Des chaînes de spins bruitées aux processus d'exclusion quantique : études de cas de systèmes quantiques étendus ouverts stochastiques

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Des chaînes de spins bruitées aux processus d’exclusion

quantique : études de cas de systèmes quantiques

étendus ouverts stochastiques

Tony Jin

To cite this version:

Tony Jin. Des chaînes de spins bruitées aux processus d’exclusion quantique : études de cas de systèmes quantiques étendus ouverts stochastiques. Physique [physics]. Université Paris sciences et lettres, 2019. Français. �NNT : 2019PSLEE033�. �tel-02879969�

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Pr épar ée à l’ ´

Ecole normale sup ´erieure

From noisy spin chains to quantum exclusion processes:

case studies of stochastic open quantum many-body

systems.

Soutenue par

Tony JIN

Le 25 Septembre 2019 ´ Ecole doctorale no₅₆₄

EDPIF

Sp ´ecialit ´e

Physique

Composition du jury : Leticia CUGLIANDOLO

Universit ´e Pierre et Marie Curie- Paris VI Pr ´esidente

Fabian ESSLER

University of Oxford Rapporteur

Tomaz PROSEN

University of Ljubljana Rapporteur

Thierry GIAMARCHI

Universit ´e de Gen `eve Examinateur

Ion NECHITA

Universi ´e de Toulouse Paul-Sabatier Examinateur

Denis BERNARD

´

Ecole normale sup ´erieure Directeur de th `ese

Michel BAUER

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Preamble

Mudada· · ·

Dio Brando This manuscript presents the work done during my PhD thesis, ranging from October 2016 to June 2019 at the ´Ecole Normale sup´erieure in Paris under the supervision of Denis Bernard. It is divided into two parts: the first one gives a general introduction to the subject of the thesis and a review of the main results obtained during its completion. The second part is a reproduction of the articles written and published during the thesis [7, 11, 6, 13].

Ref. [5] constitutes another work accomplished during the thesis but since it concerns a more distant subject, it will not be covered here.

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Remerciements-Acknowledgements

Il revient bien évidemment en premier lieu de remercier Denis. Je ne lui ai pas rendu ces trois années faciles et je le remercie chaleureusement de la pédagogie et la patience dont il aura fait preuve à mon égard tout au long de cette thèse. Je suis très heureux d’avoir pu bénéficier de son approche de la physique, à la fois cosmopolite, rigoureuse et élégante. Son attachement quotidien aux petits détails -e.g avoir des notes bien organisées- comme au tableau général -e.g quel sens ¸

ca a de faire de la physique- sera quelque chose qui me restera. On dit souvent que la thèse est un chemin de croix, pour moi ¸ca aura été surtout une passionnante exploration intellectuelle et c’est avant tout à Denis que je le dois.

Merci à Michel dont j’ai suivi avec appétit et intérêt le cours de probabilité de M2 et avec qui j’ai eu la chance de travailler durant la quasi-totalité de la thèse. Toujours disponible et souriant, ¸

c’aura été un plaisir et une source d’éclairements de discuter avec lui.

I would also like to give a warm thanks to the members of my jury, for accepting to assist to my defence. especially the reviewers for accepting the arid task of examining the manuscript in details. I’d like to thank Tomaz Prosen for the very enjoyable visit to his group in Ljubljana and his kindness in our different interactions. Merci beaucoup également à Ion pour l’invitation au séminaire et en général pour m’avoir permis de connaˆıtre la communauté mathématique de Toulouse durant ma thèse. Thanks to Fabien Essler for his benevolence during our various interactions which I really appreciated. Un grand merci à Thierry pour avoir accepté de venir à la soutenance et bien évidemment pour m’avoir accepté dans son groupe. Enfin, merci à Leticia Cugliandolo pour avoir accepté d’être présidente du jury.

Thanks to Ohad for our various discussions and interactions.

Many thanks to office E235 a.k.a the 2016 promotion of PhD s of the LPT : Duan, Ioannis, Deliang and Alexandre. I was delighted to interact with and to learn from them all. It was good to have some others pals with whom to share worries and good news with. I wish them the best for whatever will come next. Merci en particulier à Alexandre qui aura fait montre d’une grande générosité scientifique et humaine tout au long de la thèse. Son soutien pendant la stressante période de recherche de post-doc aura été précieux.

Merci à l’équipe de physique pour tous, Mathias, Louis, Alexandre et à tous ses étudiants. Sans avoir rencontré un succès éclatant, le cours aura eu le mérite d’exister durant trois ans. Je souhaite bonne chance aux repreneurs pour leurs futures batailles administratives, logistiques, pédagogiques voire idéologiques.

Merci à l’équipe de la SFJAPD et en particulier Benjamin, David, Nicolas et Paul qui auront eu le mérite de se bouger de fa¸con quasi-bi-mensuelle pendant 3 ans pour débattre de sujets triviaux ou sérieux tournant autour de la physique. J’ai beaucoup appris de ces discussions et j’espère qu’elles se poursuivront à l’avenir, malgré la distance.

I take the opportunity to salute the Balkan club, Lenart, Katja and Marko who were both great friends and great scientists to interact with.

J’ai également eu la grande chance et le grand plaisir d’enseigner au CPES. Merci à Jean-Fran¸cois, jamais avare en sympathie, pour m’en avoir donné l’opportunité, merci également à Erwan pour m’avoir fait confiance. Merci à Mohamed qui a été un collègue légendaire à bien des égards. Un grand merci bien évidemment aux élèves de ces deux années qui auront été

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exceptionnels et fait de cette première expérience d’enseignement un enjaillement total. Je leur souhaite à tous de réussir dans ce qu’ils entreprennent mais je ne me fais pas de soucis. En particulier, merci à Aurélien, Adrien et Tatiana pour m’avoir donné l’opportunité de travailler avec eux.

Je voudrais également en profiter pour remercier mes précédents maˆıtres de stage qui ont eu la tâche ingrate d’encadrer un étudiant scientifiquement immature. Je garde un excellent souvenir de toutes ces expériences sans exception. Merci à Chiara Caprini, Benoˆıt, Emmanuel, Matthieu et Yves du côté fran¸cais. Thanks to Imai Hiromitsu (and sorry again for all the insurance problems. . . ) and Anupam for the thrilling research experiences at Atsugi and Northwestern.

Je suis réellement reconnaissant à l’état fran¸cais de m’avoir permis de faire des études longues, non seulement par la gratuité des formation mais aussi grâce aux bourses diverses et variées dont j’ai pu profiter durant ma scolarité. Merci à tous les enseignants que j’ai eus la chance de croiser au lycée Hoche, à l’ESPCI et à l’ENS.

