Dynamics and thermodynamics in quantum many-body systems: from steady-state properties of open systems to scaling analysis close to quantum transitions in closed systems.

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University of Pisa

Department of Physics

Graduate Course in Physics

Ph.D. Dissertation

Dynamics and thermodynamics

in quantum many-body systems:

from steady-state properties of open systems

to scaling analysis close to quantum transitions

in closed systems

Candidate:

Davide Nigro

Supervisors:

Dr. Davide Rossini

Prof. Ettore Vicari

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Introduction

1.1 Dynamics in complex systems

In recent years, the characterization of the dynamics and properties of macroscopic systems starting from a detailed description of the fundamental processes involving their microscopic constituents is attracting a great deal of attention, not only in physics. The study of complex systems, that is structures made up of a large number of interconnected parts, is nowadays at the core of the research activity in many diﬀerent ﬁelds of science, such as chemistry, biology as well as economics.

One of the most thrilling features of complex systems is that the behavior of the unified whole, that is the properties of the macroscopic structure, may differ significantly from that of its parts. In addition, as the number of parts increases, new traits that strongly depend on the nature of microscopic interactions emerge. The appearance of macroscopic ordered states as the result of the local microscopic dynamics of billions and usually even more subparts and the observation of transitions between such different phases are without any doubt some of the most interesting features characterizing complex systems in physics. In principle, given any system made up of many microscopic parts, an exact description of all the processes taking place at microscopic scales would lead to a complete and full knowledge of its dynamics at any larger length scale. However, even today, this is far from being possible, except for a few ideal systems described by exactly solvable models. A solution to this problem for a large class of systems was found by J. Willard Gibbs. The ideas developed by Boltzmann while generalizing Maxwell’s results about the kinetic theory led Gibbs to a first formalization of what is now called statistical mechanics. To be more precise, the formalism introduced by Gibbs has led to the development of equilibrium statistical mechanics, a collection of methods suitable for the description of macroscopic properties of systems in or near the thermodynamic equilibrium. In such case, we can answer many questions concerning the macroscopic properties of a given system. Indeed, once the Hamiltonian and few other parameters are set, everything else can be derived by means of statistical mechanics.

The turning point has been the introduction of the concept of ensemble. Instead of looking at the microscopic dynamics to calculate average values of physical quantities (in most of the cases it is an impossible task), one can consider a set of suitably randomized copies of the actual system under investigation, the ensemble, and take averages on this group.

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Each element of the ensemble is a perfect copy of the actual system. However, each copy represents a diﬀerent conﬁguration of the system under investigation, satisfying all the external requirements to which the actual system is subjected. The way each state of the ensemble contributes to the averages is determined by a time-independent probability distribution in phase space. In particular, the structure of such probability distribution has a particular expression - it is represented by a Gibbs state - which depends on only the microscopic Hamiltonian and few other parameters of physical relevance i.e. the tem-perature, the chemical potential etc.

However, during the last decades some fundamental questions about Gibbs formalism and in general about theoretical procedures exploited in equilibrium statistical physics, such as the thermodynamic limit, have been raised. For instance, how is it possible that any given system in thermodynamic equilibrium is described by a Gibbs state? Are there other possibilities? More precisely, is it possible to induce the thermalization towards probability distributions not in the Gibbs form? In addition, for what concerns the thermodynamic limit procedure: how does the behavior of real systems deviate from such ideal condition obtained when both the number of degrees of freedom and the volume are let to become inﬁnite, while keeping constant their ratio?

The following pages are devoted to an analysis of some of these issues.

1.2 Summary of this thesis

This thesis is a detailed description of the research activity carried out by the author during the last three years. In particular, the novel results presented hereafter led to the following four manuscripts

• D. Nigro, Trap effects and continuum limit of the Hubbard model in the presence of an harmonic potential, Phys. Rev. A 96, 033608 (2017);

• D. Nigro, On the uniqueness of the steady-state solution of the Lindblad-Gorini-Kossakowski-Sudarshan equation, arXiv:1803.06279;

• D. Nigro, D. Rossini and E. Vicari, Scaling properties of work fluctuations after quenches at quantum transitions, J. Stat. Mech. 023104 (2019);

• D. Nigro and D. Rossini, Steady-state properties of the dissipative XYZ model in two dimensions: a numerical approach, in preparation.

In the ﬁrst part - Chapter 2 to Chapter 6 - we consider the physics of Markovian open quan-tum systems within the framework developed by Davies, Spohn and other mathematical physicists during the last century. This formalism, which provides a natural framework for studying how equilibrium conditions are approached, allows for a full quantum description of the time evolution of a given quantum system coupled to the surrounding environment, by means of the Lindblad-Gorini-Kossakowski-Sudarshan (LGKS) master equation [19, 20]. Here by environment, we mean everything else we do not directly care of, but that strongly inﬂuences the properties developed by the system under investigation during its time evo-lution. This approach can be considered as an extension of the ideas introduced by Gibbs

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1.2. Summary of this thesis 3

and in some sense sheds light on particular aspects of the theory of equilibrium statistical mechanics that are not usually discussed in standard courses of statistical physics. In equi-librium statistical mechanics one considers a system in thermodynamic equiequi-librium. The system ends up in this equilibrium condition thanks to the presence of a heat reservoir that is an ideal entity, a system, with a given temperature, infinite heat capacity and as a consequence infinite thermal energy, so that when it is placed in thermal contact - in interaction - with other bodies its temperature remains the same. In this case, the sys-tem is represented by a Gibbs state independently of (i ) the nature of the heat reservoir and (ii ) the nature of the interactions. The formalism we will discuss in the following shows that, in general, the presence of the reservoir may drive the dynamics under studies towards asymptotic configurations, called steady-states, that only in particular cases are Gibbs states. This is the case, if the quantum system is fully coupled to the reservoir [33, 34].

The possibility of driving quantum systems towards stationary states not in the Gibbs form is a very appealing prospect. The main reason is that stationary states emerging from the dynamics of open quantum systems are dynamical attractors [32], that is conﬁgurations towards which the system spontaneously evolves. In particular cases, the convergence to-wards these states is guaranteed for any starting conﬁguration of the system.

The properties of such asymptotic states strongly depend on the nature of system-reservoir interactions. However, nowadays it is possible to design system-reservoir interactions “quite” easily by modifying the interactions at microscopic scales. By design, we mean that it is actually possible to engineer both the reservoir and the system-reservoir coupling [1, 2] in such a way that the steady-states resulting from the open quantum dynamics do have desiderable features such as entanglement [3]. The stabilization of this quantum property is of crucial importance for the success of many applications in different fields of quantum physics and for the enhancement of the efficiency of a series quantum protocols. It has been shown that by exploiting entangled states it is possible to perform several tasks better and faster than with any classical technique. These improvements range for example from sub-shot-noise phase estimation in quantum interferometry and metrology, to integer factorisation [4, 5, 6, 7, 8, 9].

The second part of this thesis - Chapter 7 to Chapter 9 - is devoted to the analysis of another aspect of the physics of many-body quantum systems. As we said above, one of the most thrilling features of complex systems is that as the number of microscopic con-stituents increases, new characteristic traits appear at the macroscopic scales. In physics, such new traits lead to the identification of different phases of matter, that is macroscopic configurations having some peculiar features. In most of the cases, such phases emerge due to the interplay of different effects that modify the dynamics at a microscopic level. In addition, by only changing the external conditions, one can drive many-body systems from a phase to another one. These phenomena are called phase transitions. The crucial point is that theoretical investigations carried out during the last century have led to a clear picture of such phenomena in the limiting case where an infinite and homogeneous many-body system is considered, that is in the thermodynamic limit. However, in real experiments such ideal conditions are never met. The second part of this thesis is devoted to an analysis of deviations from the ideal critical behavior into two different situations. In the first case, we have considered deviations from the infinite size: real systems are made up of a finite number of parts, as a consequence we need to know how their properties

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deviate from the ideal case. In particular, we studied how ﬁnite size aﬀects the scaling properties of the statistics of work done on a many-body system close to quantum phase transitions, when the system is driven out of equilibrium by a sudden swicth of one of the control parameters. In recent years such kind of studies have attracted lot of attention. However, most of the research done so far in this context has mostly addressed systems in the thermodynamic limit.

