Open Quantum System dynamics: Applications to Decoherence and Collapse models

(1)

(2)

(3)

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:

(6)

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: https://arxiv.org/abs/1602.01979

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:

(7)

List of attended Schools, Workshops

and Conferences

1. September, 2017

Training Workshop at Instituto Superior Tecnico of Lisbon, Portugal Title Lisbon Training Workshop on Quantum Technologies in Space

http://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 physics

http://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 problem

https://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 Days

http://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 Malta

http://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 Physics

https://light-and-matter.github.io/autumn-school

(8)

7. May, 2016

Workshop in Pontremoli, Italy

Title Quantum control of levitated optomechanics

https://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 standard quantum 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, Italy Title Fundamental Problems in Quantum Physics

http: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

(9)

Introduction

When I look back to the time, already twenty years ago, when the concept and magnitude of the physical quantum of action began, for the first time [. . . ] the whole development [from the mass of experimental facts to its disclosure] seems to me to provide a fresh illustration of the long-since proved saying of Goethe’s that man errs as long as he strivesa_{. And the whole strenuous intellectual work}

of an industrious research worker would appear [. . . ] in vain and hopeless, if he were not occasionally through some striking facts to find that he had, at the end of all his criss-cross journeys, at last accomplished at least one step which was conclusively nearer the truth.

a_{Johann 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 typically answered by a physicist: “To start, let us approximate the problem by considering a spherical 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 syssys-tem. 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.

(10)

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.2and 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 this memory 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 CSLbetween the system and the noise

(11)

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 6m [49], while force noise measurements on

nanomechanical cantilevers [50,51] and on gravitational wave detectors give the strongest bound for rC > 10 6m [52, 53]. Albeit several tests were performed

during 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. 1we 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.

(12)

In Chap.3we 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 equation1_{. 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.4an 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.5we 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.6we 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

(13)

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 ˆ⇢SE at time t = 0 of the global system, composed by the system

of interest S and its environment E, is usually considered as uncorrelated and it evolves according to the unitary evolution ˆUt:

ˆ

⇢SE= ˆ⇢S⌦ ˆ⇢E, (1.1a)

ˆ

⇢SE(t) = ˆUt⇢ˆSEUˆt†. (1.1b)

where ˆ⇢Sand ˆ⇢Eare the system and environmental states respectively. After some

time t, the interaction between the system and its environment correlates the two, and ˆ⇢SE(t) cannot be written in the form as in Eq. (1.1a). To extract the system

properties from ˆ⇢SE(t)one needs to average over the d.o.f. of the environment

ˆ ⇢S(t) = Tr(E) ⇥ ˆ ⇢SE(t) ⇤ , (1.2)

where ˆ⇢S(t) is called reduced state of the system S, and it is obtained by taking

the partial trace (Tr(E)⇥

· ⇤) over the d.o.f. of E. This operation is performed by choosing a basis { | (E)

i i }i of the Hilbert space associated to E and applying the

(14)

This definition does not depend on the choice of the basis. The state ˆ⇢S(t)obtained

in 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)⇥_⇢_ˆ

S(t)

⇤

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

ˆ_t_{: ˆ}_⇢_S_{7! ˆ⇢}_S_{(t) = Tr}(E)⇥ ˆ

Ut⇢ˆSEUˆt†

⇤

, (1.4)

which is different from the one describing the unitary dynamics ˆUt of the global

system, 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 scheme1_:

ˆ ⇢SE unitary evolution ! ˆ Ut ˆ ⇢SE(t) = ˆUt⇢ˆSEUˆt† Tr(E) ? ? y ??yTr(E) ˆ ⇢S ˆ_t ! reduced evolution ⇢ˆS(t) = Tr (E)⇥ ˆ Ut⇢ˆSEUˆt† ⇤ . (1.5)

We can also express the reduced dynamical map as: ˆ_t_[_{· ] = T exp}✓Z t

0

dsLs

◆

[_{· ],} (1.6)

where T is the time-ordering operator and Ls is the generator of the dynamics. Ls

describes the most important dynamical equation in the theory of the open quan-tum system: the quanquan-tum master equation

dˆ⇢S(t)

dt =Lt[ˆ⇢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 ˆ⇢S, one finds:

d dtˆt[ˆ⇢S] =Lt T exp ✓Z t 0 dsLs ◆ [ˆ⇢S] =Lt ˆt[ˆ⇢S], (1.8)

which corresponds to Eq. (1.7).

