open quantum system dynamics: applications to decoherence … and may become important for the...

i

Contents

Introduction 1

1 Open Quantum Systems 51.1 Reduced state and its evolution . . . . . . . . . . . . . . . . . . . . . 5

1.1.1 Complete positivity . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Lindblad equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Decoherence 92.1 Gallis-Flemming master equation . . . . . . . . . . . . . . . . . . . . 10

2.1.1 Short wavelength limit . . . . . . . . . . . . . . . . . . . . . . 12

2.1.2 Long wavelength limit . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Rotational Decoherence . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Quantum Brownian Motion 183.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2 The Calderira-Leggett master equation . . . . . . . . . . . . . . . . . 20

3.2.1 Complete positivity problem . . . . . . . . . . . . . . . . . . 23

3.3 Non-Markovian Quantum Brownian motion . . . . . . . . . . . . . 23

3.3.1 The adjoint master equation . . . . . . . . . . . . . . . . . . . 24

3.3.2 The Master Equation for the statistical operator . . . . . . . 27

3.3.3 Complete Positivity . . . . . . . . . . . . . . . . . . . . . . . 29

3.3.4 Time evolution of relevant quantities . . . . . . . . . . . . . 31

3.3.5 Non-Gaussian initial state . . . . . . . . . . . . . . . . . . . . 34

4 Gravitational time dilation 374.1 Model for universal decoherence . . . . . . . . . . . . . . . . . . . . 38

4.2 Heat capacity for gravitational decoherence . . . . . . . . . . . . . . 39

4.3 Competing effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.3.1 Comparison of the effects . . . . . . . . . . . . . . . . . . . . 42

5 Collapse Models 455.1 Continuous Spontaneous Localization Model . . . . . . . . . . . . . 45

5.1.1 Imaginary noise trick . . . . . . . . . . . . . . . . . . . . . . . 49

5.2 Optomechanical probing collapse models . . . . . . . . . . . . . . . 49

5.3 Gravitational wave detectors bound collapse parameters space . . . 53

5.3.1 Interferometric GW detectors: LIGO . . . . . . . . . . . . . . 56

5.3.2 Space-based experiments: LISA Pathfinder . . . . . . . . . . 57

5.3.3 Resonant GW detectors: AURIGA . . . . . . . . . . . . . . . 58

5.4 Ultra-cold cantilever detection of non-thermal excess noise . . . . . 61

5.5 Hypothetical bounds from torsional motion . . . . . . . . . . . . . . 67

5.5.1 Experimental feasibility . . . . . . . . . . . . . . . . . . . . . 69

ii

6 Conclusions 71

Appendices

A Quantum Brownian Motion master equation 74A.1 Explicit form of �(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74A.2 Explicit form of the adjoint master equation . . . . . . . . . . . . . . 74

A.3 Derivation of the master equation for the states . . . . . . . . . . . . 75

A.4 Explicit expression for ⇤dif(t) and E(t) . . . . . . . . . . . . . . . . . 76

B Gravitational time dilation 78

C Collapse Models 80C.1 CSL Diffusion coefficients . . . . . . . . . . . . . . . . . . . . . . . . 80

C.2 Effective frequencies and damping constants . . . . . . . . . . . . . 81

C.3 Cantilever . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

Bibliography 92

iii

List of Publications

Published works as outcome of the doctoral project

1.

A. Vinante, R. Mezzena, P. Falferi, M. Carlesso and A. Bassi.

Improved noninterferometric test of collapse models using ultracold cantilevers.

Physical Review Letters, 119 110401 (2017).Link to paper:

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.119.110401

Link to ArXiv:

https://arxiv.org/abs/1611.09776

The most important contents of this article are reported in Sec. 5.4.

2.

M. Carlesso and A. Bassi.

Adjoint master equation for quantum brownian motion.

Physical Review A, 95 052119 (2017).Link to paper:

https://journals.aps.org/pra/abstract/10.1103/PhysRevA.95.052119

Link to ArXiv:



3.

S. McMillen, M. Brunelli, M. Carlesso, A. Bassi, H. Ulbricht, M. G. A. Paris, and

M. Paternostro.

Quantum-limited estimation of continuous spontaneous localization.

Physical Review A, 95 012132 (2017).Link to paper:

https://journals.aps.org/pra/abstract/10.1103/PhysRevA.95.012132

Link to ArXiv:


https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.119.110401https://arxiv.org/abs/1611.09776https://journals.aps.org/pra/abstract/10.1103/PhysRevA.95.052119https://arxiv.org/abs/1602.05116https://journals.aps.org/pra/abstract/10.1103/PhysRevA.95.012132https://arxiv.org/abs/1606.00070

iv

4.

M. Carlesso, A. Bassi, P. Falferi, and A. Vinante.

Experimental bounds on collapse models from gravitational wave detectors.

Physical Review D, 94 124036 (2016).Link to paper:

https://journals.aps.org/prd/abstract/10.1103/PhysRevD.94.124036

Link to ArXiv:



5.

M. Carlesso and A. Bassi.

Decoherence due to gravitational time dilation: Analysis of competing decoher-

ence effects.

Physics Letters A, 380 (31–32), pp. 2354 – 2358 (2016).Link to paper:

http://www.sciencedirect.com/science/article/pii/S0375960116302407

Link to ArXiv:


The most important contents of this article are reported in Chap. 4.

Pre-prints

6.

M. Carlesso, M. Paternostro, H. Ulbricht, A. Vinante and A. Bassi.

Non-interferometric test of the Continuous Spontaneous Localization model

based on the torsional motion of a cylinder.

ArXiv, 1708.04812 (2017).Link to ArXiv:



7.

M. Carlesso, M. Paternostro, H. Ulbricht and A. Bassi.

When Cavendish meets Feynman: A quantum torsion balance for testing the

quantumness of gravity.

ArXiv, 1710.08695 (2017).Link to ArXiv:


https://journals.aps.org/prd/abstract/10.1103/PhysRevD.94.124036https://arxiv.org/abs/1606.04581http://www.sciencedirect.com/science/article/pii/S0375960116302407https://arxiv.org/abs/1602.01979https://arxiv.org/abs/1708.04812https://arxiv.org/abs/1710.08695

v

List of attended Schools, Workshopsand Conferences

1. September, 2017

Training Workshop at Instituto Superior Tecnico of Lisbon, Portugal

Title Lisbon Training Workshop on Quantum Technologies in Spacehttp://www.qtspace.eu/?q=node/131

Organization Dr. R. Kaltenbaek, Dr. E. Murphy, Dr. J. Leitao and Dr. Y. Omar

2. June, 2017

Workshop at University of Milano, Italy

Title Fundamental problems of quantum physicshttp://www.mi.infn.it/~vacchini/workshopBELL17.html

Organization Dr. B. Vacchini

3. May, 2017

Workshop at Laboratory Nazionali in Frascati, Italy

Title The physics of what happens and the measurement problemhttps://agenda.infn.it/conferenceDisplay.py?confId=13169

Organization Dr. A. Bassi, Dr. C.O. Curceanu, Dr. B. Hiesmayr and Dr. K. Pis-cicchia

4. May, 2017

Junior Symposium in Trieste, Italy

Title Trieste Junior Quantum Dayshttp://people.sissa.it/~alemiche/junior-tsqd-2017.html

Organization Dr. A. Bassi, Dr. F. Benatti and Dr. A. Michelangeli

5. March, 2017

Conference and Working Group Meeting in Valletta, Malta

Title QTSpace meets in Maltahttp://www.qtspace.eu/?q=node/112

Organization Dr. M. Paternostro, Dr. A. Bassi, Dr. S. Gröblacher, Dr. H. Ul-bricht, Dr. R. Kaltenbaek and Dr. C. Marquardt

6. November, 2016

Autumn School at LMU in Munich, Germany

Title Mathematical Foundations of Physicshttps://light-and-matter.github.io/autumn-school

Organization Dr. D.-A. Dercket and Dr. S. Petrat

http://www.qtspace.eu/?q=node/131http://www.mi.infn.it/~vacchini/workshopBELL17.htmlhttps://agenda.infn.it/conferenceDisplay.py?confId=13169http://people.sissa.it/~alemiche/junior-tsqd-2017.htmlhttp://www.qtspace.eu/?q=node/112https://light-and-matter.github.io/autumn-school

vi

7. May, 2016

Workshop in Pontremoli, Italy

Title Quantum control of levitated optomechanicshttps://quantumlevitation.wordpress.com

Organization Dr. A. Serafini, Dr. M. Genoni and Dr. J. Millen

8. September, 2015

International Workshop at Laboratory Nazionali in Frascati, Italy

Title Is quantum theory exact? The endeavor of the theory beyond standardquantum mechanics - Second edition

http:www.lnf.infn.it/conference/FQT2015

Organization Dr. A. Bassi, Dr. C.O. Curceanu, Dr. S. Donadi and Dr. K. Pisci-cchia

9. March, 2015

International Conference at Ettore Majorana Foundation in Erice, ItalyTitle Fundamental Problems in Quantum Physicshttp:www.agenda.infn.it/conferenceDisplay.py?confId=9095

Organization Dr. A. Bassi and Dr. C.O. Curceanu

10. February, 2015

51 Winter School of Theoretical Physics in Ladek Zdroj, Poland

Title Irreversible dynamics: nonlinear, nonlocal and non-Markovian mani-festations

http:www.ift.uni.wroc.pl/~karp51

Organization Institut of Theoretical Physics in Wroclaw, Poland

https://quantumlevitation.wordpress.comhttp:www.lnf.infn.it/conference/FQT2015http:www.agenda.infn.it/conferenceDisplay.py?confId=9095http:www.ift.uni.wroc.pl/~karp51

1

Introduction

When I look back to the time, already twenty years ago, when the concept andmagnitude of the physical quantum of action began, for the first time [. . . ] thewhole development [from the mass of experimental facts to its disclosure] seemsto me to provide a fresh illustration of the long-since proved saying of Goethe’sthat man errs as long as he strivesa. And the whole strenuous intellectual workof an industrious research worker would appear [. . . ] in vain and hopeless, if hewere not occasionally through some striking facts to find that he had, at the endof all his criss-cross journeys, at last accomplished at least one step which wasconclusively nearer the truth.

aJohann Wolfgang von Goethe, Faust, 1808.