Deviations from the ideal critical behavior emerge also when the system is subjected to constraints that break its spatial homogeneity. This is the condition usually met in ex-periments with cold atoms on optical lattices, where particles are conﬁned within a given portion of the lattice structures by exploiting space-dependent potentials. Such potentials, that in most of the cases can be represented as harmonic, modify the system behavior, leading to deviations from the ideal case described by the thermodynamic limit. In par-ticular, in correspondence of the critical points i.e. the conﬁgurations in parameter space where phase transitions take place, the system shows a non-trivial scaling behavior that cannot be captured by means of the standard approach exploited in statistical physics. This thesis is organized as follows:

• Chapter 2: here we provide a self-contained introduction to the ﬁeld of Markovian open quantum systems. In particular, we introduce the mathematical structures needed for the discussion reported in the following chapters. In addition, we show how to derive the Lindblad-Gorini-Kossakowski-Sudarshan (LGKS) master equation starting from the von Neumann equation and by performing a series of successive approximations.

• Chapter 3: here we focus our attention on the characterization of LGKS equations supporting a unique steady-state configuration. Due to the results provided by Lind-blad, this characterization can be performed by studying the generator of the time evolution, that contrary to the standard approach exploited in quantum mechanics is not the Hamiltonian operator. In addition, in this chapter some theoretical results derived by the author [28] concerning a set of sufficient conditions that guarantee the uniqueness of the steady-state configuration, and as consequence its attractivity (any starting configuration will converge to it as time goes by) are discussed. • Chapter 4: this chapter is devoted to an analysis of the most common numerical

methods exploited in recent years in the characterization of the dynamics of open quantum systems. In particular, in order to keep our discussion as clear as possible, we consider how these different methods have been exploited in the study of dissipa-tive lattice models. Beside representing the perfect playground for testing numerical methods, by analyzing such models it is easy to outline what are the main difficulties while dealing with the LGKS equation. Particular attention here and in the following two chapters is paid to the dissipative XYZ model. Here, by considering this model in d = 1 and for different boundary conditions, we show first how the steady-state properties differ from its canonical counterpart i.e. the ground-state phase diagram. After that, particular attention is paid to the Corner Space Renormalization method. This method has been recently proposed for the characterization of the steady-state

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1.2. Summary of this thesis 5

properties of dissipative lattices in d > 1. In this chapter we show that in this con-text such method is not as reliable as it may seem. This analysis is carried out by considering the two-dimensional dissipative XYZ model in the presence of a uniform magnetic ﬁeld along the x direction, on a ﬁnite cluster having 8 sites (see also results reported in Chapter 6). The solution to this problem can be found by means of other numerical methods such as exact diagonalisation. As a consequence, the quality of the results produced by means of the Corner Space algorithm can be easily tested. • Chapter 5: here we discuss the properties of the dissipative XYZ model in the

Mean-Field approximation1 _{that have been previously addressed in Ref. [49]. This}

approx-imation method is one the most common methods exploited in the description of the dynamics in dissipative quantum systems. However, as we show in this chapter by considering the dynamics of the dissipative XYZ model in d = 1, 2, 3, results obtained within this approximation scheme seem not to be reliable.

• Chapter 6: this chapter is devoted to an analysis of the dissipative XYZ model in d = 2 on finite dimensional clusters. In the first part we have considered the effects of a set of unitary transformations on the LGKS equation governing the dissipative XYZ model. In this way it has been possible to identify the non-trivial structure of the zero field susceptibility tensor characterizing the steady-state configuration of such dissipative model. In particular, the structure of such tensor has been consid-ered in two different cases, that is in the presence of a uniform and in the presence of a staggered magnetic field in the xy-plane.

The second part of this chapter is devoted to a study of the von Neumann entropy and of the entanglement properties of the steady-state conﬁguration. In particu-lar, the entanglement content encoded in the steady-state conﬁguration has been characterised by studying the entanglement Negativity and the Quantum Fisher In-formation.

• Chapter 7: this chapter can be considered as a very brief introduction to the ﬁeld of critical phenomena. We ﬁrst provide the basic tools needed for the characterization of phase transitions, and then we pay attention to the collection of methods nowadays called Renormalization Group(RG). Indeed, RG methods represent the starting point for understanding the Finite-Size Scaling (FSS) and the Trap-Size Scaling (TSS) for-malisms that are at the basis of the analysis reported in the next chapters.

• Chapter 8: here we pay attention to the characterization of the work statistics close to phase transitions [132]. In particular, we consider how finite size affects the work statistics close to quantum phase transitions. A FSS Ansatz for such quantity has been put forward and its validity has been tested by means of numerical simulations. In particular, the scaling properties of the first two moments of such distribution have been investigated in the vicinity of both First-Order Quantum Transitions (FOQT) and Continuous Quantum Transitions (CQTs).

1_{The results shown in this chapter together with those shown in the following one will be collected soon}

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• Chapter 9: in this chapter we consider a d-dimensional lattice system described by Hubbard model at zero temperature. In particular, we show how the non-trivial TSS properties developed by such system at ﬁxed number of particles and in the pres-ence of a weak harmonic potential coupled to the particle density can be derived by performing a continuum limit of the original theory. The results contained in this chapter can be considered as an extension of those discussed by the author in his Master’s Thesis.

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Chapter 2

Theory of open quantum systems

In this first part we provide a self-contained introduction to the field of open quantum systems. In particular, as said in the introductory chapter, we will pay attention to the properties of Markovian open quantum systems. We first set the notation by introducing all the ingredients needed for our discussion. Then, we show by reviewing the most rel-evant papers for our purposes how the Lindblad-Gorini-Kossakowski-Sudarshan (LGKS) equation of motion has emerged during the 70s in the context of dynamical semigroups. We end this chapter by showing how the LGKS master equations can be derived starting from the unitary time evolution i.e. the von Neumann equation by performing a series of successive approximations.

2.1 Preliminaries

The ﬁnal object of our investigation is the characterization of the behavior of a quan-tum system_{S that is coupled to a larger (inﬁnite) quantum system denoted by R which is} usually called reservoir. In particular, our aim is to determine an equation of motion for the system_{S alone which is consistent with all the requirements prescribed by the} funda-mental laws of quantum mechanics. For the sake of clarity and simplicity, in the following two subsections we follow the description provided in Ref. [10] and in Ref. [11].

2.1.1 Observables and states

In general, to each quantum systemW we associate a complex Hilbert space HW. The

elements of H_W represent acceptable quantum configurations of our system_{W that for the} moment we call vectors. If ψ and ξ represent two configurations of _{W that is ψ, ξ ∈ H}_W, exploiting Dirac bra-ket notation, their inner product is represented byh ψ | ξ i. This inner product induces a norm: for any configuration ψ, its norm is _{||ψ|| ≡}p_{h ψ | ψ i.}

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Now we have to introduce two Banach spaces1. The ﬁrst is the set of all the bounded2 operators equipped with the norm _||O||_∞ = sup_||ψ||≤1_{||O ψ||, being ψ ∈ H}_W and being O an operator acting on H_W. This space, usually denoted as _B(H_W), is also an algebra, that is a set closed under the composition of its elements (multiplication) and involution (adjoint operation _{∼ †), with the following properties}

hO†ψ_{|ξi = hψ|Oξi,} (2.1)

(O1O2)†= O2†O1†, (O†)†= O, (αO1+ βO2)†= ¯αO1†+ ¯βO2†, (2.2)

||O†O_||_∞=_||O†_||2_∞=_||O||2_∞, (2.3) being O and Oj elements of B(HW), ψ, ξ ∈ HW), α, β ∈ C and being ¯α the complex

conjugate of α.