Merci à tous mes amis qui auront assisté de près ou de loin à la réalisation de cette thèse. Je souhaite en particulier saluer la légendaire coloweed et tous ses (principaux) membres, Sylvain, Arthur, Aziz, Amaury, Adrien, Théo et Thibault. Merci aux bourrins du volley d’Ulm (et courage `

a Yan Der), en particulier Markabu et Thibaut pour me parler patiemment de maths malgré mon niveau déplorable. Merci à Tiphaine, Paul et Thomas, mes fidèles compagnons de déjeuner, à Karène qui gère mes coups de déprime depuis si longtemps, Glenn a.k.a techman, Zoé qui m’a sauvé la vie à Chicago et à Jeff et Louise pour toutes ces années de gentillesse. Merci à Théophile `

a qui je pense souvent.

Merci à ma chère Olya qui a la lourde tâche de gérer mes tracas au quotidien. Merci à mon cher ami Stefan, pour être toujours là après toutes ces années.

Enfin, merci à tous les Jin’s : ma gentille tata et mon gentil tonton qui prennent soin de moi comme de leur propre enfant, à Davy qui veille sur nous de loin, à Willy et Sophie qui me rendent très fier tous les jours. Merci à ma chère Alex qui a fait de moi une personne meilleure. Merci à mon père qui, malgré tout le reste, n’est pas pour rien dans mon envie de faire des sciences.

Merci à mon cher frère Victor pour m’avoir soutenu depuis le début et enfin, merci à ma très chère mère, qui a tout donné pour nous et sans qui, bien sûr, rien de tout cela n’aura été possible.

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Introduction

This thesis is concerned with the study of open quantum many-body systems beyond their mean behaviour. The reason to study open systems is quite simple: In practice there is no such thing as an idealised closed system. Thus, the full understanding of a quantum system requires the inclusion of its surroundings, i.e one needs to consider external degrees of freedom pertaining to a bath or environment. As it is in general hopeless to solve the closed dynamics of the full system, playing the game of open quantum systems therefore consists in finding an economic formalism which describes faithfully enough the dynamics of the system explicitly and the dynamics of the environment implicitly.

Historically, the subject has attracted intensive efforts and a variety of methods have been developed. In the early days of quantum mechanics, the fact that the environment should add additional degrees of freedom that are mostly intractable partially motivated the development of the density matrix formalism by Von Neumann [73]. In the regime of weak coupling between the system and the bath, assuming that they are initially uncorrelated, Bloch and later Redfield [77] were able to derive a dynamical equation (named after the latter) describing the evolution of the system alone. Although Redfield equations reproduce correctly an asymptotically thermalized state, they do not preserve positivity of the density matrix. A simplified but more analytically tractable model was obtained later by assuming that the bath could be modelled as an infinite collection of quantum harmonic oscillators. In an early work of 1959, Magalinskii pointed out that such coupling should lead to dissipative evolution for the system [70]. Later, Feynman and Vernon, unknowing of the work of Magalinskii, attacked the problem within the path integral formulation [44]. Their ideas were later refined by Caldeira and Leggett [23]. This lead the community to name this generic type of semi-empirical model Caldeira-Leggett models. From a more formal point of view, the search of the mathematical description of open systems evolution led the community to consider more general quantum maps -i.e beyond unitaries. It is now accepted that the most general quantum maps or quantum channels are elements of completely positive semi-group. A key step was achieved with the formulation of the GKLS equation -named after the people who derived it, Gorini, Kossakowski, Sudarshan and Lindblad- which gives the general form of the generator of such maps [69]. It has then proved to be an efficient first guess to describe dynamics of open quantum systems, encountering noticeable success for instance in quantum optics.

In the classical case, in the context of linear response theory, a link has been established between the emergence of dissipative dynamics and the existence of an underlying stochastic equation of motion. This is the essence, for instance, of the fluctuation-dissipation theorem [63]. Interestingly, in quantum physics, upon suitable hypothesis of Markovianity, one can also understand the dissipation as emerging from an underlying stochastic process that is referred to as quantum noise[45].

Let’s illustrate this idea for a classical system: consider for instance an isolated particle in free space. In absence of any external environment, it will just propagate ballistically. If one adds interaction with an environment, say as random collisions with other particles, in the limit of large number of collisions per unit time, the effective description of the system-environment

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Figure 1: Illustration showing the difference between the ballistic behaviour of an isolated classical

particle versus its behaviour in an environment causing collisions with other particles.

coupling can be written in terms of a Langevin equation, m~a = −λ~v + ~η.

The first term is a viscous damping while the second one is a random variable with Gaussian probability distribution whose variance is directly proportional to the temperature of the envi-ronment. In its celebrated work on Brownian motion, Einstein [39] showed how the physical properties of such models radically differed from their isolated counterpart. For instance, the transport properties are strongly affected. Instead of the ballistic propagation, we end up with a diffusive behaviour. The diffusion constant D is directly linked to the temperature T via the Einstein relation D = µkBT where µ is the mobility and kB the Boltzmann constant. This dif-fusive behaviour typically arises when we are looking at length scales that are large compared to the mean free path, i.e the average distance travelled between two collisions.

For quantum systems, we will be interested in Caldeira-Legett [23] type models whose Hamil-tonian is written

Htot = HS+ HB+ Hint.

where HS and HB are the Hamiltonians of the system and the bath which is constituted of an infinite ensemble of quantum harmonic oscillators and H_int describes the interaction between the system and the bath. The qualitative picture is the following: If the number of degrees of freedom of the bath is large enough, the energy transferred from the system is quickly spread away into these degrees of freedom and never “comes back” to the system, leading to dissipative phenomena. We will explain in the body of the thesis how one can attribute the emergence of this dissipative dynamics to an underlying quantum stochastic process describing the evolution of the system alone. Roughly speaking, the Hamiltonian generating the unitary for an infinitesimal time step from t to t + dt can be written under the general form

dHt= U dt +

X

α

VαdBtα,

where U and V are operators acting on the system and B_tα independent quantum Brownian processes. The dynamics generated this way will be of quantum, stochastic, unitary nature and this thesis will be mainly concerned with the application of such stochastic formalisms to the study of extended or many-body quantum systems.

In the classical theory, a particular class of models where such ideas have been crowned with important theoretical successes are the exclusion processes [32, 35]. These processes describe the dynamics of hard-core particles on a lattice. For each time-step, a given particle has some probability to move to the nearest neighbouring empty sites. In the case where the displacement is isotropic in space, the transport is diffusive. Although blatantly idealised, these kind of models have attracted a lot of interest because of the possibility to obtain exact solutions from first principles. Among other things, one should mention the contribution of these solutions in the

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understanding of the more general macroscopic fluctuation theory (MFT) [15] which provides a framework to study and to understand the fluctuations around mean equilibrium and non-equilibrium states of many-body classical systems.