Max Karl Ernst Ludwig Planck

Nobel Lecture, June 2, 1920 [1]

The question: “How does a chicken move in the atmosphere?” would be typicallyanswered by a physicist: “To start, let us approximate the problem by considering aspherical chicken in vacuum. . . ”. This is for sure a strong and rough approximation,however it can be a good starting point for solving the problem and in certain

cases it is more than enough to properly describe the motion of the system of in-

terest.

Quantum mechanics is an example of a theory exhibiting a broad collection of

theoretical results in complete agreement with experimental evidence: from the

black body radiation [2–4] to the double slit experiment [5, 6], from the photo-

electric effect [7–9] to the hydrogen atom, from interference fringes in a matter-

wave interferometry experiment [10, 11] to Bose-Einstein condensates [12, 13] and

many more. In some situations, the unitary dynamics of a quantum isolated sys-

tem is not sufficient to well describe the system. One situation is of particular

importance due to its ubiquity and unavoidability. Every realistic (quantum) sys-

tem interacts with the surrounding environment and consequently is changed by

it. In such a case, phenomena like dissipation, diffusion or decoherence emerge

and may become important for the system dynamics. External influences on a

quantum system must be considered explicitly to get a better description of Na-

ture. This is the purpose of the theory of open quantum systems.

In this thesis two different research lines are considered: decoherence and col-

lapse models. Albeit conceptually they are far from each other, they both belong

to the framework of open quantum systems. Indeed, they refer to systems inter-

acting with an external entity: an environment for decoherence models, a noise

for collapse models. Although the external influence has a different origin, they

2

can be described by similar dynamical equations and, in order to confirm or fal-

sify one of these models, similar experimental tests can be performed.

Decoherence models describe the suppression of the interference fringes of a su-

perposition due to the interaction with the surrounding environment, and they

also govern other mechanisms, like diffusion and dissipation. The theory has

made important contributions in other fields, like chemistry [14, 15], condensed

matter [16, 17] and biophysics [18–21], to name a few, which are typically resolved

via numerical analysis [22, 23]. By introducing the environment, the complex-

ity of the problem grows with the level of detail one gives to the model [24–28].

Consequently, a careful balance between the reliability of the model and its math-

ematical idealization becomes a fundamental ingredient to approach the (analyt-

ical) resolution of the problem. The seminal works of Caldeira and Leggett [29]

and of Joos and Zeh [30] are milestones in this field. They model the system-

environment interaction in a very simple way, still being able to capture the most

important properties and features of the open system dynamics, cf. Chaps. 2 and

3.

When trying to solve exactly an open quantum system problem, one usually faces

several difficulties: the most intriguing example is given by the appearance of

non-Markovian features in the system dynamics. A Markovian dynamics is ruled

by equations of motion that do not depend on the past of the system: it is a mem-

oryless dynamics [24]. If, instead, the dynamics depends on the past, and thus it

has a memory, the evolution is said to be non-Markovian. In some situations thismemory is responsible for crucial changes in the behaviour of the system: the

long-living quantum coherences in a light-harvesting systems are an important

non-Markovian effect in quantum-biology [31]. In this thesis, two examples will

be discussed explicitly, respectively as an example of a Markovian [cf. Chap. 4]

and non-Markovian dynamics [cf. Chaps. 3].

The second line of research is focused on collapse models and their experimen-

tal tests. These models unify the two dynamical principles of the quantum me-

chanics (the linear and deterministic Schrödinger evolution with the non-linear

and stochastic wave-packet reduction) in an unique description. By adding non-

linear and stochastic terms to the standard Schrödinger equation, they describe

the spontaneous collapse of the wavefunction. With this modification, they re-

cover both the quantum and the classical dynamics in the microscopic and in the

macroscopic limit respectively, thus answering to the quantum-to-classical tran-

sition debate.

Among the broad collection of collapse models [32–39], we focus on a particular

collapse model, called Continuous Spontaneous Localization (CSL) model. This

model is characterised by a coupling rate �CSL

between the system and the noise

field allegedly responsible for the collapse, and a typical correlation length rC

for

3

the latter. As the CSL model is phenomenological, the values of �CSL

and rC

must

be eventually determined by experiments. By now there is a large literature on

the subject. Such experiments are important because any test of collapse mod-

els is a test of the quantum superposition principle. In this respect, experiments

can be grouped in two classes: interferometric tests and non-interferometric ones.

The first class includes those experiments, which directly create and detect quan-

tum superpositions of the center of mass of massive systems. Examples of this

type are molecular interferometry [40–43] and entanglement experiment with di-

amonds [44, 45]. Actually, the strongest bounds on the CSL parameters come

from the second class of non-interferometric experiments, which are sensitive

to small position displacements and detect CSL-induced diffusion in position

[46–48]. Among them, measurements of spontaneous X-ray emission gives the

strongest bound on �CSL

for rC

< 10�6 m [49], while force noise measurements onnanomechanical cantilevers [50, 51] and on gravitational wave detectors give the

strongest bound for rC

> 10�6 m [52, 53]. Albeit several tests were performedduring the last decade, up to date the CSL parameter space still exhibits a vast

unexplored region.

Outline

This thesis is organized as follows.

In Chap. 1 we present the basic ingredients of the theory of open quantum sys-

tems. Starting from standard quantum mechanics, we introduce the concept of

reduced state of the system and derive its evolution. Its dynamics, constructed

starting from the global system-environment evolution, needs to satisfy several

constrains in order to be a well-defined dynamical map. We discuss these con-

strains, with particular attention to complete positivity. Eventually, we introduce

the Lindblad structure for the generator of the dynamical map, which naturally

satisfies all the above constraints.

In Chap. 2 we introduce the quantum effect of decoherence, which is one of the

most important features of an open quantum system. After discussing its main

properties, we report the derivation of the Joos and Zeh master equation [30] as

an example of dynamical equation dysplaying decoherence effects on a system

prepared in a superposition of two different positions in space. In a similar way

as the Joos-Zeh master equations has been constructed, we propose the deriva-

tion of the master equation describing the decoherence effects on a system whose

state is in a superposition of angular configurations. This work becomes of inter-

est when we consider systems whose angular configuration can change, and thus

one can prepare the system in a superposition of angular configurations. Differ-

ences and similarities between the two models are underlined.

4

In Chap. 3 we describe the quantum Brownian motion, which can be safely con-

sidered as the most known and used example of an open quantum system. Caldeira

and Leggett derived the master equation describing such a dynamics in the mem-

oryless limit [29]. This dynamical equation is the dissipative extension of the

Joos-Zeh master equation

1

. Such an equation leads to decoherence and dissipa-

tion, and it brings asymptotically the system to the thermal state. However it

does not preserve the (complete-)positivity of the dynamics. This drawback can

be avoided by considering the exact solution to the problem [54, 55]. In literature

one finds different solutions which is very useful for Gaussian initial states. We

present an alternative approach to the exact master equation, based on the use of

the Heisenberg picture. Beside recovering the already known results, we show

how one can benefit from this approach when one is interested in the dynamical

evolution of non-Gaussian initial states.

In Chap. 4 an example of memoryless system-environment interaction discussed.

We analyze a recently proposed source of decoherence, based on the gravitational

time dilation [56]. We show that modifications to the proposed model are needed

in the low temperature regime, which is the most favourable one to detect such a

decoherence source. We performed a detailed analysis by comparing the “grav-

itational” decoherence to the more standard decoherence sources, like the colli-

sions with the surrounding residual gas in the vacuum chamber and the emis-

sion, absorption and scattering of thermal radiation. Eventually, we show that

the proposed source of decoherence is orders of magnitude off to be detected

with present technology.

In Chap. 5 we introduce the Continuous Spontaneous Localization (CSL) model.

We show how optomechanical systems provide a particularly promising experi-

mental setup to infer bounds on the CSL parameters �CSL

and rC

. We report the

analyzis of three examples we considered during the doctorate project. The first is

related to the gravitational wave detectors LIGO, LISA Pathfinder and AURIGA

[52]. These experiments set strong bounds on the collapse parameters, and, for

the first time, enclose the still unexplored parameter space in a finite region. The

second one reports an improved cantilever experiment where a non-thermal ex-

cess noise of unknown origin is measured [51]. In principle such a noise is com-

patible with the predictions on CSL given by Adler [57]. The last example is an

experimental proposal we recently presented in [58], which is based on the tor-

sional motion of a system affected by the CSL noise. The proposed experiment

will eventually probe the unexplored CSL parameter space, and confirm, or fal-

sify, the hypothesis that the excess noise measured in [51] is due to CSL.

In Chap. 6 we draw the conclusions of the thesis.

1

Since the work by Joos and Zeh [30] appeared two years later than the one by Caldeira and

Leggett, it is more appropriate to say that the Joos-Zeh master equation is the restriction of the

Caldeira-Leggett master equation to the regime where dissipative effects can be neglected. Notice

that the physical motivations for the two models are very different, cf. Chap. 2 and Chap. 3.

5

Chapter 1

Open Quantum Systems

The usual approach to open quantum systems consists in considering the system

plus the environment as an isolated system, which evolves under the usual uni-

tary quantum dynamics. In the most general case, the degrees of freedom (d.o.f.)

of the system plus environment are infinite and it is prohibitive to follow them

all in time. However, since we are interested in the evolution of the system only,

we focus on it, and we take into account the influence of the environment by av-

eraging over its d.o.f. In this case we speak of an open quantum system

In the following we introduce the general properties and features of an open

quantum system. For an extended description of the theory of open quantum

systems we refer to [24, 26–28].