In order to introduce the second Banach space we need for our discussion, we ﬁrst have to introduce the trace of an operator O. Given any orthonormal (complete) set for our Hilbert space, that is a set of vectors _{ψn} such that

(i) _hψm|ψni = δm n, (2.4)

and such that for any conﬁguration ψ ∈ HW, there exists a unique set of complex

coeﬃ-cients {αn} for which

(ii) ψ =X n αnψn, X n |αn|2 = 1, (2.5)

the trace of an operator, if exists, is deﬁned as Tr[O] =X

n

hψn|O|ψni. (2.6)

An operator σ∈ B(HW) is called a trace class operator if Tr[σ†σ]1/2exists. The set of trace

class operators_{T (H}_W) with the norm_||σ||1 = Tr[σ†σ]1/2 is a Banach space. In particular,

it is worth noting that given any couple of operators σ _{∈ T (H}_W) and O _{∈ B(H}_W), both σO and Oσ are elements of _{T (H}_W). This guarantees that the measurement procedure makes sense and everything is well-deﬁned.

These two Banach spaces are essential for the formulation of quantum mechanical theories. Indeed, both observables, that is measurable quantities, and quantum states, that is density operators, represent particular elements of these two sets. Observables correspond to self-adjoint operators in_B(H_W): an operator O is self-adjoint if and only if O† _{= O. Density}

operators are instead trace class operators satisfying the following constraints

(i) ρ≥ 0, (2.7)

(ii) Tr[ ρ ] = 1, (2.8)

(iii) ρ†= ρ. (2.9)

1_{A Banach space is a complete vector space, that is a vector space with a metric i.e. a rule for computing}

length of vectors and the distance between points in the space, where any Cauchy sequence converges to an element of the space.

2_{An operator O is bounded if there exists a contant M such that for any vector ψ we have that}

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2.1. Preliminaries 9

Density operators represent the natural extension to the quantum world of the concept of probability distribution introduced by Gibbs in ensemble theory [12]. This interpretation is encoded in the constraints listed here above. Eq. 2.7 means that “the occurrence of an event is non-negative”. In the quantum mechanical language, this statement is translated into the requirement that any density operator is positive semi-definite: for any configuration ψ _{∈ H}_W the inner product _{hψ|ρ|ψi ≥ 0. Eq. 2.8 means instead that “the sum over all} the possible outcomes of an event is 1”. This requirement is encoded in the normalization condition for the trace of any given density operator. The last constraint specified in Eq. 2.9 implies instead that any density operator can be expressed as a summation of mutually orthogonal projectors. In other words, for any given density operator ρ there exists a unitary transformation U such that

ρ_{→ ρ}U = U ρU†=X

n

λnPn, (2.10)

where Pn ≡ |ψnihψn| is the orthogonal projector 3 into the eigenspace generated by the

n-th eigenvector of ρ, that is ψn, and being λn the corresponding eigenvalue. We observe

also that _{ψn} is a complete orthonormal set. By exploiting the representation provided

in (2.10), from Eq. 2.7 and Eq. 2.8 it follows that 0≤ λn≤ 1 and

X

n

λn= 1. (2.11)

Density operators form a (convex) subset,_P(H_W), of_{T (H}_W): given any couple of density operators ρ1and ρ2inP(HW), also all their convex linear combinations ρβ ≡ βρ1+(1−β)ρ2,

for β ∈ [0, 1], are element of P(HW). In particular, all the elements in P(HW) can

be separated into two classes, that is pure and mixed density operators. Geometrically, pure states represent the extreme points of the convex set _P(H_W), that is those density operators that cannot be expressed as a convex combination of other density operators. Mixed density operators are those states that are not pure. An alternative deﬁnition of purity which is more suitable for explicit computations is the following

ρ is a pure density operator_{⇔ Tr[ ρ}†ρ ] = 1. (2.12) By exploiting the hermiticity of any given density operator and the diagonal representation provided in (2.10), one ﬁnds that

ρ is a pure density operator⇔ X

n

λ2_n= 1. (2.13) However, since the eigenvalues of ρ satisfy the constraints in (2.11), we have that

X n λ2_n= 1_{⇔ λ}n= 1 for some n = ¯n 0 otherwise (2.14)

In other words, ρ is a pure density operator if and only if it is an orthogonal projector, that is there exists a unitary operator U such that ρU =_|ψn¯ihψ¯n| for some ψn¯ ∈ HS.

3_{An operator P}

nis a projector if Pn2= Pn(idempotent operator ). To be orthogonal one further requires

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2.1.2 Time evolution of Hamiltonian systems and representations

We have been taught in standard courses of quantum mechanics that the time evolution of vectors is prescribed by the Sch¨odinger equation. If H(t) denotes the Hamiltonian operator for a given quantum system and _{|ψ(t)i denotes its state at t, then the time} evolution of such system is prescribed by the following diﬀerential equation (~ = 1)

−id

dt|ψ(t)i = H(t)|ψ(t)i (2.15) The solution of (2.15) with starting condition|ψ(t0)i is given by

|ψ(t)i = U(t, t0)|ψ(t0)i, (2.16)

being U (t, t0) the unitary time evolution operator. Eq. 2.15 can be expressed as a

diﬀer-ential equation for U (t, t0), that is

i∂

∂tU (t, t0) =H(t)U(t, t0), with U (t0, t0) =1

W, (2.17)

whose solution is given by

U (t, t0) = T exp −i Z t t0 ds_H(s) (2.18) being “T” the time-ordering operator and being1

W the identity operator. In particular,

we have that the mean expectation of an observable O is a time-dependent quantity given by

hO(t)i = hψ(t)|O|ψ(t)i = hψ(t0)|U†(t, t0)OU (t, t0)|ψ(t0)i. (2.19)

Eq. 2.19 can be exploited to introduce the dual representation of the Schr¨odinger picture, that is Heisenberg representation. In this representation, vectors are time-independent, while operators do depend on time. This means that vectors in Heisenberg representation do not evolve with time. In other words, if _|ψH(t)i denotes a vector in the Heisenberg

representation, we have that

d

dt|ψH(t)i = 0, (2.20)

so that for all t_|ψH(t)i = |ψH(t0)i = |ψ(t0)i. In particular, by requiring mean expectations

to be independent of the particular choice of representation, if O denotes an operator in Schr¨odinger representation, the corresponding operator in Heisenberg picture is given in (2.19) and reads

OH(t) = U†(t, t0)OU (t, t0). (2.21)

Diﬀerentiating both sides of (2.21) with respect to time, one obtains the Heisenberg equa-tion of moequa-tion for the operator OH(t)

d

dtOH(t) = i[HH(t), OH(t)], (2.22) whereHH(t) denotes the system Hamiltonian in Heisenberg representation and [A, B] ≡

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2.1. Preliminaries 11

Schr¨odinger and Heisenberg representations represent the two standard representations exploited to describe time evolution in quantum systems. However, there are cases where it is convenient to exploit another representation, called interaction representation. This is the case when the total Hamiltonian H(t) is made up of two parts: the free Hamiltonian, denoted by _H0, and a time-dependent part, V(t), containing interaction terms. In this

case, if one deﬁnes

U0(t, t0) = exp (−iH0(t− t0)) , (2.23)

then it is possible to redeﬁne the unitary time evolution operator given in (2.18) in the following way

U (t, t0)≡ U0(t, t0)UI(t, t0). (2.24)

Eq. 2.24 has to be interpreted as a deﬁnition for the operator UI(t, t0). In this new

representation both vectors and observables are time-dependent. In particular, exploiting again that the mean expectations of observables must be independent of the choice of representation, by exploiting (2.19) and the deﬁnition (2.24), we have that

OI(t) = U0†(t, t0)OU0(t, t0), (2.25)

and that

|ψI(t)i = UI(t, t0)|ψ(t0)i. (2.26)

In particular, diﬀerentiating Eq. 2.25 and Eq. 2.26 one obtains the time evolution of observables and vectors in this representation

d dtOI(t) = i[H0, OI(t)], (2.27) and id dt|ψI(t)i = VI(t)|ψI(t)i, (2.28) where_VI(t) = U0†(t, t0)VI(t)U0(t, t0).