All dynamical maps ˆt realized following the scheme in Eq. (1.5) satisfy by

con-struction some important features. The first is the linearity of the dynamical map: ˆ_t_[ˆ_⇢₁_{+ ˆ}_⇢₂_{] = ˆ}_t_[ˆ_⇢₁_{] + ˆ}_t_[ˆ_⇢₂_], _(1.9)

1_{If the initial state ˆ⇢}

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

defined in Eq. (1.4) with ˆ⇢S = Tr(B)⇥⇢ˆSE⇤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

(15)

which is fundamental to comply with the superposition principle. The second property of ˆtis its continuity:

lim

⌧!0

ˆ_t+⌧_[ˆ_⇢_S_] ˆ_t_[ˆ_⇢_S_{] = 0,} _(1.10) which naturally follows from the continuity of ˆUt. The third property is related

to the probability interpretation we give to ˆ⇢S: its diagonal terms describe the

probabilities of finding the system in a particular state. Thus it follows that, to maintain consistently this interpretation, ˆtmust preserve the positivity, the

her-miticity and the trace of ˆ⇢S:

ˆ_t_[ˆ_⇢_S_] _0, ⇣ ˆ_t_[ˆ_⇢_S_]⌘†_{= ˆ}_t_[ˆ_⇢_S_{] ,} Tr(S)⇥⇢ˆS ⇤ = 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: ˆtmust be completely positive.

1.1.1 Complete positivity

A map ˆtis Completely Positive (CP) if (ˆ1n⌦ ˆt)is a positive map 8n 2 N, where

ˆ

1_n_{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 ˆ⇢S and ˆ⇢A, respectively.

Sup-pose the environment acts independently on the two systems, with the dynami-cal maps ˆt and ˆ⇤t respectively. Then, the dynamical map describing the effect

of the environment on the two systems is constructed as

(ˆ⇤t⌦ ˆt) : ˆ⇢AS7! ˆ⇢AS(t), (1.12)

where ˆ⇢AS = ˆ⇢A⌦ ˆ⇢S, ˆ⇢AS(t) = ˆ⇢A(t)⌦ ˆ⇢S(t)with ˆ⇢A(t) = ˆ⇤t[ˆ⇢A]and ˆ⇢S(t) = ˆt[ˆ⇢S].

Then, we require the positivity of the final total state ˆ⇢AS(t) 0, i.e. the total map

(ˆ⇤t⌦ ˆt) must be a positive map. This must hold for any form of the map ˆ⇤t,

even when ˆ⇤t = ˆ1n, and for any dimension n of the Hilbert space associated to

(16)

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 Hilbert space: L [ˆ⇢S(t)] = i ~ h ˆ HS, ˆ⇢S(t) i i ~ h ˆ H, ˆ⇢S(t) i + N2 ₁ X a,b=1 Kab  ˆ La⇢ˆS(t) ˆL†_b 1 2 n ˆ L†_bLˆa, ˆ⇢S(t) o , (1.13) where the first term describes the free coherent evolution with respect to the sys-tem Hamiltonian ˆHS, as it is isolated. The last two terms result from the influence

of the environment: _Hˆ _{modifies the coherent evolution, and is called Lamb} shift. The last term in Eq. (1.13) is called dissipator and it is distinguished by its characteristic structure. Here, ˆLa are called Lindblad operators and Kab is

the 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. _H(t)ˆ _{and K}_ab_{(t). If K}_ab_(t)_{remains a} positive matrix for any time t, then the corresponding dynamical map satisfies all the 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 ˆt. In fact, there exist

cases where, for some finite time intervals, Kab(t) < 0 and still the dynamical

(17)

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 = _p1

2(|Li + |Ri), where

|Li and |Ri identify the states centered on two different positions, e.g. on the left and 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

| SEi = p1₂ |Li + |Ri ⌦ | Ei , (2.1)

where | Ei 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 the total state becomes

| SE(t)i = p1₂(|Li ⌦ | LEi + |Ri ⌦ | R

Ei), (2.2)

where | L

Ei is the state the environmental particle takes if the system is in |Li, and

| R

Ei is the state of the particle if the system is in |Ri. Let us now take the reduced

states of S associated to the states in Eq. (2.1) and Eq. (2.2), and represent them on the system basis {|Li , |Ri}. The corresponding density matrices are

⇢S = 1₂ ✓ 1 1 1 1 ◆ and ⇢S(t) = 1₂ ✓ 1 _h R E| LEi h L E| REi 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, | L

Ei and | REi

are not equal, we have | h L

E| REi |  1. Consequently, the coherences are reduced.