1.1 Reduced state and its evolution

The initial state ⇢̂SE

at time t = 0 of the global system, composed by the systemof interest S and its environment E, is usually considered as uncorrelated and itevolves according to the unitary evolution Ût:

⇢̂SE

= ⇢̂S

⌦ ⇢̂E

, (1.1a)

⇢̂SE

(t) = Ût⇢̂SEÛ †t . (1.1b)where ⇢̂

S

and ⇢̂E

are the system and environmental states respectively. After some

time t, the interaction between the system and its environment correlates the two,and ⇢̂

SE

(t) cannot be written in the form as in Eq. (1.1a). To extract the systemproperties from ⇢̂

SE

(t) one needs to average over the d.o.f. of the environment

⇢̂S

(t) = Tr(E)⇥

⇢̂SE

(t)⇤

, (1.2)

where ⇢̂S

(t) is called reduced state of the system S, and it is obtained by takingthe partial trace (Tr(E)

⇥ · ⇤) over the d.o.f. of E. This operation is performed bychoosing a basis { | (E)i i }i of the Hilbert space associated to E and applying thefollowing map

⇢̂S

(t) =X

i

h (E)i |⇢̂SE(t)| (E)i i . (1.3)

Chapter 1. Open Quantum Systems 6

This definition does not depend on the choice of the basis. The state ⇢̂S

(t) obtainedin this way preserves all the properties of a quantum state: it is an hermitian,

linear and positive operator, whose trace is equal to one (Tr(S)⇥

⇢̂S

(t)⇤

= 1). Thedynamical map determining its time evolution is given by:

�̂t : ⇢̂S 7! ⇢̂S(t) = Tr(E)⇥Ût⇢̂SEÛ †t

⇤

, (1.4)

which is different from the one describing the unitary dynamics Ût of the globalsystem, because the partial trace operation breaks the unitarity of the dynam-

ics. This map is given by the combination of two operations: the unitary evolu-

tion provided by Eq. (1.1b), and the trace over the d.o.f. of the environment as

described in Eq. (1.2). The construction of the reduced dynamical map can be

represented by the following scheme

1

:

⇢̂SE

unitary evolution��!Ût

⇢̂SE

(t) = Ût⇢̂SEÛ †tTr(E)

?

?

y

?

?

yTr(E)

⇢̂S

�̂t��!

reduced evolution

⇢̂S

(t) = Tr(E)⇥Ût⇢̂SEÛ †t

⇤

.

(1.5)

We can also express the reduced dynamical map as:

�̂t[ · ] = T exp✓

Z t

0

dsLs◆

[ · ], (1.6)

where T is the time-ordering operator and Ls is the generator of the dynamics. Lsdescribes the most important dynamical equation in the theory of the open quan-

tum system: the quantum master equation

d⇢̂S

(t)

dt= Lt[⇢̂S(t)]. (1.7)

One can easily verify the relation between Eq. (1.6) and Eq. (1.7). In fact, by con-

sidering the time derivative of Eq. (1.6) applied to ⇢̂S

, one finds:

d

dt�̂t[⇢̂S] = Lt � T exp

✓

Z t

0

dsLs◆

[⇢̂S

] = Lt � �̂t[⇢̂S], (1.8)

which corresponds to Eq. (1.7).

All dynamical maps �̂t realized following the scheme in Eq. (1.5) satisfy by con-struction some important features. The first is the linearity of the dynamical map:

�̂t[⇢̂1 + ⇢̂2] = �̂t[⇢̂1] + �̂t[⇢̂2], (1.9)

1

If the initial state ⇢̂

SE

does not present the structure described in Eq. (1.1a), the operation

defined in Eq. (1.4) with ⇢̂

S

= Tr

(B)

⇥

⇢̂

SE

⇤

in general is not a (dynamical) map. For an exaustive

description we refer to [59] and references therein, where the first attempts in constructing a

reduced dynamical map starting from a correlated initial state are reported.


which is fundamental to comply with the superposition principle. The second

property of �̂t is its continuity:

lim⌧!0

�̂t+⌧ [⇢̂S]� �̂t[⇢̂S] = 0, (1.10)

which naturally follows from the continuity of Ût. The third property is relatedto the probability interpretation we give to ⇢̂

S

: its diagonal terms describe the

probabilities of finding the system in a particular state. Thus it follows that, to

maintain consistently this interpretation, �̂t must preserve the positivity, the her-miticity and the trace of ⇢̂

S

:

�̂t [⇢̂S] � 0,⇣

�̂t [⇢̂S]⌘†

= �̂t [⇢̂S] ,

Tr(S)⇥

⇢̂S

⇤

= 1.

(1.11)

A dynamical map owning these features can be considered as well constructed.

However, since in general it is difficult to derive exactly a dynamical map start-

ing from the global unitary dynamics, approximations are often needed. This

implies that the above features are not granted, and one has to require them ex-

plicitly. While the conditions on the hermiticity and on the trace can be simply

verified, the conservation of the positivity is difficult to characterize. This is not

however a problem since, as we will see, there are physical motivations that re-

quire a stronger condition on the dynamical map: �̂t must be completely positive.

1.1.1 Complete positivity

A map �̂t is Completely Positive (CP) if (1̂n⌦ �̂t) is a positive map 8n 2 N, where1̂n identifies the identity map acting on a n dimensional Hilbert space.

The physical motivation for the request of a CP dynamical map can be simply

understood with the following example. Consider the Universe as composed by

two systems of interest immersed in a common environment: the usual system

S and an ancilla A. The corresponding states are ⇢̂S

and ⇢̂A

, respectively. Sup-

pose the environment acts independently on the two systems, with the dynami-

cal maps �̂t and ⇤̂t respectively. Then, the dynamical map describing the effectof the environment on the two systems is constructed as

(⇤̂t ⌦ �̂t) : ⇢̂AS 7! ⇢̂AS(t), (1.12)where ⇢̂

AS

= ⇢̂A

⌦ ⇢̂S

, ⇢̂AS

(t) = ⇢̂A

(t) ⌦ ⇢̂S

(t) with ⇢̂A

(t) = ⇤̂t[⇢̂A] and ⇢̂S(t) = �̂t[⇢̂S].Then, we require the positivity of the final total state ⇢̂

AS

(t) � 0, i.e. the total map(⇤̂t ⌦ �̂t) must be a positive map. This must hold for any form of the map ⇤̂t,even when ⇤̂t = 1̂n, and for any dimension n of the Hilbert space associated tothe system A. These are the requirements for a CP dynamical map.


1.2 Lindblad equation

A simple and important example of the Eq. (1.7) is given by the so called Lind-

blad equation [60, 61]. Here we report its expression for a N dimensional Hilbertspace:

L [⇢̂S

(t)] = � i~h

ĤS

, ⇢̂S

(t)i

� i~h

�Ĥ, ⇢̂S

(t)i

+N2�1X

a,b=1

Kab

L̂a⇢̂S(t)L̂†b �

1

2

n

L̂†bL̂a, ⇢̂S(t)o

�

,

(1.13)

where the first term describes the free coherent evolution with respect to the sys-

tem Hamiltonian ĤS

, as it is isolated. The last two terms result from the influence

of the environment: �Ĥ modifies the coherent evolution, and is called Lambshift. The last term in Eq. (1.13) is called dissipator and it is distinguished by

its characteristic structure. Here, L̂a are called Lindblad operators and Kab isthe Kossakowski matrix. Effects like dissipation, diffusion and decoherence are

due to this latter term, which breaks the unitarity of the dynamical map. More-

over, Eq. (1.13) represents the most general time independent generator for a trace

preserving completely positive dynamical map [24]. Thus, a generator having a

time independent structure different from, or that cannot be rewritten in terms

of, Eq. (1.13), is not a generator of a good dynamical map. In particular, given

Eq. (1.13), the complete positivity requirement is satisfied if and only if the Kos-

sakowski matrix is positive Kab � 0.

In general, one can consider extensions of the generator defined in Eq. (1.13) to

cases with time dependent coefficients, i.e. �Ĥ(t) and Kab(t). If Kab(t) remains apositive matrix for any time t, then the corresponding dynamical map satisfies allthe requirements needed. It is worth noticing that in the time dependent case, the

requirement of a positive Kossakowski matrix can be relaxed without compro-

mising the structure of the corresponding dynamical map �̂t. In fact, there existcases where, for some finite time intervals, Kab(t) < 0 and still the dynamicalmap is CP. We will come back to this in Sec. 3.3.3, where an example is explicitly

studied.

9

Chapter 2

Decoherence

One of the most interesting features of an open quantum system is decoherence.

This is an intrinsically quantum feature, appearing due to the interaction with the

surrounding environment. It cannot be completely avoided, one can only try to

screen its action and soften its effect on the system.

Consider a system S prepared in the superposition | i = 1p2(|Li + |Ri), where

|Li and |Ri identify the states centered on two different positions, e.g. on the leftand on the right of the origin, respectively. We consider these positions suffi-

ciently apart to make sure that the two states can be considered as orthonormal,

i.e. hL|Ri = 0. The total state, system S plus environment E, at time t = 0 reads|

SE

i = 1p2

�|Li+ |Ri�⌦ |�E

i , (2.1)

where |�E

i is the state of one environmental particle. After some time t, the inter-action between the system and the environment correlates S with E. Suppose thetotal state becomes

| SE

(t)i = 1p2(|Li ⌦ |�L

E

i+ |Ri ⌦ |�RE

i), (2.2)

where |�LE

i is the state the environmental particle takes if the system is in |Li, and|�R

E

i is the state of the particle if the system is in |Ri. Let us now take the reducedstates of S associated to the states in Eq. (2.1) and Eq. (2.2), and represent themon the system basis {|Li , |Ri}. The corresponding density matrices are

⇢S

= 12

✓

1 11 1

◆

and ⇢S

(t) = 12

✓

1 h�RE

|�LE

ih�L

E

|�RE

i 1◆

. (2.3)

As we can see, the populations (probabilities of finding the system on the left

or on the right), which are the diagonal elements of the density matrix, do not

change during the evolution. Conversely, the off diagonal terms, called coher-

ences and which represent the possibility of measuring interference among the

different terms of the superposition, are modified. Since, in general, |�LE

i and |�RE

iare not equal, we have | h�L

E

|�RE

i | 1. Consequently, the coherences are reduced.If we consider N environmental particles instead of one only, the coherences be-come:

h�LE

|�RE

i ! h�L1

|�R1

i h�L2

|�R2

i h�L3

|�R3

i . . . h�LN

|�RN

i . (2.4)As time passes, more and more environmental particles interact with the system

and consequently the coherences are suppressed in time: this effect is called de-

coherence.

Chapter 2. Decoherence 10

It is important to underline that decoherence is a fully quantum mechanism. It

is due to a typically quantum feature which has no classical counterpart: the en-

tanglement. In fact, when the system and the bath particle interact, their states

entangle and thus decoherence occurs [cf. Eq. (2.2)].