We conclude this section by observing that the equation of motion for vectors shown above can be exploited to determine the time evolution of density operators in the diﬀerent rep-resentations. Let us suppose to have a density operator that in Schr¨odinger representation that at t is given by

ρ(t) =X

n

pn|ψn(t)ihψn(t)|. (2.29)

Due to Eq. 2.15, it is straightforward to see that its time evolution in Schr¨odinger repre-sentation is given by the von Neumann equation, that is

d

dtρ(t) =−i[H(t), ρ(t)]. (2.30) In addition, we have that Eq. 2.20 implies _dtdρH(t) = 0, and that Eq. 2.28 implies that

in interaction representation the density operator ρI(t) = P_npn|ψn I(t)ihψn I(t)| evolves

according to the following equation of motion d

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2.2 LGKS equation: historical remarks on its derivation

Quantum Markov processes and the related concept of Markovian quantum dynamical semigroup have emerged in theoretical physics as an extension to the quantum realm of the standard formalism exploited in the ﬁeld of probability theory. Their importance in quantum physics has been understood during the 70s. The motivation is that the time evolution of quantum system can be represented through the action of dynamical maps, that is through the action of a set of time-dependent operators from the set P(HW) into

itself, and in several cases these maps are elements of a dynamical semigroup, that is a family of dynamical maps depending on a single parameter (that in physics corresponds to time) and satisfying a particular composition law.

The axiomatic deﬁnition of these mathematical structures and the characterization of their properties have been provided by Kossakowski in [13]. In general, a semigroup is a family of two-parameter maps S0 = {Λt, s; t > s ∈ R} of the set P(HW) in itself. However,

the relevant class of dynamical semigroups for the characterization of Markovian open quantum systems is given by temporally homogeneous semigroups i.e. those semigroups for which the elements Λt, s are functions only of t− s. As deﬁned by Kossakowski, a family

S₌_{Λ_t_{; t}_{≥ 0} of linear transformations is a dynamical semigroup of a quantum system} if the following constraints are satisﬁed:

(i)4 _Λ

t: V+→ V+, t≥ 0

(ii) _||Λtρ||1 =||ρ||1, ρ ∈ V+, t≥ 0

(iii) Λt is a strongly continuous function of t≥ 0

(iv) s− limt → 0Λtρ = ρ

(v) ΛtΛs= Λs+t, t, s ≥ 0.

The ﬁrst two conditions implies that Λt is positive, while (iii) and (iv)5 are conditions

which guarantee that expectation values of observables are continuous functions of time. The condition (v) represents instead the composition law.

In particular, Kossakowski realized the intimate connection between his ﬁeld of study and the theory of one parameter semigroups of operators in Banach spaces developed by Hille and Yoshida, and introduced the concept of infinitesimal generator in the context of dynamical semigroup. The generator of a dynamical semigroup is deﬁned as

L[ ρ ] ≡ s − lim

t→0

1

t(Λtρ− ρ), (2.32)

and has the property

d(Λtρ)

dt = ΛtL[ ρ ] = L[ Λtρ ], (2.33) for all ρ_{∈ D(L) i.e. in the domain of L.}

In this work, Kossakowski characterized also the action conjugate to the semigroup S, that

4_V+_{is the set of positive definite density operators.} 5_{A sequence {σ}

n} of density operators is said to converge strongly to an element σ and we write

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2.2. LGKS equation: historical remarks on its derivation 13

is the action on the algebra of operators. The properties of the dual dynamical semigroup of a quantum system, denoted as S∗, are summarized in Theorem 1, p.252 of [13].

A crucial role in the development of this ﬁeld, in particular in the determination of the structure of L, has been played by some results derived at the end of the 60s. In those years, more precisely between 1969 and 1970, a series of papers due to K. -E. Hellwig and K. Kraus6 _{[14, 15] concerning the schematisation of measurement processes and their}

properties appeared in literature. In these papers the authors pay attention to the study of operations, that is state changes in quantum systems due to external interventions. The hypothesis at the basis of their description is the assumption that any operation emerges as a manifestation of the interaction between our system (the object) and another system (the apparatus), followed by a subsequent measurement of some property of the apparatus. According to their discussion, if ρ _{∈ P(H}_S) and η _{∈ P(H}_S′) represent respectively the

density operators for the object and the apparatus alone, when the object is put in contact with the apparatus their global state in H_S⊗ HS′ is given by ρ⊗ η, where ⊗ denotes the

tensor product. Let us denote the unitary scattering operator in H_S _{⊗ H}_S′ by U . This

operator describes the interaction between object and apparatus. Then, the state of the system after the measurement of a property Q′of the apparatus, represented by a projector in H_S′, is given by ˜ ρ = wˆ Tr_S[ ˆw ], w = (ˆ 1 S⊗ Q′)U (ρ⊗ η)U†(1 S⊗ Q′). (2.34)

If the object is considered separately, its state is given by

ρ′ = Tr_S′[ ˜ρ ]. (2.35)

The crucial point is that in general one can describe the eﬀects of an operation on the object as a transformation acting only onP(HS), that is

ρ_{→ ρ}′=X k, i ciAk, iρA†_{k, i}, Ak, i ∈ B(HS) and X k, i ciA†_{k, i}Ak, i ≤ 1S (2.36)

where the constants ci and the operators Ak, i are deﬁned as follows. Let us consider a

diagonal representation for the apparatus state η, that is η =X

i

ciPi, (2.37)

being 0 _{≤ c}i ≤ 1, Pici = 1 and being Pi the orthogonal projector into the subspace

generated by φi, with {φi} complete orthonormal set for the Hilbert space HS′. Let us

pay attention to those i for which ci 6= 0. Furthermore, let us consider a second complete

orthonormal set{ψi} of HS′ containing a basis for the subspace Q′H_S′ of H_S′. The index

k in the following runs on these states. Then the operators Ak, i are deﬁned by

hψ|Ak, i|φi ≡ hψ ⊗ ψk|U|φ ⊗ φii, ∀ φ, ψ ∈ HS. (2.38)

6_{During the same years E. B. Davies [16, 17] discussed closely related problems exploiting the formalism}

of quantum stochastic processes. For an introduction to the properties of stochastic processes see e.g. Ref. [11]

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A year later, Kraus [18] realized that the ideas exploited for the description of operations could be exploited in a more general fashion to address the generic state changes in quan-tum theories. In this paper, Kraus extended the ideas introduced above for operations from P(HS) into itself, to the entire space of trace class operators. In addition, he also

considered the action of operations on the space _B(H_S). Indeed, to any operation O : T (HS) → T (HS), it is possible to associate a conjugate mapping O†:B(HS) → B(HS)7.

In particular, the conjugate mapping associated to an operation O having the form OX =X k AkXA†k, X ∈ T (HS) (2.40) is given by O†B =X k A†_kBAk, B ∈ B(HS). (2.41)

In particular, O† is such that

||O†B|| ≤ ||B||. (2.42)

As observed by Kraus, the mapping in Eq. 2.41 is positive i.e. it maps the set of non-negative hermitian operators into itself. In addition, he showed that O† is completely positive. To any linear map from_B(H_S) into itself, it is possible to associate for any k_{∈ N} a linear mapping Φ(k) fromB(HS)⊗ Ck×k into itself given by

Φ(k) = O†⊗1

k×k, (2.43)

where Ck×k represents the set of all the k by k matrices with complex coeﬃcients. The mapping O† _{is called k-positive if for any non-negative ˜}_B _{∈ B(H}

S)⊗ Ck×k, it follows that

also Φ(k)( ˜B) is non-negative. The mapping O† is completely positive if it is positive for any k_{∈ N. In particular, he found that}

Theorem. (Theorem 3.3, p. 317 [18]) Any linear mapping T :_B(H_S)_{→ B(H}_S) with ||T B|| ≤ ||B|| which is completely positive and ultraweakly continuous8_{, is of the form}

T B =X k A†_kBAk with X k A†_kAk≤1S. (2.44)

The concept of complete positivity discussed by Kraus in the context of dynamical maps was what led to the determination of the LGKS equation. Indeed, if on one hand it is true that in 1975 Lindblad [19] and Gorini, Kossakowski and Sudarshan [20] determined the expression of the generator _{L, on the other hand it is worth noting that in both papers}

7_{This is possible because the set of all the bounded operators B(H}

S) is the conjugate space of T (HS).

