reference [16]. 16 All applications of the states described
below will only consider the case of zero temperature (T=0). Before these states are
described in more detail, a brief overview is given that describes the two sorts of initial
states that will be considered:
1. A first type are initial states where the spin state is fixed into the upper eigenstate
of σz . Two modifications of this preparation will be given:
• The bath is coupled to the fixed spin for a long time and comes into equilibrium with the spin before the constraint on the spin is released.
• The bath is uncoupled to the spin and left in its thermal equilibrium. After
the bath has been coupled to the spin, the spin state will decay.
2. In a second class of states, the spin is only partially polarized by application of a
finite magnetic field coupling to the spin in presence of the system-bath coupling.
Again, after the bath has come into equilibrium with the spin, the external field
is switched off.
Among these initial states are factorizing states as well as entangled states. For technical and physical reasons, it is of importance to discuss the role of entanglement in the
context of initial states.
The role of entanglement
Very similar to the treatment of the DHO, initial states mentioned under 1) cannot
be representated in the diagonal basis, while the partially polarized states described in
2) can, as will be shown in section 7.3. As we will see, the latter states are entangled
in the physical basis but not in the diagonal basis and in case of the former states,
this property is fulfilled vice versa. Of importance is also that in the case of the DHO
entangled and factorized initial states where shown to have a very different long-time
behaviour and also very contrasting properties in context of the FDT (cf. chapter 5).
Therefore, we will furtheron classify initial states into the categories of entangled and
product initial states. Another reason that provokes these categories is that in the
considered initial states, a representation of the initial state can be given only in the
basis where the state is not entangled (cf. sections 6.4.1 and 6.4.2). In consequence, the
time evolution of operators has to be calculated either in the diagonal or the physical
basis of the operator space, depending on the basis where the initial state is given.
Important technical aspects arise in the context of the basis dependent time-evolution
of operators. We will discuss them in more detail in section (7.1).
16
Also in chapter 4, this reference has been motivation for the choice of initial states.
6.4 Preparation of the initial state
6.4.1
103
Product initial states
In a product initial state, the full initial density matrix factorizes according to W =
ρ0 WB where ρ0 is the density matrix of the spin and WB is the density matrix of the
bath. At zero temperature, such a state is typically a product of two pure states. In
many cases of applications, it is of interest to prepare the two-level system as an eigenstate of σz . Eg this corresponds to a state localized in one well in case of a double-well
problem. In electron transfer reactions, the electron would be in a definite donor or
acceptor state. Also in the important context of quantum computing, the two-state
system will be in such a pure state after the readout of a qubit.
However, a preparation of the environment is more difficult and mostly, two equilibrium
situations are of interest:
1. Either the pure spin state is prepared suddenly and so the bath stays in the
thermal equilibrium situation described by a Bose-Einstein distribution.
2. If the spin has been polarized at a large negative preparation time t0 , the bath will
have come into equilibrium with it in the limit t0 → −∞ if the bath is an ergodic
system. Then, the bath modes will be shifted out of their thermal equilibrium
distribution due to the action of the system-bath coupling.
Furtheron, the latter situation will be referred to as preparation class B and the first one
as preparation class A. The corresponding initial states are denoted by | IAi and | IBi
It is important to note that preparation class A and B yield essentially the same dynamics after a transient time of order ωc−1 since their influence functionals can be shown
to coincide in the limit ωc → ∞. In appendix B, this equivalence will be rigorously
quantified. Inspite of these very similar dynamical properties, it is of interest to treat
both classes A and B in order to provide a test of accuracy of the flow equation method.
Coupled bath
If the bath modes are intended to come into equilibrium with the polarized spin state,
the spin could be fixed by a strong negative magnetic field −hΘ(−t) with h ∆0
for all negative times t < 0. In order to prepare a non-equilibrium situation in t=0,
the magnetic field is switched off and the spin is set free to relax. An application of
such preparations is discussed for the more general preparation procedure presented
in subsection 6.4.2 . A representation of the resultant prepared initial state can be
easily obtained after writing down the Hamiltonian for the prepared system. Since
the system-bath complex is prepared in a product initial state with the spin being in
the upper eigenstate of σz , the spectrum and the eigenstates of the Hamiltonian are
retained if we set σx ≡ 0 and σz ≡ 1 and restrict the action of both operators in the
spin Hilbert space onto the upper eigenstate of σz , leading to the Hamiltonian
104
The spin-boson model I: preparations
H=
X1
k
2
λk (bk + b†k ) +
X
ωk b†k bk
(6.36)
k
Essentially this is a Hamiltonian of bosonic bath modes that are each shifted by the
displacement sk = 12 ωλkk . It is useful to introduce a basis of dressed modes in which each
bath oscillator is shifted by the amount −sk in order to diagonalize the Hamiltonian
(6.36).
The appropriate unitary operator U which diagonalizes 17 the Hamiltonian (6.36) has
also been used -with slightly modified parameters- to diagonalize the field dependent
DHO in subsection 5.1.1.
def
U = exp(−iΩ),
def
Ω =
X
k
sk
X 1 λk †
pk
=i
(
)(b − bk )
~
2 ωk k
(6.37)
k
Note that we also could use the well-known polaron transformation
X λα
1
(b†α − bα )
UP = exp(− iσz Ω), Ω = i
2
ωα
α
[9] to diagonalize the Hamiltonian (6.36), but the additional action of UP in the spin
Hilbert space leads to unnecessary notational complications in the formulation of the
initial spin state. Since the action of U diagonalizes the Hamiltonian H, one can conclude that the initial state | IBi can be written as
| IBi =| +iz ⊗ U | 0i
(6.38)
Uncoupled bath
In contrary to a coupled bath, a bath that is decoupled from the spin while the spin
state is polarized is still in its vaccum state if the temperature is set equal 0. After this
preparation, the initial state will be obviously
| IAi =| +iz ⊗ | 0i
(6.39)
This kind of initial state might be relevant in electron transfer reactions where a particular electron donor state is suddenly prepared by photoinjection. Possible observations
of a subsequent coherent electron transfer are discussed in [46]. Electron transfer in
biomolecules is an important modern application of the spin-boson model. As another
example, in quantum dots it is possible to drive a small quantum system from a coupled
to an uncoupled regime and eg to read out a single spin state [2].
Finally, it is important to stress that the considered product inital states | IAi and
| IBi are only known in state space. In time-dependent problems, it is therefore not
17
The diagonalized Hamiltonian can be formally written as U † HU .
6.4 Preparation of the initial state
105
sufficient to solve the Heisenberg equations for spin operators just in the diagonal basis
since one cannot evaluate this solution with respect to the given initial state. One
possibility would be to find a representation of the initial state in the diagonal basis.
In previous work, this has been the only approach to time-dependent problems in the
flow equation framework. In order to treat product initial states, it will be discussed in
section 7.1 how flow equations can also be used to obtain time-dependent spin operators in the physical basis representation. The technical approach will use the concept of
time-dependent flow equations that has been applied discussed for the DHO in chapter
4.
6.4.2
Non-factorizing initial state
We now consider the complete analogy of the field-dependent preparation of the DHO
from subsection 5.1.1. By introducing a field coupling term into the Hamiltonian (6.2),
a sudden perturbation can be realized if the field is switched of instantanously in t=0.
Such sudden perturbations are especially of interest in form of gate-voltage pulses applied to nano devices. Recently, Elzerman et al. [2] reported the usage of gate-voltage
pulses for a single-shot read out of the spin configuration in a single-electron transistor
in a finite magnetic field. Such devices may be suitable for quantum computing, as
they combine a long-lived quantum state with an easy read-out protocol furnished by
the coupling to the environment.
The field is assumed to be switched on at a large enough negative time so that the
system-bath complex has come into equilibrium with it in t=0. In consequence, a nonfactorizing initial state -that we denote by | IIi- is prepared for finite field strengths 18 .
In the limit of infinite field strength the initial state | IAi -where the spin is completely
polarized- is recovered that represents in contrast a factorizing initial state.
The full time-dependent Hamiltonian for this preparation procedure reads:
H(t) = −σx
X
∆0 1 X
+ σz
λk (bk + b†k ) +
ωk b†k bk + hΘ(−t)σz + E0
2
2
k
(6.40)
k
Physical observables are evaluated under knowledge of the initial state. Since in the
physical basis, the correct non-equilibrium initial state will be entangled, it is too
complicated to find its explicit representation. However, in the diagonal basis representation, the Hamiltonian (6.40) simplifies appreciably and a representation of the initial
state can be derived in the diagonal basis. For this purpose, the Hamiltonian (6.2) and
the observable σz are separately transformed into the diagonal basis by making use of
the generator (6.12). Further technical details of these transformations were discussed
in the section (6.2) and (6.3). The transformed Hamiltonian then reads:
18
For an argumentation why we assume this initial state as non-factorizing, cf. subsection 5.1.1.
106
H(B = ∞) = −
The spin-boson model I: preparations
X
X
∆∞
σx +
ωk b†k bk + h0 Θ(t)
σx χk (∞)(bk + b†k ) + E(∞) (6.41)
2
k
k
We will extract the non-equilibrium initial state | IIi in its diagonal diagonal basis
representation as the ground state of this Hamiltonian.
Discussion of the transformed Hamiltonian (6.40)
To be more precise, in (6.40) just the sum of the two separatly transformed operators
(6.2) (the equilibrium Hamiltonian) and σz is written down. Therefore the accuracy of
this transformation reduces to the accuracy of the transformation of σz where already
operators in O(λ2k ) have been truncated. In order to determine the correct groundstate of the field-dependent Hamiltonian in diagonal basis, this transformation can
be considered as very good. As mentioned before, even the full time-dependence of
the equilibrium correlation function Szz (t) = 12 h{σz (t), σz }+ i was calculated with this
transformation in ref. [15], a result that would not be reliable if the action of the
approximated σz (∞) onto the equilibrium ground state would yield significant errors.
In near-to-equilibrium situations of small field strengths h ∆0 , where the system
remains near the equilibrium ground state, the quality of the transformation of σz
does not change appreciable. Most important, truncated operators depend on the field
strength via normal ordering. Thus, the range of field strengths h ∆0 will not be left
lateron. Under neglection of normal ordered terms in O(λ2k ) the Hamiltonian (6.41)
can be diagonalized by a unitary transformation:
U † H∞ U = −
X
∆∞
σx +
ωk b†k bk + E(∞)
2
(6.42)
k
with
def
1
U = e 2 σx
P
k
µk (b†k −bk )
χk
(6.43)
ωk
Therefore, the non-equilibrium initial state in the diagonal basis representation reads
finally:
def
µk = −2h0
˜ = U | +ix ⊗ | 0i
| IIi
(6.44)
In (6.42), we omitted an unimportant shift of the energy constant E∞ . An important
difference to the product initial states | IAi and | IBi should be mentioned. Since the
state (6.44) is given in the diagonal basis, it is sufficient to solve the Heisenberg equation
for the spin operators in the diagonal basis. As mentioned above, for the product initial
states | IAi and | IBi, a transformation of this solution of the Heisenberg into the
physical basis via time-dependent flow equations is necessary.
6.4 Preparation of the initial state
6.4.3
107
Normal ordering in the context of initial states
After the discussion and definition of different non-equilibrium states, their role in the
context of normal ordering will be discussed. In order to define a normal ordering procedure with respect to an initial state | ψi, it is sufficient to write down the contractions
hψ | bk b†k0 | ψi
(6.45)
Note that in the flow equations for spin operators (cf. section 6.3), generalized contractions have been used. For a proper usage, the explicit contractions of the concerning
non-equilibrium states have to be used. Therefore, these contractions are presented in
the following.
Explicit form of non-equilibrium contractions
For a bath in zero temperature equilibrium the corresponding normal ordering procedure amounts to the trivial procedure ”creators to the left of annihilators”, since
hψ | bk b†k0 | ψi = δkk0 . This kind of normal ordering has been applied to diagonalize the
spin-boson Hamiltonian in section (6.2) and also in all publications where equilibrium
spin-spin correlation functions were calculated [15,26]. Without any change, this kind
of normal ordering can be used for the product initial state | IAi =| +iz ⊗ | 0i where
the bath is in the vaccum state. Modifications to this kind of normal ordering occur
if the correct contractions for the initial state | IBi shall be used. In contrast to the
simple case of a bath in equilibrium the choice of normal ordering is not as trivial since
the bath modes aquire a non-zero expectation value hbk i =
6 0 with respect to this state.
Nevertheless, it is straightforward to write down the correct contractions if we want to
normal order bath operators with respect to the initial state | IBi. The contractions
now read
hψ | bk b†k0 | ψi = δkk0 + Ckk0 = δkk0 +
1 λ k λk 0
4 ωk ωk 0
(6.46)
where we used the definition (6.22) of generalized contractions. They affect the flow
equations for the spin operators as a sum over k and k 0 (cf. (6.23),(6.28),(6.29)). Note
that in (6.46), the couplings λk assume their initial value λ(B = 0) at unperformed
flow and do not change the contractions during the flow.
˜ (cf. (6.44))
Likewise, the correct contractions for the field dependent initial state | IIi
read very similar
(z)
hψ | bk b†k0 | ψi = δkk0 + Ckk0 = δkk0 + h2
(z)
(z)
1 χk (∞)χk0 (∞)
4
ωk ωk 0
(6.47)
where the couplings χk (∞) have the constant value they reach in the limit of accomplished flow (B → ∞). Moreover, the contributions Ckk0 in (6.47) are dependent on the
108
The spin-boson model I: preparations
field strength h in O(h2 ). Note that the contractions (6.47) are defined with respect to
a state given in the diagonal basis, whereas the contractions (6.46) have been defined
with respect to a state given in physical basis. A more accurate normal ordering prescription would always use the correct B-dependent state | ψ(B)i that corresponds to
the basis representation defined by the flow parameter B for normal ordering. We are
not aware of any work that applied the flow equation method with such an improved
normal ordering prescription. In practice, it is usually not necessary to use such a
normal ordering prescription [27].
Normal ordering improves the accuracy of transformations, but it neither makes them
exact nor gives an answer about quantitative errors caused by approximations. Of
course one should be aware of such errors, most obviously because flow equations for
transformed observables rely on a small expansion parameter. Care has to be taken
that such expansions parameters remain sufficiently small, since they might get larger
during the flow. The next section intends to give an answer on the question where in
parameter space our transformations are reliable and where not.
6.5
Discussion of transformations
In previous work, several exact criteria have been used to validate the accuracy of
flow equations for spin operators. Since qualitative criteria will be given in chapter
7 by a comparison of physical results for the spin-boson model with results obtained
within other methods, in the present section such exact criteria will be most important.
Previous results
In [15], the Shiba-relation [57] generalized to the spin-boson model in ref. [56] was
tested. This relation holds in general only in equilibrium situations, since it connects the
static
spin susceptibility χ0 with the equilibrium spin-spin correlation function Szz (ω) =
R ∞ −iωt
1
e
2 h{σz (t), σz }+ idt. The static spin susceptibility χ0 can be extracted with a
−∞
Kramers-Kronig relation and a fluctuation dissipation theorem from the correlation
function Szz (ω):
Z ∞
Szz (ω)
χ0 =
dω
ω
0
For an Ohmic bath J(ω) = 2αω, the Shiba relation reads:
Szz (ω)
= 2α(2χ0 )2
ω→0
ω
lim
(6.48)
For dampings α ≤ 0.05, the error remained below 10% [15] and increased with damping.
For super-Ohmic baths J(ω) ∝ ω s , s > 1 it is fulfilled more accurately even for larger
dampings, indicating that reported numerical errors in the used computer routine [15]
6.5 Discussion of transformations
109
are responsible for the larger error for Ohmic baths. Since the Shiba-relation is not
valid in general in non-equilibrium, we will not discuss this relation further.
In addition, the sum rule
!
hσz2 (B)iGS = 1
(6.49)
has been checked in [15]. It provides information if contributions of truncated higher
orders in the ansatz of the transformed spin operator σz (B) yield contributions if
the truncated operator σz (B) is evaluated in the ground state. 19 More important, it is a criterion for the evaluation of spin-spin correlation functions since eg
hσz2 (B = ∞)iGS = 2Szz (t = 0). 20 It is easy to show that the sum rule (6.49)
is exactly fulfilled for transformations using the generator (6.12) with the restriction
ηkl ≡ 0. 21 We will make use of the sum rule (6.49) and other exact criteria that have
to be fulfilled by the transformed spin operators.
Tests of transformed spin operators
Three different exact relations will be checked in order to check the accuracy of the
transformed spin operators.
1. The sum rule (6.49) will be discussed for all three spin operators, also in modified
form with respect to states different from the ground state.
!
2. The sum rule hσz2 (B)iGS = 1 represents a check of the second moment of a
spin operator. A more direct check of the truncation scheme is to check the
first moment hσz (B)iGS . This ground state expectation value is evaluated for all
three spin components and compared with NRG data where no exact results are
available.
3. In addition, the SU(2) algebra [σi , σj ] = 2iεijk σk connecting all spin components
by a simple commutation relation is tested. In another sense, a test of sum rules
and the SU(2) algebra is also a necessary criterion and a check for unitary of the
transformation of the spin operators.
19
!
The sum rule hσz2 (B)iGS = 1 evaluates the operator σz2 (B in the ground state of the Hamiltonian
(6.13) in diagonal basis representation [15]. We will consider other states lateron.
20
However, the sum rule (6.49) is a necessary, but not sufficient criterion for the accuracy of the
correlation function Szz (t).
21
Since the contribution of the additional part η (2) in the generator (6.12) is only in O(λ2k ), an
inclusion of η (2) in the generator does not lead to major differences in the sum rule (6.49) for weak
bath coupling strengths α 1.
110
6.5.1
The spin-boson model I: preparations
Test of sum rules
Firstly, we check sum rules for all spin operators. Formally, a unitary transformation
would demand to fulfil the identity σi2 (B)(t) ≡ 1 for any transformed Pauli matrix
σi , i = x, y, z. It would be clearly too ambitious a requirement to check the sum rule
even in operator space since σi2 (B), i = x, y, z is a complicated superposition of bath
operators. Lateron, we want to calculate spin-spin correlation functions by making use
of the transformed spin operators σi (B), i = x, y, z. Since the sum rule shall also be a
criterion for the accuracy of spin-spin correlation functions, it makes sense to evaluate
it with respect to an initial state (IS).
!
2
hσx,y,z
(B)iIS = 1
(6.50)
The sum rule (6.50) involves products of transformed spin operators. Eg, such products
occur in all types of spin correlation functions. Products of transformed operators are
more difficult to handle than a transformation of a single operator. We want to explain
this further.
Products of transformed operators
We assume that the observable O(1) is transformed by means of flow equations according
to an ansatz
O
(1)
(B) =
N
X
(1)
λk Ok
(6.51)
k=0
(1)
with certain operators Ok that change during the flow and an expansion parameter
λ. It is conceptionally possible to truncate this ansatz in a certain order of the expansion parameter λ and neglect all further contributions in a rather controlled way
as we explained in chapter 3. This concept is not as controlled anymore if products of
transformed operators are evaluated.
Eg. in correlation functions frequently products of two transformed operators O(1) and
O(2) have to be evaluated. These operators are each transformed by an ansatz like
(6.51). The product of two ansätze O(1) (B) and O(2) (B) will then loose in general its
property of a systematic expansion in λ. This is best explained by products of operators
with an ansatz that is truncated after linear order in the parameter λ. We rearrange
the product of both ansätze to illustrate the problem.
N
X
k=0
(1)
λk Ok
N
X
k=0
(2)
λk Ok =
N
X
m=0
λm
m
hX
(1)
ON −n On(2)
i
(6.52)
n=0
(1)
In equation (6.52) the O(λ2 ) of the operator product may contain the operators O1
(2)
(1)
and O1 from the linear order of the ansätze. However also the operators O2 and
6.5 Discussion of transformations
111
(2)
O2 can contribute in O(λ2 ) in the operator product (6.52). In a linear ansatz with
truncated operators in O(λ2 ) these operators are not considered since they were truncated from the beginning. This differs from operator products where neglected higher
orders in the expansion parameter λ might yield contributions as well if they mix with