In the next sections we will discuss qualitatively and quantitatively the decoher-

ence effects, in particular we will focus on two of the most common decoherence

sources: the scattering with thermal background radiation [30, 62] and the colli-

sions with the residual gas in the vacuum chamber [29, 55]. Other decoherence

sources act and are described in a similar way. In Chap. 4 we will analyze a re-

cently proposed source of decoherence induced by gravity and we will compare

it with other common decoherence sources, which action is derived in the next

Section.

2.1 Gallis-Flemming master equation

In typical experiments, decoherence effects cannot be fully avoided. The residual

gas in a ultra-high vacuum chamber, the thermal radiation of the chamber itself,

and even the 3 K cosmic background radiation represent decoherence sources on

the system dynamics. In their seminal paper [30] Joos and Zeh set the basis for

the description of such effects. Their master equation can be obtained from the

long wavelength limit of the Gallis and Flemming master equation [63], which is

here derived.

Let us consider the following situation. A system S of mass M , described by alocalized eigenstate |xi of the position, collides with a particle of the surroundinggas, whose state is |�i and mass is m. The scattering (or collision) between thetwo particles can be schematically represented by

|xi ⌦ |�i scattering��! Ŝ (|xi ⌦ |�i) , (2.5)where Ŝ is the unitary scattering operator describing the collision process. Inorder to simplify the problem, we consider the recoil-less limit, i.e. the infinite

mass limit

1

(M � m). In this limit, we can safely assume that the scatteringoperator heavily affects only the gas particle state and not that of S. Then Eq. (2.5)can be approximated by

|xi ⌦ |�i scattering��! |xi ⌦ Ŝx

|�i , (2.6)where the scattering operator Ŝ

x

depends on the position x of the system. In

a realistic situation, the system is described, in place of |xi, by a wavepacket :1

The scattering process for finite mass has been widely discussed, see for example [64–67].


|'i = R dx'(x) |xi. Consequently, also the scattering process changes to

|'i ⌦ |�i scattering��!Z

dx'(x) |xi ⌦ Ŝx

|�i . (2.7)

In terms of the statistical operator of the reduced system, the scattering process

can be described by

⇢̂S

=R

dxR

dx0 '(x)'⇤(x0) |xi hx0| ,?

?

y

scattering

⇢̂S

(t) =R

dxR

dx0 '(x)'⇤(x0) |xi hx0| ⌘(x,x0).(2.8)

where ⌘(x,x0) = h�|Ŝ†x

0 Ŝx

|�i. Since Ŝx

is unitary, it does not change the popula-

tions of the system: ⌘(x,x0) = 1. Instead, for x 6= x0 the environmental state |�iis changed according to the position x where the scattering occur, and we have

| h�|Ŝ†x

0 Ŝx

|�i | = | h�x

0 |�x

i | < 1, (2.9)i.e., the coherences are reduced by the scattering event. ⌘(x,x0) quantifies the re-duction of coherences, and thus it is called decoherence term.

By assuming that the environment is in the thermal equilibrium, whose state

2

is

⇢̂B

, we find that ⌘(x,x0) evolves according to

⌘(x,x0) = 1� t⇤(x,x0), (2.10)where we defined [68]

⇤(x,x0) = ngas

Z

dq µB

(q)v(q)

Z

dn̂0|f(q, qn̂0)|2⇣

1� ei (q�qn̂0)·(x�x0)~

⌘

. (2.11)

Here ngas

is the number density of the gas, µB

(q) is the momentum thermal dis-tribution of the scattered particles, f(q, qn̂0) is the scattering amplitude of theprocess with an incoming (outgoing) scattered particle momentum q (qn̂0). Thephase in parenthesis is due to the translational invariance of the scattering pro-

cess. Indeed, because of it the scattering operator can be translated to the origin:

Ŝx

= e�iq̂·x~ Ŝ0ei

q̂·x~

. The velocity v(q) take different expressions depending on thescattering particle involved: for a massive bath particle we have v(q) = q/m

gas

,

while v(q) = c the light. Substituting Eq. (2.10) in Eq. (2.8) we can simply expressthe time evolution of the state:

d⇢S

(x,x0; t)

dt= �⇤(x,x0)⇢

S

(x,x0; t), (2.12)

2

We will denote the environmental state with ⇢̂

B

only if it is a thermal state. Commonly, an

environment in thermal equilibrium is referred to as a bath. This clarifies the change of notation(E!B) with respect to the one used in Chap. 1.


where ⇢S

(x,x0; t) = hx|⇢̂S

(t)|x0i. This is the Gallis and Flemming master equation[63].

It is interesting to investigate the explicit expression of Eq. (2.11) in two limiting

cases. These are related to the ratio between the average wavelength of the en-

vironmental particles �, which can be evaluated by using the de Broglie formula� = 2⇡~/ hqi [68], and the coherent separation �x = |x� x0|.

2.1.1 Short wavelength limit

Let us first consider the so-called short wavelength limit, i.e. � ⌧ �x. In such alimit the phase appearing in Eq. (2.11) oscillates very rapidly and gives no contri-

bution to the integral. Thus it can be safely neglected, and the expression becomes

independent from the coherent separation �x. Assuming that µB(q) depends onlyon the modulus of q, we can evaluate the angular part of the integral as follows:

Z

dn̂ dn̂0

4⇡|f(q, qn̂0)|2 = �

TOT

(q), (2.13)

with �TOT

(q) denoting the total cross section of momentum q, averaged over thedirections. Consequently, Eq. (2.11) becomes position independent:

⇤(x,x0) ' �TOT

= 4⇡ngas

Z

dq q2µB

(q)v(q)�TOT

(q). (2.14)

The corresponding master equation in the position representation takes the fol-

lowing form:

d⇢S

(x,x0; t)

dt= ��

TOT

⇢S

(x,x0; t), (2.15)

where �TOT

is referred as the total scattering rate. The state evolution is simply

given by

⇢S

(x,x0; t) = ⇢S

(x,x0; 0)e��TOTt, (2.16)

showing that the coherences of the system are exponentially suppressed in time,

independently from the coherent separation �x.

2.1.2 Long wavelength limit, Joos and Zeh master equation

In the opposite limit, when � � �x, the phase in Eq. (2.11) is small, allowingto Taylor expand the expression to the second order in q. By also assuming thatµ

B

(q) = µB

(q), Eq. (2.11) reduces to

⇤(x,x0) =n

gas

~2 (x� x0)2

Z

dq µB

(q)v(q)q4�eff

(q), (2.17)


where

�eff

(q) =2⇡

3

Z

d(cos ✓)|f(q, cos ✓)|2(1� cos ✓), (2.18)is the effective cross section of the process, with ✓ denoting the angle between theincoming and the outgoing scattered vectors.

The master equation corresponding to the localization rate in Eq. (2.17) reads

d⇢̂S

(t)

dt= �⌘

2[x̂, [x̂, ⇢̂

S

(t)]] , (2.19)

where the diffusion constant takes the following expression

⌘ =n

gas

~2

Z

dq µB

(q)v(q)q4�eff

(q). (2.20)

Typically one refers to Eq. (2.19) as the Joos and Zeh master equation [24, 30]. We

stress that in this case the evolution of the state depends explicitly also on the

coherent separation �x = |x� x0|:

⇢S

(x,x0; t) = ⇢S

(x,x0; 0)e�⌘�2x

t. (2.21)

In a similar way as in the short wavelength limit, the coherences are exponentially

suppressed in time and the populations are not affected by the process.

2.2 Rotational Decoherence

We consider a situation similar to the one described in Sec. 2.1: a system in super-

position interacting with the surrounding enviromnent, but now the system is in

a superposition of angular configurations. This can be the case of interest for a

system showing an anisotropy under rotations, e.g. a system more elongated in

one direction. By exploiting the derivation of Eq. (2.12), in this section we derive

the master equation describing rotational decoherence [69].

In a similar way to what was done starting from Eq. (2.8), we define the initial

state of the system as

⇢̂S

=

Z

d⌦

Z

d⌦0 '(⌦)'⇤(⌦0) |⌦i h⌦0| , (2.22)

where |⌦i is the state representing the system prepared in the angular configu-ration ⌦. Such a configuration can be prepared by starting from some referenceconfiguration |0i (chosen, for example, in a way that the anisotropy of the system


is aligned along the x axis) and applying a rotation defined by the three Euler3

angles [71, 72]. Starting from the configuration ⌦, a scattering process can bedescribed as follows

|⌦i ⌦ |�i scattering��! |⌦i ⌦ Ŝ⌦ |�i , (2.23)where the recoil-less limit is considered [cf. Eq. (2.6)] and thus the scattering oper-

ator Ŝ⌦ acts on the environmental state |�i only. Ŝ⌦ can be related to the standardscattering operator Ŝ0 acting in the ⌦ = 0 configuration through a rotation fromthe configuration |0i to |⌦i: Ŝ⌦ = R̂(⌦)Ŝ0R̂†(⌦).After the scattering process, the reduced state of the system is given by

⇢̂S

(t) =

Z

d⌦

Z

d⌦0 '(⌦)'⇤(⌦0) |⌦i h⌦0| ⌘(⌦,⌦0), (2.24)

where

⌘(⌦,⌦0) = Tr(B)⇥

⇢̂B

Ŝ†⌦0 Ŝ⌦⇤

. (2.25)

For the sake of simplicity, let us suppose that the system is in a superposition of

angular configurations obtained only from rotations around the z axis. Then, thestate of the system can be identified by |↵i = R̂z(↵) |0i, with R̂z(↵) = exp(� i~ L̂z↵)where L̂z is the angular momentum operator with respect to the z axis. The natu-ral basis for the computation of ⌘(⌦,⌦0) is given by the energy-angular momen-tum representation

⌘(↵,↵0) =

Z

dE µB

(E)+1X

l=0

lX

m=�l

hE, l, m|Ŝ†↵0 Ŝ↵|E, l, mi , (2.26)

where µB

(E) is the energy distribution of the environmental states, l and m arerespectively the eigenvalues of the total angular momentum and L̂z. Writing Ŝ↵ in

3

These three angles describe three consecutive rotations of the system, as depicted in the fol-

lowing scheme [70]:

