Minimal Fokker-Planck theory for the thermalization of mesoscopic subsystems

Two stage decoherence of optical phonons in long oligomers

Molecular vibrations are generally responsible for chemical energy transport and dissipation in molecular systems. This transport is fast and efficient if energy is transferred by optical phonons in periodic oligomers, but its efficiency is limited by decoherence emerging due to anharmonic interactions with acoustic phonons. Using a general theoretical model, we show that in the most common case of the optical phonon band being narrower than the acoustic bands, decoherence takes place in two stages. The faster stage involves optical phonon multiple forward scattering due to absorption and emission of transverse acoustic phonons, i.e., collective bending modes with a quadratic spectrum; the transport remains ballistic and the speed can be altered. The subsequent slower stage involves phonon backscattering in multiphonon processes involving two or more acoustic phonons resulting in a switch to diffusive transport. If the initially excited optical phonon possesses a relatively small group velocity, then it is accelerated in the first stage due to its transitions to states propagating faster. This theoretical expectation is consistent with the recent measurements of optical phonon transport velocity in alkane chains, increasing with increasing the chain length.

Chaos and bipartite entanglement between Bose-Josephson junctions

The entanglement between two weakly coupled bosonic Josephson junctions is studied in relation to the classical mixed phasespace structure of the system, containing symmetry-related regular islands separated by chaos. The symmetry-resolved entanglement spectrum and bipartite entanglement entropy of the system's energy eigenstates are calculated and compared to their expected structure for random states that exhibit complete or partial ergodicity. The entanglement spectra of chaos-supported eigenstates match the microcanonical structure of a Generalized Gibbs Ensemble due to the existence of an adiabatic invariant that restricts ergodization on the energy shell. The symmetry-resolved entanglement entropy of these quasistochastic states consists of a mean-field maximum entanglement term and a fluctuation correction due to the finite size of the constituent subsystems. The total bipartite entanglement entropy of the eigenstates correlates with their chaoticity. Island-supported eigenstates are macroscopic Schrödinger cat states for particles and excitations with substantially lower entanglement.

Interacting bosons in a triple well: Preface of many-body quantum chaos

Systems of interacting bosons in triple-well potentials are of significant theoretical and experimental interest. They are explored in contexts that range from quantum phase transitions and quantum dynamics to semiclassical analysis. Here, we systematically investigate the onset of quantum chaos in a triple-well model that moves away from integrability as its potential gets tilted. Even in its deepest chaotic regime, the system presents features reminiscent of integrability. Our studies are based on level spacing distribution and spectral form factor, structure of the eigenstates, and diagonal and off-diagonal elements of observables in relationship to the eigenstate thermalization hypothesis. With only three sites, the system's eigenstates are at the brink of becoming fully chaotic, so they do not yet exhibit Gaussian distributions, which resonates with the results for the observables.

Stiffness of probability distributions of work and Jarzynski relation for initial microcanonical and energy eigenstates

We consider closed quantum systems which are driven such that only negligible heating occurs. If driving only affects small parts of the system, it may nonetheless be strong. Our analysis aims at clarifying under which conditions the Jarzynski relation (JR) holds in such setups, if the initial states are microcanonical or even energy eigenstates. We find that the validity of the JR for the microcanonical initial state hinges on an exponential density of states and on stiffness. The latter indicates an independence of the probability density functions (PDFs) of work of the energy of the respective microcanonical initial state. The validity of the JR for initial energy eigenstates is found to additionally require smoothness. The latter indicates an independence of the work PDFs of the specific energy eigenstates within a microcanonical energy shell. As the validity of the JR for pure initial energy eigenstates has no analog in classical systems, we consider it a genuine quantum phenomenon.

Taking snapshots of a quantum thermalization process: Emergent classicality in quantum jump trajectories

We investigate theoretically the emergence of classical statistical physics in a finite quantum system that is either totally isolated or otherwise subjected to a quantum measurement process. We show via a random matrix theory approach to nonintegrable quantum systems that the set of outcomes of the measurement of a macroscopic observable evolve in time like stochastic variables, whose variance satisfies the celebrated Einstein relation for Brownian diffusion. Our results show how to extend the framework of eigenstate thermalization to the prediction of properties of quantum measurements on an otherwise closed quantum system. We show numerically the validity of the random matrix approach in quantum chain models.