From the quantum side, as far as we know, exact solutions were obtained for simple, often integrable isolated many-body systems in or out-of-equilibrium. Among others, one could cite spin chains models, the Bose-Hubbard model, the Lieb-Liniger model, conformally invariant field theories, vertex models, etc. (See for instance [41, 83, 27, 29, 60, 71]). The notion of quantum integrability is heuristically understood as the existence of an exact solution to the problem and the existence of an infinite number of local (or quasi-local) conserved quantities [43]. However, it occurs that most of the time, the transport in these kind of systems is ballistic (at least locally) which makes them bad candidates for a quantized version of the MFT. One possible way to go to dissipative diffusive dynamics would be to consider an open version of such models making them inherently stochastic. The hope one can have is that the formulation of these models will still be compact enough to be analytically tractable while producing the desired dissipative, diffusive structure. The main results of this thesis will be concerned with the study of such setups.

It is worth mentioning another context, as old as quantum mechanics, where open quantum systems appeared: The Von Neumann formulation of measurement. In this case, the exterior environment is the measurement apparatus put into contact with the system. The well-known aftermath is the probabilistic projection of the wave-function of the system onto one of the eigenvectors of the measured observable. A refinement of this idea was proposed by Aharonov and was later referred as weak measurement [1]. The paradigmatic situation is to consider a series of independent probes all in the same initial state that are sent to interact with the system one-by-one. A measurement is then performed on the probe instead of the system, leading to a random backaction on the latter. In the limit where the number of probes is larger and larger while the system-probes entanglement strength is weaker and weaker, there is an effective description of the evolution of the system in terms of a continuous in time stochastic process.

Weak measurement has attracted a lot of interest in the recent years, in particular thanks to its experimental feasibility. A famous example is Haroche’s cavity QED experiment which led him to the 2012 Nobel prize. In quantum circuits one can also mention the work of D´evoret and Huard [84, 21]. However, these impressive results are mostly focused on the study of single-body systems for now. We will give a modest glimpse at what could happen for many-body system under continuous monitoring.

This thesis is organised as follows:

• In the first part, we provide a smooth, rather informal, introduction to some key concepts about open quantum system and the associated stochastic description by exemplifying them on the simple example of the spin-1₂.

• In the second part, we will discuss some particular topics concerning relaxation and trans-port phenomena in many-body quantum systems that will be of interest for us.

• Part three is intended as a more formal introduction to open quantum systems.

• Part four is the most important of all, since the main results of the thesis will be reviewed here.

• Part five deals with some corollary results that have to do with continuous monitoring. • In the last part we give a short summary of the main results and also discuss some future

perspectives that might be of interest.

• Finally, there are also three appendices, the first one presents the basic mathematical tools of stochastic processes, the second one discusses fermionic Gaussian states and their properties

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and the third one illustrates ideas coming from large-deviation theory on the symmetric simple exclusion process.

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Open single-body system

The aim of this section is to provide a smooth introduction to the general ideas of the thesis by exemplifying them on a simple system, namely a spin-1₂ put in contact with an external environ-ment that causes damping. We want to study the relaxation of the spin towards equilibrium and the fluctuations in the stationary state. The environment is modelled as a collection of quan-tum harmonic oscillators oscillating at different frequencies. This setup constitutes an example of a Caldeira-Legett or spin-boson model [23, 67] which are generic semi-empirical models used to study dissipative quantum systems. To lighten the discussion, most of the results presented won’t entail particular mathematical rigour but will rather be focused on presenting the general qualitative arguments. A more rigorous and systematic presentation will be available in part 3. We precise that the approach presented here is our own take on this rather standard problem. For a more general and complete presentation, see e.g [45].

We will first show how upon certain hypothesis on the properties of the bath, we get an effective stochastic description of the system-bath interaction in terms of continuous in time random processes called quantum noises. The analogy with classical stochastic processes will be explicit via the derivation of quantum Itˆo rules. We will then proceed to solve the evolution equations for a particular choice of models and will show how to obtain the full distribution of the system state. In average, we will see that the density matrix of the spin−1₂ decays towards a diagonal state although fluctuations of the off-diagonal components encoding for superposed states will be non zero in the long time limit. As it will be useful for later discussions, we will also present an effective description of the noisy dynamics arising in the limit of strong dephasing and long times. The full distribution of the steady state will be derived using two different methods: The first method is a direct resolution of the stationary equations, the second method consists in remarking that in the long-time limit, the dynamics is noisy enough to generate all unitaries in an uniform way. In this sense, we will say that the dynamics is maximally noisy. The natural measure will thus be provided by the uniform integral over the unitary group. These ideas will be discussed again in the presentation of the main results in a more complex setting i.e involving many-body quantum systems.

For readers unfamiliar with stochastic calculus and in particular Itˆo calculus, it is advised to have a look first at app.A.

1.1 Quantum noise

1.1.1 Spin-boson

We consider the interaction of a spin-1₂ with another quantum system that we want to think of as a bath or reservoir. We will detail the particular case of the interaction with a bosonic field, the so-called spin-boson model which is one particular example of a Caldeira-Legett model [23, 67].

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The Hamiltonian of our model is Htot = HS+ Hint+ HR, with Hint = √ γ Z dω(A†aω+ Aa†ω), HR= Z dωh(ω)a†_ωaω.

HS is the Hamiltonian of the system, Hint is the interaction Hamiltonian between the system and the reservoir and HR is the Hamiltonian of the reservoir. A and A† are some yet undefined operators acting on the system. aω and a†ω are bosonic annihilation and creation operators at frequency ω and satisfy the following commutation relation [a_ω, a†_ω0] = δ(ω − ω0). We also make

the further assumption that the coupling to every bosonic mode is identical for simplification, i.e we suppose that the ω dependence of A and A† can be taken constant in some frequency range. What is expected from such a model is that if the time scale of decay of correlation of the bath is very short compared to the internal dynamics of the system, there will be an emergent effective Hamiltonian describing dissipation of energy and coherence of the system. Upon simplifying assumptions, the dynamics of the bath will be only implicit as it is the case for instance with the classical Brownian motion. On the more formal side, since the interaction of the system with the bath is contained in Hint, we then expect that in the right picture, the algebra of theRdωaω and

R

dωa†_ω will be such that they admit an intepretation in terms of stochastic quantities. We will see that this effective description is apparent when one goes to an:

1.1.2 Interacting picture

The goal of the interacting picture is to integrate out the fast oscillating dynamics of the bath by doing a change of reference frame: explicitly, instead of the observable O, we consider the rotated O(t) = eiHRt_Oe−iHRt_{. It is implicit from now on that any operator with explicit time dependence}

written down has to be considered in this frame. We immediately see that HS(t) = HS, and

HR(t) = HR. In the interacting picture the time evolution of states is given by

d |ΨIi

dt = −iHtot(t) |ΨIi

= −i(Hint(t) + HS) |ΨIi (1.1)

One can also show that

aω(t) = eih(ω)taω,

a†_ω(t) = e−ih(ω)ta†_ω, leading to the important commutation relation

[ Z dωa†_ω(t), Z dω0aω0(t0)] = Z dωe−ih(ω)(t−t0).