0

@

x0

y0

z0

1

A

arround z0��!angle ↵

0

@

x1

y1

z0

1

A

arround x1��!angle �

0

@

x1

y2

z2

1

A

arround z2��!angle �

0

@

x3

y3

z2

1

A


terms of Ŝ0 = 1̂+iT̂,where T̂ is well known T-matrix from the standard quantum-mechanical scattering theory [73], we obtain

⌘(↵,↵0) = 1�X

l,m

X

l0,m0

Z

dE µB

(E)⇣

1� e�i(m�m0)(↵0�↵)⌘

·

·Z

dE 0 | hE, l, m|T̂†|E 0, l0, m0i |2,(2.27)

where we used the relation for the T-matrix: i(T̂† � T̂) = T̂†T̂, and introduceda completeness: 1̂ =

R

dE 0P

l0,m0 |E 0, l0, m0i hE 0, l0, m0|. We proceed by evaluatingthe matrix elements of the T-matrix, which are defined in terms of to the scattering

amplitude f(p,p0) of the process:

hp|T̂|p0i = i2⇡~m

gas

�(E � E 0)f(p,p0), (2.28)

where p and p

0are respectively the incoming and outgoing momentum of the

environmental particle of mass mgas

, and �(E � E 0) accounts for the energy con-servation of the gas particle in the recoil-less limit (E = p2/2m

gas

). Due to the

modulus square in Eq. (2.27), we have

| hp|T̂|p0i |2 = t(2⇡~)3m

gas

�(p � p0)p

|f(p,p0)|2, (2.29)

where t is the interaction time and we handled the squared energy delta functionin the usual way:

�2(E � E 0) = limt!+1

t

2⇡~m

gas

p�(p � p0). (2.30)

Now, ⌘(↵,↵0) can be rewritten in terms of Eq. (2.28) by using the following repre-sentation of the energy-angular momentum states on the momentum basis:

hp|E, l, mi = ~3/2

pMp

�(E � E 0)Ylm(p̂), (2.31)

where Ylm(p̂) are the spherical harmonics with quantum numbers l and m andwith an angular dependence defined with respect to the direction p̂. By using

this representation and applying Eq. (2.30), we can express Eq. (2.27) in terms of

the scattering amplitude functions f(p,p0):

⌘(↵,↵0) = 1� t⇤(↵,↵0), (2.32)


with

⇤(↵,↵0) =2m

gas

(2⇡~)3

Z

dE EµB

(E)X

lm

X

l0m0

⇣

1� e�i(m�m0)(↵0�↵)⌘

·

·Z

dp̂

Z

dp̂0Z

dp̂00Z

dp̂000Y ⇤l0m0(p̂)Ylm(p̂0)Yl0m0(p̂

00)Y ⇤lm(p̂000)f(p,p0)f ⇤(p00,p000).

(2.33)

The angular integrations in Eq. (2.33) can be simply evaluated since the spheri-

cal harmonics form a complete set of functions and the following completeness

relation holds

1X

l=0

lX

m=�l

Y ⇤lm(p̂0)Ylm(p̂) =

�(✓ � ✓0)sin✓

�(�� 0), (2.34)

where the two direction p̂ and p̂

0are parametrized by the angles (✓,�) and (✓0,�0),

respectively [74]. By exploiting Eq. (2.34), Eq. (2.33), multiplied by the number

density ngas

of particles, becomes

⇤(↵,↵0) = ngas

Z

dpµB

(p)p

mgas

Z

dn̂0 |f(p, pn̂0)|2 (1� R(p, pn̂0,!)) , (2.35)

where

R(p,n0,!) =f ⇤(p!, pn̂0!)

f ⇤(p, pn̂0), (2.36)

where p! is the vector p rotated by an angle ! = ↵� ↵0 around the z axis. Conse-quently the master equation reads [69]

d⇢S

(↵,↵0; t)

dt= �⇤(↵,↵0)⇢

S

(↵,↵0; t), (2.37)

where ⇢S

(↵,↵0; t) = h↵|⇢̂S

(t)|↵0i.

It is interesting to notice that the expression for ⇤(↵,↵0) in Eq. (2.35) has the samestructure of Eq. (2.11), with R(p,n0,!) replaced by

R(q, n̂0,x� x0) = f⇤(q

x�x0 , qn̂0x�x0)

f ⇤(q, qn̂0). (2.38)

Here q

x�x0 is the vector q translated in space by x� x0. This result can be under-stood once we consider the expression for the scattering amplitude generated by

the potential V (r), under the Born approximation [72]:

f(q, qn̂0) = �~mgas2⇡

Z

dr e�i(q�qn̂0)·r

~ V (r). (2.39)


Implementing the translation in space: Ŝx

= e�iq̂·x~ Ŝ0ei

q̂·x~

, as it was done to obtain

Eq. (2.11), we have

f ⇤(qx�x0 , qn̂

0x�x0) = �

~mgas

2⇡

Z

dr ei(q�qn̂0)·(r+x�x0)

~ V (r), (2.40)

and by considering the ratio between the two scattering amplitudes, i.e. Eq. (2.39)

and Eq. (2.40), one obtains the known result:

f ⇤(qx�x0 , qn̂0

x�x0)

f ⇤(q, qn̂0)= ei

(q�qn̂0)·(x�x0)~ , (2.41)

which is the exponential factor in Eq. (2.11).

As recent theoretical [58, 71, 72, 75–83] and experimental [84–92] works can tes-

tify, there is a growing interest in systems exploiting rotational or torsional de-

grees of freedom of a quantum system. Thus, the master equation (2.37) becomes

particularly useful in the description of decoherence on these systems.

18

Chapter 3

Quantum Brownian Motion

Brownian motion is considered the paradigm of an open system, both in the clas-

sical and in the quantum case. Originally [93], it was observed as the motion of

pollen grains suspended in a viscous liquid, for which different classical models

were proposed [94, 95].

In the classical case, two different but related, ways to model the liquid have been

developed. The first one is the so-called collisional model where the environment

is represented by free particles in thermal equilibrium, interacting with the sys-

tem through instantaneous collisions. This is the model considered by Einstein

[94] and Langevin [95], and its description is given by the Langevin equation [95]

ẍ(t) + 2�m

ẋ(t) + 1M @xV (x) =1M F (t), (3.1)

where the evolution of the position x of the system of mass M in a external poten-tial V (x) is damped by a friction term proportional to the velocity (Stokes term),where �

m

is a damping rate. The stochastic force F (t) describes the noise inducedby the environment, and is assumed to be gaussian, i.e. fully described by its av-

erage hF (t)i and two-time correlation function hF (t)F (s)i, where h · i denotes astatistical average. For the Brownian motion they read

hF (t)i = 0,hF (t)F (s)i = 4M�

m

kB

T �(t � s), (3.2)

where kB

is the Boltzmann constant, T is the temperature of the environment.

The second model instead considers a particle S immersed in an environmentof independent harmonic oscillators in thermal equilibrium. This is called the

harmonic bath model [96, 97] and its quantum version is the main subject of this

Chapter.

3.1 The model

The model consists of a particle S of mass M , with position x̂ and momentump̂, harmonically trapped at frequency !0 and interacting with a thermal bath ofindependent harmonic oscillators, with positions R̂k, momenta P̂k, mass mk andfrequencies !k. The total Hamiltonian ĤT of system plus bath is ĤT = ĤS+ĤB+ĤI,

Chapter 3. Quantum Brownian Motion 19

where

ĤS

=p̂2

2M+

1

2M!20x̂

2, ĤB

=X

k

P̂ 2k2mk

+1

2mk!

2kR̂

2k, ĤI = x̂

X

k

CkR̂k, (3.3)

are respectively system, bath and interaction Hamiltonians. The total initial state

is assumed to be uncorrelated

⇢̂T

= ⇢̂S

⌦ ⇢̂B

, (3.4)

where ⇢̂B

is the state of the environment. A common assumption is to consider

the environment in thermal equilibrium, and its state described by a Gibbs state

with respect to the free Hamiltonian of the bath:

⇢̂B

=e��ĤB

Tr(B)⇥

e��ĤB⇤ , (3.5)

where � = 1/(kB

T ) is the inverse temperature. The characterization of the set ofcoupling constants Ck is provided by the spectral density which is defined as1

J(!) =X

k

C2k2mk!k

�(! � !k). (3.6)

In terms of the latter, we can define the two-time correlation function of the bath

operator B̂ =P

k CkR̂k:

C(t � s) = Tr(B)⇥B̂(t)B̂(s)⇢̂B

⇤

= 12D1(t � s)� i2D(t � s), (3.7)

where B̂(t) = eiĤBt/~B̂e�iĤBt/~. D1(t) and D(t) are the noise and the dissipativekernels, describing respectively the noisy action of the environment, related to the

temperature of the latter, and the corresponding dissipative effect on the system.

They read:

D1(t) = 2~Z +1

0

d! J(!) coth(�~!/2) cos(!t), (3.8a)

D(t) = 2~Z +1

0

d! J(!) sin(!t). (3.8b)

Such kernels are related through the Fluctuation-Dissipation theorem [98–102]:

Z +1

�1dt cos(!t)D1(t) = coth

✓

�~!2

◆

Z +1

�1dt sin(!t)D(t), (3.9)

1

J(!) describes the distribution in frequency of the environmental harmonic oscillators,

weighted by the corresponding coupling constant. This expression does not imply a limit where

the frequencies !

k

form a continuous spectrum.


which provides the following time symmetries

C(�t) = C⇤(t) = C(t � i~�). (3.10)This relation, known as Kubo-Martin-Schwinger (KMS) condition, allows for a

proper definition of the thermodynamical limit of the environmental state. In-

deed, one can easily see that in the (thermodynamical) infinite bath particle limit

the Gibbs formula in Eq. (3.5) becomes meaningless [103, 104] and one needs an

alternative way to describe the bath state. Typically, instead of considering the

whole state (that becomes meaningless in the limit), one considers only some of

its properties, that are expected to remain stable in the limit. The KMS condition

[cf. Eq. (3.10)] survives the limit and it provides the basis to construct a general-

ization of Eq. (3.5) valid also in the thermodynamical limit [104]. Consequently,

the spectral density J(!) must be chosen in a way that the two kernels exist andthat Eq. (3.9) holds: these are two important as well as trivial conditions one must

respect to obtain a well defined description of the environmental state in the ther-

modynamical limit.