(18)

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 decoherdecoher-ence 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.4we 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 a localized eigenstate |xi of the position, collides with a particle of the surrounding gas, whose state is | i and mass is m. The scattering (or collision) between the two particles can be schematically represented by

|xi ⌦ | i scattering! ˆS (|xi ⌦ | i) , (2.5) where ˆS is the unitary scattering operator describing the collision process. In order to simplify the problem, we consider the recoil-less limit, i.e. the infinite mass limit1 _(M _{m). In this limit, we can safely assume that the scattering}

operator heavily affects only the gas particle state and not that of S. Then Eq. (2.5) can be approximated by

|xi ⌦ | i scattering! |xi ⌦ ˆSx| i , (2.6)

where the scattering operator ˆSx depends on the position x of the system. In

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

(19)

|'i = dx '(x)_{|xi. Consequently, also the scattering process changes to} |'i ⌦ | i scattering!

Z

dx '(x)_{|xi ⌦ ˆS}x| i . (2.7)

In terms of the statistical operator of the reduced system, the scattering process can be described by ˆ ⇢S= R dxR dx0_'(x)'⇤_(x0₎_{|xi hx}0_{| ,} ? ? yscattering ˆ ⇢S(t) = R dxR dx0_'(x)'⇤_(x0₎_{|xi hx}0_{| ⌘(x, x}0_). (2.8) where ⌘(x, x0_{) =} _{h |ˆS}†

x0Sˆx| i. Since ˆSx is unitary, it does not change the

popula-tions of the system: ⌘(x, x0_{) = 1}_{. Instead, for x 6= x}0 _{the environmental state | i}

is changed according to the position x where the scattering occur, and we have | h |ˆS†x0ˆSx| i | = | h x0| _xi | < 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 state2 _is

ˆ

⇢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 d ˆn0_{|f(q, q ˆ}n0)_|2⇣1 ei(q q ˆn0)~·(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, q ˆn0₎ _{is the scattering amplitude of the}

process with an incoming (outgoing) scattered particle momentum q (q ˆn0_{). The}

phase 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: ˆ Sx = e i ˆ q·x ~ Sˆ0ei ˆ q·x

~ . The velocity v(q) take different expressions depending on the

scattering particle involved: for a massive bath particle we have v(q) = q/mgas,

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

d⇢S(x, x0; t)

dt = ⇤(x, x

0_)⇢

S(x, x0; t), (2.12)

2_{We will denote the environmental state with ˆ⇢}

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

(20)

where ⇢S(x, x0; t) =hx|ˆ⇢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 a

limit 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 only

on the modulus of q, we can evaluate the angular part of the integral as follows: Z _{d ˆ}_{n d ˆ}_n₀

4⇡ |f(q, q ˆn

0₎_|2 ₌

TOT(q), (2.13)

with TOT(q)denoting the total cross section of momentum q, averaged over the

directions. Consequently, Eq. (2.11) becomes position independent: ⇤(x, x0)_' TOT= 4⇡ngas

Z

dq q2_µ

B(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, x

0_{; 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, allowing

to Taylor expand the expression to the second order in q. By also assuming that µB(q) = µB(q), Eq. (2.11) reduces to

⇤(x, x0) = ngas

~2 (x x

0₎2Z _{dq µ}

(21)

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 the incoming and the outgoing scattered vectors.

The master equation corresponding to the localization rate in Eq. (2.17) reads dˆ⇢S(t)

dt = ⌘

2[ˆx, [ˆx, ˆ⇢S(t)]] , (2.19) where the diffusion constant takes the following expression

⌘ = ngas ~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 ⌘ 2

xt_. (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

ˆ ⇢S = Z d⌦ Z d⌦0'(⌦)'⇤(⌦0)|⌦i h⌦0_{| ,} _(2.22)

(22)

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 be described as follows

|⌦i ⌦ | i scattering! |⌦i ⌦ ˆS⌦| i , (2.23)

where the recoil-less limit is considered [cf. Eq. (2.6)] and thus the scattering oper-ator ˆS⌦acts on the environmental state | i only. ˆS⌦can be related to the standard

scattering operator ˆS0 acting in the ⌦ = 0 configuration through a rotation from

the configuration |0i to |⌦i: ˆS⌦= ˆR(⌦)ˆS0Rˆ†(⌦).