We have to fix the energy spectrum of the bath h(ω). In a realistic situation, it should be centred around a typical value ω₀ fixing the energy scale of the bath with a typical width ∆ω. For analytical simplification, we will suppose that this width is very large and can be sent to infinity meaning that there is a very broad distribution of frequencies compared to the typical energy scales of the system. We choose a linear dispersion relation h(ω) = 2πω (but in principle one could consider more complicated dependence). The previous commutator then simplifies into

[ Z

dωa†_ω(t), Z

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with δ(t) the Dirac distribution. Were we to deal with a more refined picture, we would have to introduce some interval [ω0, ω0+ ∆ω] to integrate upon and the previous commutation relations

would not be Dirac correlated in time.

In the interacting picture, the generator dHtotof infinitesimal unitary flows (|ψt+dti = e−idHtot(t)|ψti) becomes time dependent and is given by

dHtot(t) = HSdt + √

γ(A†dWt+ AdWt) (1.2)

For later purposes, it is useful to decipher the dWt, dWtcommutation relations. First introduce

Z t+∆t t Z dωa†_ω(t0)dt0 ≡ ∆Wt Z t+∆t t Z dωaω(t0)dt0 ≡ ∆Wt One then straightforwardly obtains

[∆Wt1, ∆Wt2] = [ Z t1+∆t t1 Z dωa†_ω(t0)dt0, Z t2+∆t t2 Z dω0aω0(t00)dt00] = Z t1+∆t t1 dt0 Z t2+∆t t2 dt00δ(t0− t00)

The last expression is either 0 if [t₁, t1+ ∆t] and [t2, t2+ ∆t] are disjoint or is equal to ∆t if they

are not. For the infinitesimal version this leads to the following commutation relations: [dWt, dWt0] = 0 if t6=t’

= dt if t=t’ (1.3)

Qualitatively, dWtand dWtcan be interpreted as operators creating or annihilating an excitation in the bath. The fact that these operations commute for different times mean that there are no memory effect, i.e that we have some kind of Markov property : the system at time t “does not know” about what happened at earlier times. The fact that their commutation relation at equal times scale like dt is a property reminiscent of classical stochastic process. We will make this connection clearer by examining the:

1.1.3 Quantum Itˆo rules

We will see how one can interpret dWt and dWt as increments of a stochastic, continuous in time quantum process. We are not concerned here with mathematical rigour and will insist on the heuristics. A more systematic and rigorous treatment will be presented in part.3. The main result will be that the operators Wt ≡R₀tdWt and Wt ≡R₀tdWt can be interpreted as random, continuous in time processes living in the space of operators acting on the Hilbert space of the bath. The averages of such quantities will be defined as summing over all the possible unitary evolutions of the environment, i.e by taking the partial trace over the environment. Anticipating a bit, we will refer to them as quantum noises.

We start from a factorised state for the initial density matrix of the total system ρtot(t = 0) = ρS(0) ⊗ ρR(0).

Let’s see what happens when we evolve our density matrix using (1.2) from time t₀ = 0 to an infinitesimal time t1 = t0+ dt in the interacting picture.

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Expanding:

dρtot(t1) = −i[dHint(t0), ρS(0) ⊗ ρR(0)]

| {z } I −1 2[dHint(t0), [dHint(t0), ρS(0) ⊗ ρR(0)]] | {z } II +O(dt2)

It may appear unusual to keep the terms II as they should be of higher order. We argue below why we have to. Let us expand I and II:

I = −i√γ[A†dWt0 + AdWt0, ρS(0) ⊗ ρR(0)]

II = γ((A†dWt0 + AdWt0)ρS(0) ⊗ ρR(0)(A

†_dW t0+ AdWt0) −1 2(A † dWt0+ AdWt0) 2_ρ S(0) ⊗ ρR(0) − 1 2ρS(0) ⊗ ρR(0)(A † dWt0 + AdWt0) 2₎

Evaluating the average in our language is equivalent to take the trace over the degrees of freedom of the environment. For instance, consider the situation where the bosonic bath is initially empty, i.e ρR(0) = |Ωi hΩ|. The average of II with respect to the degrees of freedom of the environment is then

E[γ((A†_dW

t0 + AdWt0)ρS(0) ⊗ ρR(0)(A

†_dW

t0 + AdWt0)]

≡ γTr_R((A†dWt0+ AdWt0)ρS(0) ⊗ |Ωi hΩ| (A

†_dW

t0 + AdWt0))

= γAρ_S(0)A†dt.

For a more general ρ_R(0), to get the increment of the average dE[ρ_tot(t₁)], we are led to make the following replacements

{dWt0ρR(0)dWt0, dWt0dWt0ρR(0), ρR(0)dWt0dWt0} ← αdt

{dWt0ρR(0)dWt0, dWt0dWt0ρR(0), ρR(0)dWt0dWt0} ← βdt

where αdt = tr(dW_t₀ρR(0)dWt0) and βdt = tr(dWt0ρR(0)dWt0). From the commutation relations

(1.3) we must have α−β = 1. Now, the full evolution is generated by a composition of infinitesimal unitaries:

ρtot(tn) = e−idHint(tn−1)...e−idHint(t0)ρs(0) ⊗ ρR(0)eidHint(t0)eidHint(tn−1).

Since the different dWt’s taken at different times commute with each other, the rule to get expectations over the environment at any time is to make the replacements

{dW_tρR(0)dWt, dWtdWtρR(0), ρR(0)dWtdWt} ← αdt {dW_tρR(0)dWt, dWtdWtρR(0), ρR(0)dWtdWt} ← βdt

Note that even if the density matrix of the reservoir evolves with time, the α and β are still only dependent of the initial distribution we choose.

In analogy with Itˆo calculus, when evaluating differential of random variables, we can sys-tematically replace all quadratic infinitesimal stochastic operators by their mean. The effective process obtained this way converges in L2 towards the initial process. A more rigorous proof will be given in part.3. In fine, the evolution of the density matrix can be written in a compact way:

dρtot(t) = −i[HS, ρtot(t)]dt + γ(α 2(A †_ρ tot(t)A − 1 2{AA †_{, ρ} tot(t)})dt + β

2(Aρtot(t)A

†₋1

2{A

†_{A, ρ}

tot(t)}))dt − i√γ[A†dWt+ AdWt, ρtot(t)]

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where {, } denotes anti commutation. This is a quantum stochastic differential equation acting on the density matrix. The first line constitutes what we will call the deterministic terms while the second one constitutes the noisy terms. To get the evolution of the averaged density matrix, we simply need to discard the latter, i.e:

d

dtE[ρtot(t)] = − i[HS,E[ρtot(t)]] (1.4)

+ γ(α 2(A

†

E[ρtot(t)]A − 1 2{AA

†

,E[ρtot(t)]}) (1.5) +β

2(AE[ρtot(t)]A

†₋ 1

2{A

†_A,_E[ρ

tot(t)]})), (1.6)

Remark the operator L(•) ≡ A†• A −1 2{A

†_{A, •} is of the Lindblad form, i.e it is the infinitesimal}

generator of a completely positive Markovian semi-group. Here we see that they emerge naturally as the averaged dynamics of a stochastic quantum evolution. Physically, the Lindblad operators generate the mean evolution induced by the interaction with the reservoir. We will provide a presentation of Lindbladians in more details in 3.