Any element B of B(HS) defines a continuous linear functional FB(X) = Tr[ BX ] on T (HS). In this way

the conjugate mapping O†is defined for arbitrary X and B by

Tr[ O†B · X ] = Tr[ B · OX ] (2.39) .

8_{The ultraweakly convergence also called *-weak convergence of operators in B(H}

S) follows from

the ultraweakly convergence in T (HS). The sequence of operators {Tn} converges to T ultraweakly if

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2.2. LGKS equation: historical remarks on its derivation 15

the constraints imposed on the families of maps {Λt; t≥ 0} and {Λ†t; t≥ 0} were slightly

diﬀerent from those prescribed by Kossakowski in [13]. In particular, in both papers the authors realized that one needs to require both Λtand Λ†t to be completely positive, because

positivity is not suﬃcient for obtaining results coherent with the requirements of quantum mechanics. Indeed, if one considers the time evolution of a quantum system S resulting by the unitary evolution of a total system_{S + R, in Heisenberg picture one ﬁnds that}

X_{→ Λ}†_t(X) = Tr_R[(ρ_{⊗ η)U}†(X_⊗1

R)U ], X ∈ B(HS) (2.45)

and the map Λ†_t : _B(H_S) _{→ B(H}_S) is completely positive. In addition, in these papers the authors addressed the problem of determining the structure of the generator L in a diﬀerent fashion and, as speciﬁed in [20], the results shown in [19] are more general. Lindblad assumed the generator _{L to be bounded. In Heisenberg picture, this assumption} corresponds to the requirement that

lim

t→0||Λ † t−1

W|| = 0. (2.46)

This constraint implies that

(i) Λ†_t = exp[ ˜_{L t] and (ii)} lim

t→0|| ˜L − t −1_(Λ†

t−1

W)|| = 0, (2.47)

where ˜L denotes the inﬁnitesimal generator in the adjoint representation i.e. in Heisenberg picture. In particular, Lindblad by exploiting some new ideas introduced by Ingarden and Kossakowski in [21] determined the following expression for the generator (Theorem 2, p. 127 [19]) ˜ L[X] = i[H, X] +X j [V_j†XVj− 1 2(V † jVjX + XVj†Vj)], (2.48)

where Vj,P_jV_j†Vj ∈ B(HW) and H† = H ∈ B(HW). Moving in Schr¨odinger

representa-tion, one obtains the following expression for generatorL L[ ρ ] = −i[H, ρ] +X j [VjρVj†− 1 2(V † jVjρ + ρVj†Vj)]. (2.49)

The representation derived by Gorini, Kossakowski and Sudarshan for ﬁnite dimensional systems has instead the following form (Theorem 2.2, p. 822 [20])

L[ ρ ] = −i[H, ρ] + 1 2 d2₋₁ X i, j=1 ci, j{[Fi, ρF_j†] + [Fiρ, F_j†]}, (2.50) where d = dim[H_S], H†₌_{H, Tr[ H ] = 0, Tr[ F} i] = 0, Tr[ Fi†Fj] = δi j and being {ci, j} a positive matrix.

We conclude this section by observing that (2.49) and (2.50) are equivalent. Indeed, since {1/

√

d; Fj, j∈ [1, d2− 1]} is a complete orthonormal set for d × d matrices, the operators

entering in (2.49) can be decomposed in this new basis as follows Vj = α0, j 1 √ d+ d2₋₁ X l=1 αl jFl, (2.51)

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where α0, j = Tr[ Vj] and αl, j = Tr[ Fl†Vj]. By plugging the representation (2.51) in the

Eq. 2.49 one ﬁnds that this equation reduces to (2.50), with the matrix _{ci, j} having

elements

ci, j =

X

k

αi, kα∗j, k. (2.52)

As said above, the next section is devoted to the determination of the LGKS starting from the microscopic theory.

2.3 LGKS master equation: a microscopic approach

In the previous section we have introduced the fundamental ingredients needed in this section to address the problem of a (open) quantum system S coupled to a larger (inﬁnite) quantum system called reservoir and in the following denoted by _{R. As in the} previous section, the description reported here follows the discussion reported in [10, 11]. In particular, starting from the von Neumann equation describing the time evolution of S + R as closed quantum system and performing a series of approximations, we derive a series of master equations. More precisely, what will show in the following will lead to an equation of motion in the form

d

dtρS(t) = Z t

t0

ds_{K(t, s)ρ}_S+R(s), (2.53) where ρ_S+R(s) is the density operator representing the total system at s∈ [t0, t], ρS(t) =

Tr_R[ρ_S+R(t)] is the reduced density operator representing _{S as an open quantum system} and being _{K(t, s) the memory kernel operator which prescribes how the time variations} of the reduced density operator ρ_S(t) are connected to the past of the total system. The approximation methods discussed in the following allow to reduce the exact time evolution encoded in the right-hand-side of (2.53) into a coarse-grained time evolution that can be associated to a Markovian dynamical semigroup. However, it is worth noting that as discussed in the past by Prigogine and Résibois [22] and Balescu [23] (see also e.g. Zwanzig [24], who compared [22] with other works of those years) while the approach to equilibrium encoded in master equations like (2.53) may differ from its Markovian limit, if one is interested in characterizing only the steady-state properties emerging asymptotically, the Markovian limit is sufficient.

For a review about the dynamics of non-Markovian open quantum systems see e.g. Ref. [25].

2.3.1 LGKS equation in the weak coupling limit

In this section we derive the LGKS exploiting the so-called weak coupling limit [10, 11]. Let us consider a quantum systemT made up of two parts: a subsystem of interest denoted by_{S and a larger quantum system R to which S is coupled. As a whole, the time evolution} ofT = S + R is unitary. Let us denote the density operator for the total system at some instant t0 by ρ(t0) ∈ P(HS ⊗ HR) and let us consider, for the sake of simplicity, a total

Hamiltonian _{H which is time-independent and with the following structure} H = HS⊗1

R+1

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2.3. LGKS master equation: a microscopic approach 17

where HS and HR denote respectively the free Hamiltonian of S and R, and HI is the

interaction term between _{S and R. Let us now move into interaction picture. In this} representation, the time evolution of ρI is prescribed by

dρI(t)

dt =−i[HI(t), ρI(t)], (2.55) whose formal solution is provided by

ρI(t) = ρI(t0)− i

Z t t0

ds[_HI(s), ρI(s)]. (2.56)

In order to derive an equation of motion for the reduced density matrix describingS alone, we have to insert (2.56) into (2.55) and then trace out the degrees of freedom associated to the reservoir. This procedure leads to the following expression

dρ_{S I}(t) dt =−i

Z t 0

dsTr_R[_HI(t), [HI(s), ρI(t) ] ], (2.57)

where we have assumed that Tr_R[HI(t)ρ(t0)] = 0 and where we set without loss of

gen-erality t0 = 0. However, the expression in (2.57) still depends on ρI(t) (right-hand side).