a corresponding lower order in an expansion of the second operator.
In t=0, the fact that truncated normal ordered terms are at least in quadratic order of
couplings λk lowers the possible damage by a backcoupling of these higher order terms
in products of transformed Pauli matrices. In contrast, regarding the sum rule for time
evolved Pauli matrices, it cannot be excluded that a time-dependent expression
2
heiH(B)t σx,y,z
(B)e−iH(B)t iIS
(6.53)
2
will be approximated in lower accuracy than the initial value hσx,y,z
(B)iIS . A short
discussion of time-dependent sum rules is given in section 7.1. In the following, only
the case t=0 will be treated.
Many applications have shown that the flow equation method works accurately with
respect to ground state expectation values and this case represents the lowest technical
challenge to establish a transformation of operators. An underlying reason is found in
the strength of our method to diagonalize Hamiltonians. In diagonal basis, the ground
state has usually the characteristics of a vacuum state in Fock space. This leads to
a suppression of many contributions in an operator expansion -if they are evaluated
in the bosonic vacuum- and also to the most trivial application of normal ordering,
namely with respect to the vacuum. For the remaining application to physical meaningful initial states far from ground state, it makes sense to expect the equilibrium
case as a minimum requirement before one thinks of applications to non-equilibrium
situations. A transformation that leads to errors in the ground state case will very
likely be even more problematic to handle in non-equilibrium states. 22 Therefore, the
discussion of sum rules starts with the sum rule for the equilibrium ground state. The
equilibrium ground state sum rules are also important for the equilibrium correlation
functions Sxx (t) and Syy (t) that will be calculated in section 7.1.
Sum rules in the equilibrium ground state
Formally, the sum rule demands:
!
2
hσx,y,z
(B)iGS = 1
(6.54)
As we already mentioned in the context of normal ordering, the ground state of the
transformed Hamiltonian H(B) would be the correct -but unknown- state to use for
22
We will check this for the state | IAi and leave a check of sum rules for more general non-equilibrium
initial states open for future work.
112
The spin-boson model I: preparations
normal ordering and it would make sense to use it also for the sum rule (6.54). Again,
we restrict us to the pragmatic choice of the ground state | G̃Si =| +ix ⊗ | 0i in
diagonal basis. In this way, the test of (6.54) is at the same time valid for the product
initial state | IAi =| +iz ⊗ | 0i. Evaluating the ansätze σi (B), i = x, y, z from section
6.3 in this way, the sum rules from (6.54) read:
hσx2 (B)iGS = (h(x) (B) + α(B))2 +
hσy2 (B)iGS = h(y)2 (B) +
X
X
(x)
!
(µk (B) − χk (B))2 = 1
k
!
(y)2
χk (B) =
1
k
hσz2 (B)iGS = h(z)2 (B) +
X
(z)2
χk
!
(B) = 1
k
(6.55)
Obviously, differentiating these expressions with respect to B and inserting the flow
equations (6.23),(6.28) and (6.29) is the straightforward way to check these rules. For
this purpose, we set ηkl (B) ≡ 0 (cf. 6.11) in the flow equations (6.23),(6.28) and (6.29).
In this way, the sum rules for σy and σz are easily proved. One can give the general
thumb rule that a usage of the full generator (6.12) with the additional contribution η (2)
has a negligible effect on the sum rules for σy and σz for Ohmic damping with α ≤ 0.2.
The reason is that the operator η (2) modifies the flow equations for spin operators only
in O(λ2k ). Numerical results for the sum rules that consider the full generator (6.12)
are given in table (6.1). They indicate that the sum rules (6.54) are violated by less
than 5% for dampings α ≤ 0.25 if we use the full generator (6.12). It has also been
checked that this accuracy is universal if the cut off ωc is changed. Increasing cut off
leads even to higher accuracy.
!
We have not discussed the sum rule hσx2 (B)iGS = 1 yet. Using the flow equations
(6.28), it becomes obvious that the derivative of hσx2 (B)iGS is nonvanishing:
d
d
hσx2 (B)iGS = 2α(B)
hσx (B)iGS
dB
dB
(6.56)
It is difficult to draw conclusions from (6.56) analytically. Instead, in figure (6.2) numerical results for the flow of hσx2 (B)iGS are depicted. For dampings α ≤ 0.2, the sum
rule is fulfilled within 5% error and the numerical data indicates a reduction of the
error with decreasing damping strength α and increasing cut off frequency ωc .
The sum rules (6.54) ensure accuracy for dynamical correlation functions like Szz (t)
in t=0, as mentioned above. A different demand on transformed spin operators arises
in the long-time limit of the expectation values hσi (t)i that we want to calculate in
6.5 Discussion of transformations
113
Table 6.1: We check the sum rules for σy and σz from (6.54) using the full generator
(6.12). For the numerical implementation we used an Ohmic bath J(ω) = 2αωΘ(ωc −ω)
and a discretization of 2000 bath modes. All numbers give the maximum deviation
from 1 during the flow that is in all cases achieved in the limit B → ∞. Interestingly,
numerical results indicate that the sum rule can be fulfilled in excellent accuracy also
for large dampings α 1. The only case where the sum rule was violated by more
than 3% occured for the operator σz in the region of intermediate damping strengths
α = 0.4 − 0.9. For a fixed number of bath modes, both sum rules are fulfilled better
with increasing cut off frequency ωc .
hσy2 iGS
ωc
10
25
50
100
α
hσz2 iGS
ωc
10
25
50
100
α
0.1
0.5
1
5
10
50
1.003
1.003
1.003
1.002
1.015
1.008
1.005
1.003
1.018
1.010
1.007
1.004
1.020
1.009
1.004
1.002
1.013
1.005
1.002
1.001
1.003
1.001
1.001
1.000
0.1
0.25
0.5
0.75
5
10
1.003
1.002
1.002
1.001
1.028
1.024
1.016
1.004
2.143
1.188
0.995
0.999
1.347
0.965
0.999
0.999
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
chapter 7. One would expect intuitively that the time dependent first momenta of the
spin operators achieve their ground state expectation value for a thermodynamical large
number of bath states. Eg, many previous results show that the magnetization hσz (t)i
relaxates to zero for several non-equilibrium preparations [41]. 23 Thus, if the long time
behaviour of these non-equilibrium expectation values shall be described accurately, it
is important to check these ground state expectation values.
6.5.2
The ground state expectation values hσi iGS
Simple symmetry arguments [9] verify that the exact ground state expectation values
of the spin-boson Hamiltonian (6.2) for σy and σz vanish: hσy i = 0 and hσz i = 0. From
the ansätze σy (B) and σz (B) (cf. 6.18 and 6.24) it becomes obvious that these operators will have a vanishing expectation value with respect to the ground state | +ix ⊗ | 0i
23
However, simple quantum mechanical arguments forbid that a non-equilibrium system achieves its
ground state at any time.
114
The spin-boson model I: preparations
in diagonal basis.
Since σx will have a finite ground state expectation value, symmetry arguments similar
to the cases of σy and σz cannot be employed. Typically, ground state poperties are
well accessible within the method of NRG. We compare our results with calculations
from Costi and Mc Kenzie [59] as a reference (figure 6.3).
Good agreement is observed in the range of small damping strengths α ≤ 0.2 where the
relative error remains below 3%. Beginning from α = 0.2 on the relative error increases
more and more and exceeds 10% for α > 0.5. This behaviour for increasing damping
indicates the breakdown of the perturbative expansion of the ansatz for σx (B). Eg the
flow equation result deviates most clearly at the quantum critical point α = 1− where
∆r → 0 and hσx iGS → 0 in the limit ∆0 /ωc → 0.
6.5.3
Remarks on the SU(2) algebra for transformed spin operators
We already used the well known property σi2 = 1 for transformed Pauli matrices to
derive a test criterion. Another interesting property of Pauli matrices is the SU(2)
algebra that relates all three Pauli matrices:
[σi , σj ] = 2iijk σk
(6.57)
It connects all three Pauli matrices and therefore can be used as a consistency check
for all three transformed spin operators. This yields interesting insights in the quality
of the truncation scheme for the transformed spin operators σi , i = x, y, z.
First of all, it is a check if the truncation scheme for transformed operators preserves
unitary of the transformation or not. A unitary transformation preserves the SU(2)
algebra. In the last subsection it was discussed that sum rules can give insights if neglected higher order terms in a truncation scheme couple back in operator products or
not. Exactly this aspect is also important for a commutator [σi , σj ] that involves such
operator products.
Comparison of operator flow
In the last subsection it became obvious that the transformed operator σx (B) according
!
to ansatz (6.25) does not fulfil the sum rule hσx2 (B)iGS = 1 exactly. Therfore, we stick
to the commutator [σy (B), σz (B)]. This commutator can also serve as a comparison
for the operator flow of σx (B) and possibly help to understand why the sum rule for
σx (B) is broken. A fulfillment of the SU(2) algebra demands for every value of the
parameter B
!
[σy (B), σz (B)] = 2iσx (B)
(6.58)
6.6 Summary
115
We insert the ansätze (6.23),(6.28),(6.28) for the operators σi (B), i = x, y, z into
equation (6.58) and want to compare the flow of both sides of equation (6.58):
X (z)
1
!
[σy (B), σz (B)] = h(y) (B)h(z) (B)σx − σz h(y)
χk [bk + b†k ]
2i
k
X (y)
X (z) (y)
†
(z)
+iσy h
χk [bk − bk ] +
χk χk
k
!
=h
(x)
σx + σz
X
k
(x)
χk (bk
+
b†k )
+ iσy
X
µk (bk −
k
b†k )
+ α(B) = σx (B)
k
(6.59)
Formally the commutator [σy (B), σz (B)] is equivalent to the ansatz for σx (B) since
it contains the same operators (cf. (6.59)). The flow of [σy (B), σz (B)] is governed
by the flow equations (6.29) and (6.23) for σy and σz . It necessarily has to coincide
with the flow of the coupling constants of σx if the SU(2)-algebra shall hold during the
flow. Differentiating both explicit expressions contained in (6.59) and inserting the flow
equations for the resultant derivatives reveals that the SU(2)-algebra is not preserved
during the flow since the derivatives differ. We postpone a more detailed discussion of
the SU(2)-algebra to appendix A.2.
6.6
Summary
In this chapter our aim was to diagonalize the spin-boson Hamiltonian and to transform
all spin operators into the diagonal basis. These technical steps serve as a preparation
for the next chapter where we will calculate the time-dependent spin operators. A
transformation of spin operators is essential in order to formulate the very advantageous Heisenberg equation in diagonal basis where the Hamiltonian is diagonal. A
diagonalization of the Hamiltonian itself is also useful to obtain representations of nonequilibirium initial states in the diagonal basis as we demonstrated in section 6.4.
All performed transformations in this chapter are based on approximations since the
flow equations for the concerning operators were not closed. The major approximation is the ground state approximation (cf. (6.9)) that was performed to diagonalize
the Hamiltonian more accurately. This approximation yields a more accurate timeevolution of operators in time-dependent problems. Motivated by previous work, we
truncated the transformed spin operators already after linear order in the couplings
λk . We showed that such a truncation procedure leads to an accurate description of
ground state expectation values both of the single spin operators and the squared spin
operators for Ohmic baths with damping strengths below α = 0.2.
Technically more challenging is a treatment of non-equilibrium initial states where
transformed spin operators have to be transformed in addition on the time axis in order
116
The spin-boson model I: preparations
to evaluate the time-dependent dynamics of physical observables. We will therefore
open the next chapter with additional technical steps that discuss the time-dependent
transformation of the spin operators given in diagonal basis representation.
6.6 Summary
117
Figure 6.2: The full flow of the expression hσx2 (B)iGS is shown for different damping
strengths α and cut off frequencies ωc . The flow for the full generator (6.12) is plotted
without symbol and the corresponding line for the restricted generator (6.4) is marked
with triangles. Unlike to the operators σy and σz , the sum rule for σx is fulfilled more
accurately for the full generator (6.12). For a low frequency ωc = 10 one notices that
the sum rule starts to break down at α = 0.4 where the error got already larger than
5 %. Interestingly, the sum rule improves upon increasing the cut off ωc .
118
The spin-boson model I: preparations
Figure 6.3: We plotted the ground state expectation value hσx iGS against damping
strength α for different frequency ratios ∆0 /ωc . NRG data is only available for Ohmic
baths where a mapping to the anisotropic Kondo model is possible. The squares represent NRG data from [59] that is more reliable than the flow equation result for larger
dampings α ≥ 0.5. In the inset the different convergency behaviour of the NRG data
at the quantum critical point α = 1 is pronounced.
Chapter 7
The spin-boson model II: spin
dynamics
This chapter treats the time evolution of observables with respect to non-equilibrium
initial states. The underlying problem is to determine the time-evolution of spin operators and will be the starting point of this chapter. A discussion of approximations
in these calculations will be given and in this context, also the equilibrium correlation functions Sxx (t) = 12 h{σx (t), σx }+ i and Syy (t) = 12 h{σy (t), σy }+ i are presented.
Afterwards, the time evolved spin operators are applied to the initial states discussed
in section 6.4 . Since these initial states are given in different basis representations,
the time evolution of spin operators will be discussed both in diagonal and physical
basis. To be more precise, we fix here the language diagonal Heisenberg picture and
physical Heisenberg picture. In sections 7.2 and 7.3 the expectation values hσi (t)i
for all Pauli matrices σi , i = x, y, z are presented in the parameter space of damping and external field strength. The two-time non-equilibrium correlation function
Szz (t, tw ) = 21 hσz (t + tw )σz (tw )i is calculated and where the results allow, the FDT is
discussed in non-equilibrium. Comparisons to the non-interacting blip approximation
(NIBA) are given in order to test results. In every case, numerical solutions are presented. Where it is possible, analytical approximations are performed and compared
with numerical results.
7.1
Time evolution of observables
def
From a preparational time point t0 = 0 on, all initial preparations that where discussed
in section 6.4 expose the system-reservoir complex to the equilibrium Hamiltonian (6.2).
Formally, the time dependence of any Pauli matrix σi , i = (x, y, z) is determined for
t > 0 by the Heisenberg equation
dσi
= [H, σi (t)]
dt
(7.1)
120
The spin-boson model II: spin dynamics
Figure 7.1: Algorithm to solve Heisenberg equation by means of flow equations
with the equilibrium spin-boson Hamiltonian H.
However, an exact solution of (7.1) is not possible in case of the spin-boson model due to
the non-trival system-environment coupling. In diagonal basis, the system-environment
coupling in H is not present anymore and the Heisenberg equation can be solved. The
initial states (cf. section (6.4)) that incorporate the non-equilibrium properties are
given partially in diagonal basis. A solution of the Heisenberg equation is then even
essential. In contrast, factorizing initial states with completely polarized spin state are
only given in physical basis. The natural way is then to transform the time evolved
operators back from diagonal basis into physical basis by means of flow equations. A
sketch of this algorithm is depicted in figure (7.1). Just like in the case of the DHO
(cf. section 4.3 ), a solution of the Heisenberg equation in both basis representations
is therefore accessible by means of flow equations. Quite different from the case study
of the DHO, we cannot expect from this method to work exactly. We will have to take
care about the accuracy of the mapping from the diagonal Heisenberg picture to the
physical Heisenberg picture, since this transformation acts on an only approximately
known operator. This is quite different to the usual flow equation transformation,
where one naturally starts from an exactly known operator representation. Subsection
7.1.3 is partially devoted to a discussion of this additional approximation.
As depicted in figure (7.1), a preliminary task in time-dependent problems will always
be a solution of the Heisenberg equation in diagonal basis, whether this is the basis
where the initial state is given or not.
7.1.1
Time evolution of transformed observables
In previous reviews [15,26], the zero temperature equilibrium correlation function Szz (t)
was calculated on all time scales with the flow equation method. Reliable results were
obtained that agree with the correct long-time behaviour Szz (t) ∝ t−2 in contrast to
the NIBA [41] and agree with the NIBA on the rest of the time axis where the NIBA
is believed to be more accurate.
7.1 Time evolution of observables
121
Basically, these calculations exploited the knowledge of the equilibrium ground state in
the diagonal basis and a straightforward solution of the Heisenberg equation in diagonal
basis for the transformed operator σz (B = ∞). Therefore, it is essential to transform
σz (B = ∞) into diagonal Heisenberg picture -following [15]- in order to obtain the
physical quantity Szz (t). In addition, the same idea can be used to evaluate this correlator also for the spin components σx and σy . Since results involving these operators
have not been derived yet by means of flow equations, these transformed operators will
be used to calculate their symmetrized equilibrium autocorrelation functions Sxx (t)
and Syy (t). These quantities will be compared to known results and agreement with
well-known properties will be shown. In this way, the accuracy of transforming σx and
σy into the diagonal Heisenberg picture is tested.
Explicitely, the time evolution of the observables σi (∞), i = x, y, z has to be calculated
with respect to the diagonalized Hamiltonian
H(∞) = −
X
∆∞
σx +
ωk b†k bk
2
(7.2)
k
and obeys the Heisenberg equation
dσi (∞, t)
= i[H(∞), σi (∞, t)]
(7.3)
dt
where of course the commutator [σi (B = ∞), H(B = ∞)] will only add trivial frequency factors to the ansatz σi (B = ∞). Integrating the observables σi (t, B = ∞) will
then modify the operators σi (B = ∞) by trivial phase factors of the form e±iωk t .
Care has to be taken about neglected normal ordered terms in the Hamiltonian (6.6)
and the transformed observables σi (∞). Especially in the frequency region of resonant
couplings λk where ωk ≈ ∆∞ , they might contribute to time evolved operators σi (t, B =
∞). This corresponds to an error on the intermediate time scale ∆10 of the inverse
resonance frequency.
1
Using the transformed observables σi (B = ∞) from section 6.3, some simple operator
algebra yields the solutions of equation (7.3):
σx (∞, t) = h(∞)σx + α(∞) + σy
X
+σz
X
gk (t)(bk + b†k ) + iσy
(y)
k
k
X
fk (t)(bk − b†k )
(y)
k
(z)
gk (t)(bk
+
b†k )
+ iσz
X
fk (t)(bk − b†k )
(z)
k
(7.4)
with
1
Most detailed, this problem was discussed in the context of the ground state approximation in
section 6.2.
122
The spin-boson model II: spin dynamics
(y)
def
(y)
def
(z)
def
(z)
def
fk (t) = µk cos(ωk t) cos(∆∞ t) + χk sin(ωk t) sin(∆∞ t)
gk (t) = µk sin(ωk t) cos(∆∞ t) − χk sin(∆∞ t) cos(ωk t)
fk (t) = µk sin(∆∞ t) cos(ωk t) − χk sin(ωk t) cos(∆∞ t)
gk (t) = µk sin(ωk t) sin(∆∞ t) + χk cos(ωk t) cos(∆∞ t)
(7.5)
Likewise, one obtains for the remaining spin components:
i
X (y) h
σy (∞, t) = iσx
χk bk e−iωk t − b†k eiωk t
(7.6)
k
σz (∞, t) = σx
X
(z)
χk
h
bk e−iωk t + b†k eiωk t
i
(7.7)
k
where (7.7) is a result already stated in [15].
As illustrated in figure 7.1, spin operators in diagonal Heisenberg picture might either be evaluated in states known in diagonal basis or serve as operators that can be
transformed into physical Heisenberg picture. This transformation is performed with
time-dependent flow equations in order to approximate a solution of the Heisenberg
equation in physical basis
dσi (B = 0, t)
= i[H(B = 0), σi (B = 0, t)]
(7.8)
dt
Besides from any particular problem, this is an intriguing task since it represents a quite
general algorithm to solve one of the two fundamental equations for time-dependent
quantum mechanical systems. Details about this time-dependent transformation are
given in the following subsection.
7.1.2
Time evolution using time-dependent flow equations
Formally, the differential equation
dσi (B, t)
= [η(B), σi (B, t)]
(7.9)
dB
is of first order in the variable B. For uniqeness of a solution σi (B, t) that is dependent
on the variable t, it has to be supplemented by an initial condition that is dependent
on this variable t. If the solution of (7.9) in B=0 shall yield the solution σi (B = 0, t)
of the Heisenberg equation (7.1), the full solution σi (B, t) has to coincide with the
corresponding observable σi (∞, t) in diagonal Heisenberg picture in the limit B → ∞.
Therefore, the observable σi (∞, t) poses the initial condition for equation (7.9).
7.1 Time evolution of observables
123
In chapter 5, this idea was applied without any approximations to the DHO. For the
non-trivial spin-boson model, only approximated solutions for σi (B = 0, t) can be given.
Like in the special case t=0, 2 an approximated ansatz for σi (B, t) has to be chosen
for an approximate solution of equation (7.9), and this ansatz has to motivated by the
operator σi (∞, t) that has to be transformed.
On the accuracy of an ansatz σi (B, t)
However, the operator representations σi (∞, t), i = x, y, z given in (7.5)-(7.7) are only
approximate representations for an exact solution σi (∞, t). The operators (7.5)-(7.7)
are accurate up to truncated operators in O(λ2k ) of the couplings. Therefore care has
to be taken about the accuracy of an ansatz for σi (B, t) that is motivated only by
an approximately known representation of σi (∞, t) 3 . Nevertheless, if σi (∞, t) shall
also be transformed by truncating normal ordered operators of O(λ2k ) in the ansatz for
σi (B, t), the ansatz is in this accuracy consistent.
We want to illuminate this aspect further and remind the reader that an ansatz for
σi (B, t) in O(λ2k ) is motivated by the commutator
[η, σi (∞, t)]
(7.10)
O(λ2k )
or higher are truncated. Since η itself
in which all normal ordered operators in