To make an explicit comparison between the classical and quantum description,

we write down the quantum Langevin equation derived form Eq. (3.3):

¨̂x(t)� 1~MZ t

0

ds D(t � s)x̂(s)� i~M⇥

V (x̂), p̂⇤

=1

MB̂(t). (3.11)

Although Eq. (3.1) describes the evolution of a classical system, while Eq. (3.11)

is the dynamical equation for a quantum one, the two equations have much in

common. The potential V appears in both Langevin equations in the usual way,and the stochastic force F (t) in Eq. (3.1) is here replaced by the bath operatorB̂(t). The classical dissipative term proportional to the velocity of the system isnow replaced by an integral term, containing the position operator at all previous

times. In this sense, Eq. (3.11) contains a memory of the history of the system,

which is weighted by the dissipative kernel D(t). In the memoryless limit, wehave D(t � s) / @s�(t � s) and we recover

¨̂x(t) + 2�m

˙̂x(t)� i~M⇥

V (x̂), p̂⇤

=1

MB̂(t), (3.12)

which is the quantum analog of Eq. (3.1).

3.2 The Calderira-Leggett master equation

The first master equation describing the model introduced in Sec. 3.1 was derived

by Caldeira and Leggett [29] by using three approximations. The Born approxi-

mation assumes a weak coupling between the system and the bath, guaranteering

that the total state can be expressed as factorized at any time t and that the bath


state remains unperturbed: ⇢̂T

(t) ⇡ ⇢̂S

(t)⌦ ⇢̂B

. The Markov approximation instead

neglects all memory effects in the dynamics, and the corresponding master equa-

tion becomes time local. In order to implement the latter approximation a spectral

density J(!) / ! must be considered. This implies that the dissipation kernel hasa memoryless structure D(t�s) / @s�(t�s) [cf. Eq. (3.12)]. However, a drawbackof this choice is that the noise kernel D1(t) defined in Eq. (3.8) diverges, and theKMS condition in Eq. (3.10) does not hold anymore. One can cure this divergence

by introducing the (third) high temperature approximation for which we obtain

D1(t) / �(t). Under these approximations, one obtains the Caldeira-Leggett (CL)master equation in the Lindblad form with constant coefficients:

d⇢̂S

(t)

dt= � i~ [ĤS, ⇢̂S(t)]�

i�m

~ [x̂, {p̂, ⇢̂S(t)}]�2M�

m

~2� [x̂, [x̂, ⇢̂S(t)]] . (3.13)

The first term describes the coherent evolution due to the system Hamiltonian,

the second governs the dissipation, while the last term determines the tempera-

ture dependent diffusive action of the bath and it is the one responsible for deco-

herence. In this sense, the CL master equation can be considered as the dissipative

extension of the Joos and Zeh master equation (2.19).

There are several approaches [105–107] one can use to derive the CL master equa-

tion in (3.13). The original one [29] is based on the Feynman-Vernon theory

[108, 109], which we now briefly describe. We consider the position represen-

tation of the total state at time t in terms of the total state at time t = 0

hx,R|⇢̂SE

(t)|y,Qi =Z

dx0Z

dy0Z

dR0Z

dQ0 K(x,R, t; x0,R0, 0)·K⇤(y,Q, t; y0,Q0, 0) hx0,R0|⇢̂

SE

|y0,Q0i ,(3.14)

where R and Q identify the positions of the environmental particles, x and ythe position of the system and K(x,R, t; x0,R0, 0) is the position representation of

e�iĤTt/~, which can be expressed via a path integral

K(x,R, t; x0,R0, 0) = hx,R|e�iĤTt/~|x0,R0i =Z x

x0Dx

Z

R

R

0DR e i~ST[x,R], (3.15)

with ST

[x,R] =R t

0 dsLT denoting the action of the total system and LT the totalLagrangian.

The reduced state of the system S in the position representation is given by

⇢S

(x, y, t) =

Z

dR hx,R|⇢̂SE

(t)|y,Ri , (3.16)


which, under the assumption of uncorrelated initial states in Eq. (3.4), can be

expressed as

⇢S

(x, y, t) =

Z

dx0Z

dy0 J(x, y, t; x0, y0, 0)⇢S

(x0, y0, 0). (3.17)

Here the propagator J(x, y, t; x0, y0, 0) takes into account the influence of the sur-rounding environment:

J(x, y, t; x0, y0, 0) =

Z x

x0Dx

Z y

y0Dy e i~ (SS[x]�SS[y])F [x, y], (3.18)

where SS

[x] is the action of the system S alone and

F [x, y] =Z

dR0Z

dQ0⇢B

(R0,Q0, 0)

Z

R

R

0DR

Z

Q

Q

0DQ e i~ (SI[x,R]�SI[y,Q]+SB[R]�SB[Q]),

(3.19)

is the influence functional. This is defined in terms of the matrix elements of the

bath state at time t = 0

⇢B

(R0,Q0, 0) =Y

k

mk!k expn

mk

!k

2⇡~ sinh(�~!k

) [(R02k + Q

02k ) cosh(�~!)� 2R0kQ0k]

o

2⇡~ sinh (�~!k),

(3.20)

and of the actions SB

and SI

derived from the bath and interaction Lagrangians

respectively. By differentiating Eq. (3.17) with respect to time we can derive, un-

der the approximations already stated, the CL master equation (3.13). Precisely,

this is done by substituting the above actions with the following expressions:

SS

[x] =

Z t

0

dsLS

,

SB

[R] =

Z t

0

dsX

k

⇣

12mkṘ

2k � 12mk!2kR2k

⌘

,

SI

[x,R] = �Z t

0

dsxX

k

CkRk,

(3.21)

where LS

is the Lagrangian of the system.

The CL master equation has two limitations. First, it is restricted to the high

temperature regime, which cannot be always fulfilled: the latest attempts to reach

the ground state [110, 111] is an opto-mechanical example. Second, the master

equation is the generator of a dynamical map which is not CP [112, 113], i.e. it

does not map all quantum states ⇢̂S

into quantum states. Accordingly, one needs

to “reduce” the class of initial states in order to avoid gross miscalculations [54,

114, 115]. This will be discussed in the next subsection.


3.2.1 Complete positivity problem

The CP problem in the CL model can be handled by modifying the master equa-

tion by introducing suitable correcting terms. Consider the CL master equation

written in the form displayed in Eq. (1.13) and here reported:

L [⇢̂S

(t)] = � i~h

ĤS

, ⇢̂S

(t)i

� i~h

�Ĥ, ⇢̂S

(t)i

+2X

a,b=1

Kabh

L̂a⇢̂S(t)L̂†b � 12

n

L̂†bL̂a, ⇢̂S(t)oi

,

(3.22)

As already mentioned, for a master equation with a time independent Lindblad

structure, the positivity of the Kossakowski matrix is sufficient to satisfy the re-

quirement of completely positivity of the corresponding dynamical semigroup

[24]. In the case of the Caldeira-Leggett master equation, the Lamb shift reads

�Ĥ = �m2 (x̂p̂ + p̂x̂), the Lindblad operators are L̂1 = x̂ and L̂2 = p̂ and the Kos-sakowski matrix takes the form

Kab =

✓

4M�m

/~2� �i�m

/~i�

m

/~ 0

◆

. (3.23)

As one can see, the Kossakowski matrix has a negative determinant and conse-

quently the dynamical map is not CP. However, we can modify the Kossakowski

matrix in such a way that its determinant is zero, this is the minimally invasive

modification [113]. We do so by adding a term to the master equation propor-

tional to a double commutator in p̂

d⇢̂S

(t)

dt= � i~ [ĤS, ⇢̂S(t)]�

i�m

~ [x̂, {p̂, ⇢̂S(t)}]�2M�

m

~2� [x̂, [x̂, ⇢̂S(t)]]��

m

�

8M

⇥

p̂, [p̂, ⇢̂S

(t)]⇤

.

(3.24)

With this modification Eq. (3.13) can be written in the form in Eq. (3.22) with

L̂ =

s

4M�m

�~2 x̂ + ir

�m

�

4Mp̂. (3.25)

So there is only one Lindblad operator, and the Kossakowski matrix becomes

K = 1, and the dynamics satisfies the CP condition. For high temperatures,� ! 0, the term we added by hand is small compared to the others and its actionis negligible. On the other hand, for low temperatures one obtains different pre-

dictions from the one given by the CL master equation. This will be discussed in

detail in Sec. 3.3.4.

3.3 Non-Markovian Quantum Brownian motion

The main contributions in overcoming the limitations given by the CL model

were given by Haake and Reibold [54] and later by Hu, Paz and Zhang [55], who

provided the exact master equation for the particle S given the total Hamiltonian


ĤT

:

d⇢̂S

(t)

dt= � i~ [Ĥ(t), ⇢̂S(t)]�

i�(t)

~ [x̂, {p̂, ⇢̂S(t)}]� h(t) [x̂, [x̂, ⇢̂

S

(t)]]� f(t) [x̂, [p̂, ⇢̂S

(t)]] , (3.26)

where Ĥ(t) and the coefficients �(t), h(t) and f(t) now are time dependent. Werefer to this model as to the Quantum Brownian Motion (QBM) model. Contrary

to the CL master equation, which is valid only for the specific ohmic spectral

density (J(!) / !), Eq. (3.26) is valid for arbitrary spectral densities J(!) andtemperatures T . The explicit form of the coefficients, beyond the weak couplingregime, was provided by Haake and Reibold [54] and later by Ford and O’Connell

in [116].

The generality of such a solution is outstanding; however, as noticed in [116],

solving the time-dependent master equation is in general a formidable problem.

In [116] the authors show that the dynamics of the system can be more easily

solved by working with the Wigner function of the system and bath at time tand then averaging over the degrees of freedom of the bath. According to their

procedure, the reduced Wigner function W at time t can be expressed in terms ofthat at time t = 0 as follows

W (x, p, t) =

Z +1

�1dr

Z +1

�1dqP (x, p; r, q; t)W (r, q, 0), (3.27)

where P describes the transition probability of Gaussian form [116]. The draw-back of such a procedure is the limited set of initial states ⇢̂

S

for which the Wigner

function is analytically computable. For Gaussian states this is not a problem;

however there exist physically relevant situations where this is not the case [117–

119]. An example is provided by a system initially confined in an infinite square

potential [cf. Sec. 3.3.5].