The weak coupling limit or Born approximation is based on the assumption that if the interactions between S and R are suﬃciently weak, then the reservoir will be negligibly aﬀected by changes taking place in _{S. Therefore, one can assume that starting from a} factorized density operator

ρI(t)≈ ρS I(t)⊗ ρR, (2.58)

S will evolve and the reservoir will remain in ρR forever. Plugging this expression for the

density operator in (2.57) one obtains the following equation of motion dρ_{S I}(t)

dt =−i Z t

0

dsTr_R[_HI(t), [HI(s), ρS I(s)⊗ ρR] ]. (2.59)

The main problem in (2.59) is that the state at t still has memory of its past, that is it depends on all its previous conﬁgurations. In order to reduce (2.59) to a Markovian equation of motion, one needs to work a little more. If the typical time scale τ_S over which the state of the system S varies appreciably is large compared to the scale τR at which

the reservoir correlations decay, then (2.59) reduces to the following expression (Markov approximation) dρ_{S I}(t) dt =−i Z _+∞ 0 dsTr_R[HI(t), [HI(t− s), ρS I(t)⊗ ρR] ], (2.60)

where the time integral has been extended from [0, t] to [0, ∞].

However, as pointed out by Davies [26, 27], it is not guaranteed that the weak coupling pro-cedure leading to (2.60) deﬁnes the generator of a dynamical semigroup.9 _{Therefore, one}

9_{A sufficient condition for this to happen is that the relavant (see in the following) two-point reservoir}

correlation functions, here denoted by Gi, j(t), satisfy the following constraint

Z ∞ 0

dt|Gi, j(t)|(1 − t)ǫ< ∞, (2.61)

for some ǫ > 0. In general, this may happen if the reservoir has infinite volume: in the case of finite volume the relevant two-point correlation functions are periodic functions of t.

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performs the so called rotating-wave approximation. The idea at the basis of this technique is to neglect those contributions fastly oscillating, and to retain only low-frequency con-tributions because on long time scales only the latters lead to non-vanishing mean terms. This goal can be achieved in the following way. Without loss of generality, we can assume to have an interaction term in the form (in Schr¨odinger representation)

HI =

X

k

Sk⊗ Rk, S_k†= Sk ∈ B(HS), R†k= Rk ∈ B(HR). (2.62)

Indeed, if_HI is for example in the following form [30]

HI = X k X_k†_{⊗ Y}k+ Xk⊗ Y_k† , Xk∈ B(HS), Yk ∈ B(HR), (2.63)

by exploiting the following decomposition Xk= Xk(a)+ i X (b) k , Yk= Yk(a)+ i Y (b) k , with X (l) k = Xk(l) †, Y (l) k = Yk(l) †( l = a, b ) (2.64)

it is possible to reduce (2.63) in the form (2.62).

In addition, let us suppose to have a free Hamiltonian HS with a discrete spectrum here

denoted by sp(_H_S) = _{λn}, being λn the eigenvalues of HS. Let us now introduce the

following operators

Sk(ω) =

X

λn−λm=ω

PnSkPm, (2.65)

where the summation runs on the couples of eigenvalues whose diﬀerence is equal to ω and being Pn is the projector into the n− th eigenspace corresponding to λn. These operators

have some interesting properties. They are eigenoperators ofHS

[_H_S, Sk(ω)] =−ωSk(ω) and [HS, Sk†(ω)] = +ωS†k(ω). (2.66)

By moving into interaction representation, they only get a phase factor eiHSt_S

k(ω)e−iHSt= e−iωtSk(ω) and eiHStS_k†(ω)e−iHSt= eiωtS_k†(ω) (2.67)

In addition, we also have that

[HS, Sk†1(ω)Sk2(ω)] = 0 and S † k(ω) = Sk(−ω), (2.68) and that X ω Sk(ω) = X ω S_k†(ω) = Sk. (2.69)

In particular, due to the properties summarized above and due to the completeness of the projectors Pk, we can rewrite the interaction term in the following way

HI = X k, ω Sk(ω)⊗ Rk= X k, ω S_k†(ω)⊗ Rk†. (2.70)

The motivation for the introduction of the representation (2.70) is that moving into inter-action representation one has that_HI becomes

HI(t) = X k, ω e−iωtSk(ω)⊗ Rk(t) = X k, ω eiωtS_k†(ω)_{⊗ R}†_k(t), (2.71)

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2.3. LGKS master equation: a microscopic approach 19

being Rk(t) = eiHRRke−iHR. Exploiting the representation for the operators Sk(ω) and

Rk in interaction picture provided above, one ﬁnds that the time evolution reported in

(2.60) becomes dρ_{S I}(t) dt = X ω,ω′ X k, l ei(ω′−ω)tΓk, l(ω) Sk(ω)ρS I(t)Sl†(ω′)− Sl†(ω′)Sk(ω)ρS I(t) + h.c., (2.72) where h.c. the Hermitian conjugate. The quantity Γk, l(ω) denotes the one-sided Fourier

transform of the reservoir two-point correlation functions Gk, l(t, s)

Γk, l(ω)≡ Z _∞ 0 ds eiωs_Gk, l(t, s), (2.73) beingGk, l(t, s) given by Gk, l(t, s) =hR†_k(t) Rl(t− s)i = TrR[R†k(t) Rl(t− s) ρR]. (2.74)

The fact that (2.73) does not depend on t is related to the fact that by choosing a stationary density operator for the reservoir, for example a Gibbs state at some temperature, the correlation functionsGk, l(t, s) are actually invariant under time translations, so that

Gk, l(t, s) =hR†_k(t) Rl(t− s)i = hR†_k(s) Rl(0)i ≡ ˜Gk, l(s) (2.75)

As said above, the approximation we have exploited is valid if τ_S, the time scale over which the properties of _{S change appreciably, is small compared to τ}_R, that is the time over which reservoir correlation decay signiﬁcantly. The scale τ_S is usually related to the eigenvalues of HS, in particular τS ≈ |ω′− ω|−1 for some ω′ 6= ω. If τS ≫ τR, the terms

involving large energy diﬀerences that is those terms with ω′_{6= ω are fastly oscillating and} can be neglected. As a consequence, Eq. 2.72 reduces to

dρ_{S I}(t) dt = X ω X k, l Γk, l(ω) Sk(ω)ρS I(t)S†l(ω′)− Sl†(ω′)Sk(ω)ρS I(t) + h.c. (2.76) However, (2.76) it is not in the LGKS form yet. To achieve this goal one has to introduce the functions γk, l(ω) and σk, l(ω), that are related to the one-sided Fourier correlation

functions Γk, l(ω) by the following relations

γk, l(ω) = Γk, l(ω) + Γ∗l, k(ω) and σk, l(ω) =

1

2i(Γk, l(ω)− Γ∗l, k(ω)) (2.77) In particular, the matrix γk, l(ω) is hermitian and positive. By exploiting the decomposition

reported in (2.77) in Eq. 2.76 one obtains the expression of the master equation describing the time evolution of the open quantum systemS in interaction representation

dρ_{S I}(t)

dt =−i[HLS, ρS(t)] +D[ρS(t)], (2.78) where_HLS =Pω

P

k, lσk, lS_k†(ω)Sl(ω) denotes the Lamb shift Hamiltonian and being

D[ρS I(t)] = X ω X k, l γk, l(ω) Sl(ω)ρS ISk†(ω)− 1 2 S_k†(ω)Sl(ω)ρS I+ ρS ISk†(ω)Sl(ω) (2.79)

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the dissipator of the master equation.

The expression of the equation of motion (2.78) in Schr¨odinger picture can be easily ob-tained by exploiting the properties of the operators Sk(ω) and by rotating back to this

representation the density matrix ρ_{S I}(t). In the end, one obtains that in Schr¨odinger representation the time evolution of _{S alone is given by}

dρ_S(t)

dt =−i[H

′

S, ρS(t)] +D[ρS(t)], (2.80)

being_H′_S=_H_S+_HLS.

