Closed quantum many-body systems out of equilibrium pose several long-standing problems in physics. Recent years have seen a tremendous progress in approaching these questions, not least due to experiments with cold atoms and trapped ions in instances of quantum simulations. This article provides an overview on the progress in understanding dynamical equilibration and thermalisation of closed quantum many-body systems out of equilibrium due to quenches, ramps and periodic driving. It also addresses topics such as the eigenstate thermalisation hypothesis, typicality, transport, many-body localisation, universality near phase transitions, and prospects for quantum simulations.Comment: 7 pages, review and perspectives article, updated to journal version after embarg

The dynamics of quantum phase transitions pose one of the most challenging problems in modern many-body physics. Here, we study a prototypical example in a clean and well-controlled ultracold atom setup by observing the emergence of coherence when crossing the Mott insulator to superfluid quantum phase transition. In the 1D Bose-Hubbard model, we find perfect agreement between experimental observations and numerical simulations for the resulting coherence length. We, thereby, perform a largely certified analog quantum simulation of this strongly correlated system reaching beyond the regime of free quasiparticles. Experimentally, we additionally explore the emergence of coherence in higher dimensions, where no classical simulations are available, as well as for negative temperatures. For intermediate quench velocities, we observe a power-law behavior of the coherence length, reminiscent of the Kibble-Zurek mechanism. However, we find nonuniversal exponents that cannot be captured by this mechanism or any other known model. ultracold atoms | optical lattice | Mott insulator | nonequilibrium dynamics | quantum simulation P hase transitions are ubiquitous but rather intricate phenomena, and it took until the late 19th century until a first theory of classical phase transitions was established. Quantum phase transitions (QPTs) are marked by sudden drastic changes in the nature of the ground state on varying a parameter of the Hamiltonian. They constitute one of the most intriguing frontiers of modern quantum many-body and condensed matter physics (1-4). Although it is typically possible to adiabatically follow the slowly changing ground state in a gapped phase, where the lowest excitation is separated from the ground state by a finite energy, these spectral gaps usually close at a QPT. Because the correlation length simultaneously diverges, adiabaticity is bound to break down, and several important questions emerge. How does a state dynamically evolve across the QPT (i.e., how does the transition literally happen?)? To what extent can the static ground state of a gapless phase be prepared in a realistic finitetime experiment? When entering a critical phase associated with an infinite correlation length, such as superfluid or ferromagnetic order, at what rate and by what mechanism will these correlations build up? Despite the fundamental importance of these questions, satisfactory answers have not been identified so far. Although the intrinsic complexity of the underlying nonintegrable models hinders numerical studies in most cases, the progress in the field of ultracold atoms now enables quantitative experiments in clean, well-isolated, and highly controllable systems.Here, we study the quantitative dynamics of a transition into a gapless, superfluid phase in the regime of short and intermediate quench times, finding complex behavior outside the scope of any current theoretical model. As a prototypical many-body system with a QPT, we use the transition from a Mott insulator to a superfluid in the Bose-Hubbard model (...

The phenomenon of many-body localisation received a lot of attention recently, both for its implications in condensed-matter physics of allowing systems to be an insulator even at non-zero temperature as well as in the context of the foundations of quantum statistical mechanics, providing examples of systems showing the absence of thermalisation following out-of-equilibrium dynamics. In this work, we establish a novel link between dynamical properties -a vanishing group velocity and the absence of transport -with entanglement properties of individual eigenvectors. Using Lieb-Robinson bounds and filter functions, we prove rigorously under simple assumptions on the spectrum that if a system shows strong dynamical localisation, all of its manybody eigenvectors have clustering correlations. In one dimension this implies directly an entanglement area law, hence the eigenvectors can be approximated by matrix-product states. We also show this statement for parts of the spectrum, allowing for the existence of a mobility edge above which transport is possible.The concept of disorder induced localisation has been introduced in the seminal work by Anderson [1] who captured the mechanism responsible for the absence of diffusion of waves in disordered media. This mechanism is specifically well understood in the single-particle case, where one can show that in the presence of a suitable random potential, all eigenfunctions are exponentially localised [1]. In addition to this spectral characterisation of localisation there is a notion of dynamical localisation, which requires that the transition amplitudes between lattice sites decay exponentially [2,3].Naturally, there is a great interest in extending these results to the many-body setting [4,5]. In the case of integrable systems that can be mapped to free fermions, such as the XY chain, results on single particle localisation can be applied directly [6,7]. A far more intricate situation arises in interacting systems. Such many-body localisation [8,9] has received an enormous attention recently. In terms of condensed-matter physics, this phenomenon allows for systems to remain an insulator even at non-zero temperature [10], in principle even at infinite temperature [11]. In the foundations of statistical mechanics, such many-body localised systems provide examples of systems that fail to thermalise. When pushed out of equilibrium, signatures of the initial condition will locally be measurable even after long times, in contradiction to what one might expect from quantum statistical mechanics [4,12,13].Despite great efforts to approach the phenomenon of manybody localisation, many aspects are not fully understood and a comprehensive definition is still lacking. Similar to the case of single-particle Anderson localisation, there are two complementary approaches to capture the phenomenon. On the one hand probes involving real-time dynamics [14,15] have been discussed, showing excitations "getting stuck", or seeing suitable signatures in density-auto-correlation functions [16], le...