Here we report an alternative approach to the master equation in Eq. (3.26) de-

rived in the Heisenberg picture [120]. The approach exploited is valid for a gen-

eral environment at arbitrary temperatures, regardless of the strength of the cou-

pling and of the form of the initial state. The master equation we derive is exact

and equivalent to that in [54, 55], however it can be use also for non-Gaussian

states.

3.3.1 The adjoint master equation

We start by deriving the adjoint master equation for QBM. This is the dynamical

equation describing the time evolution of a generic operator Ô of the system S,once the average over the bath is taken.


For reasons that will be clear later, let us consider the von Neumann representa-

tion [121, 122] of the operator Ô, defined, at time t = 0, by the following relation:

Ô =

Z

d� dµ O(�, µ)�̂(�, µ, t = 0), (3.28)

where O(�, µ) is the kernel of the operator Ô and �̂(�, µ, t = 0) = exp[i�x̂+ iµp̂] isthe generator of the Weyl algebra, also called characteristic or Heisenberg-Weyl

operator [122]. The reduced operator Ô at time t is obtained by taking the unitarytime evolution of the extended operator Ô ⌦ 1̂

B

with respect to the total Hamil-

tonian ĤT

of the system plus bath, where 1̂B

is the bath identity operator, and by

tracing over the degrees of freedom of the bath. In terms of the von Neumann

representation, this reads

Ôt =

Z

d� dµ O(�, µ)�̂t, (3.29)

where we introduced the characteristic operator at time t:

�̂t = Tr(B)

⇥

⇢̂B

⇣

Û †t (�̂(�, µ, 0)⌦ 1̂B)Ût⌘

⇤

,

= Tr(B)⇥

⇢̂B

ei�x̂(t)+iµp̂(t)⇤

,(3.30)

and Ût = exp(� i~ĤTt), x̂(t) and p̂(t) are the position and momentum operators ofthe system S evolved by the unitary evolution Ût and ⇢̂B is defined in Eq. (3.4).

In order to obtain the explicit expression of x̂(t) and p̂(t), we rewrite the bathand interaction Hamiltonians defined in Eq. (3.3) in terms of the creation and

annihilation operators b̂†k and b̂k of the k-th bath oscillator: ĤB =P

k ~!kb̂†kb̂k and

ĤI

= �x̂B̂(0), where B̂(t) is defined as

B̂(t) = �X

k

Ck

r

~2mk!k

⇣

b̂ke�i!

k

t + b̂†kei!

k

t⌘

. (3.31)

We solve the Heisenberg equations of motions for x̂(t) and p̂(t) by using theLaplace transform:

x̂(t) = G1(t)x̂ + G2(t)p̂

M+

1

M

Z t

0

ds G2(t � s)B̂(s), (3.32a)

p̂(t) = MĠ1(t)x̂ + Ġ2(t)p̂ +

Z t

0

ds Ġ2(t � s)B̂(s), (3.32b)

where x̂ and p̂ denote the operators at time t = 0, and the two Green functionsG1(t) and G2(t) are defined as

G1(t) =d

dtG2(t), G2(t) = L�1

M

M(s2 + !2R

)� L[D(t)](s)/~�

(t), (3.33)


where L denotes the Laplace transform, and D(t) is the dissipation kernel definedin Eq. (3.8). Given Eqs. (3.32), since the operators of the system and of the bath

commute at the initial time, it follows that:

�̂t = ei↵1(t)x̂+i↵2(t)p̂ Tr(B)

⇥

⇢̂B

�̂�(t)⇤

, (3.34)

where ↵1(t) and ↵2(t) are defined as

↵1(t) = �G1(t) + µMĠ1(t), (3.35a)

↵2(t) = �G2(t)/M + µĠ2(t), (3.35b)

and the operator �̂B

(t) refers only to the degrees of freedom of the environment:

�̂B

(t) = exp

i

Z t

0

ds B̂(s)↵2(t � s)�

. (3.36)

Under the assumption of a thermal state for the bath [cf. Eq. (3.5)], the trace over

�̂B

(t) gives a real and positive function of time Tr(B)⇥

⇢̂B

�̂B

(t)⇤

= e�(t), where the ex-plicit form of �(t) can be obtained exploiting the definition of the spectral densityin Eq. (3.6). In Appendix A.1 we present the explicit form of �(t), written as thesum of three terms: �(t) = �2�1(t) + µ2�2(t) + �µ�3(t). The time derivative of �̂tgives

d�̂tdt

=h

i↵̇1(t)x̂ + i↵̇2(t)p̂ +i~2

⇥

↵̇1(t)↵2(t)� ↵1(t)↵̇2(t)⇤

+ �̇(t)i

�̂t; (3.37)

after substituting this expression in:

d

dtÔt =

Z

d� dµ O(�, µ)d�̂tdt

, (3.38)

we obtain the adjoint master equation for the operator Ôt. We underline that �̂tdepends also on the two parameter � and µ.

The integral in Eq. (3.38) depends on the choice of the kernel O(�, µ). On the otherhand, we want an equation that can be directly applied to a generic operator Ôwithout having first to determine its kernel. This means that we want to rewrite

Eq. (3.37) in the following time-dependent form

d�̂tdt

= L̃⇤t [�̂t] =i

~

h

ˆ̃Heff

(t), �̂ti

+2X

a,b=1

K̃ab(t)

L̂a�̂tL̂†b �

1

2

n

L̂aL̂†b, �̂t

o

�

, (3.39)

where the effective Hamiltonian

ˆ̃Heff

(t), the hermitian Kossakowski matrix K̃ab(t)and the Lindblad operators L̂a do not depend on the parameters � and µ. Thus,the linearity of Eq. (3.38) will allow to extend Eq. (3.39) to any operator Ôt. Toreach this goal we must rewrite the the coefficients ↵i and �(t), appearing inEq. (3.37), in a way that do not depend on the parameters � and µ. To achieve


this, let us consider the commutation relations among x̂, p̂ and �̂t:

⇥

�̂t, x̂⇤

= ~↵2(t)�̂t and⇥

�̂t, p̂⇤

= �~↵1(t)�̂t. (3.40)Exploiting Eqs. (3.35), we can express ��̂t and µ�̂t as a linear combination of theabove commutators. By using this result, we can rewrite Eq. (3.37) in the structure

given by Eq. (3.39), where

ˆ̃Heff

(t) =p̂2

2M+

�A(t)

2(x̂p̂ + p̂x̂) +

1

2M�A(t)x̂2, (3.41)

the Lindblad operators are L̂1 = x̂ and L̂2 = p̂. The time dependent function �A(t),�A(t), D1(t) and the elements of the Kossakowski matrix K̃ab(t) are reported inAppendix A.2. An important note: one of the elements of the Kossakowski ma-

trix vanishes K̃22(t) = 0, and the corresponding term proportional to [p̂, [p̂, ⇢̂S]] isabsent. In the case of Caldeira-Leggett master equation [29], this implied that the

dynamics was not completely positive. In the case under study, complete posi-

tivity is instead automatically satisfied, as it is explicitly shown in Sec. 3.3.3. This

result is in agreement with previous results [54, 55, 116, 123].

Since Eq. (3.39) is linear in �̂t and does not depend on � and µ, it holds for anyoperator Ôt:

ddtÔt = L̃

⇤t [Ôt]. This is the adjoint master equation for QBM and L̃⇤t

is the generator of the dynamics for the operators. The corresponding adjoint

dynamical map is given by

�⇤t [ · ] = T exp✓

Z t

0

ds L̃⇤s◆

[ · ]. (3.42)

The result here obtained is very general and depends only on: the form of the total

Hamiltonian ĤT

together with the separability of the initial total state [cf. Eq. (3.4)],

but does not depend on the particular initial state of the system S. We now de-rive the master equation for the density matrix, starting from the adjoint master

equation, and we show that we recover known results in the literature.

3.3.2 The Master Equation for the statistical operator

Let us consider the dynamical map �t for the states:

�t : ⇢̂S(0) 7! ⇢̂S(t), (3.43)which is the adjoint map of �⇤t defined in Eq. (3.42), and similarly to it, the map�t can be written as

�t[ · ] = T exp✓

Z t

0

dsLs◆

[ · ]. (3.44)


The adjointness, denoted here by the ⇤-symbol, has to be understood in the fol-lowing sense:

h�̂ti = Tr(S)⇥

�⇤t [�̂(0)] ⇢̂S(0)⇤

= Tr(S)⇥

�̂(0)�t [⇢̂S(0)]⇤

. (3.45)

Let us consider the time derivative of h�̂ti and let us express it as follows:d

dth�̂ti = Tr(S)

⇥

⇤⇤t [�̂(0)] ⇢̂S(0)⇤

= Tr(S)⇥

�̂(0)⇤t [⇢̂S(0)]⇤

. (3.46)

The above equation defines the two maps ⇤t and ⇤⇤t :

⇤t [⇢̂S(0)] = Lt [⇢̂S(t)] = Lt ��t [⇢̂S(0)] , (3.47a)⇤⇤t [�̂(0)] = L̃⇤t [�̂t] = L̃⇤t ��⇤t [�̂(0)] . (3.47b)

If one considers a time independent adjoint master equation, switching to the

master equation for the states is straightforward: the two dynamical maps �⇤tand �t, respectively defined in Eq. (3.42) and Eq. (3.44), reduce to exp(tL̃⇤) andexp(tL). In particular, the map �⇤t and its generator L̃⇤ commute, yielding to

⇤⇤t [�̂(0)] = �⇤t � L̃⇤ [�̂(0)] . (3.48)

By taking the adjoint of the latter expression, and comparing it with the definition

given in Eq. (3.47a), we obtain

L = L̃ ��t ��t�1 = L̃. (3.49)A similar procedure is applied in the time dependent case. We start by consid-

ering the adjoint of the definition given in Eq. (3.47b), which takes the following

form

⇤t [⇢̂S(0)] = �t � L̃t [⇢̂S(0)] , (3.50)and we compare it the definition in Eq. (3.47a). Thus, we obtain

Lt = �t � L̃t ��t�1. (3.51)In terms of this latter expression, Eq. (3.47b) becomes

⇤⇤t [�̂(0)] = �⇤t � L⇤t [�̂(0)] . (3.52)