We will be particularly interested in a special limit we call the classical limit for the noise, i.e the limit where tr(dWtdWtρR) = tr(dWtdWtρR) i.e α = β. Since we also have α − β = 1, this only happens when α → ∞. For this limit to make sense, we need to rescale the coupling to the environment: γ ← γ_α. This amounts to take the limit of weak coupling of the system with the reservoir while the mean number of particles in the reservoir diverges. One then gets from (1.4).

dρtot(t) = −i[HS, ρtot(t)]dt − γ[A†[A, ρtot(t)]]dt − i √

γ[(A†+ A), ρtot(t)]dBt. (1.7) We replaced the noise operators dW_√t

α and dW_√ t

α by dBt which is the infinitesimal increment of a Brownian process B_t. We can do so because in the classical limit their mean quadratic expecta-tions, i.e, their Itˆo rules coincide. Remark also that if we forge that dB_t was once an operator, the above equation can be interpreted as an evolution on the system alone. With a slight abuse of notation, we will later forget about the environment density matrix and simply identify ρtot→ ρS in the classical noise limit.

Finally let us notice another way to write this equation that makes the unitary nature of the evolution explicit:

ρS(t + dt) = e−i(HSdt+(A

†_+A)dB

t)_ρ

S(t)ei(HSdt+(A

†_+A)dB

t) _(1.8)

Let’s end with some remarks:

• When dealing about expectation of physical observables, there are now two type of averages of fundamentally different nature. The quantum expectation of an observable O is obtained by taking the trace with respect to ρS: Tr(ρSO). On the other the expectation of an observable with respect to the stochastic noise is obtained by taking the averages over the different random processes B_t and will be denoted throughout the manuscript asE(O). • Because we are now working with stochastic theories, quantum expectations are still random

variables with respect to the different realizations of the noise. For instance,E[Tr(ρ_SA)Tr(ρSB)] 6=E[Tr(ρSA)]E[Tr(ρSB)] in general. We will see throughout this thesis that these fluctua-tions are in general non trivial and worth studying. The relafluctua-tionship between the different averages is summed up on fig.1.1.

• Even if we are dealing with random evolutions, for each realization of the noise, the evolution is still unitary. So the dissipative aspects will be visible only after taking stochastic averages. For instance, starting from a pure state, we will remain in a pure state for each realization of the quantum noise. This is no longer true if we are interested in the average evolution of this pure state. In this case, we immediately evolve towards a statistical mixture.

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Figure 1.1: Relationships between different types of averages.

1.2 Another point of view on the quantum noise

An other -maybe more intuitive- way to see the quantum noise is as the emergent description of the interaction of a quantum system (in the present case, the spin-1₂) with infinite series of auxiliary quantum ancilla bits. The ancilla are all decoupled at the beginning and their numbering has to be thought of a time index determining in which order they are sent to interact with the system. The operators acting on this auxiliary chain will be the quantum noise operators that we introduced in the limit where the number of ancillas go to infinity while the amplitude of the operators is sent to zero. The qualitative picture is that the infinite chain of ancillas form the reservoir that we introduced previously. The Markov property is enforced “by hand” by imposing that the ancillas are initially uncorrelated and are sent to interact one by one.

Let’s make this construction explicit (but again with no mathematical rigour).

First, consider the infinite tensor product of Hilbert spaces describing N individual spin-1₂. This is the Hilbert space where the chain lives: HN_chain ≡ H₁⊗ H₂⊗ ... ⊗ H_N where H_j is the Hilbert space of a spin-1₂. Let’s define the following operators acting on the chain:

ΣN₊ ≡ σ₊(1)+ σ₊(2)+ .. + σ₊(N ) ΣN₋ ≡ σ₋(1)+ σ₋(2)+ .. + σ₋(N ) ΣN z ≡ σ (1) z + σz(2)+ .. + σz(N )

The superscript refers to the corresponding Hilbert space. These are operators that respectively create and annihilate excitations on the whole spin chain and count the total magnetization of the chain. There is an interesting parallel to draw with classical stochastic processes. One important concept coming from classical theory is that of a filtration notedF. A filtration is a concatenated collection of σ-algebras (F_t)_t∈T indexed by time such that F_s ⊂ F_t for s < t and the process at time t is measurable on Ft. For our system with the ancilla, the natural extension of this idea is to consider the “quantum filtration”

F ≡ (Fn)n∈[1,N ],

with F_n≡ H₁⊗ H₂⊗ ... ⊗ H_n. The operators Σn_a are naturally “measurable” on F_nin the sense that they only act on this space. To read a realization of the process at the step n one then has to look at the state of the ancilla with index n0< n.

We now want to consider the continuous limit where N goes to infinity. In general, these operators will diverge so we need to rescale them by a small factor . We call the limit where

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N → ∞ and → 0 while N ≡ t is kept fixed the continuous limit (c.l). One can show by calculating the corresponding commutation relations that the following scalings make sense:

At≡ lim c.l √ ΣN₊, A†_t ≡ lim c.l √ ΣN₋, Tt≡ lim c.l Σ N z , Rt≡ lim c.l 2(N − Σ N z ) They fulfill the commutation relations

[A_t, A†_t] = T_t, [R_t, A_t†] = A†_t, [R_t, At] = −At.

The operator T_t is central meaning that it commutes with all the other in the continuous limit. We will therefore, interpret it as the time and simply denote it t. As in the discrete case, the operators At and A†t can then be seen as creating an excitation between time 0 and time t.

For the initial state of the spin chain, we take each spin to be decoupled from the others and we suppose that all spins are described by the same density matrix i.e, in the discrete setting: ρtot= ρ(1)⊗ ρ(2)... ⊗ ρ(N ). The average of operators acting on HN_chainare evaluated with respect to

this state: E[•] ≡ Tr(ρ_chain•). An important remark is that the statistical properties of operators acting locally at different times are uncorrelated due to the choice of a factorized density matrix ρtot. This is equivalent to the Markov hypothesis of the previous section. This means in particular that the statistical properties of the local increments dA_tand dA†_t are always the same no matter how the chain was modified for time s < t. Physically, one can think of these increments as creating or annihilating an excitation on the ancilla of the chain that interact with the system between time t and t + dt.

The increments dAtand dA†tagain fulfill Itˆo rules. They depend explicitly on the initial choice of the density matrix of each individual spin. For the following choice ρ = 1 − η 0

0 η

!

, we have the following Itˆo rules:

dA†_tdAt= 1 − η, dAtdA†t = η,

dA†_tdt = dAtdt = dtdt = 0.

In the discrete setting, we write the evolution generated by the interaction of the jth probe with the system during a time interval ∆t as

∆U = e−i( √ VS⊗σ−(j)+ √ V_S†⊗σ+(j)_)∆t ,

with V_S and V_S† operators acting on the system. In the continuous limit this gives dU = e−i(VS⊗dA†t+V

†

S⊗dAt).

