Emergence of fluctuating hydrodynamics in chaotic quantum systems
Julian F. Wienand, Simon Karch, Alexander Impertro, Christian Schweizer, Ewan McCulloch, Romain Vasseur, Sarang Gopalakrishnan, Monika Aidelsburger, Immanuel Bloch
Abstract A fundamental principle of chaotic quantum dynamics is that local subsystems eventually approach a thermal equilibrium state. The corresponding timescales increase with subsystem size as equilibration is limited by the hydrodynamic build-up of fluctuations on extended length scales. We perform large-scale quantum simulations that monitor particle-number fluctuations in tunable ladders of hard-core bosons and explore how the build-up of fluctuations changes as the system crosses over from integrable to fully chaotic dynamics. Our results indicate that the growth of large-scale fluctuations in chaotic, far-from-equilibrium systems is quantitatively determined by equilibrium transport coefficients, in agreement with the predictions of fluctuating hydrodynamics. This emergent hydrodynamic behaviour of subsystem fluctuations provides a test of fluctuation–dissipation relations far from equilibrium and allows the accurate determination of equilibrium transport coefficients using far-from-equilibrium quantum dynamics.
Abstract A fundamental principle of chaotic quantum dynamics is that local subsystems eventually approach a thermal equilibrium state. The corresponding timescales increase with subsystem size as equilibration is limited by the hydrodynamic build-up of fluctuations on extended length scales. We perform large-scale quantum simulations that monitor particle-number fluctuations in tunable ladders of hard-core bosons and explore how the build-up of fluctuations changes as the system crosses over from integrable to fully chaotic dynamics. Our results indicate that the growth of large-scale fluctuations in chaotic, far-from-equilibrium systems is quantitatively determined by equilibrium transport coefficients, in agreement with the predictions of fluctuating hydrodynamics. This emergent hydrodynamic behaviour of subsystem fluctuations provides a test of fluctuation–dissipation relations far from equilibrium and allows the accurate determination of equilibrium transport coefficients using far-from-equilibrium quantum dynamics.
We analytically describe the decay to equilibrium of generic observables of a nonintegrable system after a perturbation in the form of a random matrix. We further obtain an analytic form for the time-averaged fluctuations of an observable in terms of the rate of decay to equilibrium. Our result shows the emergence of a fluctuation-dissipation theorem corresponding to a classical Brownian process, specifically, the Ornstein-Uhlenbeck process. Our predictions can be tested in quantum simulation experiments, thus helping to bridge the gap between theoretical and experimental research in quantum thermalization. We test our analytic results by exact numerical experiments in a spin chain. We argue that our fluctuation-dissipation relation can be used to measure the density of states involved in the nonequilibrium dynamics of an isolated quantum system.
A large number of multifaceted quantum transport processes in molecular systems and physical nanosystems can be treated in terms of quantum relaxation processes which couple to one or several fluctuating environments. A thermal equilibrium environment can conveniently be modelled by a thermal bath of harmonic oscillators. An archetype situation provides a two-state dissipative quantum dynamics, commonly known under the label of a spin-boson dynamics. An interesting and nontrivial physical situation emerges, however, when the quantum dynamics evolves far away from thermal equilibrium. This occurs, for example, when a charge transferring medium possesses nonequilibrium degrees of freedom, or when a strong time-dependent control field is applied externally. Accordingly, certain parameters of underlying quantum subsystem acquire stochastic character. Herein, we review the general theoretical framework which is based on the method of projector operators, yielding the quantum master equations for systems that are exposed to strong external fields. This allows one to investigate on a common basis the influence of nonequilibrium fluctuations and periodic electrical fields on quantum transport processes. Most importantly, such strong fluctuating fields induce a whole variety of nonlinear and nonequilibrium phenomena. A characteristic feature of such dynamics is the absence of thermal (quantum) detailed balance.