After the scattering process, the reduced state of the system is given by ˆ ⇢S(t) = Z d⌦ Z d⌦0'(⌦)'⇤(⌦0)_{|⌦i h⌦}0_{| ⌘(⌦, ⌦}0), (2.24) where ⌘(⌦, ⌦0) = Tr(B)⇥⇢ˆBˆS†_⌦0Sˆ⌦ ⇤ . (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, the state of the system can be identified by |↵i = ˆRz(↵)|0i, with ˆRz(↵) = exp( _~iLˆz↵)

where ˆLz 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) +1 X l=0 l X m= l hE, l, m|ˆS†↵0Sˆ↵|E, l, mi , (2.26)

where µB(E)is the energy distribution of the environmental states, l and m are

respectively the eigenvalues of the total angular momentum and ˆLz. Writing ˆS↵in

3_{These three angles describe three consecutive rotations of the system, as depicted in the}

(23)

terms of ˆS0 = ˆ1+iˆT,where ˆT 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 dE0_{| hE, l, m|ˆT}†_|E0, l0, m0_{i |}2, (2.27)

where we used the relation for the T-matrix: i(ˆT† _{T) = ˆT}ˆ †_{T, and introduced}ˆ a completeness: ˆ1 = R dE0P

l0_,m0|E0, l0, m0i hE0, l0, m0|. We proceed by evaluating

the matrix elements of the T-matrix, which are defined in terms of to the scattering amplitude f(p, p0₎_{of the process:}

hp|ˆT|p0i = i 2⇡~mgas

(E E0)f (p, p0), (2.28) where p and p0 _{are respectively the incoming and outgoing momentum of the}

environmental particle of mass mgas, and (E E0)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|ˆT|p0_{i |}2 ₌ t (2⇡~)3_m gas (p p0) p |f(p, p 0₎_|2_, _(2.29)

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

2_(E _E0_{) = lim} t!+1 t 2⇡~ mgas p (p p 0_). _(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 p M p (E E 0_)Y lm( ˆp), (2.31)

where Ylm( ˆp) are the spherical harmonics with quantum numbers l and m and

with an angular dependence defined with respect to the direction ˆp. 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_):

(24)

with ⇤(↵,↵0) = 2mgas (2⇡~)3 Z dE EµB(E) X lm X l0_m0 ⇣ 1 e i(m m0)(↵0 ↵)⌘_· · Z d ˆp Z d ˆp0 Z d ˆp00 Z d ˆp000Y_l⇤0m0( ˆp)Ylm( ˆp0)Yl0_m0( ˆp00)Y_lm⇤ ( ˆp000)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 1 X l=0 l X m= l Y_lm⇤ ( ˆp0)Ylm( ˆp) = (✓ ✓0) sin✓ ( 0_), _(2.34)

where the two direction ˆpand ˆp0are 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 dp µB(p) p mgas Z d ˆn0_{|f(p, pˆ}n0)_|2(1 R(p, p ˆn0, !)) , (2.35) where R(p, n0, !) = f⇤(p!, p ˆn0!) f⇤_{(p, p ˆ}_n0₎ , (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↵|ˆ⇢S(t)|↵0i.

It is interesting to notice that the expression for ⇤(↵, ↵0₎_{in Eq. (2.35) has the same}

structure of Eq. (2.11), with R(p, n0_{, !)}_{replaced by}

R(q, ˆn0, x x0) = f

⇤_(q

x x0, q ˆn0_{x x}0)

f⇤_{(q, q ˆ}_n0₎ . (2.38)

Here qx 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, q ˆn0) = ~mgas 2⇡

Z

(25)

Implementing the translation in space: ˆSx = e i ·x

~ Sˆ₀ei ~·x, as it was done to obtain

Eq. (2.11), we have f⇤(qx x0, q ˆn0_{x x}0) = ~m gas 2⇡ Z dr ei(q q ˆn0)·(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, q ˆn0_{x x}₀)

f⇤_{(q, q ˆ}_n0₎ = e

i(q q ˆn0)_~·(x x0)_, _(2.41)

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

(26)

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]

¨

x(t) + 2 m˙x(t) + _M1 @xV (x) = _M1 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 mis a damping rate. The stochastic force F (t) describes the noise induced

by 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 a statistical average. For the Brownian motion they read

hF (t)i = 0,

hF (t)F (s)i = 4M mkBT (t s),

(3.2) where kBis the Boltzmann constant, T is the temperature of the environment.