In this work, we present a result on the non-equilibrium dynamics causing equilibration and Gaussification of quadratic non-interacting fermionic Hamiltonians. Specifically, based on two basic assumptions -clustering of correlations in the initial state and the Hamiltonian exhibiting delocalizing transport -we prove that nonGaussian initial states become locally indistinguishable from fermionic Gaussian states after a short and well controlled time. This relaxation dynamics is governed by a power-law independent of the system size. Our argument is general enough to allow for pure and mixed initial states, including thermal and ground states of interacting Hamiltonians on and large classes of lattices as well as certain spin systems. The argument gives rise to rigorously proven instances of a convergence to a generalized Gibbs ensemble. Our results allow to develop an intuition of equilibration that is expected to be more generally valid and relates to current experiments of cold atoms in optical lattices.Despite the great complexity of quantum many-body systems out-of-equilibrium, local expectation values in such systems show the remarkable tendency to equilibrate to stationary values that do not depend on the microscopic details of the initial state, but rather can be described with few parameters using thermal states or generalized Gibbs ensembles [1][2][3]. Such behavior has been successfully studied in many settings theoretically and experimentally, most notably in instances of quantum simulations in optical lattices [2,4,5].By now, it is clear that, despite the unitary nature of quantum mechanical evolution, local expectation values equilibrate due to a dephasing between the eigenstates [3,[6][7][8][9][10][11]. So far it is, however, unclear why this dephasing tends to happen so rapidly. In fact, experiments often observe equilibration after very short times which are independent of the system size [5,12], while even the best theoretical bounds for general initial states of concrete systems diverge exponentially [2,11]. This discrepancy poses the challenge of precisely identifying the equilibration time, which constitutes one of the main open questions in the field [1][2][3].What is more, only little is known about how exactly the equilibrium expectation values emerge. Due to the exponentially many constants of motion present in quantum manybody systems, corresponding to the overlaps with the eigenvectors of the system, there seems to be no obvious reason why equilibrium values often only depend on few macroscopic properties such as temperature or particle number. In short: It is unclear how precisely the memory of the initial conditions is lost during time evolution.To make progress towards a solution of these two problems, it is instructive to study the behavior of non-interacting particles captured by so-called quadratic or free models. In these models the time evolution of so called Gaussian states, which are fully described by their correlation matrix, is particularly simple to describe. While studying...

The experimental realization of large-scale many-body systems in atomic-optical architectures has seen immense progress in recent years, rendering full tomography tools for state identification inefficient, especially for continuous systems. To work with these emerging physical platforms, new technologies for state identification are required. Here we present first steps towards efficient experimental quantum-field tomography. Our procedure is based on the continuous analogues of matrix-product states, ubiquitous in condensed-matter theory. These states naturally incorporate the locality present in realistic physical settings and are thus prime candidates for describing the physics of locally interacting quantum fields. To experimentally demonstrate the power of our procedure, we quench a one-dimensional Bose gas by a transversal split and use our method for a partial quantum-field reconstruction of the far-from-equilibrium states of this system. We expect our technique to play an important role in future studies of continuous quantum many-body systems.