Entanglement and magic are resources that reveal complementary aspects of quantum many-body systems. Their interplay is captured by nonlocal magic, the magic that survives arbitrary local changes of basis. Yet their markedly different dynamical behavior leaves open how this irreducible component of magic spreads. Here we connect nonlocal magic to the capacity of entanglement, a tractable quantity measuring fluctuations of the entanglement Hamiltonian. This connection enables analytically controlled predictions across a wide range of many-body dynamics, from chaotic to integrable systems, which we investigate also using large-scale numerical simulations. In chaotic systems, nonlocal magic exhibits a transient buildup, with logarithmic growth in time followed by decay to a size-independent value. For integrable systems, we develop a quasiparticle picture in quantitative agreement with numerics, showing that the same initial growth instead leads to saturation at a value logarithmic in subsystem size. These contrasting behaviors have an operational consequence for entanglement embezzlement: the strongest scramblers embezzle only transiently, whereas free-fermionic dynamics can sustain universal embezzlement in the steady state.
The question of how statistical physics can be seen to emerge from unitary quantum dynamics goes back to Schrödinger and von Neumann, however has been reaffirmed as a topic of importance due to recent advances in experimental capabilities, allowing the observation of the unitary evolution of many-particle systems over long periods of time. Understanding why such a system evolves to the specific thermal equilibrium state, effectively described by a relevant thermodynamic ensemble, has thus seen significantly more interest in recent years. Furthermore, observations of the dynamics of such systems open questions on the route to equilibrium, and subsequent fluctuations. In this thesis I will develop an approach to this problem, taking as a starting point the non-integrability of the system under study, which leads to a generic model in terms of random matrix theory (RMT), and more generally, chaotic wavefunctions. In the formulation of these methods, special attention is paid to the form of local observables of quantum systems, and an approach to their description is developed in terms of correlation functions of chaotic wavefunctions. From this approach a key conjecture in the understanding of thermalization, the eigenstate thermalization hypothesis (ETH), is derived in full, and the dynamical behaviour of such observables is obtained analytically. Further, I will show that emergent classical behaviour can be observed in the form of a fluctuation-dissipation theorem (FDT) of Brownian processes. This is exploited to develop an experimental proposal to measure the ‘complexity’ of a quantum device, by experimentally obtaining its effective Hilbert space dimension in terms of a fully connected system. Finally, quantum jump trajectories, describing stroboscopic projective measurements of local observables are studied, and shown to display classical Brownian dynamics in the form of a Markov process. Throughout this thesis, exact diagonalization calculations of realistic quantum spin systems are used to confirm analytical results.
Isolated many-body quantum systems quenched far from equilibrium can eventually equilibrate, but it is not yet clear how long they take to do so. To answer this question, we use exact numerical methods and analyze the entire evolution, from perturbation to equilibration, of a paradigmatic disordered many-body quantum system in the chaotic regime. We investigate how the equilibration time depends on the system size and observables. We show that if dynamical manifestations of spectral correlations in the form of the correlation hole (``ramp'') are taken into account, the time for equilibration scales exponentially with system size, while if they are neglected, the scaling is better described by a power law with system size, although with an exponent larger than what is expected for diffusive transport.
The statistics of fluctuations on large regions of space encodes universal properties of many-body systems. At equilibrium, it is described by thermodynamics. However, away from equilibrium such as after quantum quenches, the fundamental principles are more nebulous. In particular, although exact results have been conjectured in integrable models, a correct understanding of the physics is largely missing. In this paper, we explain these principles, taking the example of the number of particles lying on a large interval in one-dimensional interacting systems. These are based on simple hydrodynamic arguments from the theory of ballistically transported fluctuations, and, in particular, the Euler-scale transport of long-range correlations. Using these principles, we obtain the full counting statistics in terms of thermodynamic and hydrodynamic quantities, whose validity depends on the structure of hydrodynamic modes and on the fluctuations in the initial state. In fermionic-statistics interacting integrable models with a continuum of hydrodynamic modes, such as the Lieb-Liniger model for cold atomic gases, the formula reproduces previous conjectures, but is in fact not exact: It gives the correct cumulants up to, including, order 5, while long-range correlations modify higher cumulants. In integrable and nonintegrable models with two or less hydrodynamic modes, the formula is expected to give all cumulants.