The second model instead considers a particle S immersed in an environment of 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 ˆx and momentum ˆ

p, harmonically trapped at frequency !0 and interacting with a thermal bath of

independent harmonic oscillators, with positions ˆRk, momenta ˆPk, mass mkand

(27)

where ˆ HS = ˆ p2 2M + 1 2M ! 2 0xˆ2, HˆB = X k ˆ P2 k 2mk +1 2mk! 2 kRˆ2k, HˆI = ˆx X k CkRˆk, (3.3)

are respectively system, bath and interaction Hamiltonians. The total initial state is assumed to be uncorrelated

ˆ

⇢T = ˆ⇢S⌦ ˆ⇢B, (3.4)

where ˆ⇢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:

ˆ ⇢B =

e HˆB

Tr(B)⇥e HˆB⇤, (3.5)

where = 1/(kBT )is the inverse temperature. The characterization of the set of

coupling constants Ckis provided by the spectral density which is defined as1

J(!) =X k C2 k 2mk!k (! !k). (3.6)

In terms of the latter, we can define the two-time correlation function of the bath operator ˆB =P_kCkRˆk:

C(t s) = Tr(B)⇥ ˆB(t) ˆB(s)ˆ⇢B

⇤

= 1₂D1(t s) ₂iD(t s), (3.7)

where ˆB(t) = ei ˆHBt/~_Beˆ i ˆHBt/~. D

1(t) and D(t) are the noise and the dissipative

kernels, 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 1 dt cos(!t)D1(t) = coth✓ ~! 2 ◆ Z +1 1 dt 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

(28)

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 and that 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):

¨ˆx(t) 1 ~M Z t 0 ds D(t s)ˆx(s) i ~M ⇥ V (ˆx), ˆp⇤ = 1 MB(t).ˆ (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 operator

ˆ

B(t). The classical dissipative term proportional to the velocity of the system is now 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, we have D(t s) / @s (t s)and we recover

¨ˆx(t) + 2 m˙ˆx(t) i ~M ⇥ V (ˆx), ˆp⇤ = 1 MB(t),ˆ (3.12)

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

3.2 The Calderira-Leggett master equation

(29)

state remains unperturbed: ˆ⇢T(t)⇡ ˆ⇢S(t)⌦ ˆ⇢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 approximaequa-tion a spectral density J(!) / ! must be considered. This implies that the dissipation kernel has a memoryless structure D(t s) / @s (t s)[cf. Eq. (3.12)]. However, a drawback

of this choice is that the noise kernel D1(t)defined in Eq. (3.8) diverges, and the

KMS 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: dˆ⇢S(t) dt = i ~[ ˆHS, ˆ⇢S(t)] i m ~ [ˆx,{ˆp, ˆ⇢S(t)}] 2M m ~2 [ˆx, [ˆx, ˆ⇢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|ˆ⇢SE(t)|y, Qi = Z dx0 Z dy0 Z dR0 Z dQ0K(x, R, t; x0, R0, 0)_· K⇤(y, Q, t; y0, Q0, 0)_hx0, R0_|ˆ⇢SE|y0, Q0i , (3.14)

where R and Q identify the positions of the environmental particles, x and y the position of the system and K(x, R, t; x0_{, R}0_{, 0)}_{is the position representation of}

e i ˆHTt/~, which can be expressed via a path integral

K(x, R, t; x0, R0, 0) =_{hx, R|e} i ˆHTt/~_|x0_{, R}0_{i =} Z x x0 Dx Z R R0 DR e i ~ST[x,R]_, _(3.15) with ST[x, R] = Rt

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

Lagrangian.

The reduced state of the system S in the position representation is given by ⇢S(x, y, t) =

Z

(30)

which, under the assumption of uncorrelated initial states in Eq. (3.4), can be expressed as ⇢S(x, y, t) = Z dx0 Z dy0J(x, y, t; x0, y0, 0)⇢S(x0, y0, 0). (3.17)

Here the propagator J(x, y, t; x0_{, y}0_{, 0)} _{takes into account the influence of the}

sur-rounding environment: J(x, y, t; x0, y0, 0) = Z x x0 Dx Z y y0 Dy 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 dR0 Z dQ0⇢B(R0, Q0, 0) Z R R0 DR Z Q Q0 DQ 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!kexp n mk!k 2⇡~ sinh( ~!k)[(R 02 k + Q02k) 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 ⇣ 1 2mkR˙ 2 k 12mk! 2 kR2k ⌘ , SI[x, R] = Z t 0 dsxX k CkRk, (3.21)

where LSis 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 ˆ⇢Sinto quantum states. Accordingly, one needs

(31)

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

L [ˆ⇢S(t)] = i ~ h ˆ HS, ˆ⇢S(t) i _i ~ h ˆ H, ˆ⇢S(t) i + 2 X a,b=1 Kab h ˆ La⇢ˆS(t) ˆL†b 12 n ˆ L†_bLˆa, ˆ⇢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