This expression is similar to the one derived for the Caldeira-Legett model. The tensored Hilbert space with which we defined our filtration naturally extends to the continuous limit and can be interpreted as the natural space on which At and A†t act on.

1.3 Illustration: Single-qubit undergoing strong dephasing.

In this section, we illustrate the formalism introduced previously on a specific example, namely: we will consider a spin-1₂ driven by a magnetic field along the x axis. In addition, we suppose that there is an interaction with an external environment which induces random dephasing of the component of the spin along the z axis. We will work in the previously introduced classical limit for the noise.

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As explained before, what one expects is that the coupling to the external environment will induce a dissipative evolution of the system.

A point we will detail is the existence of a slow mode dynamics for this kind of system. Qualitatively, the evolution induced by the random dephasing will tend to make the system collapse in mean towards a submanifold made of states invariant under this random action. However, if the internal dynamics of the system does not commute with this noisy action, one can expect to have at time long compared to the collapse type a non-trivial dynamics on this slow-mode manifold. As it will be useful for later purposes, we will decipher what this effective slow dynamics is.

Even though the evolution is stochastic, it is also unitary, meaning that realization by realiza-tion, quantum coherences of the system should be preserved. We then expect to get dissipative, decoherence effect in mean but coherent behaviour for each realization of the noise. We will see how this impacts the fluctuations of the stationary regime.

We consider the dynamical evolution equation dρt= −iν[σx, ρt]dt − γ 2[σ z_{, [σ}z_{, ρ} t]]dt − i √ γ[σz, ρt]dBt. It is generated by the stochastic Hamiltonian dHt= νσx+

√

γσzdBt. The first term describes the internal unitary dynamics which is a rotation of the spin while the following two terms describe the noisy interaction with the environment. Let us parametrize the density matrix by ρ_t= 1₂(1+ ~St.~σ) with ~St in the Bloch ball. Projecting the above equation over Pauli matrices yield

dSz_t = −2νSy_tdt,

dS_tx= −2γS_txdt + 2√γS_tydBt,

dS_ty = 2νS_tzdt − 2γS_tydt − 2√γS_txdBt. (1.9) For the noiseless case γ = 0 we’d have

dS_tz = −2νS_tydt, dS_tx= 0,

dS_ty = 2νS_tzdt,

which encodes rotations of the vector ~St around the x-axis as is well-known. The oscillation has an infinitely long life time as it should for a system whose energy is conserved.

Let us first look at the mean flow for γ = 0. We define ¯Sa

t =E[Sta]. We have: d dt ¯ S_tz = −2ν ¯S_ty, d dt ¯ S_tx = −2γ ¯S_tx, d dt ¯ S_ty = 2ν ¯S_tz− 2γ ¯S_ty. The solution is ¯ S_tx= ¯Sx₀e−2γt, ¯ S_ty = ¯ S₀y(eλ+t_λ +− eλ−tλ−) + 2ν ¯S₀z(eλ+t− eλ−t) (λ+− λ−) , ¯ S_tz = 2ν ¯S y 0(eλ−t− eλ+t) + ¯S0z(eλ−tλ+− eλ+tλ−) (λ+− λ−) , with λ± = −γ ± p

γ2_{− 4ν}2 _{the two eigenvalues of the linear problem. We see that in average,}

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of the Bloch ball i.e towards a classical mixture of up and down states. This is a decoherence effect which is intuitively expected for the mean evolution of a system state in contact with an environment. On a more mathematical side, we have seen that the mean dynamics was generated by the Lindblad operators. A well-known property of such operators is that their eigenvalues have necessary a negative real part. Hence, at long-time the system state relaxes in mean towards eigenstates who have zero real values (there can be non zero imaginary part though). More details in part.3. Even though the mean dynamics leads to the decoherence of the state, the evolution is still unitary for every realization of the noise. We will see that fluctuations beyond this mean behaviour entail a richer and non trivial structure. Before that we first describe the :

1.3.1 Slow-mode dynamics

For large γ, we have λ−' −2γ and λ+' −2ν2/γ. From this we see that

¯

S_ty ' ¯Sy₀e−2γt → 0, S¯_tz ' ¯S₀ze−2ν2(t/γ),

asymptotically in γ. That is: only the component along the z-axis survives in the large γ limit with a non trivial dynamics w.r.t. the time s = t/γ. This encourages us to look for an effective description of the model that will capture what we call the slow-mode dynamics, i.e to look at the scaling γ → ∞, t → ∞ while s is kept constant. We will see that if we look at the dynamics not in mean but for each realization, there is an effective stochastic dynamics that is expressed in terms of the “slow-time” s. If there were only the dephasing noise and no internal dynamics, a glimpse at (1.9) tells us that the component on the z-axis of the spin is preserved while the other components undergo a random rotation. In the long-time regime, the x and y components of the spin will be uniformly distributed on the circle described by the rotation. If the dephasing is strong, this stationary distribution is reached in a time ≈ 1/γ. Now the effect of putting an internal evolution that does not commute with the action of the noise will tend to “push out” the system out of the stationary circle. Roughly speaking, one has to wait longer that the collapse time to witness an interesting dynamics, hence the proposed slow-time scaling.

We now proceed to derive this effective dynamics. Recall the evolution equation for the density matrix:

ρt+dt= e−idHtρteidHt

with dH_t= νσxdt +√γσzdBt. Let us go to an interacting picture:

ρt≡ ei √ γσz_B t_ρ te−i √ γσz_B t_.

The time-evolution of ˆρt is given by

ρt+dt= e−idHtρteidHt, with dHt= ei √ γσzBt_νσx_{dt e}−i √ γσzBt = ν(e2i √ γBt_σ+_{+ e}−2i √ γBt_σ−_)dt.

Making use of the scaling properties of Brownian motion, i.e B_γs=√γBs, we get to

dHs= γν(e2iγBsσ++ e−2iγBsσ−)ds Let’s have a look at statistical property of W_s ≡ Rs

u=0γe2iγBudu and Ws ≡Ru=0s γe−2iγBudu in the limit of large γ. First remark that in this limit they are martingales. A martingale Xt is a continuous in time random process that respects the following property:

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for t0 ≤ t where F_t is the natural filtration associated to the process and E[•|F_t0] denotes the

conditioned expectancy w.r.t this filtration. We have E[Ws+s0|F_s] = Z s 0 γe2iγBu_{du +}E[ Z s+s0 s

γe2iγBu_du|F

s] = W_s+ e2iγBsE[

Z s0

0

γe2iγBu_du]

where we used that W_s is an F_s-measurable process, the additivity property of Brownian pro-cesses: Bu+s = Bu + Bs and the independence of Rs+s

0

s γe2iγBudu and Fs. We can explicitly calculate the average of the second term:

E[Z s

0

γe2iγBu_{du] =} Z s0 0 du Z dx√γ 2πue 2iγx_e−x2 2u = 1 − e −2γ2_s0 2γ2

we see that in the limit γ → ∞, E[W_s+s0|F_s] = W_s and we have the Martingale property.