is given as an operator in O(λk ), even the truncated operators in O(λ2k ) in an ansatz
σi (B, t) are already determined by the operator σi (∞, t) that contains all operators of
order O(λk ) in the couplings.
Following these ideas, it is now possible to formulate time- dependent flow equations
in O(λ2k ) in the couplings in complete analogy to the time-independent flow equations
formulated in section (6.3).
Backtransformation of the Pauli matrices σi (∞, t), i=x,y,z
A formal ansatz for σz (B, t) motivated by (7.10) reads:
σz (B, t) = σx
X
(iαk (B, t)(bk − b†k ) + ᾱk (B, t)(bk + b†k )) + σz z(B, t) + σy y(B, t) (7.11)
k
Interestingly, this ansatz is formally the sum of the ansätze (6.18) and (6.24) for the
i (B)
This case had been treated in chapter 6 in form of the time-independent transformations dσdB
=
[η(B), σi (B)]
3
i (B)
This is different from the time-independent transformation dσdB
= [η(B), σi (B)], where the exact
representation of the initial operators σi was known.
2
124
The spin-boson model II: spin dynamics
transformed operators σy (B) and σz (B). As a consequence, also σy (∞, t) will transform according to the ansatz (7.11) and thus, with exactly the same flow equations. In
addition these flow equations will have the structure of two decoupled systems of coupled differential equations. Each system corresponds to either the set of flow equations
(6.29) for σy or (6.23) for σz , respectively. However, the solutions of the flow equations
(6.23) and (6.29) discussed in section 6.3 will only be particular solutions of the flow
equations for the ansatz (7.11), since they are time-independent.
Motivated by the commutator [η, σx (∞, t)], we formulate the ansatz for σx (B, t).
x(B, t)σx + α(B, t) + σz
X
ᾱk (B, t)(bk + b†k ) + iσz
αk (B, t)(bk − b†k )
k
k
+iσy
X
X
βk (B, t)(bk −
b†k )
+ σy
X
β̄k (B, t)(bk + b†k )
(7.12)
k
k
It is important to emphasize that the generator
η(B) = iσy
X
ηk (bk + b†k ) + σz
(y)
k
X
ηk (bk − b†k ) + η (2)
(z)
(7.13)
k
makes use of the ground state approximation (cf. subsection 6.2.3) in order to diagonalize the Hamiltonian more accurately, yielding a more accurate approximation of the
observables σi (∞, t) in diagonal Heisenberg picture. To derive flow equations for the
time-dependent ansätze σi (B, t), the generator has to be the same that transformed
into the diagonal basis and therefore, again the full generator (6.12) is used.
Formulation of the flow equations
Before the time-dependent flow equations for the spin operators are actually applied to
a certain initial state, again the notation of generalized contractions Ckk0 introduced
in subsection 6.3.1 is used. In case of an application to a definite initial state, these
contractions will be specified with respect to the initial state. With the above choice
of generator, the time-dependent flow equations finally read:
For the ansatz σx (B, t):
7.1 Time evolution of observables
125
X
X (z)
X (y)
dx(B, t)
(y)
= −2
ᾱk ηk − 2
ηk βk − 8
ηk ᾱk0 Ckk0
dB
k
k
kk0
X
X (y)
dα(B, t)
(z)
=2
ᾱk ηk + 2
ηk βk
dB
k
k
X
dᾱk (B, t)
(y)
= 2x(B, t)ηk + 2
ᾱl ηkl
dB
l
X
dαk (B, t)
= −2
ηlk αl
dB
l
dβ̄k (B, t)
=2
ηkl β̄l
dB
l
X
dβk (B, t)
(z)
= −2
ηlk βl + 2x(B, t)ηk
dB
X
l
(7.14)
For the ansatz σz (B, t):
X
dαk (B, t)
(z)
= −2y(B, t)ηk − 2
αl (B, t)ηlk
dB
l
X
dᾱk (B, t)
(y)
= −2z(B, t)ηk + 2
ᾱl (B, t)ηkl
dB
l
X
X (y) (z)
dz(B, t)
(y)
=2
ᾱk (B, t)ηk + 8
ηk ᾱk0 Ckk0
dB
k
kk0
X
X (z) (y)
dy(B, t)
(z)
=2
αk (B, t)ηk − 8
ηk αk0 Ckk0
dB
0
k
(7.15)
kk
Comments on the flow equations (7.14) and (7.15)
Interestingly the flow equations (7.14) for σx (B, t) are equivalent to the time-independent
flow equations (6.28) for σx , if the contribution η (2) (cf. equation (6.12)) to the full generator is neglegted. Regarded as an approximation, η (2) ≡ 0 influences only resonant
couplings αk (0, t) (cf. subsection 6.2.3) and ᾱk (0, t) in a solution σx (0, t). Since these
couplings are time-dependent, they might influence all time scales. This approxmation
would need therefore for a careful discussion.
It should be reemphasized that the flow equations (7.15) are exactly the same for both
of the observables σy and σz . Since the solutions for the operators σy (0, t) and σz (0, t)
are not equivalent, the difference between these operators originates from the different
126
The spin-boson model II: spin dynamics
initial conditions (7.6) and (7.7) that have to be obeyed in the limit B → ∞ for every
single point in time.
Initial conditions
By comparing the ansätze (7.11) and (7.12) with the representations (7.5)-(7.7) in
diagonal Heisenberg picture, the initial conditions for all coeffients in these ansätze are
obtained. For σy (B, t), the conditions are:
αk (∞, t) = cos(ωk t)χk
ᾱk (∞, t) = sin(ωk t)χk
z(∞, t) ≡ y(∞, t) ≡ 0
(7.16)
whereas for σz (B, t), they read:
αk (∞, t) = − sin(ωk t)
ᾱk (∞, t) = cos(ωk t)
z(∞, t) ≡ y(∞, t) ≡ 0
(7.17)
It is worth to note that for numerical evaluations with a finite number of bath modes,
y(∞) and z(∞) remains finite and the initial conditions for σy (B, t) and σz (B, t) modify
according to:
y(∞, t) = h(y) (∞) cos(∆∞ t)
z(∞, t) = h(z) (∞) sin(∆∞ t)
(7.18)
for σy (B, t) and
y(∞, t) = −h(y) (∞) sin(∆∞ t)
z(∞, t) = h(z) (∞) cos(∆∞ t)
(7.19)
for σz (B, t).
7.1.3
Discussion of transformations
Already in section 6.3, approximations for transformed spin operators were discussed,
but these operators were not time-dependent. Time-dependent transformed spin operators have several properties that deserve their own discussion.
7.1 Time evolution of observables
127
1. Firstly, the accuracy of normal ordering inthe context of time-dependent spin
operators is discussed.
2. In addition, the question if the sum rules (6.54) are also valid for time-dependent
spin operators is treated.
3. Finally, the accuracy of approximated spin operators in diagonal Heisenberg picture is checked by calculating their equilibrium autocorrelation functions and
comparing them to known results obtained by other methods.
Discussion of normal ordering for time-dependent observables
In any previous application of flow equations, normal ordering was always used with
respect to time-independent initial states. In general, the application of a unitary time
evolution transforms normal ordered operators :O: into operators O(t) that are not
normal-ordered anymore.
eiHt : O : e−iHt = O(t)
(7.20)
Moreover, this statement also includes the case when the Hamiltonian H is normal
ordered with respect to the correct initial state since products of normal ordered operators are usually not normal ordered. Therefore, the time evolved normal ordered
operator (7.20) 4 has to be carefully examined and their possible contribution to the
expectation value hψ | O(t) | ψi has to be considered even Therefore, it is not possible
to savely truncate normal ordered terms in a time-dependent problem without using
time-dependent contractions Ckk0 (t) that are calculated with respect to the correct time
evolved state | ψ(t)i. A proper definition of these contractions reads:
def
Ckk0 (t) = hψ(t) | bk b†k0 | ψ(t)i
(7.21)
Since the flow equation method allows not to obtain a representation of time evolved
states, it is difficult to evaluate such time dependent contractions. Regarding these
circumstances, normal ordering with respect to the initial state seems the best choice.
Then a normal ordered operator :O: will be evaluated in the initial state and has vanishing contribution h: O :i = 0 to a physical observable at least in t=0. As argued
above, such normal ordered operators will in general contribute to physical observables
for positive times and it should be ensured that this contribution is negligible.
We saw that time-dependent operators diminish the accuracy of a truncation scheme
since normal ordering loses accuracy. In addition, it should be checked if the sum rules
hσi2 (B, t)i are fulfilled also accurately enough for time-dependent spin operators.
4
Although these terms are truncated in the ansatz, they formally contribute to the time evolved
observable.
128
The spin-boson model II: spin dynamics
Check of the sum rule for time-dependent spin operators
We want to check the sum rules for the time-dependent spin operators σi (B, t), i =
x, y, z both in dependence of time and the parameter B.
!
hσi2 (B, t)iGS = 1
(7.22)
Most important, theses rules should be fulfilled in the limit B → 0 that corresponds
to the physical basis where the ansätze σi (B, t), i = x, y, z shall be transformed to.
In physical basis, eg the non-equilibrium initial states | IAi and | IBi were given (cf.
section 6.4). In the following, only the state | IAi =| +iz ⊗ | 0i is regarded since
the state | IBi turns out to be difficult to treat analytically. However, the state | IAi
leads to the same sum rule as the equilibrium ground state | G̃Si in diagonal basis does.
In dependence of coefficients, the formal expression (7.22) can be made explicit as:
hσz2 (B, t)iGS = y 2 (B, t) + z 2 (B, t) +
X
k
αk2 (B, t) +
X
ᾱk2 (B, t)
(7.23)
k
Since σy (B, t) and σz (B, t) obey the same flow equations (7.15), (7.23) is also the valid
expression for hσy2 (B, t)i. The corresponding sum rule for σx (B, t) will not be discussed,
since correlation functions of this operator will be evaluated only in the ground state
in diagonal basis 5 . Using the representations (7.6) and (7.7), one concludes that
hσy2 (∞, t)iGS ≡ hσy2 (∞)iGS and hσz2 (∞, t)iGS ≡ hσz2 (∞)iGS .
Differentiating the expression (7.23) with respect to B and inserting the flow equations
dhσ 2 (B,t)i
dhσ 2 (B,t)i
y
y
(7.15) with the restricted generator (6.4), one shows that
≡
≡ 0,
dB
dB
2
2
2
2
leading to hσy (B, t)i ≡ hσy (∞)i and hσz (B, t)i ≡ hσz (∞)i. Already in section 6.5 it
was argued that the full generator (6.12) yields contribution to the sum rule only in
O(λ2k ) and therefore, hσy2 (∞)i and hσz2 (∞)i are equal 1 in very good accuracy for small
damping strengths α ≤ 0.2. In order to neglect higher orders in the couplings λk in an
ansatz, the damping strength α has to be small anyway. The same arguments can be
given for the sum rules (7.22) (excluding the sum rule for σx (B, t))and therefore, the
sum rule (7.22) will be fulfilled for σy (B, t) and σz (B, t) in high accuracy for not too
large dampings α < 0.2.
After two different technical technical aspects of approximated time-dependent spin operators have been discussed, these operators will now be discussed with respect to their
physical content. A qualitative criterion will be if they reproduce the known physics of
equilibrium correlation functions.
The equilibrium correlation function Sxx (t)
5
For the diagonal basis, the discussion of the sum rules from section 6.5 is sufficient, while for
time-dependent spin operators in the physical basis, the present discussion is needed.
7.1 Time evolution of observables
129
Using the operator representation (7.4) for σx (∞, t) in diagonal Heisenberg picture, the
symmetrized zero temperature equilibrium correlation function
def
Sxx (t) =
1
h{σx (t), σx }+ iGS
2
(7.24)
can be evaluated in the ground state | G̃Si =| +ix ⊗ | 0i in diagonal basis.
1
(7.25)
Sxx (t) = h+ |x ⊗h0 | {σx (∞, t), σx (∞)}+ | 0i⊗ | +ix
2
Such correlations will appear when the spin-boson model is coupled to a probe which
induces correlations in the tunneling rate ∆0 , like changing the capacitance in a SQID.
Although the feasibility of such measurements is questionable, it illustrates the fact
that the effects of dissipation are rather different on the various correlation functions
associated with the spin-boson model.
Before we start the discussion of physical observables of the spin-boson model with the
first example, we introduce a definition. In principle, physical observables might show
a dependence on the cut off frequency of the bath spectrum. We shall refer to an observable which is a function of ∆r 6 without any other ωc dependence as universal. Vice
versa, any extra dependence on the cut off ωc is called non-universal. The universality
or scaling limit is ωc → ∞ with ∆r fixed.
In [58], NRG calculations where used to discuss the behaviour of the correlator Sxx (t)
for Ohmic baths. A flow equation calculation of Sxx (t) can therefore test the accuracy
of the time evolved operator σx (∞, t) (7.4) by a comparison to NRG results from [58].
Two aspects of these results can serve as a check:
1. Sxx (t) is a non-universal observable. It depends sensitively on the choice of cut
off frequency ωc , since much of its spectral weight is situated in the cut off range
ω ≈ ωc . In consequence, it decays on a time scale of ωc−1 to the squared ground
state expectation value hσx i2GS .
def R ∞
2. In the low frequency regime, the Fourier transform Sxx (ω) = −∞ e−iωt Sxx (t)dt
has a delta peak in ω = 0 and shows the asymptotic frequency dependency
ω3
Sxx (ω) ∼ ∆
, ω ∆r where ∆r is the renormalized tunneling matrix element
r
mentioned in subsection 6.1.1.
Formally, the correlation function Sxx (t) reads
Sxx (t) = (α(∞) + h(∞))2 +
X
(z)
(y)
(χk − µk )(gk (t) − fk (t))
(7.26)
k
6
For Ohmic baths the renormalized frequency ∆r depends on the cut off ωc according to subsection
6.1.1. In modified form, ∆r shows a dependence on ωc also for other bath types.
130
The spin-boson model II: spin dynamics
Figure 7.2: We plotted the equilibrium correlation function Sxx (t) for an Ohmic bath
with different cut off frequencies ωc . A damping strength α = 0.05 and a discretization
with 2000 bath modes were chosen.
where all coefficients and unknown functions were defined in (6.25). This function
decomposes into an incoherent part (α(∞) + h(∞))2 = hσx i2GS and into a coherent
P
(z)
(y)
background k (χk − µk )(gk (t) − fk (t)) that decays in the limit t → ∞. In this way,
the flow equation treatment may shed additional light on the known results mentioned
under 2). Most important, the incoherent part of (7.26) equals the squared ground
state expectation value hσx i2GS . Thus, the incoherent part of (7.26) is responsible for
the delta peak of Sxx (ω) at zero frequency.
In figure (7.2), the non-universal behaviour of Sxx (t) is clearly visible. From the numerical results it becomes obvious that Sxx (t) decays on a time scale ωc−1 to its asymptotic
value hσx i2GS . We can compare the asymptotic value limt→∞ Sxx (t) indicated by the
numerical results to results for hσx i2GS calculated by flow equations. This comparison
shows that Sxx (t) reaches the asymptotic value hσx i2GS within a relative error below
0.2% at a time t ≈ 50.
A so called dressed coherence correlation function cures this problem of non-universality
def
by using a polaron-transformed tunneling operator σ̃x = UP σx UP−1 with the polaron
transformation UP defined in subsection 6.4.1 [59]. We leave this topic open for possible
future work.
Analogous to the calculation of Sxx (t), we apply the transformed operator σy (∞, t) to
calculate its symmetrized equilibrium autocorrelation function Syy (t). It has interesting relations to the correlation function Szz (t) that is related to the level population of
7.1 Time evolution of observables
131
Figure 7.3: We depict the current-current correlation function Syy (t) for different damping strengths. An Ohmic bath with a cut off frequency ωc = 10 was discretized with
2000 modes.
the two-state system.
The equilibrium correlation function Syy (t)
In context of the spin-boson model, the spin operator σy is also called ”current operator”
1 dhσz (t)i
since it fulfills the relation hσy (t)i = − ∆
[9], and thus, hσy (t)i is proportional to
dt
dhσz (t)i
the tunneling current dt . The corresponding current-current correlation function
def
Syy (t) =
1
h{σy (t), σy }+ iGS
2
(7.27)
was discussed in [50] in context of the zero temperature transition from coherent to
incoherent tunneling for Ohmic baths at the critical damping strength α = 12 [41]. In
the case of coherent tunneling, the position correlation function Szz (t) has a damped
oscillatory background that vanishes if tunneling becomes fully incoherent. It was argued that the correlation time τ of the current correlation function Syy (t) remains finite
for incoherent tunneling and goes to zero at the critical damping strength α = 21 where
tunneling becomes incoherent. The correlation time τ gives a measure how long in time
two succeeding measurements of an observable can be considered as correlated. It was
in [50] formally defined as the first zero of Syy (t) and it was shown that τ decreases to
zero at the transition point α = 21 .
2
We can exploit that Syy (t) and Szz (t) are related by the simple relation d Sdtzz2 (t) =
def R ∞
−4Syy (t) [50]. The Fourier transformed correlator Syy (ω) = −∞ e−iωt Syy (t)dt there-
132
The spin-boson model II: spin dynamics
fore obeys
Syy (ω) = 4ω 2 Szz (ω)
(7.28)
Eg this yields an ∝ ω 3 , ω ∆r low-frequency behaviour for Syy (ω) 7 and thus,
an algebraic t−4 long time decay of Syy (t). According to (7.28), spectral weight of the
function Syy (ω) will therefore be shifted to higher frequencies -in comparison to Szz (ω)since we assume ∆ ωc . For α < 1, this will lead to coherent oscillations with higher
frequency than in the correlator Szz (t). A final representation of Syy (t) is evaluated
with help of the identity
1
Syy (t) = hG̃S | {σy (∞, t), σy (∞)}+ | G̃Si
2
(7.29)
where σy (∞, t) is the operator (7.6) in diagonal Heisenberg picture. For Ohmic damping
with α < 1, the property y(∞) = 0 was proved in appendix A.1 and therefore, the
correlator Syy (t) is evaluated as
Syy (t) =
X
(y)2
χk
(∞) cos(ωk t)
(7.30)
k
In section 6.3 the approximate solutions
ωk λk (0)
ωk2 − ∆2r
(y)
χk (∞) =
(y)
for the coupling constants χk (∞) were derived. 8 Employing these approximate
solutions, the asymptotic behaviour of Syy (ω) reads:
( J(ω)
Syy (ω) =
,
ω2
ω 2 J(ω)
,
∆2r
ω ∆r
ω ∆r
(7.31)
For Ohmic baths, J(ω)ω 2 ∝ ω 3 and therefore, we derived the correct algebraic long-time
behaviour Syy (t) ∝ t−4 of the current correlation function. For many observables, eg
equilibrium correlation functions, the NIBA predicts the wrong long-time behaviour at
zero temperature and therefore flow equations describe this behaviour more accurately.
Numerical data for Syy (t) shows not t−4 behaviour up to times of O(100∆−1 ).
A further interesting test for Syy (ω) would be to check the exact equation (7.28) using
data from [15] for the correlator Szz (ω). We leave this topic open for possible future
work.
7
The correlation function Szz (ω) has a low-frequency behaviour Szz (ω), ω ∆r .
These solutions where shown to be valid in the limits ωk ∆r and ωk ∆r , where ωk is far from
the resonant frequency ∆r .
8
7.2 Application to a product initial state
7.2
133
Application to a product initial state
As stated in section 6.4, two different product initial states shall be considered in order
to investigate their non-equilibrium dynamics. For this purpose, the expectation values
hσi (t)i and the non-equilibrium correlator Szz (t, tw ) = 12 h{σz (t + tw )σz (tw )}+ i will be
discussed.
7.2.1
Formulation of the problem
If the spin state is initially prepared as an eigenstate of the operator σz , the bath can
either be decoupled from the spin system or not. If there is no system-environment
coupling present, at zero temperature the bath will simply be given by a bosonic vacuum
state.
| IAi =| +iz ⊗ | 0i
(7.32)
However, the bath might as well have come into equilibrium with the spin eigenstate
that tends to shift the bath modes in dependence of the coupling coefficients λk . In
the limit of zero temperature, the initial state will be
| IBi = U | +iz ⊗ | 0i
(7.33)
with
U = exp(−iΩ),
Ω=
X
k
sk
X 1 λk †
pk
=i
(
)(b − bk )
~
2 ωk k
(7.34)
k
Obviously, both states are excited states that cannot be representated in diagonal basis. Therefore, they show up to be typical examples for a conceptional problem of the
flow equation formalism that has not been solved before. If an initial state cannot be
expressed in the diagonal basis but only in the physical basis, operators have to be time
evolved also in physical basis in order to calculate observables.
Non-equilibrium initial states are in general difficult to express in a diagonal basis. In
most cases it is not possible to construct a unitary representation of the transformation that transforms into diagonal basis. However, this would be the only possibility
to perform a transformation in state space, since the differential formulation of this
operation in form of flow equations was designed only to work in operator space. In
some cases, it is possible to construct the non-equilibrium state in diagonal basis out
of the diagonal ground state by a unitary transformation. An example for these states
was presented in section 6.4 and section 7.3 will be devoted to the dynamics of these
states.
For these reasons, the initial states (7.32) and (7.33) are of great conceptional importance in order to apply the formalism of time-dependent flow equations. Inspite of the
134
The spin-boson model II: spin dynamics
quite different experimental preparations of these initial states 9 , they yield essentially
the same dynamics. Influence functionals describing the different environmental influence of the two preparation classes can be shown to coincide after a transient time of
order ωc−1 . In appendix B, it will be rigorously discussed under which conditions these
states yield the same dynamics.
Note that within the ansatz (7.11) for σz (B, t), the resultant expression for hψ | σz (t) |
ψi is formally always the same, if ψ is a product initial state with a spin in the upper
eigenstate of σz . Eg, the bath could be prepared in any state and the formal result
would be always hψ | σz (t) | ψi = z(0, t), where z(0, t) is the coefficient of the operator
σz that is at first sight independent of the initial bath state. This is not necessarily a
contradiction because if one wants to treat an initial state properly within our method,
one would have to use normal ordering with respect to the initial state. Normal ordering influences the flow equations for σz (B, t). In this way, the initial state modifies the
operator coefficient z(0, t) and the physical observable hψ | σz (t) | ψi.
Normal ordering and the initial state
In general, normal ordering of bosonic operators can be performed with respect to any
kind of non-equilibrium initial state | ψi if it is possible to calculate its contractions
hψ | bk b†k0 | ψi. To the extend such a normal ordering procedure is applicable, our
method can be extended to a broader class of non-equilibrium preparations. 10 In
the above example for the observable hψ | σz (t) | ψi one can see that the subtle influence of initial preparations enters in this case completely through the choice of normal
ordering. Therefore, it is important to note that in order to distinguish dynamics of
hψ | σz (t) | ψi for different non-equilibrium states, only different choices of normal