How to probe the microscopic onset of irreversibility with ultracold atoms

The microscopic onset of irreversibility is finally becoming an experimental subject. Recent experiments on microscopic open and even isolated systems have measured statistical properties associated with entropy production, and hysteresis-like phenomena have been seen in cold atom systems with dissipation (i.e. effectively open systems coupled to macroscopic reservoirs). Here we show how experiments on isolated systems of ultracold atoms can show dramatic irreversibility like cooking an egg. In our proposed experiments, a slow forward-and-back parameter sweep will sometimes fail to return the system close to its initial state. This probabilistic hysteresis is due to the same non-adiabatic spreading and ergodic mixing in phase space that explains macroscopic irreversibility, but realized without dynamical chaos; moreover this fundamental mechanism quantitatively determines the probability of return to the initial state as a function of tunable parameters in the proposed experiments. Matching the predicted curve of return probability will be a conclusive experimental demonstration of the microscopic onset of irreversibility.

Ballistic and diffusive vibrational energy transport in molecules

Energy transport in molecules is essential for many areas of science and technology. Strong covalent bonds of a molecular backbone can facilitate the involvement of the molecule's high-frequency modes in energy transport, which, under certain conditions, makes the transport fast and efficient. We discuss such conditions and describe various transport regimes in molecules, including ballistic, diffusive, directed diffusion, and intermediate regime cases, in light of recently developed experimental and theoretical approaches.

Chaotic Dynamics in a Quantum Fermi-Pasta-Ulam Problem

We investigate the emergence of chaotic dynamics in a quantum Fermi—Pasta—Ulam problem for anharmonic vibrations in atomic chains applying semi-quantitative analysis of resonant interactions complemented by exact diagonalization numerical studies. The crossover energy separating chaotic high energy phase and localized (integrable) low energy phase is estimated. It decreases inversely proportionally to the number of atoms until approaching the quantum regime where this dependence saturates. The chaotic behavior appears at lower energies in systems with free or fixed ends boundary conditions compared to periodic systems. The applications of the theory to realistic molecules are discussed.

Adiabatic Passage through Chaos

We study the process of nonlinear stimulated Raman adiabatic passage within a classical mean-field framework. Depending on the sign of interaction, the breakdown of adiabaticity in the interacting nonintegrable system is not related to bifurcations in the energy landscape, but rather to the emergence of quasistochastic motion that drains the followed quasistationary state. Consequently, faster sweep rate, rather than quasistatic variation of parameters, is better for adiabaticity.

Molecules and the Eigenstate Thermalization Hypothesis

We review a theory that predicts the onset of thermalization in a quantum mechanical coupled non-linear oscillator system, which models the vibrational degrees of freedom of a molecule. A system of N non-linear oscillators perturbed by cubic anharmonic interactions exhibits a many-body localization (MBL) transition in the vibrational state space (VSS) of the molecule. This transition can occur at rather high energy in a sizable molecule because the density of states coupled by cubic anharmonic terms scales as N³, in marked contrast to the total density of states, which scales as exp(aN), where a is a constant. The emergence of a MBL transition in the VSS is seen by analysis of a random matrix ensemble that captures the locality of coupling in the VSS, referred to as local random matrix theory (LRMT). Upon introducing higher order anharmonicity, the location of the MBL transition of even a sizable molecule, such as an organic molecule with tens of atoms, still lies at an energy that may exceed the energy to surmount a barrier to reaction, such as a barrier to conformational change. Illustrative calculations are provided, and some recent work on the influence of thermalization on thermal conduction in molecular junctions is also discussed.

Stiffness of probability distributions of work and Jarzynski relation for non-Gibbsian initial states

We consider closed quantum systems (into which baths may be integrated) that are driven, i.e., subject to time-dependent Hamiltonians. Our point of departure is the assumption that if systems start in non-Gibbsian states at some initial energies, the resulting probability distributions of work may be largely independent of the specific initial energies. It is demonstrated that this assumption has some far-reaching consequences, e.g., it implies the validity of the Jarzynski relation for a large class of non-Gibbsian initial states. By performing numerical analysis on integrable and nonintegrable spin systems, we find the above assumption fulfilled for all examples considered. Through an analysis based on Fermi's golden rule, we partially relate these findings to the applicability of the eigenstate thermalization ansatz to the respective driving operators.