Furthermore, one can obtain from direct computation that lim γ→∞E[WsWs 0] = min(s, s0) lim γ→∞E[WsWs 0] = lim γ→∞E[WsWs 0] = 0 Consider now M_s≡ Ws_√+Ws 2 . SinceE[M 2

s] = s, by the L´evy characterization of Brownian motion,

Ms is then a Brownian motion. We can use exactly the same argument to prove that iWs√−W₂ s is also a Brownian motion. Eventually, we can write

lim γ→∞dWs= dB_s1+ idB_s2 √ 2 , lim γ→∞dWs= dB1 s_√− idBs2 2 ,

where dB_s1, dB_s2 two independent Brownians real-valued Brownians.

In what follows, we will assume that the large γ limit has already been taken i.e that we already have dW_s= dB1s_√+idBs2

2 . We end up with

dHs= ν(σ+dWs+ σ−dWs). (1.10)

We then see that in the slow-time limit, the effective dynamics can be interpreted as a

spin-1

2 undergoing random jumps between its two-levels. We now proceed to study the stationary

distribution towards which the system converges:

1.3.2 Stationary distribution

We are interested in characterizing the long-time stationary state of the density matrix of the system. Since it is a random variable, a complete description will either mean to obtain the probability distribution of the density matrix or to compute all momenta. We choose to do the latter in following. The evolution equation of ρs is given by (setting ν = 1):

dρs = − 1 2[dHs, [dHs, ρs]] − i[dHs, ρt] = −1 2([σ +_{, [σ}−_{, ρ} s]] + [σ−, [σ+, ρs])dt − i[σ+, ρs]dWs− i[σ−, ρs]d ¯Ws,

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where to go from the first line to the second line we applied the Itˆo rules: dW_sdWs= dWsdWs=

dt, dW_s2 = dW2_s = 0. In terms of the spin-component ~Ss, (ρs = 1₂(I + ~Ss.~σ)), the evolution equations are given by

dS_sx= −S_sxds −√2S_szdB_s2 dS_sy = −S_syds −√2S_szdB_s1

dS_sz= −2S_szds +√2(S_sxdB_s2+ S_sydB_s1)

As in the finite γ case, the mean,E(~S_s) decays towards ~0 so that the mean density matrix in the stationary state is simply given by ¯ρ∞= 1₂I. More interesting things happen if we look at higher

momenta.

First, let us see what happens to second order correlations functions i.e, h(S_sx)2i,h(Sy s)2i, h(Sz

s)2i,hSsxSsyi,hSsySszi,hSsxSszi, where hi denotes average over different stochastic realizations of the process. Using Itˆo calculus rules it is easy to get

d dsh(S x s)2i = −2h(Sxs)2i + 2h(Ssz)2i, d dsh(S y s)2i = −2h(Ssy)2i + 2h(Ssz)2i, d dsh(S z s)2i = −4h(Ssz)2i + 2h(Ssx)2i + 2h(Ssy)2i, d dshS x sSsyi = −2hSsxSsyi, d dshS y sSszi = −5hSsySszi, d dshS x sSszi = −5hSsxSszi, and we see that in the stationary state ;

h(S_∞x )2i = h(S_∞y )2i = h(S_∞z )2i = C/3, hS_∞xS_∞y i = hS_∞y S_∞z i = hS_∞xS_∞z i = 0,

where C is the conserved quantity h(S_sx)2+ (S_sy)2+ (Sz_s)2i. We already learn an important fact: Even though in mean, the components of ~Sscollapse to 0 in the long-time limit, it is not true for the fluctuations around the center of the Bloch sphere. This fact is illustrated on fig.1.2 where we have plotted the time evolution for one realization of the noise of the real and imaginary component of S_sx. We took as initial condition a spin entirely polarized along the z-axis, i.e S₀x = S₀y = 0 and S₀z = 1. Even though the x-axis component is zero in the initial state, the amplitude of the fluctuation around this value are finite at all time.

Let us see now look at higher moments or more generally, at the expectation of any function hF (~Ss)i of the coordinates. The time evolution of such an object is governed by a dual Fokker-Planck operator D;

∂shF (~Ss)i = DhF ( ~Ss)i. (1.11) It can be shown by direct computation that D = −(L2_X + L2_Y) with LX = i(Sz∂Sy − Sy∂_Sz),

LY = i(Sx∂Sz − Sz∂_Sx) the generators of rotation around the X and Y axis in the 3D space.

Or alternatively D = L2 − L2

Z with L2 = L2X + L2Y + L2Z and LZ = i(Sy∂Sx + Sx∂_Sy). Let

us restrain ourselves on polynomial functions of the {S_sx, S_sy, S_sz} for simplification i.e: F (~Ss) =

P∞

k=0axk(Ssx)k+ a y

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Figure 1.2: Illustration of the stochastic dynamics of the real (in blue) and imaginary (in red)

component along the x-axis component of a spin 1/2 undergoing random unitary random jumps for a single realization of the noise. The long-time behaviour is characterized by finite amplitude fluctuations. coordinates ~xs = ~ Ss R with R = q (Sx₀)2_{+ (S}y 0)2+ (S0z)2. E[F (~Ss)] = ∞ X k=0 Rk(ax_kxk_s + a_kyy_sk+ az_kz_sk) ≡ ∞ X k=0 Rkfk,s(θs, φs),

with θ, φ the canonical spherical coordinates ; x = sin θ cos φ, y = sin θ sin φ, z = cos θ. Now a direct application of (1.11) gives

Rk= constant ∂sfk,s = Dfk,s

to solve the equation on the f ’s we can decompose it in the spherical harmonic basis Y_lm(θ, φ) indexed in the canonical way by the eigenvalues l and m. (L2Y_lm = l(l + 1), L_zY_lm = mY_lm) : fk,s =P∞`=0

P`

m=−`fl(k,t)m Ylm(θ, φ). The general solution is then given by ;

E[F (~Ss)] = ∞ X k=0 Rk ∞ X `=0 ` X m=−` f_l(k,s=0)m e−s(l(l+1)−m2)Y_lm(θ, φ) (1.12)

All the modes decay exponentially fast except those for which m = 0 and l = 0 i.e for t → ∞ we have ; E∞[F ( ~S∞)] = ∞ X k=0 Rkf_0(k,t=0)0 Y₀0(θφ),

where E∞[•] denotes the average in the stationary state. We see that even in the long-time

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The fact that only the modes of 0 angular momenta survives tell us that in the long-time limit only the modes invariant under rotation survive, i.e the dynamics is “noisy” enough so as to not favor any particular direction. We will see in the next subsection how this fact can be understood from the point of view of group theory, making use of the invariants of SU (2).