ˆ H = m

2 (ˆxˆp + ˆpˆx), the Lindblad operators are ˆL1 = ˆx and ˆL2 = ˆpand 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 ˆp

dˆ⇢S(t) dt = i ~[ ˆHS, ˆ⇢S(t)] i m ~ [ˆx,{ˆp, ˆ⇢S(t)}] 2M m ~2 [ˆx, [ˆx, ˆ⇢S(t)]] m 8M ⇥ ˆ p, [ˆp, ˆ⇢S(t)] ⇤ . (3.24) With this modification Eq. (3.13) can be written in the form in Eq. (3.22) with

ˆ L = s 4M m ~2 x + iˆ r m 4Mp.ˆ (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 action is 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

(32)

ˆ HT: dˆ⇢S(t) dt = i ~[ ˆH(t), ˆ⇢S(t)] i (t) ~ [ˆx,{ˆp, ˆ⇢S(t)}] h(t) [ˆx, [ˆx, ˆ⇢S(t)]] f (t) [ˆx, [ˆp, ˆ⇢S(t)]] , (3.26)

where ˆH(t)and the coefficients (t), h(t) and f(t) now are time dependent. We refer 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(!) and temperatures T . The explicit form of the coefficients, beyond the weak coupling regime, 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 t and 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 of that at time t = 0 as follows

W (x, p, t) = Z +1 1 dr Z +1 1 dqP (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 ˆ⇢Sfor 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

(33)

For reasons that will be clear later, let us consider the von Neumann representa-tion [121,122] of the operator ˆO, defined, at time t = 0, by the following relation:

ˆ O =

Z

d dµO( , µ)ˆ( , µ, t = 0), (3.28) where O( , µ) is the kernel of the operator Ôand ˆ( , µ, t = 0) = exp[i ˆx + iµˆp] is the generator of the Weyl algebra, also called characteristic or Heisenberg-Weyl operator [122]. The reduced operator Ôat time t is obtained by taking the unitary time evolution of the extended operator Ô_{⌦ ˆ1}B with respect to the total

Hamil-tonian ˆHT of the system plus bath, where ˆ1B 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

ˆ Ot=

Z

d dµO( , µ)ˆt, (3.29)

where we introduced the characteristic operator at time t: ˆt= Tr(B) ⇥ ˆ ⇢B ⇣ ˆ Ut†( ˆ( , µ, 0)⌦ ˆ1B) ˆUt ⌘⇤ , = Tr(B)⇥⇢ˆBei ˆx(t)+iµˆp(t) ⇤ , (3.30)

and ˆUt = exp( _~iHˆTt), ˆx(t) and ˆp(t) are the position and momentum operators of

the system S evolved by the unitary evolution ˆUtand ˆ⇢B is defined in Eq. (3.4).

In order to obtain the explicit expression of ˆx(t) and ˆp(t), we rewrite the bath and interaction Hamiltonians defined in Eq. (3.3) in terms of the creation and annihilation operators ˆb†

kand ˆbk of the k-th bath oscillator: ˆHB =

P k~!kˆb†kˆbkand ˆ HI = x ˆˆB(0), where ˆB(t)is defined as ˆ B(t) = X k Ck r ~ 2mk!k ⇣ ˆbke i!kt+ ˆb†_kei!kt ⌘ . (3.31)

We solve the Heisenberg equations of motions for ˆx(t) and ˆp(t) by using the Laplace transform: ˆ x(t) = G1(t)ˆx + G2(t) ˆ p M + 1 M Z t 0 ds G2(t s) ˆB(s), (3.32a) ˆ p(t) = M ˙G1(t)ˆx + ˙G2(t)ˆp + Z t 0 ds ˙G2(t s) ˆB(s), (3.32b)

where ˆx and ˆp denote the operators at time t = 0, and the two Green functions G1(t)and G2(t)are defined as

(34)

where L denotes the Laplace transform, and D(t) is the dissipation kernel defined in 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:

ˆt = ei↵1(t)ˆx+i↵2(t)ˆp Tr(B) ⇥ ˆ ⇢Bˆ (t) ⇤ , (3.34)

where ↵1(t)and ↵2(t)are defined as

↵1(t) = G1(t) + µM ˙G1(t), (3.35a)

↵2(t) = G2(t)/M + µ ˙G2(t), (3.35b)

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

ˆB(t) = exp  i Z t 0 ds ˆB(s)↵2(t s) . (3.36)

Under the assumption of a thermal state for the bath [cf. Eq. (3.5)], the trace over ˆB(t)gives a real and positive function of time Tr(B)

⇥ ˆ ⇢BˆB(t)

⇤

= e (t)_{, where the}

ex-plicit form of (t) can be obtained exploiting the definition of the spectral density in Eq. (3.6). In AppendixA.1we present the explicit form of (t), written as the sum of three terms: (t) = 2

1(t) + µ2 2(t) + µ 3(t). The time derivative of ˆt

gives d ˆt dt = h i ˙↵1(t)ˆx + i ˙↵2(t)ˆp + i~ 2 ⇥ ˙↵1(t)↵2(t) ↵1(t) ˙↵2(t) ⇤ + ˙(t)iˆt; (3.37)

after substituting this expression in: d

dtOˆt= Z

d dµO( , µ)d ˆt

dt , (3.38)

we obtain the adjoint master equation for the operator ˆOt. We underline that ˆt

depends also on the two parameter and µ.