ordering can describe differences in the dynamics of observables.
In this stage of our discussion, the important role of normal ordering in non-equilibrium
situations becomes pronounced. However, for the product initial states | IAi and | IBi
it can be expected that normal ordering plays only a role on short time scales, since
these states show the same dynamics after a short transient time of order ωc−1 . Even
more general, intuitive arguing would expect that differences between non-equilibrium
initial bath states decay during the dissipation process leading to smaller and smaller
differences between the corresponding time-dependent contractions of the bath operators. More detailed research on this topic would be necessary. In previous literature
concerning flow equations, normal ordering was in general used only with respect to
the equilibrium ground state. The importance of discussed examples in non-equilibrium
shows that in non-equilibrium, it would be useful to use normal ordering in a more so9
cf. section 6.4
Not every kind of normal-ordering yields accurate physical results, e.g. if contractions diverge with
increasing control parameters, like the magnetic field in (6.47).
10
7.2 Application to a product initial state
135
phisticated way. This problem seems to be an interesting topic for future work.
7.2.2
The expectation values hσi (t)i
In order to calculate the observables hσi (t)i, i = x, y, z, a numerical solution of the flow
equations is possible without any conceptional problems. For the components hσy (t)i
and hσz (t)i, an analytical treatment of the time-dependent flow equations (7.15) is
possible by exploiting their formal equivalence to the time-independent flow equations
(6.23) and (6.29). We will not calculate any results for the observable hσy (t)i in this
chapter since the more important observable hσz (t)i can be evaluated with exactly the
same technical steps. Eg we can use the same time-dependent flow equations (7.15).
Analytical relationships to time-independent flow equations
Any linear superposition of solutions of the flow equations (6.29) for σy and (6.23) for σz
is a solution of the time-dependent flow equations (7.15). As far as the flow equations
(7.15) induce a unitary transformation from physical to diagonal basis, these properties
can be used to derive analytical exact expressions for both coefficient functions y(0, t)
and z(0, t). Explicit steps of this calculation are given in appendix A.3. We mention
the results for the coupling constants in σz (B, t) that can be obtained in a similar way
also for the ansatz σy (B, t).
P
(z)
(y)
χk (∞)χk (∞) sin(ωk t)
P (y)2
k χk (∞)
P (z)2
χk (∞) cos(ωk t)
z(0, t) = k P
(z)2
k χk (∞)
k
y(0, t) = −
(7.35)
(7.36)
However, unitarity of the transformation is only ensured if the flow equations close
without any truncation procedure. As we truncated the operator expansion for all spin
operators σi after linear order in the coupling constants λk , we cannot expect an exact
fulfilment of unitarity. For a quantitative description of possible deviations from unitarity, further analysis of the concerned flow equations would be needed.
Concerning the physical problem of calculating hσz (t)i for both of the product initial
states | IAi and | IBi one obtains therefore the astonishing result
P
P (t) = z(0, t) =
k
(z)2
χk (∞) cos(ωk t)
Szz (t)
= P (z)2
P (z)2
k χk (∞)
k χk (∞)
(7.37)
In case of the preparation | IAi, this yields the identity P (t) = Szz (t). 11 This identity
alures to the well-known P (t) ≡ Szz (t) result obtained within the NIBA [41] for the
11
Here, the sum rule
P
k
(z)2
χk
(∞) = 1 has been assumed.
136
The spin-boson model II: spin dynamics
Figure 7.4: We compare the equilibrium correlation function Szz (t) with the expectation
value P (t) = hσz (t)i for the initial state | IAi. We used an Ohmic bath with a cut
off frequency ωc = 10 and a damping strength α = 0.05. To calculate P (t) with the
time-dependent flow-equations (7.15) we used 500 bath modes. The correlator Szz (t)
P (z)2
was calculated according to the representation Szz (t) = k χk (∞) cos(ωk t) [15] for
2000 bath modes.
7.2 Application to a product initial state
137
initial state | IBi and known to be wrong in the long time limit t ∆−1 . However,
for the initial state | IBi we used non-equilibrium contractions in the flow equations
(z)
(6.23) for χk (B) and the expression
P
k
(z)2
χk
P
k
(∞) cos(ωk t)
(z)2
χk
(∞)
is therefore not equivalent to
the dynamics of Szz (t) that is derived with equilibrium contractions. Since the long
time dynamics (t ∆−1
∞ ) of the states | IAi and | IBi is known to be equal, the
identity P (t) = Szz (t) for the state | IAi is therefore an artefact. For this reason, the
above demanded care when making use of unitary in a flow equation transformation
has importance.
The numerical results depicted in figure 7.4 show a deviation of P(t) and Szz (t) of over
10% for times t ≥ 20∆−1
0 . Nevertheless, improving the flow equation expansion of the
spin operators into higher orders of the couplings λk might give new insights into the
failure of the NIBA by identifying the contributions to P(t) that cause the deviation
from dynamics of Szz (t).
In the following we will solve time-dependent flow equations only numerically.
The expectation value hσx i(t)
Using the representation (7.12) for the transformed operator σx (B = 0, t) in the physical
Heisenberg picture, formal expressions for the time-dependent observable hσx i(t) read:
For the initial state | IAi:
hσx i(t) = α(B = 0, t)
(7.38)
For the initial state | IBi:
hσx i(t) = α(B = 0, t) +
X λk
k
ωk
ᾱk (B = 0, t)
(7.39)
P
One should not identify the term k ᾱk (B = 0, t) ωλkk in equation (6.38) as the difference between the dynamics of hσx i(t) for the two different product initial states, since
hσx i(t) is also dependent on the initial state via different non-equilibrium contractions
that influence the flow equations (7.14).
Discussion of the results
Numerical evaluations (figure 7.5) show that in the limit t → ∞, hσx (t)i tends to reach
the ground state expectation value hσx iGS calculated by Costi and Kieffer with NRG
[49] and confirmed by the method of flow equations for damping strengths α ≤ 0.5 (cf.
fig. 6.3). Note that at a time scale up to t = 2π∆−1
r the observable hσx (t)i shows small
oscillations with a frequency that equals the cut off frequency ωc . We interpret this as
a band edge effect. Remarkably, the observable hσx (t)i reaches the expectation value
138
The spin-boson model II: spin dynamics
Figure 7.5: We compare the relaxation of hσx (t)i for different damping strengths α
with a fixed cut off ωc = 10 (left figure) and different cut off frequency with a fixed
damping strength α = 0.1( right figure). A discretization of 500 bath modes was used
for all curves. For a cut off frequency ωc = 80, this discretization does not allow to
resolve times scales larger than ∆0 t = 20 (cf. appendix C.2).
hσx iGS with a relative error below 0.1% at a time scale t = 100∆−1
0 .
In addition our numerical results for hσx (t)i indicate the correct non-universal behaviour
[59]
lim hσx (t)i ≡ 0
ωc →∞
(compare right hand side of figure 7.5).
We compare this highly accurate description of the equilibration process to the equilibrium expectation value with results from the NIBA. At zero temperature and weak
damping, the NIBA predicts the asymptotic value [9]
hσx (t → ∞)i =
∆
||
(7.40)
The parameter describes a bias that is contained in an additional contribution σz to
the spin-boson Hamiltonian (6.2). For the unbiased Hamiltonian 6.2, the result (7.40)
leads to an unphysical divergency of hσx (t → ∞)i in the corresponding limit → 0. It
is known that the NIBA result for hσx (t)i is inconsistent for T=0, zero bias and weak
damping also at finite times [9].
The expectation value hσz i(t)
Formally for a product initial state with a spin in the upper eigenstate of σz , the
7.2 Application to a product initial state
139
expectation value hσz i(t) reads:
hσz i(t) = z(B = 0, t)
(7.41)
However, it cannot be expected that both initial states | IAi and | IBi will lead exactly
to the same time-dependent observable hσz i(t), at least for small times of order ωc−1
(cf. appendix B). In the following, these time-dependent differences between the two
product initial states are discussed and a comparison to results from the NIBA is given.
Before we do so, we briefly repeat important results derived within the NIBA.
Comparison to NIBA results
For an Ohmic bath in the regime T=0, 0 ≤ α ≤ 21 the NIBA provides an analytical
benchmark for our result. Arguments on its validity in the whole parameter space of
damping and the frequency relation ω∆c can be found in appendix D of [41]. Eg it is
shown that in the scaling limit ω∆c → 0, the NIBA is accurate in O(α2 ) of the damping
strength. We briefly summarize the results of the NIBA, outlined in [41]. Within the
def
NIBA the function P NIBA (t) = hσz (t)i with P NIBA (t ≤ 0) = 1 and the symmetrized
def
equilibrium correlation function C NIBA (t) = 21 h{σz (t), σz }+ i yield the same expression.
ω
For Ohmic baths J(ω) = 2αωe− ωc it has been evaluated in ref. [41] with the result
PNIBA (t) = E2(1−α) (−y 2(1−α) )
(7.42)
P
zk
where Eν (z) is the Mittag-Leffler function Eν (z) = ∞
k=0 Γ(νk+1) and y stands for the
dimensionless time scale ∆eff t with
1
∆eff ≡ (cos(πα)Γ(1 − 2α)) 2(1−α) ∆
∆
α
1−α
(7.43)
ωc
The series representation of the Mittag-Leffler function can be summed up using Hankel’s contour integral representation for the reciprocal of the Γ function and evaluating
the resultant integral by deformation of the contour [41]. One obtains
P NIBA (t) = Pcoh (t) + Pcoh (t)
(7.44)
1
with the coherent contribution Pcoh (t) = 1−α
cos(Ωt)e−γt where the oscillation freπα
πα
quency Ω is Ω = ∆eff cos( 2(1−α) ) and the dephasing rate is given by γ = ∆eff sin( 2(1−α)
).
The incoherent contribution is given by
Z
sin(2πα) ∞
z 2α−1 e−zy
Pinc (t) = −
dz 2
(7.45)
π
z + 2z 2α cos(2πα) + z 4α−2
0
For 0 ≤ α < 12 the function Pcoh (t) describes damped coherent oscillations. In the
exactly solvable case α = 21 , the exact result
P (t) = Pcoh (t) = e−γt
(7.46)
140
The spin-boson model II: spin dynamics
Figure 7.6: We compare the observable hσz (t)i for the different initial states | IAi and |
IBi (upper figure, the red curves correspond to the initial state | IBi) and the results for
the state | IBi with the NIBA (lower figure, the red curves represent the NIBA results).
All results are presented for different damping strengths α = 0.025, 0.05, 0.1, 0.15. We
used an Ohmic bath with cut off ωc = 10 and 500 bath modes.
7.2 Application to a product initial state
141
2
with γ = π∆
2ωc is recovered. It can be obtained by an exact mapping from the Kondo
model, where it corresponds to the exactly solvable Toulouse point. For 12 < α < 1
only the incoherent contribution Pinc (t) remains, leading to a fully incoherent decay
of P NIBA (t). A localization P NIBA (t) ≡ 1 takes place for all α > 1 [47]. If α 6= 21 ,
the incoherent part Pinc (t) dominates the long-time behaviour as P (t) ∝ t−2(1−α) . The
sluggish decay of this incoherent part fails to reproduce the correct long-time behaviour
that is given by [60]
hσz (t)i ∝ e−γt ,
t ∆−1
r
(7.47)
where the relaxation rate γ depends only on damping strength and the cut off frequency
ωc and is explicetely given in ref. [60]. In addition to the wrong long-time behaviour,
the NIBA predicts the asymptotic behaviour
hσz (t → ∞)i = − tanh
~ε 2kB T
(7.48)
[9] and thus at T=0, strict localization hσz (t → ∞) ≡ 1 even for an infinitesimal small
bias ε.
Below we present a comparison with the NIBA result for different values ωc and different damping strengths α. We want to emphasize that the NIBA is an approximation to
formally exact path integral representations. Numerical methods (real time QMC [52])
allow for a numerical exact evaluation of those exact path integral expressions. These
evaluations are restricted to time scales of O(10∆−1
r ) due to the dynamical sign problem
[52]. It would be interesting to have also comparisons with real time QMC calculations.
Indeed, the observable hσz (t)i has a quantitatively equivalent behaviour for both initial
states | IAi and | IBi for dampings below α = 0.05 where the relative deviation is
below 3% on all time scales. For larger dampings α = 0.1 and α = 0.15 the relative
deviation increases above 10% during the first oscillation period and decreases below
5% afterwards. This behaviour indicates that the quantitative difference between the
two different normal ordering procedures decays with increasing time. It also agrees
with the fact that the influence functionals for both initial states | IAi and | IBi differs by a phase factor with a phase proportional to the damping strength α. For larger
dampings the transient time that passes until the influence functionals coincide will
therefore increase proportional to damping strength. Since the contractions used for
the state | IBi grow ∝ α2 , the higher relative deviation for larger dampings is within
the flow equation technique a consequence of the different normal ordering schemes.
A comparison to the NIBA result (7.42) yields high accuracy for small dampings
α ≤ 0.05 with a relative error below 5%. For larger damping strengths α > 0.05
the NIBA starts to deviate from our results by more than 10% relative error and well
above any numerical errors. Eg the oscillation frequency of the coherent decay pre-
142
The spin-boson model II: spin dynamics
dicted by the NIBA deviates more and more with increasing damping. It tends to a
faster oscillation than the flow equation result predicts and oscillates eg. 20% faster at
a damping α = 0.15.
Preparational effects
Figure 7.7: The observable hσz (t)i is evaluated both for the localized state | loci where
hσz (t)i = z(0, t) and the initial state | deloci where hσz (t)i = y(0, t). The four plots
differ only in the damping strength α. The numerical results for an Ohmic bath with
cut off ωc = 10 and a discretization with 500 bath modes indicate that the delocalized
state behaves different from a localized state. Eg for larger dampings α ≥ 0.1 and a
time scale of t ≥ 20∆−1
0 the delocalized state decays faster than a localized state (cf.
lower two plots).
The ansatz (7.11) suggests to prepare the two state system in an initial state that has
a non-vanishing expectation value hσy i since then, the previously uninvolved function
7.2 Application to a product initial state
143
y(0, t) can start to take part in the dynamics of hσz (t)i.
An apropriate initial preparation can be realized by an application of a large magnetic
field coupling term Hf ield = −hy σy − hz σz with h ∆0 . For non-zero field strength,
h
Hf ield has non-degenerate eigenstates and thus, the ratio hyz controls the initial state.
A finite field strength hy tends to delocalize the initial state and it will have the general
form
λ1
hy hy | loci + λ2
| deloci
hz
hz
(7.49)
where | loci is the upper eigenstate | +iz 12 of the position operator σz and | deloci is
the delocalized state | +iz + i | −iz with hdeloc | σz | deloci = 0. The coefficients λ1
h
h
and λ2 are dependent on the ratio hyz and related to hσy i and hσz i as hσy i =| λ1 ( hyz ) |2
h
and hσz i =| λ2 ( hyz ) |2 . In dependence of field direction the decay of the initial state
will show a different time-dependence:
hσz (t)i = hσz iz(0, t) + hσy iy(0, t)
(7.50)
The formal expression (7.50) suggests a numerical comparison of the coefficient functions y(0,t) and z(0,t) that control the relaxation process. Indeed, figure 7.7 indicates
that the function y(0, t) that corresponds to a delocalized state decays for larger dampings α ≥ 0.1 faster to the equilibrium value hσz iGS than the localized state does. We
suggest that the initial preparation of a spin in the y-z plane can controll the relaxation
rate of the spin out of the initial state at time scales t ∆−1
0 .
7.2.3
How does a spin decay?
The time-dependent relaxation process of a spin in a dissipative environment is highly
nontrivial due to the quantum nature of a spin. Flow equations make a visualization of
this process in three spatial coordinates possible since the time-dependence of all three
spin operators has been evaluated in section 7.1.
In figure 7.8 the numerical integration of the time-dependent flow equations (7.14) and
(7.15) with respect to the initial state | IAi =| +iz ⊗ | 0i is depicted. We resolved
an Ohmic bath (damping strength α = 0.1, cut off ωc = 10) with 1000 bath modes.
All time scales up to the approach to the equilibrium ground state expectation values
hσi i, i = x, y, z have been resolved. These asymptotic values are approached with an
excellent accuracy of below an absolute error of 10−2 . Up to a time of approximately
2π∆−1
r the curve shows oscillations of the frequency ωc that decay for larger times. We
interpret these oscillations as a band edge effect.
12
In the context of the double well system (cf. figure 6.1), a localized state | loci represents a quantum
mechanical state that is restricted to one well.
144
7.2.4
The spin-boson model II: spin dynamics
The correlation function Szz (t, tw )
We want to discuss the two-time non-equilibrium correlation function
1
Szz (t, tw ) = h{σz (t + tw ), σz (tw )}+ i
(7.51)
2
for the product initial states | IAi and | IBi. In the equilibrium initial state, this
correlation function is the relevant quantity for the neutron scattering characteristics
of the system [9]. Experimental results for this correlator were obtained for hydrogen
trapped by oxigen in niobium [61]. The proton is tunneling between two trap sites
in the N b(OH)x samples. It constitutes a two-level system which seems to be ideally
suited for studying the influence of dissipation on quantum systems. Eg the damping
parameter is very small [62], namely α = 0.05 and it is possible to apply flow equations
to investigate this damping regime.
In earlier calculations [41] of the equilibrium correlation function Szz (t) the systembath correlations in the initial state were often neglected [41]. As discussed in [62],
an inclusion of initial system-bath correlations can lead to important differences in
the long-time decay of Szz (t). Correspondingly, the low-frequency behaviour of the
structur factor is also qualitatively influenced by the correlations in the initial state. In
previous work, flow equations were already used to calculate the equilibrium correlation
function Szz (t) for the correct entangled ground state [15]. We use the time-dependent
flow equations (7.15) to evaluate the non-equilibrium correlation function Szz (t, tw ) for
product initial states and obtain the formal expressions
Szz (t, tw ) = z(t + tw )z(tw ) + y(t + tw )y(tw ) +
X
[ᾱk (t + tw )ᾱk (tw ) + αk (t + tw )αk (tw )]
k
(7.52)
for the initial state | IAi and for the initial state | IBi, the formal representation reads
Szz (t, tw ) = z(t + tw )z(tw ) + y(t + tw )y(tw ) +
X
[ᾱk (t + tw )ᾱk (tw )
k
+ αk (t + tw )αk (tw )]
X
λ k λk 0
λ k λk 0
+ αk0 (t + tw )αk (tw )
]
+
[ᾱk0 (t + tw )ᾱk (tw )
ωk ωk 0
ωk ωk 0
0
kk
(7.53)
Interestingly the correlation function Szz (t, tw ) turns out to be completely independent
of the initial spin state and is always of the form (7.52) if the bath is initially in its
vaccum state. Eg the spin states | +iy or | +iy could be chosen instead of the state | +iz .
Discussion of numerical results
7.3 Application to an entangled initial state
145
Theoretically, we expect a cross over behaviour:
In tw = 0, Szz (t, tw = 0) shows the exponential long-time decay of the observable
hσz (t)i due to the identity Szz (t, tw = 0) ≡ hσz (t)i 13 . In the limit tw → ∞ we expect
that Szz (t, tw ) becomes equal to the equilibrium correlation function Szz (t, tw ).
Numerical results for the product initial state | IAi show that the correlator Szz (t, tw )
shows only weak dependence on the waiting time tw . We draw the conclusion that
the initial preparation of the system reservoir-complex at zero temperature effects the
function Szz (t, tw ) very weakly.
7.3
Application to an entangled initial state
In the discussion of the symmetrized spin-spin correlation function Szz (t, tw ), we saw
that entangled initial states might lead to a quite different behaviour in comparison
to a product initial state that neglects initial correlations between system and environment. All previously treated inititial states prepared a pure spin state that lead to
a product initial state. In subsection 6.4.2 we introduced the entangled initial state
| IIi that is prepared by an external magnetic field that is small enough to lead only
to a partially polarized spin projection 0 <| hσz i |< 1. In consequence the initial state
remains entangled with the environment.
Intermediate field strengths h = O(∆0 ) are most difficult to treat within the flow
equation technique. For larger field strengths h ∆0 , the initial state becomes a
product initial state with polarized spin projection. For smaller field strengths h 1
field-dependent terms in higher orders of the flow equation expansion become negligible 14 . This is therefore the limit of interest in this section. In order to drive the
system appreciable from its ground state, the field strength should be of order the tunnel splitting ∆0 . In most experimental relevant cases, a magnetic field couples only to
magnetic moments of O(µK ) (in case of a nuclei) or O(µB ) (in case of an electron) with
µK = 5.05 × 10−27 JT −1 and µB = 9.274078 × 10−24 JT −1 . Therefore the limit h << 1
is even for very large magnetic field strengths of O(105 Tesla) excellently fulfilled and
applications to NMR and ESR problems are possible.
According to these circumstances, two possible treatments of the problem can be considered:
• If the system is driven not far from its equilibrium ground state, normal ordering
with respect to the equilibrium ground state is useful. It allows to formulate
analytical expressions that are based on the corresponding equilibrium solutions.
13
The operator σz acts here on an eigenstate
The field dependence of operators in higher order in the couplings λk is introduced via normal
ordering with respect to the field-dependent initial state.
14
146
The spin-boson model II: spin dynamics
• A numerical solution can employ the more accurate normal ordering procedure
with respect to the correct non-equilibrium initial state.
Analogous to the previous calculations for product initial states, we calculate the symmetrized spin-spin correlation function Szz (t, tw ) and the expectation values hσi (t)i, i =
x, y, z. A usage of normal ordering with respect to the ground state will allow us to
check the fluctuation dissipation theorem since the resultant expressions for correlation
functions are related to the concerning equilibrium correlation functions.