Semiclassical theory of strong localization for quantum thermalization

We introduce a semiclassical theory for strong localization that may arise in the context of many-body thermalization. As a minimal model for thermalization we consider a few-site Bose-Hubbard model consisting of two weakly interacting subsystems that can exchange particles. The occupation of a subsystem (x) satisfies in the classical treatment a Fokker-Planck equation with a diffusion coefficient D(x). We demonstrate that it is possible to deduce from the classical description a quantum breaktime t^{*} and, hence, the manifestations of a strong localization effect. For this purpose it is essential to take the geometry of the energy shell into account and to make a distinction between different notions of phase-space exploration.

Least-rattling feedback from strong time-scale separation

In most interacting many-body systems associated with some "emergent phenomena," we can identify subgroups of degrees of freedom that relax on dramatically different time scales. Time-scale separation of this kind is particularly helpful in nonequilibrium systems where only the fast variables are subjected to external driving; in such a case, it may be shown through elimination of fast variables that the slow coordinates effectively experience a thermal bath of spatially varying temperature. In this paper, we investigate how such a temperature landscape arises according to how the slow variables affect the character of the driven quasisteady state reached by the fast variables. Brownian motion in the presence of spatial temperature gradients is known to lead to the accumulation of probability density in low-temperature regions. Here, we focus on the implications of attraction to low effective temperature for the long-term evolution of slow variables. After quantitatively deriving the temperature landscape for a general class of overdamped systems using a path-integral technique, we then illustrate in a simple dynamical system how the attraction to low effective temperature has a fine-tuning effect on the slow variable, selecting configurations that bring about exceptionally low force fluctuation in the fast-variable steady state. We furthermore demonstrate that a particularly strong effect of this kind can take place when the slow variable is tuned to bring about orderly, integrable motion in the fast dynamics that avoids thermalizing energy absorbed from the drive. We thus point to a potentially general feedback mechanism in multi-time-scale active systems, that leads to the exploration of slow variable space, as if in search of fine tuning for a "least-rattling" response in the fast coordinates.

Statistics of the stochastically forced Lorenz attractor by the Fokker-Planck equation and cumulant expansions

We investigate the Fokker-Planck description of the equal-time statistics of the three-dimensional Lorenz attractor with additive white noise. The invariant measure is found by computing the zero (or null) mode of the linear Fokker-Planck operator as a problem of sparse linear algebra. Two variants are studied: a self-adjoint construction of the linear operator and the replacement of diffusion with hyperdiffusion. We also access the low-order statistics of the system by a perturbative expansion in equal-time cumulants. A comparison is made to statistics obtained by the standard approach of accumulation via direct numerical simulation. Theoretical and computational aspects of the Fokker-Planck and cumulant expansion methods are discussed.

Numerical evidence for approximate consistency and Markovianity of some quantum histories in a class of finite closed spin systems

Closed quantum systems obey the Schrödinger equation, whereas nonequilibrium behavior of many systems is routinely described in terms of classical, Markovian stochastic processes. Evidently, there are fundamental differences between those two types of behavior. We discuss the conditions under which the unitary dynamics may be mapped onto pertinent classical stochastic processes. This is first principally addressed based on the notions of "consistency" and "Markovianity." Numerical data are presented that show that the above conditions are to good approximation fulfilled for Heisenberg-type spin models comprising 12-20 spins. The accuracy to which these conditions are met increases with system size.

Macroscopically deterministic Markovian thermalization in finite quantum spin systems

A key feature of nonequilibrium thermodynamics is the Markovian, deterministic relaxation of coarse observables such as, for example, the temperature difference between two macroscopic objects which evolves independently of almost all details of the initial state. We demonstrate that the unitary dynamics for moderately sized spin-1/2 systems may yield the same type of relaxation dynamics for a given magnetization difference. This observation might contribute to the understanding of the emergence of thermodynamics within closed quantum systems.

Dynamics of the central-depleted-well regime in the open Bose-Hubbard trimer

We study the quantum dynamics of the central-depleted-well (CDW) regime in a three-mode Bose Hubbard model subject to a confining parabolic potential. By introducing a suitable set of momentum-like modes we identify the microscopic variables involved in the quantization process and the dynamical algebra of the model. We describe the diagonalization procedure showing that the model reduces to a double oscillator. Interestingly, we find that the parameter-space domain where this scheme entails a discrete spectrum well reproduces the two regions where the classical trimer excludes unstable oscillations. Spectral properties are examined in different limiting cases together with various delocalization effects. These are shown to characterize quantum states of the CDW regime in the proximity of the borderline with classically unstable domains.