The integral in Eq. (3.38) depends on the choice of the kernel O( , µ). On the other hand, we want an equation that can be directly applied to a generic operator ˆO without having first to determine its kernel. This means that we want to rewrite Eq. (3.37) in the following time-dependent form

d ˆt dt = ˜L ⇤ t[ ˆt] = i ~ h ˆ˜ Heff(t), ˆt i + 2 X a,b=1 ˜ Kab(t)  ˆ LaˆtLˆ†b 1 2 n ˆ LaLˆ†b, ˆt o , (3.39)

where the effective Hamiltonian ˆ˜Heff(t), the hermitian Kossakowski matrix ˜Kab(t)

and the Lindblad operators ˆLa 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 ˆOt. To

reach this goal we must rewrite the the coefficients ↵i and (t), appearing in

(35)

this, let us consider the commutation relations among ˆx, ˆp and ˆt: ⇥ ˆt, ˆx ⇤ =~↵2(t) ˆt and ⇥ ˆt, ˆp ⇤ = ~↵1(t) ˆt. (3.40)

Exploiting Eqs. (3.35), we can express ˆt and µˆtas a linear combination of the

above commutators. By using this result, we can rewrite Eq. (3.37) in the structure given by Eq. (3.39), where

ˆ˜ Heff(t) = ˆ p2 2M + A_(t) 2 (ˆxˆp + ˆpˆx) + 1 2M A_(t)ˆ_x2_, _(3.41)

the Lindblad operators are ˆL1 = ˆxand ˆL2 = ˆp. The time dependent function A(t),

A_{(t), D}

1(t) and the elements of the Kossakowski matrix ˜Kab(t) are reported in

AppendixA.2. An important note: one of the elements of the Kossakowski ma-trix vanishes ˜K22(t) = 0, and the corresponding term proportional to [ˆp, [ˆp, ˆ⇢S]]is

absent. 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 ˆtand does not depend on and µ, it holds for any

operator ˆOt: _dtdOˆt = ˜L⇤t[ ˆOt]. This is the adjoint master equation for QBM and ˜L⇤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 ˜L⇤_s ◆ [_{· ].} (3.42)

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

t: ˆ⇢S(0)7! ˆ⇢S(t), (3.43)

which is the adjoint map of ⇤

(36)

The adjointness, denoted here by the ⇤-symbol, has to be understood in the fol-lowing sense: hˆti = Tr(S) ⇥ _⇤ t[ ˆ(0)] ˆ⇢S(0) ⇤ = Tr(S)⇥_ˆ(0) t[ˆ⇢S(0)] ⇤ . (3.45)

Let us consider the time derivative of hˆti and let us express it as follows:

d dthˆti = Tr (S)⇥_⇤⇤ t [ ˆ(0)] ˆ⇢S(0) ⇤ = Tr(S)⇥_ˆ(0)⇤ t[ˆ⇢S(0)] ⇤ . (3.46)

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

⇤t[ˆ⇢S(0)] =Lt[ˆ⇢S(t)] =Lt t[ˆ⇢S(0)] , (3.47a)

⇤⇤_t[ ˆ(0)] = ˜L⇤_t[ ˆt] = ˜L⇤t ⇤t [ ˆ(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 ⇤ t

and t, respectively defined in Eq. (3.42) and Eq. (3.44), reduce to exp(t˜L⇤)and

exp(tL). In particular, the map ⇤

t and its generator ˜L⇤commute, yielding to

⇤⇤t [ ˆ(0)] = ⇤t L˜⇤[ ˆ(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 = ˜L t t 1 = ˜L. (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[ˆ⇢S(0)] = t L˜t[ˆ⇢S(0)] , (3.50)

and we compare it the definition in Eq. (3.47a). Thus, we obtain

Lt= t L˜t t 1. (3.51)

In terms of this latter expression, Eq. (3.47b) becomes

⇤⇤_t [ ˆ(0)] = ⇤_t L⇤_t[ ˆ(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 AppendixA.3and the final result is:

(37)

where ˆ Heff(t) = pˆ 2 2M A_(t) 2 (ˆxˆp + ˆpˆx) + 1 2M A_(t)ˆ_x2_, _(3.54)

and the elements of Kab(t) are reported in Appendix A.3. Finally, by merging

Eq. (3.46) with Eq. (3.52), we obtain d dthˆti = Tr (S)⇥ L⇤ t[ ˆ(0)] t[ˆ⇢S(0)] ⇤ = Tr(S)⇥ˆ(0)⇤t[ˆ⇢S(0)] ⇤ , (3.55)

where ⇤t[ˆ⇢S(0)] = _dtd⇢ˆS(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:

dˆ⇢S(t) dt = i ~ h ˆ Heff(t), ˆ⇢S(t) i + 2 X a,b=1 Kab(t)  ˆ L†_b⇢ˆS(t) ˆLa 1 2 n ˆ LaLˆ†b, ˆ⇢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 by

˜ L⇤

t defined in Eq. (3.39). The action of this dynamical map on the generic operator

ˆ Oof the system S is ⇤ t[ Ô] = Ôt = Tr(B) ⇥ ˆ ⇢B ⇣ ˆ Ut†( Ô⌦ ˆ1B) 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, a

(38)

The action of the dynamical map ton the characteristic operator ˆ is ⇤

t : ˆ(0) = exp

⇣

i_h⇠|Ri⌘_{7! ˆ}t = exp (ih⇠|Xt|Ri) exp 1₂h⇠|Yt|⇠i , (3.58)

where Xtand Yt are 2 ⇥ 2 matrices describing the evolution of the characteristic

operator Xt= ✓ G1(t) G2(t)/M M ˙G1(t) G˙2(t) ◆ , Yt = ✓ 2 1(t) 3(t) 3(t) 2 2(t) ◆ , (3.59) and h⇠| = ( , µ) and hR| = (ˆx, ˆp). In terms of Xt, Ytand 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 tto be completely positive (CP) is the positivity of tfor any t > 0. Since

the matrix tis a 2 ⇥ 2 matrix, the request of its positivity reduces to the request

of 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 check explicitly 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.

(39)

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, which is 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 only on the properties of the adjoint evolution t.

For example, by plugging the expression of ˆx2_(t) _{(obtained from Eq. (3.32)) in}

Eq. (3.39) we obtain an equation for the expectation value hˆx2 ti: d dthˆx 2 ti = 2 ˙G1(t) ˙G2(t)hˆx2i + 2 ˙G1(t) ˙G2(t)hˆp2i /M2+ + (G1(t) ˙G2(t) + ˙G1(t)G2(t))h{ˆx, ˆp}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 directly solved without having to consider the full system of differential equations (con-sisting of d

dthˆx

2

ti, dtd hˆp 2

ti and dtd h{ˆxt, ˆpt}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) =} _hˆx2

ti hˆxti2 and the average energy of the system E(t) =

hˆp2

ti /2M +12M !S2hˆx2ti, whose explicit expressions are given in AppendixA.4. We

also consider the decoherence function dec_{(t), which is defined as follows. We}

nor-malization constant. The probability density in position at time t is [24]:

P(x, t) = N2 ( X a=↵ ⇢aa(x, t) + 2 q

⇢↵↵(x, t)⇢ (x, t) exp [ dec(t)] cos

⇥ '(x, t)⇤ ) , (3.63) where ⇢↵ (x, t) = hx| Tr(B) ⇥ Ut(|↵i h |)Ut† ⇤

|xi. The latter expression has two contri-butions: the incoherent sumP_a=↵⇢aa(x, t) which describes the two populations

Open Quantum System dynamics: Applications to Decoherence and Collapse models

Contents

List of Publications

Published works as outcome of the doctoral project

Pre-prints

List of attended Schools, Workshops

and Conferences

Introduction

Outline

Chapter 1

Open Quantum Systems

1.1 Reduced state and its evolution

1.1.1 Complete positivity

1.2 Lindblad equation

Chapter 2

Decoherence

2.1 Gallis-Flemming master equation

2.1.1 Short wavelength limit

2.1.2 Long wavelength limit, Joos and Zeh master equation

2.2 Rotational Decoherence

Chapter 3

Quantum Brownian Motion

3.1 The model

3.2 The Calderira-Leggett master equation

3.2.1 Complete positivity problem

3.3 Non-Markovian Quantum Brownian motion

3.3.1 The adjoint master equation

3.3.2 The Master Equation for the statistical operator

3.3.3 Complete Positivity

3.3.4 Time evolution of relevant quantities