1.3.3 Another point-of-view: Invariant measure of SU (2)

Let us derive again the previous result from another point-of-view that will make the invariant properties of the stationary state visible. Recall the infinitesimal Hamiltonian

dHs= ν(σ+dWs+ σ−dWs).

One key remark is that the terms weighting the noises in the infinitesimal generator (1.10) form a system of simple root generators of the lie algebra su(2). Being in the stationary state means that the distribution is invariant under the action of dH_s. Because of the presence of the simple roots, this will be equivalent to invariance under the unitary group i.e the iterated products

e−idHsn_{· · · e}−idHs2_e−idHs1_,

will cover densely SU (2) for any collection of time increments dt_k. Averaging the flow over different realizations in the long-time limit must then be equivalent to averaging over all the possible flows overs the unitary group. This is what is meant to be maximally noisy.

If we define the generating function of moments of elements of ρ, Z(A) ≡ E∞[etr(Aρ)], the

previous statement translates into the following important result: Z(A) =

Z

SU (2)

dη(V )etrAV†ρ0V

where η is the Haar measure and ρ0 ≡ ρ(t = 0). This is the Harish-Chandra-Itzikson-Zuber

integral [58] on the special unitary group SU (2) (normalized to unit volume). For an exact proof of this claim, we refer to [6].

The exact expression for a general group SU (L) is given by

Z(A) = ( L−1 Y k=1 k!) det eaigjL i,j=1 ∆(a)∆(g) ,

(ai)Li=1 and (gi)Li=1 are the spectrum of A and G0 respectively and ∆(a) (resp. ∆(g)) are the

Vandermonde determinants of A (resp. G₀), ∆(a) =Q

i<j(ai− aj).

Another expression is in terms of the irreducible representations of the group SU (L). These can be indexed by Young tableaux Y . The generating function Z(A) can be expanded in charac-ters of the unitary group

Z(A) =X Y 1 m(Y )! σY dY χY(A)χY(ρ0) (1.13)

where m(Y ) is the number of boxes in Y , σY the dimension of the representation of the permuta-tion group S_{m(Y )}associated to Y and dY, χY(A) are respectively the dimension and the character of the representation of SU (L) indexed by Y . This sum is graded because the character χ_Y(A) are polynomials in A of degree m(Y ). The first few terms are

Z(A) = 1 + 1 L (A) (ρ0) + 1 2( 2 L(L + 1) (A) (ρ0) + 2 L(L − 1) (A) (ρ0)) + .... (1.14) The characters of the linear group up to m(Y ) = 3.

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Y χY (A) dY σY tr(A) L 1 1 2(trA) 2_{+ trA}2₎ 1 2L(L + 1) 1 1 2(trA) 2_{− trA}2₎ 1 2L(L − 1) 1 1 6(trA) 3_{+ 2trA}3_{+ 3trAtrA}2₎ 1 6L(L + 1)(L + 2) 1 1 3(trA) 3_{− trA}3 1 3L(L + 1)(L − 1) 2 1 6(trA) 3_{+ 2trA}3_{− 3trAtrA}2₎ 1 6L(L − 1)(L − 2) 1

We now may rederive our previous results from this new perspective. For instance, by solv-ing the equation of stationarity explicitly, we had that h(S_∞x )2i = C/3. Let’s introduce the parametrization A = a11 a12 a21 a22 ! . Then h(S_∞x)2i = (1 2∂ 2 a12+ 1 2∂ 2 a21+ ∂a12∂a21)F (A)|A=0 = 1 3(trρ 2₋1 2) = C 3.

And the two approaches are indeed consistent with each other. This last approach will appear to be a more powerful and systematic one, especially when it comes to dealing with extended quantum systems as we will see in section 4.2.

1.4 Sum up

We end this part by summing the main ideas we presented. We have studied the dynamical and long-time stationary behaviour of a spin-1₂ undergoing dissipative dynamics. The dissipation arised from the interaction with an infinite set of harmonic oscillators. It turned out that upon some assumption on the spectral properties of the bath, there was an effective description of the effect of the bath on the system in terms of stochastic markovian processes that we called quantum noises.

We then looked at a specific example with a quantum noise whose effect was to induce a random dephasing on the z-axis while the internal dynamics was a Rabi oscillation around the x-axis. Even though decoherence was at play for the mean behaviour of the system, there are still fluctuations of quantum nature around the long-time steady state.

We additionally showed that there was an effective long-times dynamics in terms of slow-modes that can be interpreted as a spin-1₂ undergoing random jumps between up and down levels. The full steady state distribution was characterized by a systematic derivation of all momenta. Interestingly, a simple ergodic assumption stating that at long-times, the stochastic Hamiltonian generates all unitaries of SU (2) was enough to recover the right results and this independently of the details of the dynamics.

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Chapter 2

Relaxation and transport in

integrable and non-integrable

many-body quantum systems: an

artless discussion

As explained in the introduction, one of the main goal of this thesis is to study the behavior of extended many-body quantum systems when put into contact with an external environment. In this section, we briefly discuss the heuristic behavior that one expects when dealing with isolated and/or integrable many-body quantum systems versus their open and/or non-integrable counterparts. Of course, this is an extremely wide and rich topic and we will only restrict ourselves to a tiny set of topics and results. This is in no means an expert’s presentation and we apologise in advance for the omissions of major topics. We will focus on two topics that will be most relevant for our purposes: relaxation and transport. When dealing with these questions, an important distinction is to be made between systems that are said to be integrable versus non-integrable. The definition of the notion of quantum integrability is a field of research in itself but for us we will be contempt by saying that a system will be integrable if there exists a systematic procedure to construct a basis of eigenfunctions which solves the dynamics at least formally. This notion of exact solvability is in general tightly associated with the existence of an infinite number of local or quasi-local conserved charges for the system in consideration. Another possible definition concerns the algebraic structure of the theory. It is commonly admitted that a model will be integrable if the dynamics is reducible to a two-body interaction, i.e it satisfies a Yang-Baxter equation.

A paradigmatic example of integrable model that will also be of particular interest for us is given by spin chains. Historically, introduction of quantum spin chain models is attributed to Heisenberg as a simplified minimal model of quantum many-body physics aiming at capturing the properties of magnetism in solide-state physics. Although the model is a very rough simplification of the physics that actually take place in many-body systems and the derivation of the model from minimal principles is a problem of its own, it is believed that it captures important qualitative features of real magnetic materials such as the existence of phase-transitions, transport properties, etc. Exactly solvable quantum many-body systems are not only restricted to spin chains and one could non exhaustively also include the Hubbard model, Tomonaga-Luttinger liquids, the Lieb-Liniger model, Sinh Gordon model, etc. In the recent decades, such model have been realized experimentally in quasi-isolated setups such as cold atoms, optical lattices or cavity experiments. At the other side of the spectrum where our work will sit as well lie the non-integrable models. The name speaks for itself: This comprises all the models that are not considered to be integrable. More precisely, when we say non-integrable models, we have in mind the following situations: