tems in presence of both driving and dissipation mechanisms represent natural platforms to understand and explore such dynamical phases. A well know example is provided by exciton-polariton condensates where superfluidity has an order parameter oscillating in time [ 30 , 31 , 70 , 122 ]. Yet the oscillating condensate is successfully described by semi- classical theories such as **driven**-**dissipative** Gross-Pitaevski equations that are valid in the regime of weak interactions. More recently the attention has shifted toward strongly correlated **quantum** lattice models with drive and dissipation, where several works have re- vealed the existence of limit cycles, i.e. non-stationary solutions of the **quantum** dynamics for a macroscopic order parameter, at least at the mean field level [ 16 , 17 , 20 , 93 , 187 – 189 ]. In this chapter we focus on a paradigmatic model of **driven**-**dissipative** interacting bosons on a lattice, which is directly relevant for the upcoming generation of circuit QED arrays experiments [ 61 , 62 , 86 ]. We argue that a dynamical susceptibility of such an open **quantum** **many** **body** system, which in thermal equilibrium is finite and small since non-zero frequency modes are typically damped by interactions, can display a genuine singularity at finite frequency, as a result of strong interactions and non-equilibrium effects. The critical frequency is non-trivial and set by a competition of interactions with drive and dissipation. Eventually, the system undergoes a dynamical phase transition where the order parameter emerges with a finite oscillation frequency and in the broken symmetry phase oscillates in time without damping, thus breaking the continuous time-translational symmetry. This stationary-state instability is controlled by both **dissipative** and coherent couplings, in particular by the ratio between hopping and local interaction, thus providing the strongly correlated analogue of weak coupling non-equilibrium bosons condensation.

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In Chapter I we introduce the class of Spin-Boson models and focus in particular on the Rabi model, the ohmic spinboson model, and their lattice versions. The Rabi model considers a two-level system coupled to a quantized harmonic oscillator and de- scribes the simplest interaction between matter and light. Its lattice version describing a set of interacting light-matter **systems** opens the door to **many**-**body** physics with light. The ohmic spinboson model was first introduced to describe **dissipative** effects on a two-level system. In this description dissipation is modelled by a bath of quantized harmonic oscillators, and **many**-**body** effects of the bath notably induce a dissipa- tive **quantum** phase transition at large spin-bath coupling (see “**Many**-**Body** **Quantum** Electrodynamics Networks: Non-Equilibrium Condensed Matter Physics with Light”). In Chapter II, we derive a Stochastic Schr¨odinger Equation (SSE) describing the time evolution of the spin-reduced density matrix for Spin-boson problems. We test this framework and recover known results for the dynamics of the Rabi model (see “**Quantum** dynamics of the **driven** and **dissipative** Rabi model”) and the ohmic spinbo- son model. We also compare the SSE approach to other known methods and present the limitations related to the SSE.

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Among our models, the most tractable is certainly the case of non-interacting fermions (strictly speaking, fermionic lattice **systems** are not covered here, but this could easily be remedied). It is therefore surprising that the phenomenon of ’localization in energy’ has barely been rigorously studied in the absence of interaction, i.e. for the one-particle case. In [12], localization was proved for periodically kicked operators. The authors of [37] considered the disordered Anderson model with a local time-periodic perturbation and proved the stability of localization. In [6], the same is achieved for a quasi-periodic perturbation, which can be viewed as the case of multiple frequency dimensions. Very recently, [15] considered the problem from a very similar point of view as in the present paper: [15] proves stability of Anderson localization with a periodic driving term, provided that the driving frequency is not too small. High-frequency asymptotics (instead of strict localization) in periodically **driven** **quantum** **systems** have been investigated in [16, 34, 39] by techniques similar to ours, but not applicable to the **many**-**body** problem.

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mentally observed in a nearly integrable one-dimensional Bose gas 10 .
In this paper, we establish some general properties of dynamics of periodically **driven** **many**-**body** **systems** (Floquet **systems**). Periodic driving in **quantum** **systems** has recently attracted much theoretical and experimental attention, because, amongst **many** applications, it pro- vides a tool for inducing effective magnetic fields, and for modifying topological properties of Bloch bands 11 – 14 . Indeed, since periodic driving is naturally realized in cold atomic **systems** by applying electromagnetic fields, topologically non-trivial Bloch bands (Floquet topologi- cal insulators) in non-interacting **systems** have been ob- served experimentally 15 – 17 . However, since periodic driv- ing breaks energy conservation, **driven** ergodic (**many**- **body**) **systems** are expected to heat up, eventually evolv- ing into a featureless, infinite-temperature state 18 – 21 . Thus, **many**-**body** effects are expected to generally make such Floquet **systems** unstable. Below, we derive gen- eral bounds for energy absorption rates in periodically **driven** **many**-**body** **systems**, which can be applied for in- stance to understand the lifetimes of Floquet topological insulators.

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An environment can also serve for bath (or dissipation) engineering. We have illustrated this point by studying Kondo physics of light in Josephson circuits in the microwave limit, emergent Kondo physics in **quantum** RC circuits, multi-channel Kondo fixed points in topological Josephson junctions with Majorana fermions, and hybrid **systems** comprising a cQED environment coupled to a biased mesoscopic system. A **dissipative** environment engenders fluctuations and disorder in time; we have analyzed this aspect by building connections between Kondo and **Quantum** Brownian motion physics. These **quantum** impurity **systems** can also be useful as sensors of **quantum** **many**-**body** phenomena. We have illustrated this concept in ladder **systems** where the Rabi dynamics of a spin-1 /2 can probe the Mott-Superfluid transition. We have also studied the relaxation of particle densities and currents, in the language of spin-boson **systems**. Studying the fast dynamics in **quantum** materials has also attracted some attention recently, and our work could stimulate new frontiers.

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Another implication of our results is that MBL does not rely on global conservation laws. Further, MBL phase is robust under sufficiently weak periodic driving, and there exists a finite driving threshold above which transport is restored, and the system ultimately delo- calizes. This may serve as an experimental signature of the **many**-**body** localization. An interesting subject for future research, relevant for experiments in disordered solid-state **systems**, is to study periodically **driven** MBL system weakly coupled to a thermal bath (we note that spectral properties of a static MBL system coupled to a bath were recently considered in Refs. [ 45 , 46 ]).

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is sufficiently large. For the specific case of the Ising model, the operators γ k † γ k have
been found to respect these conditions in [ 187 ]. The relaxation of different integrable models to the GGE has been tested explicitly in **many** different models as: Luttinger Liquids [ 189 , 190 ], free bosonic theories [ 191 , 192 ], hardcore bosons [ 187 ], Lieb-Lininger model [ 193 , 194 ], spin models [ 195 , 196 , 197 , 198 ] and Hubbard-like models [ 199 , 200 ]. The previous discussion is more an argument than a real theorem, even if thermalization has been found in some classes of models, a general theorem that states how different **systems** thermalize is still missing. Moreover, the exploration of long-time dynamics of **many**-**body** interacting **quantum** **systems** is, from the numerical point of view, extremely challenging. The possibility to simulate different Hamiltonians in experimental cold atomic gases plays then a key role in the exploration of the thermalization problem in such **systems**. As we said in Sec. 1.3 , cold atomic gases can be used to engineer different Hamiltonians and to drive them out of equilibrium. Moreover it is also possible to constraint **systems** in reduced dimensionality, as one or two dimensional geometries. Integrability is in fact an extremely delicate characteristic of the system, a fine tuning of all parameters of the Hamiltonian is needed to observe it. Extra terms in the Hamiltonian drive the system away from integrability and produce drastically different results. In classical physics the role of perturbations in an integrable theory is well understood using the KAM theory [ 3 ]. At the **quantum** level, however, this is not the case. Even small perturbations produce pre-thermalisation plateau [ 201 , 148 , 202 ] described by the GGE before that real thermalization occurs. Anyway, it is still not known how to define “small” and “large” in the context of perturbation around the integrable case.

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Also in the-two dimensional problem discussed in the present paper, the correlations among the particles do not affect the macroscopic dynamics of the system (this explains why the coupling constant in front of the nonlinearity in (1.6) is just the integral of the potential). On the contrary, the correlation structure would be very important in the study of two-dimensional **systems** in the Gross-Pitaevskii scaling limit (where the scattering length of the interaction potential is exponentially small in the number of particles). In [23], Lieb, Seiringer, and Yngvason proved that, in this limit, the ground state energy per particle can be obtained by the minimization of the so called Gross-Pitaevskii energy functional. In [22], it was then shown by Lieb and Seiringer that the ground state vector, in the Gross-Pitaevskii limit, exhibits complete Bose Einstein condensation. In order to prove these two results, it was very important to identify the short scale correlation structure in the ground state wave function (the energy of factorized wave functions, with absolutely no correlations, is too large by a factor of N ). Unfortunately, we are not yet able to study the dynamics of Bose-Einstein condensates in the two-dimensional Gross- Pitaevskii scaling limit; nevertheless, since the infinite hierarchy which is expected to describe the time-evolution of the limiting densities {γ ∞,t (k) } k≥1 is still given by (2.9) (with a different coupling

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(c † i c i+1 + hc) (1)
with n i = c † i c i , where the on-site random field is drawn
from a box distribution h i ∈ [−W, W ] and we set J = 1/2
as our unit of energy, to map exactly on the XXZ spin chain after Jordan-Wigner. Notice that FEs are a rather general and flexible approach which can be applied to other **quantum** disordered problems (see discussion for future applications). The basic idea of the FE approach is to iteratively diagonalize the Hamiltonian of the sys- tem by a CUT U (l) parametrized by a scale l and gen- erated by an anti-Hermitian operator η(l), such that U (l) = T l exp R η(l)dl . The flow of any operator O(l)

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Although we elaborated on the basic principles underlying the construction, we mainly concentrated on analysing the stochastic Heisenberg XXZ spin chain. Of course **many** ques- tions remain to be studied –transport, finite size **systems** with or without boundary injection, boundary effects, robustness to perturbations, etc (see ref.[28]). We dealt with the effective theory at large friction –but studying the sub-leading contributions could also be interest- ing as they generate a non-linear diffusion constant [29]. In this limit the effective **quantum** stochastic dynamics that we identified are natural quantizations of the fluctuating discrete hydrodynamic equations. They could now be directly taken as starting points for modelling **quantum** diffusive transports and their fluctuations, but the detour we took through the large friction limit justified their precise structures –and part of them, say the dressing of the hop- ping operators in the case of the XXZ spin chain, would had been difficult to guess without this detour. It is interesting to notice that stochasticity within conformal field theory has recently been considered in [13].

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2 The MCTDHF ansatz
As remarked above convenient approximations for a system of N interacting Fermions should not rely on a mean field hypothesis. N may be large, but a N → ∞ mean field limit in the weak coupling scaling does in general not correspond to the physics. The TDHF method also has the important disad- vantage that by definition it cannot catch ”correlations”, a crucial concept for N particle **quantum** **systems** (see e.g. [26, 27]). However, by approximating Ψ by linear combinations of Slater determinants, an approximation hierarchy called Multiconfiguration time dependent Hartree Fock (MCTDHF) is obtained that allows for very precise and numerically tractable models of correlated few **body** **systems** (see e.g. [11, 43]). The MCTDHF ansatz involves a finite number

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MATHIEU LEWIN, PHAN TH ` ANH NAM, AND NICOLAS ROUGERIE
Abstract. We consider a 2D **quantum** system of N bosons in a trapping potential |x| s ,
interacting via a pair potential of the form N 2β−1 w(N β x). We show that for all 0 < β <
(s + 1)/(s + 2), the leading order behavior of ground states of the **many**-**body** system is described in the large N limit by the corresponding cubic nonlinear Schr¨ odinger energy functional. Our result covers the focusing case (w < 0) where even the stability of the **many**-**body** system is not obvious. This answers an open question mentioned by X. Chen and J. Holmer for harmonic traps (s = 2). Together with the BBGKY hierarchy approach used by these authors, our result implies the convergence of the **many**-**body** **quantum** dynamics to the focusing NLS equation with harmonic trap for all 0 < β < 3/4.

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PACS numbers: 05.30.Rt, 72.15.Rn, 72.20.Ee
I. INTRODUCTION
It is now almost mathematically proven that **many**- **body** localization, i.e., the absence of long-range trans- port in a thermodynamic **many**-**body** system, occurs in certain one-dimensional **quantum** lattice models at any energy density if sufficiently strong quenched disorder is present [1]. In this case, **many**-**body** localization (MBL) comes along with a complete set of conserved quasi- local quantities [2–4]. However, it remains less clear whether the originally predicted localization transition at finite temperature [5, 6] exists as a genuine dynam- ical phase transition defining a sharp **many**-**body** mo- bility edge in energy density. Even though several nu- merical investigations in small one-dimensional (1D) sys- tems have reported such mobility edges [7–9], studies in larger **systems** did not find similar evidence [10, 11] and, moreover, linked-cluster analysis [12] of the numer- ical data hint that the extent of the localized phase has been vastly overestimated. Furthermore, recent theoret- ical considerations [4, 13–16] have raised doubts about non-perturbative effects which might reduce the putative transition to a crossover. A related open issue concerns the **many**-**body** analog of Mott’s argument, which forbids the coexistence of localized and delocalized states at the same energy in single-particle problems.

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Understanding the inter-play of quenched disorder, in- teractions and **quantum** fluctuations has been a central theme of hard condensed matter for **many** years. Activ- ity on this topic boomed recently, in particular after that Basko, Aleiner and Altshuler (BAA) showed by using the self-consistent Born approximation that interacting and isolated **quantum** **systems** can fail to thermalize due to Anderson localization in Fock space [1]. This phe- nomenon, called **Many** **Body** Localization (MBL), repre- sents a new kind of ergodicity breaking transition, which is purely dynamical—indeed it can take place even at infi- nite temperature by increasing the amount of disorder— and which results from the interplay of disorder, interac- tions and **quantum** fluctuations [2, 3]. One of the most surprising results is that even the delocalized phase is unusual in a wide range of parameters already before the MBL transition. In fact both in numerical simulations [4– 8] and in experiments [9–11] it was found that transport appears to be sub-diffusive and that out-of-equilibrium relaxation toward thermal equilibrium is slow and power- law-like with exponents that gradually approach zero at the transition. Several works explained this behavior in terms of Griffiths regions, i.e., rare inclusions of the lo- calised phase which impede transport and relaxation [12– 15]. However, also quasi-periodic 1d and disordered 2d **systems**, in which Griffiths effects should be absent or milder [14, 15], do display analogous unusual transport and relaxation [10, 11, 16, 17]. It is therefore important to look for other explanations that might hold beyond the particular case of 1d disordered **systems**. Moreover, it is interesting to complement the real space Griffiths perspective to one directly based on **quantum** dynamics in Fock-space. These are the aims of our work.

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Laboratoire de Simulation Atomistique (L Sim), SP2M, INAC, CEA-UJF 17 Av. des Martyrs, 38054 Grenoble, France.
Even though organic **systems** offer an extremely large flexibility in terms of available molecules and architec- tures, the rather low **quantum** efficiency (a few per- cent), as compared to standard silicon-based solar cells, and the problems of stability under UV radiations, are serious limitations that need to be addressed. It has been shown however that significant gains [5,6] can be achieved by tuning the absorption spectrum, band gap and band offsets of the active molecules/polymers in standard donor/acceptor hetero-structure cells [7–9]. Using computer **quantum** simulations to predict such spectroscopic parameters for actual size molecules or polymers would therefore be of much interest.

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In the past, atomic **systems**, optical **systems** or trapped ions **systems** have been investigated as a basis to implement such **quantum** machines. However, since the first demonstration of a **quantum** superconducting bit, more than twenty years ago [5, 6], the domain of mesoscopic artificial atoms built out of superconducting **quantum** circuits became one of the most promising physical system to implement **quantum** machines. Such circuits can be used to create resonators which store individual microwave photons as well as superconducting **quantum** bits. All of these circuit elements are intrinsically **quantum** mechanical, with quantized energy levels with spacing much greater than the energy associated with the temperature (for cryogenic temperatures which are achievable in the laboratory) and in order to properly predict their properties, both the current and voltages should be represented by non-commuting operators. The coherence time of these circuits have improved rapidly and steadily over a few orders of magnitude, now reaching a few milliseconds for the best **systems** [7, 8, 9, 10, 11]. Contrary to the atomic physics based **systems**, such superconducting circuits provide greater flexibility in the range of exploitable parameters and in the complexity of the Hamiltonian they can implement. This flexibility let us explore a whole new realm of operating regimes. Moreover, the microwave signals, contrary to the optical ones, are very well controlled with commercial electronic equipment developed for radio-frequency engineering. The nanofabrication processes required to fabricate such **quantum** superconducting circuits are more and more controlled, so that it becomes possible to fabricate multiple complex **systems** in parallel, leading to the hope that it will soon be possible to scale such **quantum** processors with large numbers of qubits and leading to the recent development of **many** commercial interests (from Google, IBM, Intel or Rigetti Computing).

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The iterative methods to diagonalize matrices and **many**-**body** Hamiltonians can be reformulated as flows of Hamiltonians towards diagonalization **driven** by unitary transformations that preserve the spectrum. After a comparative overview of the various types of discrete flows (Jacobi, QR- algorithm) and differential flows (Toda, Wegner, White) that have been introduced in the past, we focus on the random XXZ chain with random fields in order to determine the best closed flow within a given subspace of running Hamiltonians. For the special case of the free-fermion random XX chain with random fields, the flow coincides with the Toda differential flow for tridiagonal matrices which is related to the classical integrable Toda chain and which can be seen as the continuous analog of the discrete QR-algorithm. For the random XXZ chain with random fields that displays a **Many**- **Body**-Localization transition, the present differential flow should be an interesting alternative to compare with the discrete flow that has been proposed recently to study the **Many**-**Body**-Localization properties in a model of interacting fermions (L. Rademaker and M. Ortuno, Phys. Rev. Lett. 116, 010404 (2016)).

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The exponential complexity of **quantum** **many**-**body** **systems** is at the heart of **many** remarkable phenomena. Advances in correlated materials and recently developed synthetic **quantum** **systems** – e.g. atomic gases [ 1 ], trapped ions [ 2 ], and nanoelectronic devices [ 3 – 6 ] – have allowed **many**-**body** states to be characterized and controlled with unprecedented precision. The latest of these **systems**, **quantum** computing chips, are highly engineered out-of- equilibrium **many**-**body** **systems**, where the interacting dynamics performs computational tasks [ 7 ]. However, our understanding of these **many**-**body** **systems** is limited by their intrinsic complexity. While uncontrolled approxi- mations can give insight into possible behaviors, there is a growing effort to develop controlled, high-precision methods [ 8 ], especially ones that apply far from equilib- rium [ 9 – 11 ]. These allow us to make quantitative pre- dictions about the physics of **many**-**body** **systems** and to uncover qualitatively new effects at strong coupling.

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These lecture notes are addressed to PhD student and/or researchers who want a general overview of microscopic approaches based on mean-field and applied to nuclear dynamics. Our goal is to provide a good description of low energy heavy-ion collisions. We present both formal aspects and practical applications of the time-dependent Hartree-Fock (TDHF) theory. The TDHF approach gives a mean field dynamics of the system under the assumption that particles evolve independently in their self-consistent average field. As an example, we study the fusion of both spherical and deformed nuclei with TDHF. We also focus on nucleon transfer which may occur between nuclei below the barrier. These studies allow us to specify the range of applications of TDHF in one hand, and, on the other hand, its intrinsic limitations: absence of tunneling below the Coulomb barrier, missing **dissipative** effects and/or **quantum** fluctuations. Time-dependent mean-field theories should be improved to properly account for these effects. Several approaches, generically named ”beyond TDHF” are presented which account for instance for pairing and/or direct nucleon-nucleon collisions. Finally we discuss recent progresses in exact ab-initio methods based on the stochastic mean-field concept.

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in the context of MBPT by employing NCSM reference states [ 22 ]. A second option of present interest is to exploit the concept of spontaneous symmetry breaking, breaking e.g.
U (1) symmetry associated to the particle number conservation in order to capture the
superfluid character of singly-open-shell nuclei. Doubly-open-shell nuclei can be addressed as well via the breaking of SU (2) symmetry associated with the angular-momentum conservation, allowing for nuclei to deform. Breaking U (1) symmetry allows one to deal with Cooper pair’s instability and capture the dominant effect of the infrared source of non- perturbativeness already at the level of the reference state. Via the use of a more general Bogoliubov vacuum, the degeneracy of an open-shell Slater determinant with respect to particle-hole excitations is lifted and commuted into a degeneracy with respect to symmetry transformations of the symmetry-broken group (U (1) in this case). As a consequence, the ill-defined (i.e. singular) expansion of exact quantities around a Slater determinant is replaced by a well-behaved one around a Bogoliubov state. Symmetry breaking has been used for decades by the EDF community [ 41 – 44 ], i.e. at the mean field level. In the last ten years, novel ab initio **many**-**body** methods have been developed on top of symmetry-broken reference states, e.g. the Gorkov SCGF (GSCGF) framework [ 45 – 47 ], the Bogoliubov CC (BCC) formalism [ 48 , 49 ], the Bogoliubov MBPT (BMBPT) [ 50 , 51 ] and even the Bogoliubov CI (BCI) [ 52 , 53 ], the last one not being an expansion method. A difficulty encountered by these methods is that the symmetry breaking cannot actually occur in finite **quantum** **systems** and results of **many**-**body** calculations carry a contamination associated with contributions of non-targeted particle numbers, i.e. of several irreducible representations of the broken symmetry group.

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