7.3.1
Analytical results in the limit of small field strength h
For small field strengths h 1, the non-equilibrium initial state
h
˜ =| +ix ⊗ U | 0i, U = e− 2
| IIi
P
k
χk (∞)
(bk −b†k )
ωk
˜ + O(h2 ). If normal order˜ =| GSi
˜ + h P χk (∞) (bk − b† ) | GSi
can be rewritten as | IIi
k
k
ω
ing is used with respect to the ground state, the corresponding bosonic contractions are
wrong in O(h2 ) in comparison to contractions with respect to the correct initial state
˜
| IIi.
For small field strengths h 1 this will lead to negligible corrections. In order to
calculate physical observables, we use the time-dependent operator σz (∞, t) from (7.7)
˜ in diagonal basis representation.
and evaluate it in the non-equilibrium initial state | IIi
The expectation values hσi (t)i
We obtain the following results for the non-equilibrium expectation values hσi (t)i i =
x, y, z:
hσy (t)i = −2h
X
hσx (t)i ≡
(y)
(z)
χk (∞)χk (∞)
hσz (t)i = −2h
sin(ωk t)
ωk
k
X χ(z)2 (∞)
k
k
hσx iGS
ωk
cos(ωk t)
(7.54)
Note that the result hσx (t)i ≡ hσx iGS is unphysical and will be improved in subsection
6.3.2 by using the correct non-equilibrium contractions. The analytic expressions for
hσy (t)i and hσz (t)i can be rewritten by using the approximate analytical solutions (6.32)
(y)
(z)
and (6.33) for the coupling constants χk (∞) and χk (∞)
(y)
χk (∞) =
ωk λk (0)
ωk2 − ∆2r
(z)
χk (∞) =
∆r λk (0)
ωk2 − ∆2r
7.3 Application to an entangled initial state
147
that are valid for couplings away from the resonant frequency ∆r (cf. section 6.3). By
employing these couplings in the expressions (7.54), we can extract the low-frequency
behaviour of the Fourier transformed observables hσi i(ω), i = y, z.
Z ∞
def
dte−iωt hσi (t)i, i = y, z
(7.55)
hσi i(ω) =
−∞
We obtain the behaviour
hσy i(ω) ∝ J(ω), ω ∆r
J(ω)
hσz i(ω) ∝
, ω ∆r
ω
(7.56)
For Ohmic baths with J(ω) ∝ ω we obtain a very contrasting behaviour in the long-time
decay of both observables. While the observable hσz (t)i decays exponentially for long
times, the observable hσy (t)i decays only algebraically ∝ t−2 , t ∆−1
r . Moreover, for
s
super-Ohmic baths J(ω) ∝ ω , s > 1 the decay of hσz (t)i changes into an algebraic
∝ t1−s long-time behaviour.
The correlation function Szz (t, tw ) and the FDT
In an completely analogous way to the calculations of the observables hσi (t), i = x, y, zi
we evaluate the function Czz (t, tw ) = hσz (t + tw )σz (tw )i with the result
Czz (t, tw ) =
X
(z)2 −iωk t
χk
e
+
k
hσz (t + tw )ihσz (tw )i
(7.57)
cum (t, t ) is equal to the corresponding symmetrized equilibObviously the cumulant Szz
w
eq
rium correlation function Szz
(t).
def
cum
Szz
(t, tw ) =
X
1
eq
h{σz (t+tw ), σz (tw )}+ i−hσz (t+tw )ihσz (tw )i =
χ2k cos(ωk t) = Szz
(t)
2
k
(7.58)
def
Moreover, the anti-symmetrized spin-spin correlation function Azz (t, tw ) = 12 h[σz (t +
tw ), σz (tw )]i as the real part of (7.57) is equal to its equilibrium version. We conclude
that the response function χzz (t, tw ) = 2iΘ(t)A(t, tw ) is the equilibrium reponse function. Together with (7.58) this proves that the fluctuation dissipation theorem remains
fulfilled via
cum
Szz
(ω) ≡ sgn(ω)χ00zz (ω)
(7.59)
148
The spin-boson model II: spin dynamics
The correlation function Szz (t, tw ) can show a different cross over behaviour to the
eq
equilibrium correlation function Szz
(t), dependent on the behaviour of the product
hσz (t + tw )ihσz (tw )i. For Ohmic baths, this approach is exponentially in the long-time
s
limit tw ∆−1
r , while for super-Ohmic baths with J(ω) ∝ ω , s > 1 the long-time
2(1−s)
approach becomes algebraically ∝ tw
.
Discussion of approximations:
In general, normal ordering with respect to the ground state will yield good results compared with normal ordering with respect to the correct non-equilibrium initial state.
The corresponding contractions are field-dependent only in O(h2 ). A comparison of
numerical results could indicate for which field strengths the two normal ordering prescriptions will start to deviate. In order to check the validity of the results for the
˜ | σ 2 (B) | GSi
˜ def
= 1 provides a necessary critecorrelator Czz (t, tw ) the sum rule hGS
z
rion.
˜ = U | +ix ⊗ | 0i
The sum rule in the non-equilibrium initial state | IIi
P
We use an expansion of the operator U = exp(− h2 k χkω(∞)
(bk − b†k )) in the field
k
strength h to obtain an asymptotic expansion for the sum rules.
˜ | σ 2 (B) | IIi
˜ = 1 + h2
hII
z
hX χ(z)2 (B) i2
X χ(z)2 (B)
2
k
k
4
− h (B)
ωk
ωk2
k
!
+ O(h4 ) (7.60)
k
and for σy2 (B)
˜ | σ 2 (B) | IIi
˜ = 1 − h2 h2 (B)
hII
y
X χ(z)2 (B)
k
k
ωk2
+ O(h4 )
(7.61)
Therefore the sum rule will have a validity range of small field strengths h 1. For
larger field strengths, neglected higher order terms in the flow equation expansion of
the operators -that also are of higher order in the field strength h- σi (t) will start
contributing to the sum rules and lead to a violation.
7.3.2
Numerical results for an improved normal ordering scheme
In this subsection an improved normal ordering scheme is used that is more accurate
than normal ordering with respect to the vacuum, yet has to be treated numerically.
We calculate the contractions hb0k b†k i now with respect to the correct non-equilibrium
˜
initial state | IIi.
In consequence, this contraction contains now an additional field
dependent component:
7.4 Summary
149
Table 7.1: A calculation of the expectation value hσx i in dependence of external field
strength. We discretized an Ohmic bath with 2000 bath modes up to a cut off frequency
ωc = 10 and chose a tunneling matrix element ∆0 .
hσx i
α
0.025
0.1
h
0
0.01
0.05
0.1
0.15
0.963
0.857
0.963
0.856
0.946
0.831
0.897
0.765
0.825
0.676
hbk0 b†k i
= δkk0 +
(z)
(z)
2 χk (∞)χk0 (∞)
h
ωk ωk0
(7.62)
(z)
It is important to mention that the coefficients χk (∞) theirselves should be also obtained by using this normal-ordering procedure. A possible way out is to first calculate
them with the usual vaccum normal ordering, plug them into the expressions for the
field-dependent contractions and iterate then the resulting equations until convergence
has been reached.
Discussion of the results
We give a brief discussion of numerical results that use the improved contractions
(7.62). In figure 7.10 we depict the observable hσz (t)i for different field strengths and
two damping strengths α. All qualitative features of the curves are independent of the
chosen field strengths h. Eg. the hσz (t)i shows a coherent decay at a frequency of
approximately the frequency ∆2π∞ .
We argue that an improved normal ordering procedure leads not to a higher accuracy
for an evaluation of the correlator Szz (t, tw ) since also the sum rule (7.60) is only
valid in O(h2 ) and we give no further numerical implementation of this correlator.
In contrast the influence of the external field onto the observable hσx (t)i ≡ hσx i can
be described only by the correct non-equilibrium contractions. Table 7.1 shows the
significant reduction of the expecation value hσx i with increasing field strength h. This
field-dependent behaviour could not be described with a simplistic normal ordering with
respect to a vacuum state since this approximation would drop any field dependence
of the observable hσx (t)i.
7.4
Summary
In this chapter it was shown how the flow equation method can be employed to treat
non-equilibrium problems in the non-trivial spin-boson model. The fundamental step
150
The spin-boson model II: spin dynamics
was to calculate the time evolution of the spin operators both in diagonal and physical
basis. In this way, both product and entangled initial states could be treated. We
checked the approximated representations of the time-evolved spin operators in the
diagonal basis by a calculation of the symmetrized equilibrium correlation functions
Sxx (t) and Syy (t). These correlators showed the correct long-time behaviour known
from previous results. Afterwards, we applied these operators to an entangled initial
state. In the limit of small field strengths, we derived an exponential long-time decay of the observable hσz (t)i and a fulfillment of the fluctuation dissipation theorem
for the symmetrized spin-spin correlation function Szz (t, tw ) independent of the time tw .
We used time-dependent flow equations for all spin operators to investigate the timedependent decay of two product initial states. It became obvious that normal ordering
is an essential technique to describe the influences of different initial bath preparations.
A numerical implementation of the spin components hσi (t)i shows that flow equations
agree qualitatively with results from the NIBA at intermediate times t = O(∆−1
0 ) and
where
the
NIBA
often
fails.
yield accurate behaviour in the long-time limit t ∆−1
0
Finally, the flow equation method has shown to be able to evaluate correlation functions on all time scales for different non-equilibrium preparations.
Several approximations that were performed could be avoided if all operators are transformed with ansätze that truncate only operators of third order in the couplings. Such
a procedure is possible and would avoid the ground state approximation and the significant influence of normal ordering onto physical observables. We leave this topic open
for future work.
7.4 Summary
Figure 7.8: Three dimensional visualization of a Bloch vector
151
152
The spin-boson model II: spin dynamics
Figure 7.9: We depicted the non-equilibrium correlation function Szz (t, tw ) for the
product initial state | IAi. The damping strength α = 0.025 and different waiting
times tw were chosen. The Ohmic bath was discretized with 500 modes up to the cut
off frequency ωc = 10.
Figure 7.10: We depict the observable hσz (t)i for different field strenghts h and two
damping strengths α. It shows the typical coherent decay known from the product
initial states discussed in section 7.2. We discretized an Ohmic bath with 2000 bath
modes up to a cut off frequency ωc = 10 and chose a tunneling matrix element ∆0 = 1.
Chapter 8
Comparative view on different
models
Throughout this thesis, it might have become apparent to the reader that only few
results exist that lead to an understanding of non-equilibrium many body systems. In
a recent series of publications by Kehrein and Lobaskin [16,17], new results for the
time-dependent Kondo model were derived. Very close relations to many topics of this
thesis can be drawn. In chapter 6, the choice of initial states was motivated by these
calculations. Furthermore, at the Toulouse point [63] the Kondo model can be mapped
onto the Ohmic spin-boson model. Since only the Ohmic spin-boson model shows this
direct relationship to the Kondo model we will discuss only Ohmic baths.
Some out of very few exact results that can be used to test numerical methods were
obtained at the Toulouse point, especially the first exact analytic result for the spin
correlator Szz (t, tw ) [16]. It will be our aim to discuss these results and compare their
qualitative properties against those that were obtained within the present thesis for two
different models. We attempt to identify properties that can be observed in different
models. Our discussion concentrates on three physical properties of non-equilibrium
many body systems:
1. The observables hx(t)i and hσz (t)i describe the experimentally most relevant properties of either a diffusion process, the magnetization of an impurity or the tunneling motion of a two level system, dependent on the model.
2. The two time correlation functions of these observables describes the influence
of damping caused by either fermionic or bosonic baths. The decay of these
correlation functions gives an answer to the question how fast the environment
destroys a pure initial state and causes decoherence.
3. The fluctuation dissipation theorem is a suitable theorem to identify properties
that are caused by a non-equilibrium situation.
154
Comparative view on different models
Each of these topics is discussed for all three models. Firstly, we briefly summarize in
a few sentences what has been achieved in the thesis of D. Lobaskin [39]1 .
8.1
Results for the time-dependent Kondo model
All problems treated in [39] used the Kondo Hamiltonian to describe dynamical processes of an impurity spin.
H=
X
k,α
εk c†kα ckα +
X
i
Ji
X
c†0α Si σiαβ c0β
(8.1)
α,β
The Kondo Hamiltonian describes an impurity coupled to a conduction band of elec~ and a localized electron orbital with fermionic operators
trons. A spin operator S
†
c0α and c0α describe the impurity. For this Hamiltonian two different non-equilibrium
problems were formulated. These non-equilibrium situations are defined by two product
initial states that are prepared at zero temperature.
1. The impurity spin is fixed by a large magnetic field term h(t)Sz that is switched
of at t=0: h(t) TK for t < 0 and h(t) = 0 for t ≥ 0. The impurity is coupled
to the conduction band all the time.
2. The impurity spin is decoupled from the bath degrees of freedom for all negative
times. Then the coupling is switched on in t=0 - Ji (t) = Θ(t), Ji > 0 - and is
time-independent for t ≥ 0. A certain projection hSz (t ≤ 0)i = 21 is assumed
which is realized by an infinitesimal small magnetic field.
Both of these conditions allow for an experimental preparation up to some point in
time when the system is left under the conditions described by the Kondo Hamiltonian (8.1). After lifting the external experimental constraints, the impurity spin will
dissipate energy to the fermionic bath and relaxate towards a state that eventually has
properties of the equilibrium ground state of the Kondo Hamiltonian.
For both initial preparations the magnetization P (t) = hσz (t)i and the symmetrized
spin-spin correlator Szz (t, tw ) were calculated. In addition, the fluctuation dissipation
theorem was discussed for this spin-spin correlator.
The parameter regime of the couplings
Ji was restricted to two important cases. At the
√
Toulouse point Jk = 2π(2 − 2), bosonization and refermionization techniques yield
exact results on all time scales. Of physical relevance is the Kondo limit where the conduction band is isotropically and weakly coupled to the impurity spin (Ji ≡ J 1). In
the Kondo limit, the flow equation method was used to diagonalize the Kondo Hamiltonian. 2 The diagonal Kondo Hamiltonian was related to the refermionized Kondo
1
2
All figures presented in this chapter were taken from [39] with a kind agreement of D. Lobaskin.
This work was done by Hofstetter and Kehrein in 2001 [64].
8.2 The expectation values hx(t)i and hσi (t)i
155
Hamiltonian from the Toulouse point analysis and in this way, all further technical
steps were the same than in the Toulouse point analysis.
It turned out that the observables P(t) and Szz (t, tw ) behave very similar both at the
Toulouse point and in the Kondo regime and were independent of the initial preparation. Eg. both observables showed an exponential long-time 3 decay to equilibrium
expectation values. Furthermore it turned out that the fluctuation dissipation theorem
for the spin-spin correlation fucntion Szz (t, tw ) is violated for tw > 0 and is recovered
exponentially fast in the limit tw → ∞.
8.2
The expectation values hx(t)i and hσi (t)i
Firstly, we want to summarize the long-time behaviour of the spin expectation value
hσz (t)i for the spin-boson model and the Kondo model. We will conclude that both
models yield for all treated initial states an exponential long-time decay. This behaviour can be compared to the dynamics of the quantity hx(t)i for the DHO. Inspite
of being a trivial model, it turned out in chapter 5 that it can show both exponential
and algebraic long-time decay of the observable hx(t)i.
Kondo model
The spin-boson model and the Kondo model can be mapped onto each other at the
K
Toulouse point, leading to the exponential decay P (t) = e−2t∆ , ∆ = Tπw
in the Kondo
4
model . This result was for both initial preparations the same. It was also argued
that this expectation value is totally independent of the initial bath preparation.
In the Kondo limit, the purely exponential decay of the Toulouse point result was recovered for both initial preparations as long-time asymptotics P (t) ∼ e−2t∆ , t ∆−1
and confirmed the exact result of Lesage and Saleur [67]. On short time scales the
decay is even faster [39].
Spin-boson model
For Ohmic baths in the damping regime 0 ≤ α ≤ 1/2, Lesage and Saleur obtained the
exact asymptotic long-time decay P (t) ∝ cos(Ωt) exp(−γt) 5 for a spin-boson model
that is prepared in the product initial state | IBi from section 6.4. As argued in appendix A.1, this result is also valid for the product initial state | IAi. Obviously the
3
Note that the correlator Szz (t, tw ) shows this equilibration behaviour in the long-time limit of the
time tw .
4
For a definition of the Kondo temperature TK and the Wilson number w, see [39].
5
For expressions of the frequency Ω and the dephasing rate γ see ref. [9].
156
Comparative view on different models
Kondo model shows not this coherent oscillation in the long-time limit but a similar
decay of the envelope. An exponential long-time decay was also derived for the entangled initial states | IIi as discussed in section 7.3. It would be interesting to check if
this long-time behaviour coincides with that from the product initial states. For superohmic baths this long-time behaviour changed to an algebraic long-time decay. We
do not compare this behaviour with the Kondo model since only for Ohmic baths, the
spin-boson model can be mapped onto the Kondo model.
It is worth to mention that all results for the magnetization P (t) in the Kondo model
were totally independent of the initial bath preparation. We compare this behaviour
with the spin-boson model. It is known that the preparations | IAi and | IBi of the
spin-boson model will differ only on a transient time scale ∝ ω1c . This behaviour was
confirmed numerically by the flow equation method. Moreover, flow equations for spin
operators always are influenced by the initial bath preparations through the contractions of the bath operators (cf. chapters 6 and 7). We conclude that the spin-boson
model is sensitive to the initial bath preparation for small times t ∆−1
r whereas the
Kondo model is not.
The majority of all results discussed above can be summarized in the observation that
the long-time decay of an impurity spin showed in both models always exponential
long-time decay. As we have seen in chapter 5, the DHO is more sensitive to the initial
preparation and can also show algebraic long-time decay.
Dissipative harmonic oscillator
Firstly, we discussed an entangled initial state with shifted bath modes that is similar
to the state (6.44) that has been treated for the spin-boson model. It shows also an
exponential long-time decay of the position expectation value hx(t)i of the central osscillator. Unlike to the spin-boson model, with increasing damping strength this decay
remains coherent until the Hamiltonian becomes unbounded from below at a critical
damping strength (cf. (4.17)). Interestingly the exponential long-time decay can be
stabilized into an algebraic long-time decay by the application of a time-dependent
external field h(t), t ≥ 0.
Coherent initial states showed very contrasting properties. A coherent state controls
inital position and momentum of the central oscillator. If the initial momentum vanishes, the preparation is very similiar to that of the entangled state (6.44) where an
external field fixed the initial position and the initial momentum is zero. Surprisingly,
it turned out that the coherent state decays only algebraically ∝ t−3 in the long-time
limit. Obviously, a coherent state interacts different with the bath modes and decays
slower. For a coherent state with finite initial momentum, it turned out that hx(t)i
8.3 Correlation functions
157
decays even slower ∝ t−2 in the long-time limit t ∆−1
r .
Conclusions
We saw a quite different sensitivity of all three models on the initial preparation of the
system for the treated initial state. In the Kondo model, the time-dependent decay
of the observable P (t) was completely independent of the initial bath preparation.
The spin-boson model turned out to be more sensitive to the initial preparation and
differences in the intial bath preparation decay in the observable hσz (t)i only after a
r
transient time. This transient time vanishes only in the scaling limit ∆
ωc → 0. Both
models had in common that the long-time decay of these observables was exponentially.
Most sensitive to a preparation of the initial state was the DHO. An entangled initial
state showed exponential long-time decay similar to an analogous initial state in the
spin-boson model. In contrast, a coherent state as an example for a product initial state
decayed only algebraically in the long-time limit. The exponent of this decay was even
sensitive to the inital momentum of the central mode. Finally, this example showed
that an exponential long-time decay of observables in dissipative quantum systems is
not generic at zero temperature.
8.3
Correlation functions
We will discuss the symmetrized correlation function Szz (t, tw ) = 21 h{σz (t+tw ), σz (tw )}+ i
both for the Kondo model and the spin-boson model. This correlation function describes decoherence effects that destroy pure spin states. The behaviour of the spinspin correlations is compared against the symmetrized displacement correlation function Sqq (t, tw ) of the DHO. Emphasis is put on the cross over behaviour from nonequilibrium correlations to equilibrium correlations with increasing tw . All examples
treated in this thesis showed this cross over behaviour in the limit tw → ∞.
Kondo model
In tw = 0, the correlator Szz (t, tw ) is equivalent to the magnetization P(t) if the initial
spin state is an eigenstate of σz . An algebraic long-time decay of the equilibrium correlator Szz (t) is then contrasting the exponential long-time decay of P(t) that describes
the correlations in the non-equilibrium situation. Both at the Toulouse point and in
the Kondo limit the cross over to the equilibrium correlation function is exponentially
−1
fast in tw for large values of tw /tK [17]. The inverse Kondo temperature tK = TK
is the relevant time scale for this decay to the equilibrium correlations. In the Kondo
limit, the approach to equilibrium for small times is faster than at the Toulouse point.
Pictorially, the crossover from exponential to algebraic long-time decay is related to a
reduction of the finite offset in Szz (ω = 0, tw ) (compare figure 8.1). For finite tw the
function Szz (ω, tw ) has a finite slope in ω = 0 that indicates the ∝| ω | behaviour at
158
Comparative view on different models
Figure 8.1: The correlation function Szz (ω, tw ) at the Toulouse point. In tw = 0 this
function has a vanishing slope in ω = 0 and shows an exponential long-time decay.
For finite tw , the slope is ∝ ω. Already at a time scale of O(tK ) convergency to the
equilibrium correlation function is visible. In the Kondo limit, all qualitative features
of Szz (ω, tw ) described above remain the same [39].
small frequencies, leading to the algebraic ∝ t−2 long-time decay.
Spin-boson model
If the initial spin state is an eigenstate of σz , the spin-spin correlation function Szz (t, tw )
shows generically exponential long-time decay in tw = 0. Similar to the Kondo model,
a cross over from exponential long-time decay to the known equilibrium algebraic longtime decay ∝ t−2 could be observed in the limit tw → ∞. Numerical calculations from
section 7.2 show that Szz (t, tw ) depends very weakly on tw . It is difficult to identify
the cross over from non-equilibrium decay to equilibrium behaviour. In addition no
analytical results exist that describe this cross over in dependence of tw away from the
Toulouse point α = 0.5.
In subsection 7.3 we treated initial spin preparations with only partially polarized
expectation value | hσz i |< 1. We showed that in the limit of weak field strengths
h 1, the correlation function Szz (t, tw ) relaxates to equilibrium with increasing tw
proportional to the observable hσz (tw )i.
8.3 Correlation functions
159
eq
Szz (t, tw ) = Szz
(t) − hσz (t + tw )ihσz (tw )i
For large times tw ∆−1 , hσz (tw )i decays exponentially fast and Szz (t, tw ) approaches
the ∝ t−2 long-time behaviour of Szz (t). However, this long-time behaviour always
dominates the decay of Szz (t), also in tw = 0. We can conclude that the correlator
Szz (t, tw = 0) will show an algebraic long-time decay. If the field strength becomes
large enough to fully polarize hσz i, the perturbative expansion in h used in section
(7.3) is not valid anymore and Szz (t, tw = 0) will instead decay eponentially for large
times. Not that in all these examples for the spin-boson model, entangled initial states
(eg also the ground state) show an algebraic long-time decay in tw = 0, while product
initial states (eg the states | IAi and | IBi defined in section 6.4) usually show an
exponential long-time decay.
All analytical results discussed above indicate that zero temperature non-equilibrium
correlation functions decay for long times exponentially in tw to the related equilibrium
correlation functions. It turns out that the DHO behaves anormalously in this aspect.
Dissipative harmonic oscillator
Non-equilibrium correlation functions of the DHO can approach the equilibrium correlation function either algebraically or exponentially for large tw ∆−1 . Firstly, we
discuss the example that shows an unusual algebraic approach to equilibrium.
For a coherent inital state the dispclacement corelation function Sqq (t, tw ) is given by
the real part of equation (5.40).
Sqq (t, tw ) =
N
1 X
[C0,n + D0,n ][C0,n0 + D0,n0 ] ×
2 0
n,n =0
N X
−Ck,n Dk,n0 cos[(ω̄n0 + ω̄n )tw + ω̄n t] − Dk,n Ck,n0 cos[(ω̄n0 + ω̄n )tw + ω̄n t] +
k=0
Ck,n Ck,n0 cos[(ω̄n0 − ω̄n )tw ] − ω̄n t] + Dk,n Dk,n0 cos[(−ω̄n0 + ω̄n )tw + ω̄n t]
!
+hx(t + tw )ihx(tw )i(8.2)
It is obvious that this function can approach equilibrium only algebraically for tw ∆−1
0 . Any dependence on the eigenvalue α is contained in the product hx(t+tw )ihx(tw )i
−6
which decays either ∝ t−4
w if α has an imaginary part or ∝ tw if it is real. Numerical
−1
results from section 5.3 confirm a fast (tw = O(10∆ )) cross over to equilibrium correlations if α = 0. This algebraic approach to equilibrium is qualitatively very different
160
Comparative view on different models
to all examples discussed in the Kondo model and the spin-boson model. In addition,
such an algebraic behaviour is not universal for all initial states and changes if the
DHO is prepared in the initial state | IIi discussed in section 5.2.
In presence of a finite field h coupling to the position operator of the central oscillator,
eq
the equilibrium correlation function Szz
(t) is extended by the product hx(t+tw )ihx(tw )i
(cf. section 5.2).
eq
Sqq (t, tw ) = Sqq
(t) + hx(t + tw )ihx(tw )i
(8.3)
This result is exact even if the external field has an arbitrary time dependence h(t) for
positive times. A special case is considered if the field is switched off instantanously
in t=0 (h(t) = hΘ(−t)). Since hx(tw )i decays exponentially in the long-time limit
eq
t ∆−1
0 , the equilibrium correlation function Sqq (t) will be also restored exponentially
−1
fast for tw ∆0 . In presence of time-dependent fields, the cross over behaviour to
equilibrium is more complicated.
hx(tw )i will be modified by an additional component r(tw ). For the Drude type bath
from section 5.5.2 the low frequency behaviour of the field transform h(ω) will lead
to different types of algebraic long-time decay of r(t). We conclude that also timedependent fields lead to an algebraic cross over to equilibrium correlations that will
change into an exponential decay if the field h is switched of instantaneously.
Conclusions
Like for the observables hx(t)i and hσi (t)i, we saw that correlation functions show a
cross over behaviour that depends on the model. In the Kondo model, this cross over
of the correlator Szz (t, tw ) occurs exponentially fast for large tw and eg changes from
an exponential long-time decay in tw = 0 to an algebraic long-time decay for any finite
tw > 0.
In principle the spin-boson model should show the same behaviour for the product
initial states | IAi and | IBi (cf. section 6.4). A lack of analytical results prevents us
to justify this conjecture. In contrast, the entangled initial state showed the interesting
property that the function Szz (t, tw ) decays also in tw = 0 algebraically in the long-time
eq
limit. The long-time approach to the equilibrium correlation function Szz
(t) occurs in
this case exponentially fast in tw .
Finally the DHO oscillator turned out to behave anormalously and shows both exponential and algebraic long-time approach in tw to the equilibrium correlation function
eq
Sqq
(t).
8.4 The fluctuation dissipation theorem
8.4
161
The fluctuation dissipation theorem
In general, the fluctuation dissipation theorem is not fulfilled for non-equilibrium correlation functions.
cum
χ00zz (ω, tw ) 6= sgn(ω)Szz
(ω, tw )
(8.4)
Usually, correlation functions show a cross over to equilibrium correlation functions in
the limit tw → ∞, leading to a fulfillment of the FDT. Of central interest is to analyze
the time-dependent crossover on all frequency scales. Most important is the equilibration process at zero frequency. This process defines the time-dependent evolution of
the effective temperature Teff as defined in (5.48). Notice that in general ’the effective
temperature’ depends on the observable chosen for its definition [65].
Kondo model
For both initial preparations, the fluctuation dissipation theorem
cum
χ00zz (t, tw ) = sgn(ω)Szz
(t, tw )
(8.5)
is not fulfilled in the Kondo model.
Both in the Kondo limit as at the Toulouse point the same qualitative features occur.
For zero waiting time tw = 0 the FDT is trivially fulfilled since the system is prepared
in an eigenstate of Sz and therefore both functions in (8.5) vanish identically. For increasing waiting time the curves start to differ and indicate the violation of the FDT
in non-equilibrium. At large waiting times -compared to the Kondo time scale tk - the
curves will approach each other exponentially in tw coincide again. This behaviour indicates that the spin-spin correlations reach equilibrium behaviour for tw → ∞ where the
FDT will hold. From figure 8.2 one concludes that the maximum violation of the FDT
occurs at zero frequency, while it becomes fulfilled more rapidly at higher frequencies.
In addition to these qualitative descriptions, the effective temperature Teff was analyzed.
Its behaviour is depicted in figure 8.3 for the Toulouse point. The Kondo temperature
goes up very quickly as a function of tw and reaches a maximum of 0.4 ≈ TK at
tw ≈ 0.1tK and goes to zero exponentially fast in the limit tw → ∞. Some quantitative changes occur in the Kondo limit where Teff reaches its maximum even faster
for tw = 0.03tK . Furthermore, it was discussed in [17] that this behaviour of the effective temperature can be understood by energy diffusion and heating effects in the
conduction band. We will not discuss the effective temperature for the DHO and the
spin-boson model in detail since it turned out that a reliable numerical calculation of
Teff was not possible in these models.
162
Comparative view on different models
Figure 8.2: At the Toulouse point, the FDT is violated for a finite value tw > 0. For
tw = 10tK , the spin-spin correlation function Szz (ω, tw ) (black curve) is already indistinguishable from the imaginary part χ00zz (ω, tw ) of the response function (red curve).
8.4 The fluctuation dissipation theorem
163
Figure 8.3: Effective temperature Teff as a function of the waiting time tw at the
Toulouse point. The inset shows the same curve on a linear scale to illustrate how fast
the initial heating occurs.
Spin-boson model and dissipative harmonic oscillator
We obtained only one result for the fluctuation dissipation theorem in the spin-boson
model. It predicts a fulfilment of the fluctuation dissipation theorem for the entagled
initial states | IIi for weak fields h 1. This result might be changed when the linear
response regime h 1 is left. Again, the DHO showed very contrasting properties.
The fluctuation dissipation theorem was fulfilled exactly for arbitrary field strength
that prepares an entangled initial state. Even for arbitrary time-dependent fields acting after the initial preparation this property did not change. In contrast, a product
initial state with a coherent state of the central oscillator violated the fluctuation dissipation theorem for all waiting times tw . Unlike to the Kondo model (compare figure
8.2) the violation of the FDT was not quantitatively most intense in ω = 0. Rather this
violation turned out to be small in ω = 0 and even negative for some values and caused
difficulties in the evaluation of the effective temperature that was besides independent
of the excitation strength of the coherent state. It turned out that the fluctuation
dissipation theorem is fulfilled fast after a time of tw = 10∆−1
0 and this behaviour gives
a signature of exponential fast fulfillment of the FDT.
Conclusions
An examination of all three models shows that non-equilibrium dissipative quantum
164
Comparative view on different models
systems can behave quite different in their properties concerning the FDT. This is
indicated by the puzzling behaviour of the DHO that fulfills the FDT for arbitrary
time-dependent fields but violates it for a coherent initial state totally independent of
the eigenvalue of the coherent state. Interestingly, all treated entangled initial states
fulfilled the FDT, while product initial states turned out to violate the FDT. We leave
it as an open question if entanglement enhances a fulfilment of the FDT.
8.5
Conclusions
At the beginning of our comparative discussion we cited results for the Kondo model
that suggest a generic behaviour of the decay of non-equilibrium observables. This
behaviour suggested an exponential long-time decay of single observables like hσz (t)i
and their correlators like Szz (t, tw ) in the time variables t and tw repectively to the
eq
equilibrium observables hσz iGS and Szz
(t). Eg. this exponential relaxtation of the correlator Szz (t, tw ) suggests a violation of the FDT that vanishes exponentially fast for
large tw .
Our comparison with two other quantum impurity systems at zero temperature lead
to the conclusion that it is necessary to consider a more diverse behaviour, eg it is
difficult to formulate generic laws. Already the spin-boson model can show a cross
over behaviour from exponential long-time decay to algebraic long-time decay of the
observable hσz (t)i. We demonstrated this in section 7.3.2 for a change from an Ohmic
bath to a super-Ohmic bath.
Most clearly the diverse behaviour of quantum impurity systems became obvious from
the DHO was shown to be fulfilled for several non-equilibrium preparations. In addition
both exponential or algebraic long-time decay of the observables hx(t)i and Sqq (t, tw )
to equilibrium behaviour occured in dependence of the initial preparation. We close
with the important conclusion that the decay of non-equilibrium observables cannot
be fully understand from the coupling mechanism to the environment but also has to
consider the initial preparation of the system as an important aspect.
Chapter 9
Conclusions and outlook
Summary
In this work the flow equation method was used as a non-perturbative method to investigate two time-dependent dissipative quantum systems. For the dissipative harmonic
oscillator we were able to indicate the potential of our method and derived several exact
results that are typical for low temperature dissipative quantum systems but usually
not accessible by often used approximations like the Born or Markow approximation.
Eg we derived the correct algebraic long time decay of the displacement correlation
function Sqq (t).
We claim that our method is a powerful analytical method to treat non-equilibrium
dissipative quantum systems in general and demonstrated this by an application to the
ubiquitous spin-boson model that can only be treated with approximative methods.
We showed that the flow equation method provides a controlled approximation scheme
that is able to treat all time scales of non-equilibrium processes. Eg. we were able to
analyze the quantum equilibration of all spin expectation values to their equilibrium
values on all time scales for two product initial states.
Most important, in this thesis the flow equation method was conceptionally extended
to allow for a treatment of arbitrary initial states by introducing the concept of timedependent flow equations.
Future potential of our method and suggested work
It is desirable to test the flow equation method against an exact analytical result in
order to analyze the approximation scheme for time-dependent problems in greater
detail. After this thesis was nearly finished, we became aware of reference [69] where
an exact result for the observable hσx (t)i in the spin-model model was presented. We
166
Conclusions and outlook
suggest this result as an interesting test that can help to understand our approximation
scheme in more detail.
Concerning future applications of our method, it is of great interest to apply the flow
equation method to models of coupled qubits in order to achieve a better understanding
of decoherence processes in quantum computing. Eg. our method can be extended
to treat continous time-dependent variations of model parameters that occur during
calculational steps and read-out processes in a quantum computer.
Appendix A
Properties of flow equations for
spin operators
A.1
Asymptotic decay into bath operators
In chapter 6 we transformed the operators σy and σz into diagonal basis by means of flow
P (z)
equations. These observables transform into the operators h(z) (B)σz +σx k χk (B)(bk +
P (y)
b†k ) and h(y) (B)σ(y) + iσx k χk (B)(bk − b†k ) during the flow. It was mentioned in
section 6.3 that the initial operators σy and σz decay completely in the limit of accomplished flow. Here, we give a prove that limB→∞ h(y) (B) = 0 under certain conditions.
A prove for the property limB→∞ h(z) (B) = 0 was given in [26] in a similar way.
P (z)
(y)
†
k ηk (bk −
k ηk (bk +bk )+σz
def
†
2
bk ) from (6.4). For this generator the sum rule hσy (B)iGS = 1 is exactly fulfilled (cf.
(y)
section 6.5). We will use the flow equations for the couplings λk (B) and χk (B).
The prove is formulated using the generator η(B) = iσy
P
dλk
= −(ωk − ∆(B))2 λk (B)
dB
(y)
dχk (B)
(y)
= −2ηk (B)h(z) (B)
dB
(y)2
h(y)2 (B) + χk (B) ≡ 1
(A.1)
(A.2)
(A.3)
In addition we formulated the sum rule (6.55) in its explicit form (A.2). The couplings
λk (B) can be formally integrated.
Z
λk (B) = λk (0) exp −
B
(ω − ∆(B 0 ))2 dB 0
0
We integrate equation (A.1 ) and insert the expression (A.4).
(A.4)
168
Properties of flow equations for spin operators
(y)
χk (B)
Z
B
=−
0
Z B0
ωk − ∆
00
∆
(ωk − ∆)2 dB 0 dB 00
λk (0)h(B ) exp −
ωk + ∆
0
(A.5)
For convenience, we introduce the function f˜(ωk , B).
def
f˜(ω, B) = −
Z
0
B
Z B 00
ω−∆
(y)
00
∆
λk (0)h (B ) exp −
(ω − ∆)2 dB 0 dB 00
ω+∆
0
We insert f˜(ω, B) and A.5 into the sum rule A.2 and take the limit B → ∞.
Z ∞
2
Z ∞
h(B)f˜(ω, B)dB dω = 1
h(y)2 (∞) +
J(ω)
(A.6)
(A.7)
0
0
It is now necessary to ensure that ∆∞ >
R ∞0. For Ohmic baths this requires α < 1. If
we now assume h(∞) 6= 0, the integral 0 h(B)f˜(ω, B)dB diverges like (ω − ∆∞ )−1 .
(cf. [26]). Obviously the principal value of (A.7) cannot exist then due to the pole of
order two in ω = ∆∞ . Therefore we conclude that h(∞) = 0 for Ohmic baths with a
damping strength α < 1. The same line of arguments was drawn in [26] by employing
the sum rule for the operator σz to show that limB→∞ h(z) (B) = 0 if α < 1 .
A.2
Flow equations for Pauli matrices and the SU(2)symmetry
We give a closer analysis of the reason why the SU (2) symmetry fails for the transformed spin operators σi (B), i = x, y, z. In a first step the transformation of the
commutator [σy (B), σz (B)] is discussed without any explicit representation for the operators σy (B) and σz (B). Then the truncated ansätze for these operators are inserted.
It becomes clear then in which way the truncated operators influence the violation of
the SU(2) algebra.
For convenience, we define the formal operator σ̃x (B).
def
σ̃x (B) =
1
[σy (B), σz (B)]
2i
(A.8)
Now the derivative of σ˜x (B) is calculated for an unspecified generator η(B).
σ̃x (B)
1
dσz (B)
1 dσy (B)
= [σy (B),
]+ [
, σz (B)]
dB
2i
dB
2i dB
1
1
= [σy (B), [η, σz (B)]] + [[η, σy (B)], σz (B)]
2i
2i
1
= [η, [σy (B), σz (B)] = [η, σ̃x (B)]
2i
(A.9)
(A.10)
(A.11)
(A.12)
A.2 Flow equations for Pauli matrices and the SU(2)-symmetry
169
i (B)
In equation (A.10) only the formal differential equations dσdB
= [η, σi (B)], i = y, z
were employed. Every single step in equations (A.9)-(A.11) is therefore exact. It
is now of interest to insert the truncated ansätze (6.18) and (6.24) for σy (B) and
σz (B) into (A.10). After this approximation the flow of σ˜x (B) is described by the two
independents flows of the truncated operators σy (B) and σz (B). We argued in section
6.5 that this approximation will break the SU(2) symmetry. The reason is that in the
P (y)
ansätze (6.18) and (6.24) the normal ordered terms : [η, iσx k χk (bk − b†k )] : and
P (z)
: [η, σx k χk (bk + b†k )] : were truncated. If these terms are considered in (A.7), they
(B)
yield two additional contributions to σ̃xdB
.
[: [η, iσx
X
χk (bk − b†k )] :, σz (B)]
(A.13)
χk (bk + b†k )] :]
(A.14)
(y)
k
[σy (B), : [η, σx
X
(z)
k
We evaluate the sum of (A.13) and (A.14) explicitely using the generator (6.4).
−4ih(z) σx
X
ηk0 χk : (bk0 − b†k0 )(bk − b†k ) :
(z) (y)
kk0
(y)
+ 4ih
σx
X
ηk χk0 : (bk + b†k )(bk0 + b†k0 ) :
(y) (z)
kk0
+ 4iσz
X
−4σy
X
ηl χl
(z) (y)
X
(y) (z)
ηl χl
X
l
l
χk (bk + b†k )
(z)
k
χk (bk − b†k )
(y)
k
(A.15)
All terms in (A.15) are neglected in (A.10) due to the truncated operators σy (B) and
σz (B). Obviously the last two terms in (A.15) would yield additional contributions to
the flow of σ˜x (B) since such terms were already contained in the ansatz (6.25) for σx (B).
Conclusion
We analyzed the relation [σy (B), σz (B)] = 2iσx for operators σi (B), i = x, y, z with
ansätze that are truncated after linear order in the couplings λk . We finally conclude
that this relation is not fulfilled exactly. It was shown that the relation [σy (B), σz (B)] =
2iσx is violated up to normal ordered operators that are at least in O(λ2k ) of the
couplings (cf. A.15). The latter property holds in general for the relation [σi , σj ] =
2iijk σk . For weak couplings α 1 it can be expected that the SU(2) algebra is fulfilled
in good approximation.
170
A.3
Properties of flow equations for spin operators
Analytical treatment of time-dependent flow equations
In section 7.3 the result of an analytical treatment of the time-dependent flow equations
(7.15) was presented, leading to the results (7.35) and (7.36). Using the analogy of the
time-dependent flow equations (7.15) for the observable σz to their time-independent
formulation (6.23), we give a derivation of these results. It will become clear that these
coefficients can be obtained in a completely analogous way for the observable σy (0, t).
Analogy between time-dependent and time-independent flow equations
In order to illuminate the relations between the time-independent and the time-dependent
flow equations for the observable σz , we briefly repeat some basic definitions. In section
7.1, the ansatz for the transformed observable σz (B, t) was given as
σz (B, t) = y(B, t)σy + z(B, t)σz + iσx
X
αk (B, t)(bk − b†k ) + σx
X
ᾱk (B, t)(bk + b†k )
k
k
(A.16)
In the limit B → ∞ it has to obey a condition that was given in (7.7).
σz (∞, t) = σx
X
χk (∞) cos(ωk t)(bk − b†k )
(z)
(A.17)
k
This condition was derived with the time-independent ansatz σz (B) (cf. section 7.1)
σz (B) = z(B)σz + σx
X
χk (B)(bk + b†k )
(z)
(A.18)
k
that was used to solve the Heisenberg equation in the diagonal basis.
Any solution σz (B) of the time-independent flow equations (6.23) for the ansatz (A.18)
is also a solution of the time-dependent flow equations (7.15) (cf. section 7.1). We exploit this relationship and construct a solution for the coupling coefficient z(0,t) out of
particular solutions of the time-independent flow equations. For convenience we define
a mapping that allows us to represent solutions of the time-dependent flow equations
(7.15) as real vectors. This vector representation is suitable to formulate superpositions
of particular solutions and also to define orthogonality relations.
Mapping onto a real vector space
The full operator space O = OS ⊗ FB space of system and bath degrees of freedom
can be decomposed into the operator space OS of a spin 12 and the Fock space FB of
all bosonic bath modes. We assume that all of the operators σy , σz , iσx (bk − b†k ) and
A.3 Analytical treatment of time-dependent flow equations
171
σx (bk + b†k ) in (A.16) are elements of an operator basis of the operator space O. 1
The subspace that is spanned by these operators is defined as Oσz . In this way, the
transformed operator σz (B, t) will be element of the subspace Oσz . It is convenient to
identify operators of this subspace with real vectors. For this purpose we now define
a canonical mapping from Hσz onto a real vector space V. The mapping is defined by
identifying the operator (A.16) with a real vector in the following way:


z(B, t)
ᾱ1 (B, t)




..


.


ᾱk (B, t)




..
X
X


.
†
†

y(B, t)σy +z(B, t)σz +iσx
αk (B, t)(bk −bk )+σx
ᾱk (B, t)(bk +bk ) 7−→ 
 y(B, t) 


k
k
α1 (B, t)




..


.


αk (B, t)


..
.
(A.19)
As a further convention we denote any solution opertator σz (B, t) of time-dependent
flow equations by the corresponding real vector ~v (B, t).
Vector representation of a solution σz (B, t)
We want to obtain an analytical solution for σz (0, t). As we shall see, this is only
possible for the most important coefficients y(0,t) and z(0,t). We give the derivation
only for the coefficient z(0,t) since the coefficient y(0,t) can be obtained in a completely
analogous way. According to the vector representation (A.19) the coefficient z(0,t) can
be decomposed from the full solution vector ~v (0, t) in the following way:

 
0
z(0, t)
 0  ᾱ1 (0, t)


 
 0   .. 




~v (0, t) = 
+ . 
.
.
 .  ᾱk (0, t)
 


..
..
.
.

(A.20)
With respect to the standard scalar product the vector decomposition (A.20) is orthogonal. Our aim is to represent the vector equation (A.20) in the diagonal basis where
we want to eliminate the representation of the unknown vector
1
We assume existence of this basis and do not care about uniqueness. Eg, a basis for the operator
space O of this type has been given in [66].
172
Properties of flow equations for spin operators


0
ᾱ1 (0, t)


 .. 
 . 


ᾱk (0, t)


..
.
(A.21)
which we denote by the vector µ
~ (t). From the introduction of this section we know
P (z)
that in the diagonal basis the operator σz is representated as k χk (bk + b†k ) and the
operator σz (∞, t) is representated according to equation (A.17). With these operators
in mind, we reformulate the vector equation (A.20) in the diagonal basis representation.



0
0
χ(z) (∞)
χ(z) (∞) cos(ω1 t)
 1

 1




..
..
! 




.
.
z(0, t) 
~ (t) = 
+µ

χ(z) (∞)
χ(z) (∞) cos(ω t)
 k

 k
k 
..
..
.
.

(A.22)
In order to eliminate the unknown vector µ
~ (t) from equation (A.22), we will assume
that it is orthogonal to the vector

0
χ(z) (∞)
 1



..


.


χ(z) (∞)
 k

..
.

(A.23)
Before we do so, we discuss the justification of this assumption.
Assumption of unitary
All flow equations derived for the spin-boson model use truncations of operator expansions. It cannot be assumed that these flow equations establish unitary transformations
in the strict sense. We make an approximation and assume in the following that all
flow equations for spin operators lead to unitary transformations in the vector space
V. 2
In consequence the vector decomposition in equation (A.22) will be orthogonal since the
orthogonal vectors from (A.20) were mapped unitarily onto these vectors. Thus, we are
2
If flow equations shall map unitary in V it is necessary that also the mapping from the operator
space O to the vector space V defined in (A.19) is unitary. It can be easily extended to a unitary
mapping by a suitable definition of a scalar product in the operator space O.
A.3 Analytical treatment of time-dependent flow equations
173
finally able to eliminate the unknown vector µ
~ (t) from equation (A.22) by calculating
the scalar product of eq. (A.22) with the vector

0
χ(z) (∞)
 1



.
.


.


χ(z) (∞)
 k

..
.

(A.24)
As a result the coefficient function z(0,t) is now expressed as a series in the couplings
(z)
χk (∞).
z(0, t) =
X
(z)2
χk
(∞) cos(ωk t)
(A.25)
k
P (z)2
In the expression (A.25) we made use of the sum rule k χk (∞) = 1 (cf. section 6.5).
Note that the coefficients y(0,t) and z(0,t) can be obtained for the observable σy (0, t)
in a completely analogous way since they are solutions of the same time-dependent flow
equations (7.15). Eg. it is again possible to project these coefficients out of the full
solution by orthogonality relations.
174
Properties of flow equations for spin operators
Appendix B
Conditions for an equivalence of
two initial preparations
In section (6.4) two product initial states were discussed that prepare the two state
system in a pure initial state. In an experiment the bath might have either come into
equilibrium with the pure state or it is decoupled from the two state system and left
in its thermodynamical equilibrium. This appendix intends to give an exact criterion
under which assumptions these two different initial states | IAi and | IBi lead to the
same dynamics.
Within the path integral formalism, all effects of the bath preparation onto the dynamics of the system are completely contained in so called influence functionals. These
are functionals of the system path. Formal representations of physical quantities of a
dissipative quantum system can be written as functional integrals over all system paths
where the functional contains all environmental influences via the influence functional
[9,30]. The initial states | IAi and | IBi have influence functionals that differ by a
time-dependent phase factor [9]:
1Z t
d
exp i
dt0 [σ(t0 ) − σ 0 (t0 )] Q00 (t0 )
2 0
dt
(B.1)
The above phase factor will be integrated over all spin paths σ(t0 ) and σ 0 (t0 ) of the two
state system. A spin path σ(t0 ) jumps forth and back between the values -1 and +1.
In t=0, the preparations A and B lead to the constraint σ(t = 0) = σ 0 (t = 0).
Besides from the spin paths the phase factor (B.1) depends also on the imaginary part
Q00 (t) of the bath correlation function Q(t) [9].
00
Z
Q (t) =
dω
0
ω
∞
J(ω)
sin(ωt)
ω2
(B.2)
For an Ohmic bath J(ω) = 2αωe− ωc the function Q00 (t) can be evaluated in closed
form.
176
Conditions for an equivalence of two initial preparations
Q00 (t) = 2α arctanh(ωc t)
(B.3)
We will show that the exponent of (B.1) vanishes for the physical relevant spin paths in
α
−1
r
the scaling limit ∆
ωc → 0, ∆r fixed proportional to ωc . After a transient time of O(ωc )
both influence functionals will coincide. This transient time is decreased proportional
to the damping strength α.
We approximate the phase of the exponent (B.1) by
Z t
d
α (σ(t0 ) − σ 0 (t0 )) 0 arctanh(ωc t0 )dt0
dt
t0
(B.4)
This amounts in neglecting spin flip processes for times t t0 . Spin flip processes are
−1
−1
relevant only on time scales t ≥ O(∆−1
r ). We set ωc t0 ∆r in order to neglect
also band edge effects. An asymptotic expansion of the term (B.4) in ωc1t0 shows that
it vanishes in the limit lim ωc t0 → ∞.
Z t
d
2α |
(σ(t0 ) − σ 0 (t0 )) 0 arctanh(ωc t0 ) |
dt
t0
h 1
1 i
≤ 4α(arctanh(ωc t) − arctanh(ωc t0 )) ≤ 4α
−
+ O(ωc t0 )−2
(B.5)
ω c t0 ω c t
r
We finally conclude that the phase factor vanishes in the scaling limit ∆
ωc → 0, ∆r fixed
under neglection of fast spin flip processes σ(t = 0) = 1 −→ σ(t = t0 ) = −1 on a time
scale t0 ∆−1
r .
Appendix C
Used numerical approaches
C.1
Numerical approaches to solve flow equations
Two different types of flow equations turned out to be not solvable by analytical means.
Several numerical results were discussed in chapter 7 that are based on a numerical implementation of these flow equations. A given set of flow equations has to be converted
into a formulation of a numerical problem. This numerical problem is conceptionally
very dependent on the distinction between time-dependent and time-independent flow
equations. As we know, time-dependent flow equations have to obey initial conditions
that are obtained by a numerical implementation of the concerning time-independent
flow equations. A solution of time-dependent flow equations is numerically expensive
and we discuss therefore also the aspect of numerical complexity.
For each type of flow equation, we discuss first the numerical problem and then its
implementation.
C.1.1
Time-independent flow equations
Numerical problem
Flow equations without any time-dependence represent coupled differential equations
that describe the flow of all coupling coefficients of a transformed operator O(B). This
transformation aims at a representation of the operator O in the
P diagonal basis. If
eg. the operator O will be transformed into the operator OP= k µk (B = ∞)Ok an
˜ | Ok | psii.
˜
observable is evaluated in diagonal basis as hψ | O | ψi = k µk (∞)hpsi
Therefore the numerical problem is to calculate all coefficients of the transformed operator in the limit B −→ ∞. Formally, for this purpose the underlying flow equations
have to be solved on the interval [0, ∞] of the flow parameter B. In a numerical solution
it is impossible to implement such a limit and the hope is that numerical convergence
178
Used numerical approaches
is achieved fast enough at a finite value of the parameter B. All flow equations depend
on the transformed coupling coefficients of the Hamiltonian. In general the decay of
these couplings during the flow determines how quick a solution of flow equations will
converge to its asymptotic behaviour in the limit B −→ ∞.
Implementation
We used an adaptive step-size 4th order Runge-Kutta algorithm [55] as an adequate
algorithm to reduce the computational time spent for convergence by an adaptive increasing of the integration step size. This algorithm was implemented in the commonly
used language C. It turned out that an algebraic decay ∝ √1B of the couplings leads
to a very slow convergence for operators of the DHO. In the next section we present
a diagonalization procedure for the DHO using a linear mapping that is much less expensive.
Only a small fraction of computational time -in comparison to the DHO- has to be
spent for an integration of flow equations for the spin-boson model since its couplings
decay asymptotically exponentially like λk (B) ∝ exp(−B(ωk − ∆∞ )2 ). Only the resonant couplings with ωk ≈ ∆∞ with little spectral weight decay algebraically.
We formulated the numerical problem and chose a method of implementation on a computer. Furthermore, we add some details that illuminate the numerical complexity and
essential aspects that influence the accuracy of a numerical implementation. From here
on we restrict our discussion to the more important flow equations for the spin-boson
model.
When Pauli matrices are transformed, couplings to bath modes have always derivatives
that are proportional to λk (B). For nonresonant couplings their flow is asymptotically
proportional to λk (B) ∝ exp(−B(ωk − ∆∞ )2 ). It is possible to set these derivatives
equal zero if −B(ωk − ∆∞ )2 > 5 since they will have already decayed to less then
1% of their initial values. The smallest energy difference that has to be decoupled
during the flow is the bath discretization ∆ω = ωNc . Resonant couplings decay only
algebraically and their spectral weight increases strongly during the flow. A safety
2
factor of 1000 multiplied to the squared inverse of the smallest energy difference ωNc
leads to a convergency of transformed operators with an accuracy of less then 0.1%
error in its coefficients.
It is therefore sufficient to integrate all flow equations only up
2
to B = 1000 ωNc .
Numerically most expensive is an evaluation of the derivative for all couplings. The
numerical complexity is O(N 2 ) and is expressed through the number N of bath modes.
C.1 Numerical approaches to solve flow equations
179
On a standard workstation flow equations for 1000 bath modes can be integrated in
less than one hour. It is important to increase the number of bath modes up to a
value when convergence of physical observables is observed. For the spin-boson model
it turned out that results for 500 bath modes differ from calculations with only 200
modes by less than 1% relative error.
The numerical problem related to an integration of time-independent flow equations is
slightly different. Its implementation is based on the implementation of time-independent
flow equations and includes some additional algorithmic details.
C.1.2
Time-dependent flow equations
Numerical problem
Time-dependent flow equations are in general related to a transformation of a timedependent operator O(∞, t). This operator is obtained by a solution of
Pthe Heisenberg equation in diagonal basis for the transformed operator O(∞) = k µk (∞)Ok .
The
P time-evolved operator O(∞, t) is transformed according to an ansatz O(B, t) =
k µk (B, t)Ok . Its coefficients are the solution of flow equations. Obviously this leads
to a different formulation of the numerical problem. A solution for the coefficients
µk (B, t) for B=0 has to be calculated. Therefore a parameter dependent set of coupled
differential equations has to be solved. For every single point in time this parameter
changes and leads to a different set of coupled differential equations. The solution of
these differential equations is determined by the asymptotic behaviour limB→∞ µk (B, t)
that follows from the known operator O(∞, t). Obviously the numerical complexity
has now in addition to be multiplied with the number of time steps that discretize the
parametrization of time.
Implementation
We use the same Runge Kutta algorithm as above to integrate the time-dependent
flow equations. For any time t the initial condition for the integration of the coeffi2
cients µk (B, t) is set as µk (1000 ωNc , t). These coefficients depend -apart from exactly
known physical parameters- only on the coefficients µk (∞). After solving the timeindependent flow equations the coefficients µk (∞) are stored in a file and linked to
the time-dependent algorithm via data import. We integrate µk (B, t) therefore on the
2
intervall [0, 1000 ωNc ] since µk (B) converged well at this upper bound. The time
dependence of the flow equations is implemented within an additional loop that discretizes the time axis.
180
Used numerical approaches
Figure C.1: The observable P (t) = hσz (t)i is depicted on the time scale of the inverse
level spacing frequency 2πN
ωc . We discretized an Ohmic bath with 200 and 500 bath
modes up to the cut off ωc = 10 for a weak damping strength α = 0.025.
In all numerical implementations we used a constant frequency spacing ∆ω = ωNc for the
bath discretization. It is important to note that eg. the operators σz (∞, t) and σy (∞, t)
are then periodic in time with periodicity τ = 2π
ωc . Slightly, different, the operator
σx (∞, t)leads to observables that represent uniformly almost periodic functions (cf.
Besicovitch [67]). Consequently for any ≥ 0 one can find a time T > 0 such that any
interval of length T contains at least one translation time τ . These translation times
are defined by
sup
−∞<t<∞
| hσx i(t + τ ) − hσx i(t) |≤ (C.1)
Since only the renormalized system mode deviates from the constant frequency spacing
of the bath modes σx (∞, t) will be nearly periodic in the inverse level spacing ∆ω −1 .
We discretize the time axis from t=0 up to a time of order τ . This inverse level spacing
of the bath discretization plays, roughly spoken the role of the period of hOi(t) and can
be compared with the Poincaré recurrence time. This last one is defined as the time
in which a dynamical system recurs within a desired degree of accuracy. In the given
system this period tends to infinity if the distance between the frequencies ωk tends
to zero. Using time-dependent flow equations, all exact periodicities in time can be
diminished by two effects. Firstly, time translational invariant operators are not transformed exactly unitary by time-dependent flow equations of the spin-boson model. In
addition, small numerical errors in the time-dependent coupling coefficients µk (B, t)
could have significant effects on a periodic time-dependence. In figure C.1 we depicted
periodicity effects for the observable hσz (t)i in the initial state | IBi. This observable
shows an accurate periodicity on exactly the time scale of the inverse level spacing ∆ω
what has been confirmed by two different numbers N of bath modes.
C.1 Numerical approaches to solve flow equations
181
Figure C.2: The routine forspinboson.c integrates time-independent flow equations and
stores the flow of the coupling constants in the file flow.dat. The routine stiff adaptiv.c
reads the file flow.dat and uses the data to integrate the time-dependent flow equations.
In principle the adaptive step size algorithm can be used to integrate time-dependent
2
to B=0. It turned out that more than 50% of the
flow equations from B = 1000 ωNc
couplings λk (B) decayed already to less than the minimum double precision of 2·10−308
2
when convergence of all coupling constants has been reached at B = 1000 ωNc . This
leads to large relative numerical errors in the evaluation of derivatives. We implemented
a much more accurate integration routine that uses the previously calculated flow of
the couplings λk (B), the tunneling matrix element ∆(B) and the discrete steps of the
flow parameter B. During a solution of the time-independent flow equations after every
adaptive step these functions are stored in matrices and finally exported into a file.
The discretized flow of λk (B), ∆(B) and B itself is then read in via data import before
the integration routine for the time-dependent flow equations starts (compare figure
C.2). Instead of calculating the flow of ∆(B), λk (B) and B via adaptive stepping and
an evaluation of their derivatives, it is taken from the stored file.
The adaptive stepsize routine from [55] was therefore rewritten into a routine that uses
a stiff integration mesh. Its discretization is read in from the stored flow of the parameter B. Accuracy of this approach has been tested by an integration of the flow
equations in t=0, since it can be checked against exact initial conditions in B=0 that
were used for the inverse integration routine. More accurate results were achieved by
switching from the stiff integration mesh back to the adaptive routine at approximately
182
Used numerical approaches
a flow parameter B = ∆−2
∞ that compares to the inverse squared resonance frequency.
Accuracy
An exact test of accuracy is eg. to transform σz with the routine forspinboson.c and
transform this observable back for t=0 with the routine stiff adaptiv.c. The coefficient
z(B=0,t=0) from the ansatz (7.11) should be equal one after an application of these
routines. Tests show that the numerical error depends on damping strength. For dampings α ≤ 0.15 the error was reduced to less than 10−3 for small enough discretization
of all adaptive step sizes.
C.2
Normal mode transformation of the DHO
In sections 5.3 and 5.4 we made use of a normal mode transformation in order to
calculate time-dependent physical observables for a given initial state in physical basis.
In the normal mode representation, the Hamiltonian will be of the form
N
X
H=
ω̄k a†k ak
(C.2)
k=0
All definitions for the normal mode transformation were already stated in section 5.3.
C.2.1
Definition of the normal mode transformation
The normal modes are related to the physical modes via
def
ak =
N
X
Ank bk + Bnk b†k
(C.3)
n=0
with real matrices A and B. An inverse transformation is defined as
def
bk =
N
X
Ckn an + Dkn a†n
(C.4)
n=0
with real matrices C and D. A similar transformation has been formulated already by
Ullersma [5]. It transforms momenta and positions instead of creators and annihilators.
Using these definitions, it is easy to express the time-dependence of the system operator
b(t). We use definition (5.38) to express b through the operators an and a†n . These
operators are eigenoperators of the operation of commuting them with the Hamiltonian.
[H, an ] = −ω̄n an , 0 ≤ n ≤ N
(C.5)
[H, a†n ] = ω̄n a†n , 0 ≤ n ≤ N
(C.6)
C.2 Normal mode transformation of the DHO
183
These eigenoperators lead to the trivial solution an (t) = e−iω̄n t an of the Heisenberg
equation for t ≥ 0. After inserting the time-dependent eigenoperators into (C.4) and
reexpressing them through physical operators via (5.39), a formal solution for b(t) is
obtained.
b(t) =
N X
C0n e−iω̄n t
n=0
hX
i
hX
i
Ank bk + Bnk b†k + D0n eiω̄n t
Ank b†k + Bnk bk
k
(C.7)
k
All physical observables related to position and momentum of a particle in a given
initial state can be calculated using the operator (C.7). The matrix elements of all
matrices A,B,C,D and the eigenfrequencies ω̄n need to be determined from an additional
eigenvalue problem.
C.2.2
Solution of an eigenvalue problem
Firstly, we want to calculate the renormalized eigenfrequencies ω̄n . These normal frequencies can in principle be obtained by a typical eigenvalue problem. We postpone the
formulation of an eigenvalue problem to the actual calculation of the matrix elements
of the matrices A,B,C,D that can be obtained as eigenvectors of a real matrix.
Calculation of the normal frequencies ω̄n
Our strategy is to calculate the commutator [H, b†k ] for the equilibrium Hamiltonian ()
in two different ways and then to compare the coefficients of all bosonic operators.
1. Firstly, we calculate [H, b†k ] directly, using only bosonic operators from the physical basis. To enable a comparison with a second representation of the commutator,
the result is expressed through eigenoperators via (5.38).
2. The second possibility is to commute the ansatz (5.39) with H.
These strategies are implemented in equations (C.8) and (C,9) for all bath operators
bk , k ≥ 1.
[H, b†k ] = ωk b†k +λk (b+b† ) =
N n
o
X
ωk Ckn a†n +ωk Dkn an +λk (C0n +D0n )(an +a†n ) (C.8)
n=0
[H,
N
X
n=0
(Ckn a†n + Dkn an )] =
N
X
(ω̄n Ckn a†n − ω̄n Dkn an )
(C.9)
n=0
We compare the coefficients of all eigenoperators ak in C.8 and C.9 and obtain two
matrix relations.
184
Used numerical approaches
λk
(C0,n + D0,n ), 1 ≤ k ≤ N
ω̄n − ωk
λk
(C0,n + D0,n ), 1 ≤ k ≤ N
=−
ω̄n + ωk
Ckn =
Dkn
(C.10)
An analogous treatment of the commutator [H, b] yields two additional matrix relations.
PN
C0n =
k=1 λk (Cn,k
+ Dn,k )
ω̄n − ∆0
k=1 λk (Cn,k + Dn,k )
ω̄n + ∆0
PN
D0n = −
(C.11)
We insert Cnk and Dnk from equations (C.10) into the sum of equations (C.11) to
eliminate all matrix elements. The result is a transcendent equation for the normal
frequencies ω̄n .
N
X
∆20 − ω̄n2
2ωk
=
λ2k 2
2∆0
ωk − ω̄n2
(C.12)
k=1
If the boundary condition (4.17) is obeyed, equation (C.12) posesses N+1 positive solutions ω̄n2 and determines all normal frequencies. We solved equation (C.12) numerically.
Formulation of the eigenvalue problem
Formally, we have to find a linear combination of bosonic operators that represents
an eigenoperator of the operation of commutation with the Hamiltonian H. In order
to solve the eigenvalue problem, it is useful to define a matrix representation of the
superoperator [·, H] that commutes operators in the full Fock space of system and
bath modes with the Hamiltonian. By convenience, the representation is given in the
physical basis of system and bath operators. Each bosonic bath mode is mapped onto
a canonical basis vector of the real vector space RN (cf. table (C.1)). The resultant
matrix is diagonal up to transitions from system operators to bath operators.
C.2 Normal mode transformation of the DHO
operator
basis vector
b0
~e1
...
...
bN
b†0
~eN +1
~eN +2
185
...
...
b†N
~e2N +2
Table C.1: In each column an operator is identified with a basis vector in a real vector
space of dimension 2N+2. Creators and annihilators are linear independent objects in
the full bosonic Fock space of system and bath modes.

−∆0 −λ1 . . . −λN




−λ1 −ω1




..
..


.
.



−λN
−ωN
M=
−λ1 . . . −λN
 0




−λ
1



..


.



−λN
0
λ1
..
.

λ1 . . . λ N 












λN
∆0 λ1 . . . λ N
λ1 ω1
..
..
.
.
λN
ωN













Dotted lines in the matrix M abbreviate coefficients that have to be distinguished only
by their indices. All matrix elements that are not occupied with any symbol are equal
zero.
P
†
The superoperator [·, H] has exactly the eigenoperators an = N
n=0 Ank bk + Bnk bk . We
conclude that the set of eigenvectors of the matrix M will be given by the 2N+2 vectors



Bn0
An0
 .. 
 .. 
 . 
 . 






AnN 

 , BnN 
 Bn0 
 An0 




 .. 
 .. 
 . 
 . 
BnN
AnN

0≤n≤N
(C.13)
The highly sparsed structure of the matrix M allows to formulate simple analytical
expressions that determine all eigenvectors. We insert the eigenvectors (C.13) into the
homogenous equation M − ω̄n id = 0 and obtain 2N+1 equations.
186
Used numerical approaches
∆0 − ω̄n
∆0 + ω̄n
∆0 + ω̄n
λ1
(1 +
)Bn,0
=−
ω1 + ω̄n
∆0 − ω̄n
..
.
Bn,0 = −An,0
Bn,1
λN
(1 +
ωN + ω̄n
λ1
(1 +
=
ω1 − ω̄n
Bn,N = −
An,1
An,N =
∆0 + ω̄n
)Bn,0
∆0 − ω̄n
∆0 + ω̄n
)Bn,0
∆0 − ω̄n
..
.
λN
∆0 + ω̄n
(1 +
)Bn,0
ωN − ω̄n
∆0 − ω̄n
(C.14)
These equations determine all eigenvectors of the matrix M. In order to fulfil the bosonic
commutation relation [an , a†n ] = 1 of the eigenoperators, these eigenvectors have to fulfil
a normalization condition.
2
2
[an , a†n ] = 1 ⇔ A2n,0 + ... + A2n,N − Bn,0
− ... − Bn,N
=1
(C.15)
From the equation array (C.14) we conclude that the bosonic commutation relation is
fulfilled if Bn,0 is chosen as
1
Bn,0 = q
∆0 +ω̄n 2
(∆
) − 1 + (1 +
0 −ω̄n
λk
∆0 +ω̄n 2 PN
2
k=1 {( ωk −ω̄n )
∆0 −ω̄n )
k
− ( ωkλ+ω̄
)2 }
n
(C.16)
Up to now the transformation matrices C and D are not determined, yet they contribute in the expression (C.7). We will show that they can easily be obtained from
the matrices A and B.
Relations between the matrices A,B and C,D.
We substitute the physical modes in the expression bk + b†k by the expression (C.4) and
reexpress all normal modes using the inverse transformation (C.3). In the resultant
expression
bk +
b†k
=
N X
N
X
{(Ck,n + Dk,n )(An,p + Bn,p )(bp + b†p )} ; 0 ≤ k ≤ N
p=0 n=0
we compare all coefficients and obtain a matrix relation.
C.2 Normal mode transformation of the DHO
N
X
(Ck,n + Dk,n )(An,p + Bn,p ) = δk,p
187
(C.17)
n=0
In an exact analogy we rearrange the representation for bk − b†k and obtain
N
X
(Ck,n − Dk,n )(An,p − Bn,p ) = δk,p
(C.18)
n=0
We conclude that A-B and A+B are the inverse matrices of C-D and C+D, respectively.
In addition, we also rearrange the commutation relations
[bk , b†k0 ] = δk,k0 ,
[bk , bk0 ] = 0
(C.19)
Two helpful matrix identities can be derived.
N
X
(Ck,n Ck0 ,n − Dk,n Dk0 ,n ) = δk,k0 ⇔ CC † − DD† = 1
(C.20)
n=0
N
X
(Ck,n Dk0 ,n − Dk,n Ck0 ,n ) = 0 ⇔ CD† − DC † = 0
(C.21)
n=0
Finally, we combine (C.20) and (C.21) to obtain two matrix identities that are similar
to equations (C.17) and (C.18).
(C + D)(C † − D† ) = (C − D)(C † + D† ) = 1
(C.22)
Obviously equations (C.17) and (C.18) can be related to equation (C.22) in order to
prove that
C = A†
,
D = −B †
(C.23)
All transformation matrices are now determined.
C.2.3
Calculation of physical quantities
We evaluate the correlator Cqq (t, tw ) = hx(t + tw )x(tw )i with respect to the initial state
| Ii =| αi⊗ | 0i ⊗ ...⊗ | 0i and express the result using the formal expression (C.7) for
the operator b(t).
188
Used numerical approaches
Cqq (t, tw ) =
N
1 X
[C0,n + D0,n ][C0,n0 + D0,n0 ] ×
2 0
n,n =0
N
X
−Ck,n Dk,n0 e−i[(ω̄n0 +ω̄n )tw +ω̄n t] − Dk,n Ck,n0 ei[(ω̄n0 +ω̄n )tw +ω̄n t] +
k=0
Ck,n Ck,n0 ei[(ω̄n0 −ω̄n )tw ]−ω̄n t] + Dk,n Dk,n0 ei[(−ω̄n0 +ω̄n )tw +ω̄n t] +
!
4α2 [C0,n C0,n0 − C0,n D0,n0 − D0,n C0,n0 + D0,n D0,n0 ] cos(ω̄n (t + tw )) cos(ω̄n0 tw ) (C.24)
Again, this correlator represents a uniformly almost periodical function (cf. Besicovitch
[67] and Ullersma [5]). The normal frequencies are only slightly renormalized with
respect to the bath mode frequencies due to a N1 effect (cf. (4.15)). At time scales of
the averaged inverse normal mode spacing ωNc unphysical periodicity effects show up.
C.2.4
Checks of numerical accuracy
We employ two tests against exact results in order to check the accuracy of all numerical calculations that made use of the normal mode transformation. In section 5.3 we
derived the exact analytical result (5.36) for the expectation value hx(t)i with respect
to a coherent initial state. By employing the representation (C.7) for the operator b(t)
this result can be expressed through the matrix elements of the normal mode representation. A comparison to the exact result (5.36) shows on which time scale periodicity
effects can be neglected. In addition it can be checked if the result converges on shorter
time scales with an increasing number of bath modes. All errors in the time domain
will influence the accuracy of Fourier transformations. Eg if the time axis is cut off at
the recurrence time scale ωNc the low-frequency behaviour of the Fourier transformed
function has to be checked.
Accuracy in the time domain
We use the exact result (5.36) for the expectation value hx(t)i and compare it against
the numerical solution obtained from the operator representation (C.7) of b(t). The
comparison of both results is depicted in figure (C.3) for the damping strength α = 0.01
and the tunneling matrix element ∆0 = 1. For the numerical result we discretized an
Ohmic bath with 200 modes up to a cut off frequency ωc = 10. The eigenvalue of
the coherent state was adjusted to the initial value x( t = 0)) =0 .5. We observe excellent
agreement to the analytical result with a relative error below 1% for times t ≤ 50∆−1
0
that increases up to 2% at a time t = 100∆−1
0 (inset). This time corresponds approximately to the inverse bath discretization frequency 200 2π
ωc .
C.2 Normal mode transformation of the DHO
189
Figure C.3: Test of the normal mode transformation routine against the exact result
Figure C.4: Check of a numerical Fourier transform calculated via a normal mode
transformation against the exact result
190
Used numerical approaches
Accuracy in the frequency domain
We use the function hx(t)i obtained via a normal mode transformation and calculate its
Fourier transform numerically. The result is checked against the exact Fourier transform (5.37). All coefficients were chosen the same as in the time domain test, eg. the
bath was discretized with 200 bath modes. In order to perform the Fourier transformation, we integrated the time domain up to the inverse bath frequency 200 2π
ωc .
We observe an excellent accuracy also in the frequency domain (figure (C.4)). The
relative error remains below 1% in the peak region where hxi(ω) remains below half of
its maximum value. The relative error increases up to 10% in the spectral range where
ω
ω
−3 absolute value.
∆0 < 0.2 or ∆0 < 0.2. In ω = 0 the offset remains well below 10
Thus, we conclude that the offset in the violation of the FDT (cf. section 5.3) in ω = 0
is of physical origin since it exceeds an absolute value of 0.02 for several waiting times
tw and two different damping strengths.
Appendix D
Evaluation of a time ordered
commutator
We present all steps for the exact evaluation of the expression
Ũ † x̃Ũ = x̃ + Ũ † [x̃, Ũ ]
(D.1)
where the propagator Ũ the position operator x̃ and the Hamiltonian H̃(t) are representated in the diagonal basis.
Ũ = T> e−i
H̃(t) = ∆∞ b† b +
Rt
X
0
H̃(t0 )
ωk b†k bk + x̃h(t)
k
1 X
x̃ = √
2αk [bk + b†k ]
2 k
For our calculation, it suffices to evaluate the commutator [x̃, Ũ ]. For this purpose, it
is useful to employ the commutator
1 X
[x̃, H̃(t)] = √
2αk ωk [bk − b†k ]
2 k
(D.2)
Firstly, we rearrange Ũ into a series representation
T> e−i
Rt
0
H̃(t0 )dt0
Z
Z 0
(−i)2 t 0 t 00
dt
dt H(t0 )H(t00 ) +
2!
0
0
0
Z
Z 0
Z 00
(−i)3 t 0 t 00 t
dt
dt
dt000 H(t0 )H(t00 )H(t000 ) + ...
3!
0
0
0
Z
= 1 + (−i)
t
H(t0 )dt0 +
(D.3)
192
Evaluation of a time ordered commutator
This series representation is used to calculate the commutator [x̃, Ũ ]. We commute the
operator x̃ successively past the operator H̃(t) and employ (D.2).
[x̃, T> e−i
Rt
H̃(t0 )dt0
]=
Z t
X
1
√
[2αk ] (−i)
dt0 [ωk + H(t0 )][bk − b†k ] +
2 k
0
(−i)2
2
Z
t
0
Z
t0
dt
0
(−i)3
3!
Z
0
t
0
dt00 {ωk2 + ωk [H(t0 ) + H(t00 )] + H(t0 )H(t00 )}[bk + b†k ] +
Z
t0
Z
t00
h
dt000 ωk3 + ωk2 [H(t0 ) + H(t00 ) + H(t000 )] +
0
0
0
i
†
0
00
0
000
00
000
0
00
000
ωk [H(t )H(t ) + H(t )H(t ) + H(t )H(t )] + H(t )H(t )H(t ) [bk − bk ] + ...
0
dt
dt
00
(D.4)
All contributions in (D.4) can be rearranged as components of two exponential series
and finally lead to
i
Rt
√ X h −iω t
0
0
[x̃, T> e−i 0 H̃(t )dt ] = 2
αk (e k − 1)bk + (eiωk t − 1)b†k
(D.5)
k
List of Tables
4.1
Definition of the physical and the diagonal basis in Fock space . . . . .
32
5.1
Definition of physical and normal modes . . . . . . . . . . . . . . . . . .
62
6.1
Sum rules for σy and σz in ground state . . . . . . . . . . . . . . . . . . 113
7.1
The field dependent expectation value hσx i . . . . . . . . . . . . . . . . 149
C.1 Matrix representation of a superoperator. . . . . . . . . . . . . . . . . . 185
194
LIST OF TABLES
List of Figures
3.1
Algorithm to solve Heisenberg equation by means of flow equations . . .
23
4.1
4.2
4.3
Sketch of a dissipative harmonic oscillator exposed to an external field .
The function K(ω) for different bath types . . . . . . . . . . . . . . . .
Sketch of the algorithm to solve Heisenberg equations by means of flow
equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
39
Decay of the observable hx(t)i for an entangled initial state . . . . . . .
The symmetrized equilibrium correlation function Sqq (t) of the DHO . .
Decay of hx(t)i for a coherent state . . . . . . . . . . . . . . . . . . . . .
cum (t, t ) for coherent states . . . . . . . . . .
The correlation function Sqq
w
Analytical result for the violation of the FDT in tw = 0 . . . . . . . . .
Violation of the FDT for coherent states . . . . . . . . . . . . . . . . . .
Time-dependent population of the central oscillator for a coherent state
The real part (dispersion) of the dynamical response function χqq (ω) of
the DHO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison of the observable hx(t)i for an entangled initial state and a
product initial state . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
54
60
64
66
67
72
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
6.1
6.2
6.3
A double-well system in the ”two-state” limit . . . . . . . . . . .
!
Flow of the sum rule hσx2 (B)iGS = 1 . . . . . . . . . . . . . . . .
Comparison of the ground state expectation value hσx iGS against
data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1
7.2
Algorithm to solve Heisenberg equation by means of flow equations . . .
The equilibrium correlation function Sxx (t) in dependence of cut off frequency ωc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The current correlation function Syy (t) for different damping strengths .
Test of an analytical solution of time-dependent flow equations . . . . .
The expectation value hσx (t)i in the parameter space of damping strength
α and bath cut off frequency ωc . . . . . . . . . . . . . . . . . . . . . . .
Comparison of the observable hσz (t)i to results from the NIBA. . . . . .
Comparison of the observable hσz (t)i for a localized and a delocalized
initial spin preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3
7.4
7.5
7.6
7.7
42
79
81
. . . . 84
. . . . 117
NRG
. . . . 118
120
130
131
136
138
140
142
196
LIST OF FIGURES
7.8
7.9
Three dimensional visualization of a decaying Bloch vector . . . . . . . 151
The non-equilibrium correlation function Szz (t, tw ) for a product initial
state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
7.10 The observable hσz (t)i for an entangled initial state . . . . . . . . . . . . 152
8.1
8.2
8.3
The spin-spin correlation function Szz (ω = 0, tw ) for the time-dependent
Kondo model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
Violation of the FDT at the Toulouse point of the Kondo model . . . . 162
Time-dependent behaviour of the effective temperature Teff at the Toulouse
point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
C.1 Periodicity effects for finite numbers of bath modes . . . . . . . . . . . .
C.2 A sketch of the numerical algorithm used to integrate time-dependent
flow equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.3 Test of the normal mode transformation routine against the exact result
C.4 Check of a numerical Fourier transform calculated via a normal mode
transformation against the exact result . . . . . . . . . . . . . . . . . . .
180
181
189
189
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ERKLÄRUNG
Hiermit erkläre ich, dass ich die vorliegende Arbeit selbständig und nur unter Verwendung der angegebenen Quellen und Hilfsmittel angefertigt habe.
Diese Arbeit hat in gleicher oder ähnlicher Form noch keiner anderen Prüfungsbehörde
vorgelegen.
Augsburg, den 07.06.2006
Andreas Hackl
Danksagung
An dieser Stelle möchte ich mich bei ganz herzlich bei Prof. Dr. Dieter Vollhardt dafür
bedanken, dass er mir durch die Aufnahme an seinem Lehrstuhl die Arbeit an einem
aktuellen und interessanten Forschungsgebiet ermöglicht hat. Über die Bearbeitung
der Diplomarbeit hinaus wurde mir auf seine Initiative mehrfach Unterstützung durch
den SFB 484 zuteil. Neben einer Förderung meiner Arbeit im Rahmen eines Projekts
des SFB 484 wurde mir auch die Teilnahme an der DPG Frühjahrstagung ermöglicht.
In gleichem Maße bedanke ich mich bei Prof. Dr. Stefan Kehrein, der während meiner
Diplomarbeit und des vorangegangen Fortgeschrittenenpraktikums immer Zeit für Fragen und Diskussionen aller Art hatte. Seine langjährige Erfahrung mit der Flußgleichungsmethode hat viel zum Verständnis meines Themas beigetragen. Ganz besonders
gratuliere ich ihm zu seiner Berufung auf eine Professur an der Ludwig-MaximiliansUniversität München.
Weiterhin danke ich allen Mitgliedern des Lehrstuhls für viele interessante Gespräche
und allgemein dafür, dass sie jeden Tag zu einer angenehmen Arbeitsatmosphäre beitragen. Ein spezieller Dank geht an D. Lobaskin der mir die in Kapitel 8 gezeigten Originalabbildungen zur Verfügung gestellt hat.
Nicht zuletzt geht ein Dank an meine Eltern für ihre fortwährende Unterstützung
meines Studiums und vieles mehr.