2.3.2 LGKS equation in the strong coupling limit

In the previous subsection we have shown how starting from the von Neumann equation it is possible to derive under a series of hypothesis the LGKS equation. Here, we consider the opposite limit that is the strong coupling limit or singular coupling limit. The validity of the discussion reported in the previous subsection is related to the fact that in the weak coupling limit, non-unitary effects leading to dissipation are slow if compared to the free evolution of the quantum system of interest. This fact allows neglecting fast oscillating terms and to reduce the time evolution to the form reported in (2.80). In the strong coupling limit, one performs a rescaling of the reservoir Hamiltonian so that the typical time scale τ_R at which correlations in the reservoir decay becomes smaller and eventually vanishes in the limit of scaling parameter going to zero. This scaling, in order to induce a finite effect, must be accompanied by a rescaling of the interaction term. In such a way, one can reduce the problem to one similar to that discussed in the previous section, with the only difference that in this case the rotating-wave approximation is no longer needed. If one considers an Hamiltonian in the form

H = HS⊗1 R+1 S⊗ 1 ǫ2HR+ 1 ǫHI, (2.81)

by following the same steps discussed in the first part of the previous section, one finds that the time evolution of the reduced density operator ρ_S(t) can be cast in Schrödinger representation in the following form

dρ_S(t) dt =−i[HS, ρS(t)] + ˜K[ρS(t)], (2.82) being ˜ K[ρS(t)] =− Z _∞ 0 dsTr_R[_HI, [e−iHRtHIe+iHRt, ρS(t)⊗ ρR]] (2.83)

Using the properties of two-point correlation functions discussed in the previous section (2.83) reduces to the LGKS equation

dρ_S(t) dt =−i[H ′ S, ρS(t)] + X k, l γk, l SlρS(t)Sk− 1 2(SkSlρS(t) + ρS(t)SkSl) , (2.84) where_H′_S=_H_S+P_{k, l}σk, lSkSl.

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Chapter 3

Lindbladians with a single

steady-state

This chapter is devoted to the characterization of the properties of Markovian dy-namical semigroups. Thanks to the results provided by Lindblad which guarantee that a generic element of the semigroup can be expressed exploiting the exponential map, this characterization can be performed by considering the spectral properties of the inﬁnites-imal generator L. In particular, we will pay attention to the properties of Lindbladian with a single steady-state. For this particular class of Lindbladians a series of general re-sults related to representation of the LGKS discussed in the previous chapter have been derived during the 70s. These results will be connected to more recent characterizations appeared in papers of the last decade. The last section of this chapter is instead devoted to the discussion of a series of theoretical results derived by the author which guarantee the uniqueness of the steady-state conﬁguration. For a discussion about the properties of Lindbladians with more than a single steady-state and their applications in quantum physics see e.g. Ref. [29].

3.1 Spectral properties of Lindbladians

In the previous chapter we have discussed how the LGKS equation has emerged in the context of dynamical semigroups as a definition for the infinitesimal generator of the time evolution. In particular, we have discussed how the constraints provided by Kossakowski in [13] have been modified by Lindblad in [19] in order to derive the explicit structure of the generator L. In particular, Lindblad, by working in Heisenberg representation, observed (Theorem 1, p. 125 [19]) that the dual map

Λ†(t) = eLt˜ t_{≥ 0} (3.1) is completely positive with Λ†_(t)

1W = 1W if and only if the inﬁnitesimal generator ˜L

is completely dissipative. As for complete positivity, complete dissipativity is deﬁned in terms of the behavior of a given operator, let us say ˜Φ :_{B → B, and its natural extensions}

˜

Φ(k)_{= ˜}_Φ_⊗

1k:B ⊗ C

k×k_{→ B ⊗ C}k×k_{, being k a positive integer. In particular, according}

to the deﬁnition given by Lindblad (Sec. 3, [19]), an operator ˜Φ satisfying ˜

Φ[1

W] = 0 and Φ[X˜ †] = ˜Φ[X]† (3.2)

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is dissipative if for any X inB, the operator

D( ˜Φ; X, X) = ˜Φ[X†X]− ˜Φ[X†]X− X†Φ[X]˜ ≥ 0 (3.3) that is D( ˜Φ; X, X) is a non-negative operator1_{. An operator is called completely dissipative}

if its extension ˜Φ(k) is dissipative for any k∈ N.

Since the dual generator ˜_{L is dissipative, the dynamical semigroup {Λ(t); t ≥ 0} is a} contraction semigroup (Theorem 2.5, p. 13 [30]) and in particular thanks to the Hille-Yoshida theorem (Theorem 2.4, p. 11 [30]) the eigenvalues of its inﬁnitesimal generator in Schr¨odinger representation, that is _{L, have non-positive real part. In addition, the} spectrum of a generic_{L, sp(L), is invariant under conjugation. This fact follows from the} structure of the LGKS equation which guarantees that for any operator ˜ρ we have

L[ ˜ρ ]†=L[ ˜ρ†]. (3.4) As a consequence, by denoting the eigenoperator associated to the eigenvalue λi by ρi,

that is the operator satisfying

L[ρi] = λiρi, (3.5)

it is straightforward to see that its adjoint ρ†_i satisﬁes the following equation

L[ρ†i] = λ∗iρ†i, (3.6)

which means that both λi and λ∗i are eigenvalues ofL.

However, it is worth noting that

(i) since L is not hermitian, eigenoperators corresponding to diﬀerent eigenvalues are not orthogonal;

(ii) the eigenoperators of _{L are not in general elements of P(H}_W) that is density opera-tors.

The ﬁrst observation implies that in general it is not possible to choose, as in the case of Hamiltonian quantum systems, an orthonormal set of eigenoperators of the inﬁnitesimal generator as a basis for our representation for states. Explicitly, this means that

Λ(t)ρ6=X

i

ci(t)ρi. (3.7)

As discussed in [31], this is instead the case if_{L is diagonalizable. The second observation} means instead that, if on one hand it is true that

Λ(t)ρi = eLtρi = eλitρi, (3.8)

on the other hand it is not true a priori that ρi corresponds to a density operator. This

can be seen this way. The spectrum sp(L) is the union of four sets, that is

sp(L) = l1(L) ∪ l2(L) ∪ l3(L) ∪ l4(L), (3.9)

1_{Since ˜}_{Φ satisfies (3.2), the operator D( ˜}_{Φ; X, X) is hermitian. As a consequence, it is diagonalizable}

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3.1. Spectral properties of Lindbladians 23 being l1(L) = {λi ∈ sp(L) : Re[λi]6= 0, Im[λi] = 0}, (3.10) l2(L) = {λi ∈ sp(L) : Re[λi]6= 0, Im[λi]6= 0}, (3.11) l3(L) = {λi ∈ sp(L) : Re[λi] = 0, Im[λi]6= 0}, (3.12) and l4(L) = {λi ∈ sp(L) : Re[λi] = 0, Im[λi] = 0}. (3.13)

The decomposition (3.9) induces a decomposition of the set of all the eigenoperators of _L, that is ǫ(L), in the following form

ǫ(L) = ǫ1(L) ∪ ǫ2(L) ∪ ǫ3(L) ∪ ǫ4(L), (3.14)

being

ǫj(L) = {ρi ∈ ǫ(L) : λi∈ lj(L)}. (3.15)

A representation of the spectrum of a generic _{L in the complex plane that keeps into} account the properties and the decomposition discussed above is given in Fig. 3.1.

Re[λi]

Im[λi]

0

Figure 3.1: Representation of the spectrum of a generic inﬁnitesimal generator _{L in the} complex plane. Since _{L is the generator of a contracting semigroup, its eigenvalues λ}i =

Re[λi] + iIm[λi] (circles) satisfy the constraint Re[λi] ≤ 0. Circles with diﬀerent colors

correspond to eigenvalues belonging to the diﬀerent sets li(L): red circles represent l1(L),

green circles represent l2(L), blue circles represent l3(L) and the black circle in the origin

represent l4(L).

Let us consider for example an eigenoperator ρi belonging to ǫ1(L). Under the action of

Λ(t) deﬁned in (3.8), we have that this eigenoperator is mapped into

ρi(t) = e−|Re(λi)|tρi, (3.16)

and by taking the trace it is easy to see that the trace of ρi(t),

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goes to zero as t goes to inﬁnity meaning that ρi(t) is not a density operator. The same

happens for the elements of ǫ2(L) and to their linear combinations. In order to see that

also the elements of ǫ3(L) do not represent proper density operators it is suﬃcient to note

that they are not self-adjoint. Indeed, if one assumes that ρi ∈ ǫ3(L) is such that ρi= ρ†i,

by exploiting (3.4), one obtains that

i Im[λi]ρi =L[ ρi] =L[ ρ†_i] =L[ ρi]†=−i Im[λi]ρi, (3.18)

which implies Im[λi] = 0. However, since λi ∈ l3(L) this is absurd, meaning that ρi cannot

be self-adjoint.

Let us now pay attention to the set ǫ4(L), and in particular let us consider a ρi ∈ ǫ4(L).

In addition, let us suppose ρi∈ P(HW). We racall that this implies that

(i) ρi ≥ 0, (ii) ρi = ρ†i, (iii)Tr[ ρi] = 1. (3.19)

Contrary to the cases discussed above, while considering the elements of ǫ4(L), the

con-straints prescribed in (3.19) do not lead to an absurd. Indeed, we have that

0 =L[ ρi] =L[ ρ†i] =L[ ρi]†= 0, (3.20)

and, most importantly, that

ρi(t) = Λ(t)ρi= eLtρi = ρi, ∀ t ≥ 0. (3.21)

Eq. 3.21 has a series of consequences that are extremely relevant in the study of open quantum systems. If one is interested in characterizing the equilibrium conﬁgurations of an open quantum system i.e. the steady-states, under a series of assumptions this is equiv-alent to characterizing the set containing the fixed points of the dynamical map Λ(t), that is those states that are invariant under the action of Λ(t). As shown above, these states correspond to elements of ǫ4(L), that is the Kernel (or null space) of L, here denoted by

Ker[_{L]. The following sections are devoted to the study of the conditions under which the} characterization of Ker[_{L] is equivalent to the characterization of the equilibrium} conﬁgu-rations of the dynamical semigroup generated byL.

3.2 Fate of open quantum systems: steady-states and limit

cycles

In the previous section we have discussed in a general fashion the spectral properties of the infinitesimal generator L. In particular, we have shown that the eigenoperators belonging to Ker[_{L] can be (and actually are) proper density operators, and that they} correspond to the fixed points of the dynamical map Λ(t). However, it worth noting that it is not guaranteed a priori that for any initial condition the system will end up in such configurations. By regarding the LGKS equation as a dynamical system in _P(H_W) [32], any density operator ρss in Ker[L] has its own basin of attraction, that is a region I(ρss)

in the space of the density operators such that any element belonging to the basin has ρss

as steady-state. More explicitly, we have that the basin of attraction of ρss is deﬁned as

I( ρss) ={ ρ : lim

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3.2. Fate of open quantum systems: steady-states and limit cycles 25

In this case, ρss is called attractor. However, depending on the structure of Zr[L] ≡

ǫ3(L) ∪ Ker[L], it is possible to have regions of the space of density operators that do not

lie into any of the basins of attraction of the states belonging to Ker[_{L]. In this case, such} regions are mapped by the dynamical semigroup into the so called limit cycles, that is time-dependent closed conﬁgurations having the following structure

ρLC(t) = ρss+ X i δi eλit_ρ i+ eλ ∗ itρ† i , (3.23)

where ρss∈ Ker[L], δi ∈ R, λi∈ l3(L) and ρi∈ ǫ3(L). This scenario is depicted in Fig. 3.2.

P(H

W

)

ρB ρLC(t) I(ρss) ρss ρA

Figure 3.2: Representation of a portion of the spaceP(HW) in the vicinity of a steady-state

ρss (red circle). Density operators belonging to the basin of attraction I(ρss) (grey region

in the middle) like ρA evolve towards ρss. Trajectories starting from points outside I(ρss)

(gray circles) like ρBafter some time spent inP(HW) (dashed blue lines) enter in the limit

cycle conﬁguration denoted by ρLC(t) (blue solid line).

Physically, the presence of limit cycles is associated to the impossibility of a given open quantum system to reach a proper equilibrium configuration, that is a stationary state whose properties are time-independent. Now the question is: Are there criteria which guarantee that the fate of a given open quantum system is to approach asymptotically configurations that are fixed points of the time evolution? During the last forty years a great deal of attention has been paid to this problem and several useful results for the characterization of the time evolution of open quantum systems have been derived. The following section is devoted to an analysis and a detailed description of such criteria.

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3.3 Convergence to fixed points

After the derivation of the LGKS master equation several theoretical results concerning the general properties of open quantum systems have been published. In particular, a lot of attention has been paid to the derivation of theoretical criteria capable of providing information about the asymptotic time behavior of such systems. However, in the major part of these papers, especially those published during the 70s, this issue is addressed in a formal way. As a consequence, it is often complex to truly understand their importance and moreover to understand the general nature of some results. Here, we will consider the most relevant for our purposes, that is those providing criteria which guarantee that in the large time limit the system conﬁguration is represented by a ﬁxed point of the dynamics i.e. a density operator in Ker[_{L]. In particular, what we are going to perform is} an explicit analysis: we will discuss in detail the theoretical assumptions on which these results are based and moreover how they are related to each other. This goal is pursued by considering the LGKS equation governing the open dynamics of a two-level system coupled to a reservoir of harmonic oscillators at temperature T .

3.3.1 A lesson from the simplest open quantum system

We consider a two-level system coupled to a reservoir of harmonic oscillators and de-scribed by the following Hamiltonian [11]

H = HS+HR+HI, (3.24) where HS = ǫ 2σ z_, _H R= X k ωkb†_kbk, HI = X k gkσ+bk+ g∗kσ−b†k , (3.25) describe respectively a system with two levels (| ↑i, | ↓i) separated in energy by ǫ (ǫ↑−ǫ↓ =

ǫ), a reservoir of harmonic oscillators of frequency ωkwith creation (annihilation) operators

b†_k(bk) and system-reservoir interaction with coupling constants gk, and being

σz= 1 0 0 −1 , σ+= 0 1 0 0 , σ−= 0 0 1 0 . (3.26)

The Hamiltonian (3.24) can be used to mimick several physical systems, in particular it efficiently describes the time evolution of a two-level atom interacting with the radiation field. In this case, the meaning of the Hamiltonian terms (3.24) is clear: the first two terms describe the free evolution of the atom and the radiation field respectively, while the latter describes absorption and emission processes which involve both matter and photons. In the Born-Markov approximation, by assuming in correspondence of the temperature T a mean-thermal occupation of the k-mode (~ = 1, kB = 1)

N (ωk)≡ hb†_kbki =

1

eωk/T − 1, (3.27)

the reduced time evolution of the two-level atom can be cast in the following form d dtρS =− i ǫ′ 2 [σ z_{, ρ} S] + γ↓ σ−ρ_Sσ+−1 2 σ +_σ−_ρ S+ ρSσ+σ− + +γ_↑ σ+ρ_Sσ−−1 2 σ −_σ+_ρ S+ ρSσ−σ+ , (3.28)

Dynamics and thermodynamics in quantum many-body systems: from steady-state properties of open systems to scaling analysis close to quantum transitions in closed systems.

University of Pisa

Department of Physics

Graduate Course in Physics

Ph.D. Dissertation

Dynamics and thermodynamics

in quantum many-body systems:

from steady-state properties of open systems

to scaling analysis close to quantum transitions

in closed systems

Candidate:

Davide Nigro

Supervisors:

Dr. Davide Rossini

Prof. Ettore Vicari

Contents

Chapter 1

Introduction

1.1

Dynamics in complex systems

1.2

Summary of this thesis

Chapter 2

Theory of open quantum systems

2.1

Preliminaries

2.2

LGKS equation: historical remarks on its derivation

2.3

LGKS master equation: a microscopic approach

Chapter 3

Lindbladians with a single

steady-state

3.1

Spectral properties of Lindbladians

3.2

Fate of open quantum systems: steady-states and limit

cycles

P(H

)

3.3

Convergence to fixed points