Accordingly, when the generator is time dependent, in order to construct the mas-

ter equation for the states one needs to derive explicitly the form of L⇤t . The ex-plicit calculations are reported in Appendix A.3 and the final result is:

L⇤t [�̂(0)] =i

~

h

Ĥeff

(t), �̂(0)i

+2X

a,b=1

Kab(t)

L̂a�̂(0)L̂†b �

1

2

n

L̂aL̂†b, �̂(0)

o

�

, (3.53)


where

Ĥeff

(t) =p̂2

2M� �

A(t)

2(x̂p̂ + p̂x̂) +

1

2M�A(t)x̂2, (3.54)

and the elements of Kab(t) are reported in Appendix A.3. Finally, by mergingEq. (3.46) with Eq. (3.52), we obtain

d

dth�̂ti = Tr(S)

⇥

L⇤t [�̂(0)]�t [⇢̂S(0)]⇤

= Tr(S)⇥

�̂(0)⇤t [⇢̂S(0)]⇤

, (3.55)

where ⇤t[⇢̂S(0)] =ddt ⇢̂S(t) This can be simply done by exploiting the cyclic prop-

erty of the trace Tr(S)⇥ · ⇤, that applied on the expression in Eq. (3.53) leads to the

master equation for the states of the system S:

d⇢̂S

(t)

dt= � i~

h

Ĥeff

(t), ⇢̂S

(t)i

+2X

a,b=1

Kab(t)

L̂†b⇢̂S(t)L̂a �1

2

n

L̂aL̂†b, ⇢̂S(t)

o

�

, (3.56)

This is the desired result, which coincides with the master equation in (3.26).

3.3.3 Complete Positivity

We now discuss the complete positivity of the dynamical map �⇤t generated byL̃⇤t defined in Eq. (3.39). The action of this dynamical map on the generic operatorÔ of the system S is

�⇤t [Ô] = Ôt = Tr(B)

⇥

⇢̂B

⇣

Û †t (Ô ⌦ 1̂B)Ût⌘

⇤

, (3.57)

which is the combination of two completely positive maps: the unitary evolution

provided by the total Hamiltonian of system plus environment, and the trace

over the environment. Therefore, by construction the dynamical map is com-

pletely positive. However, in a situation where approximations are needed in

order to compute explicitly the coefficients of the (adjoint) master equation, the

verification of the complete positivity of the dynamics becomes a fundamental

point of interest.

When the generator L of the dynamics is time independent, the sufficient and nec-essary condition for the complete positivity of the dynamical map is the positivity

of the Kossakowski matrix [61, 124]. For a time dependent generator Lt, instead, apositive Kossakowski matrix is only a sufficient condition for complete positivity.

An example is precisely the HPZ model under study, whose Kossakowski matrix

is not positive for all times, nevertheless the dynamics is completely positive. For

a time dependent generator, a necessary and sufficient condition instead is given

by the following theorem [125, 126], under the assumption of a Gaussian channel.


The action of the dynamical map �t on the characteristic operator �̂ is

�⇤t : �̂(0) = exp⇣

i h⇠|Ri⌘

7! �̂t = exp (i h⇠|Xt|Ri) exp��12 h⇠|Yt|⇠i

�

, (3.58)

where Xt and Yt are 2 ⇥ 2 matrices describing the evolution of the characteristicoperator

Xt =

✓

G1(t) G2(t)/MMĠ1(t) Ġ2(t)

◆

, Yt =

✓�2�1(t) ��3(t)��3(t) �2�2(t)

◆

, (3.59)

and h⇠| = (�, µ) and hR| = (x̂, p̂). In terms of Xt, Yt and of the symplectic matrix⌦ =

�

0 1�1 0

�

, we can define the following matrix t:

t = Yt +i~2⌦� i~

2Xt⌦X

T

t . (3.60)

The theorem states that the necessary and sufficient condition for the dynamical

map �t to be completely positive (CP) is the positivity of t for any t > 0. Sincethe matrix t is a 2⇥ 2 matrix, the request of its positivity reduces to the requestof positivity of its trace and determinant:

Tr⇥

t⇤

= �2⇣

�1(t) + �2(t)⌘

, (3.61a)

det⇥

t⇤

= 4�1(t)�2(t)� �23(t)�1

4

⇣

~� F (t)⌘2

. (3.61b)

The condition of positivity of the trace, Eq. (3.61a), is easily verified for all phys-

ical spectral densities: the spectral density is positive by definition [cf. Eq. (3.6)]

and this implies the negativity of �1(t) and �2(t) [cf. Eqs. (A.1)] for any tempera-ture. On the other hand, the second condition, Eq. (3.61b), cannot be easily ver-

ified in general. Once a specific spectral density J(!) is chosen, one can checkexplicitly whether det[ t] � 0. For example, the spectral density J(!) / !, orig-inally chosen in [29] to describe the quantum brownian motion, does not satisfy

the above condition also in the simple case of no external potentials. It is in fact

well known that the Caldeira-Leggett master equation is not CP.

As already remarked, the QBM model automatically guarantees complete posi-

tivity. On the other hand, in particular cases one is not able to compute explicitly

the time dependent coefficients of the Kossakowski matrix. Approximations are

needed, in which case complete positivity is not automatically guaranteed any-

more. This can be checked in a relatively easy way by assessing the positivity of

det( t).


3.3.4 Time evolution of relevant quantities

The original QBM master equation (3.26) is expressed in terms of functions, whose

explicit expression is not easy to derive, even if one considers the solution given

in [116]. They are solutions of complicated differential equations, difficult to solve

except for very simple situations. More important, expectation values are not

easy to compute: one has to determine the state of the system at time t, whichis in general a formidable problem also in a particularly simple situation. In our

derivation, instead, the adjoint master equation provides a much easier tool for

the computation of expectation values. The evolution is expressed in the Heisen-

berg picture, therefore it does not depend on the state of the system S but onlyon the properties of the adjoint evolution �t.

For example, by plugging the expression of x̂2(t) (obtained from Eq. (3.32)) inEq. (3.39) we obtain an equation for the expectation value hx̂2t i:

d

dthx̂2t i = 2Ġ1(t)Ġ2(t) hx̂2i+ 2Ġ1(t)Ġ2(t) hp̂2i /M2+

+ (G1(t)Ġ2(t) + Ġ1(t)G2(t)) h{x̂, p̂}i /M � 2�̇1(t).(3.62)

The time dependence in the right hand side of the latter equation is only in the

functions G1(t), G2(t) and �1(t): the expectation values of the operators here ap-pearing are computed at time t = 0. This implies that Eq. (3.62) can be directlysolved without having to consider the full system of differential equations (con-

sisting of

ddt hx̂2t i, ddt hp̂2t i and ddt h{x̂t, p̂t}i), as it is necessary when one deals with

the problem in the Schrödinder picture [24], as well as for the case of the Wigner

function approach [116, 127, 128].

To make an explicit example, we provide the general solution of some physi-

cal quantities of interest for a specific spectral density. We consider: the diffu-

sion function ⇤dif(t) = hx̂2t i � hx̂ti2 and the average energy of the system E(t) =hp̂2t i /2M + 12M!2S hx̂2t i, whose explicit expressions are given in Appendix A.4. Wealso consider the decoherence function �dec(t), which is defined as follows. Weconsider a particle which, at time t = 0, is described by a state | (t = 0)i =N [|↵i+ |�i], where |↵i and |�i are two gaussian wave packets with spread equalto �0, centered respectively in x↵ = h↵|x̂|↵i and x� = h�|x̂|�i and N is the nor-malization constant. The probability density in position at time t is [24]:

P(x, t) = N 2(

�X

a=↵

⇢aa(x, t) + 2q

⇢↵↵(x, t)⇢��(x, t) exp [�dec(t)] cos

⇥

'(x, t)⇤

)

,

(3.63)

where ⇢↵�(x, t) = hx|Tr(B)⇥Ut(|↵i h�|)U †t

⇤|xi. The latter expression has two contri-butions: the incoherent sum

P�a=↵ ⇢aa(x, t) which describes the two populations

of the system, and the last term which determines the interference pattern of the

superposition. The latter is modulated by the phase '(x, t) and the reduction ofthe interference contrast is determined by the decoherence function �dec(t) < 0,


which takes the following form:

�dec(t) = �4�40�

2p + ~2�2x~2 ·

M2�1(t)

8M2�20�1(t)� ~2G22(t)� 4M2�40G21(t). (3.64)

Here �x and �p are the distances between the two guassians in position andmomentum, and the function �1(t) is defined in Eq. (A.1a).

As a concrete example, we consider the case of the Drude-Lorentz spectral den-

sity, which is commonly used for example to describe light-harvesting systems

[129, 130]:

J(!) =2

⇡M�

m

⌦2!

(!2 + ⌦2), (3.65)

where ⌦ is the characteristic frequency of the bath. The corresponding dissipationand noise kernels, defined in Eq. (3.8), are:

D(t) = 2M�m

~⌦2e�⌦|t|sign(t), (3.66a)

D1(t) =2M�

m

~⌦2⇡

�L

✓

e�2⇡|t|�~ , 1,��~⌦

2⇡

◆

+ �L

✓

e�2⇡|t|�~ , 1,

�~⌦2⇡

◆�

+

+ 2M�m

~⌦2e�⌦|t| cot✓

�~⌦2

◆

,

(3.66b)

where the function �L is the Hurwitz-Lerch function �L(z, s, a) =P+1

n=0 zn(n +

a)�s. The two Green functions are:

G2(t) =3X

i=1

(⌦+ Ci)eCit

Di, (3.67)

and G1(t) =ddtG2(t), where C1, C2 and C3 are the complex roots of the polynomial

y(s) = (y2+!2S

+2�⌦)(y+⌦)�2�⌦2 and Di =Q3

j=1,j 6=i(Ci�Cj). In terms of thesefunctions, we can compute the functions �i(t) with the help of Eqs. (A.1) as wellas the three relevant quantities previously discussed, whose explicit expressions

are displayed in Eq. (3.64), Eq. (A.17) and Eq. (A.18).

Fig. 3.1 and Fig. 3.2 show the evolution of the diffusion function ⇤dif(t), of theenergy E(t) and of the decoherence function �dec(t), and we compare their timeevolution according

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