Entanglement Dynamics and Classical Complexity.

We study the dynamical generation of entanglement for a two-body interacting system, starting from a separable coherent state. We show analytically that in the quasiclassical regime the entanglement growth rate can be simply computed by means of the underlying classical dynamics. Furthermore, this rate is given by the Kolmogorov-Sinai entropy, which characterizes the dynamical complexity of classical motion. Our results, illustrated by numerical simulations on a model of coupled rotators, establish in the quasiclassical regime a link between the generation of entanglement, a purely quantum phenomenon, and classical complexity.

Interplay of decoherence and relaxation in a two-level system interacting with an infinite-temperature reservoir.

We study the time evolution of a single qubit in contact with a bath, within the framework of projection operator methods. Employing the so-called modified Redfield theory, which also treats energy conserving interactions nonperturbatively, we are able to study the regime beyond the scope of the ordinary approach. Reduced equations of motion for the qubit are derived in an idealistic system where both the bath and system-bath interactions are modeled by Gaussian distributed random matrices. In the strong decoherence regime, a simple relation between the bath correlation function and the decoherence process induced by the energy conserving interaction is found. It implies that energy conserving interactions slow down the relaxation process, which leads to a Zeno freezing if they are sufficiently strong. Furthermore, our results are also confirmed in numerical simulations.

Numerically probing the universal operator growth hypothesis.

Recently, a hypothesis on the complexity growth of unitarily evolving operators was presented. This hypothesis states that in generic, nonintegrable many-body systems, the so-called Lanczos coefficients associated with an autocorrelation function grow asymptotically linear, with a logarithmic correction in one-dimensional systems. In contrast, the growth is expected to be slower in integrable or free models. In this paper, we numerically test this hypothesis for a variety of exemplary systems, including one-dimensional and two-dimensional Ising models as well as one-dimensional Heisenberg models. While we find the hypothesis to be practically fulfilled for all considered Ising models, the onset of the hypothesized universal behavior could not be observed in the attainable numerical data for the Heisenberg model. The proposed linear bound on operator growth associated with the hypothesis eventually stems from geometric arguments involving the locality of the Hamiltonian as well as the lattice configuration. We derive and investigate a related geometric bound, and we find that while the bound itself is not sharply achieved for any considered model, the hypothesis is nonetheless fulfilled in most cases.

Integral fluctuation theorem and generalized Clausius inequality for microcanonical and pure states.

Fluctuation theorems are cornerstones of modern statistical mechanics and their standard derivations routinely rely on the crucial assumption of a canonical equilibrium state. Yet rigorous derivations of certain fluctuation theorems for microcanonical states and pure energy eigenstates in isolated quantum systems are still lacking and constitute a major challenge to theory. In this work we tackle this challenge and present such a derivation of an integral fluctuation theorem (IFT) by invoking two central and physically natural conditions, i.e., the so-called "stiffness" and "smoothness" of transition probabilities. Our analytical arguments are additionally substantiated by numerical simulations for archetypal many-body quantum systems, including integrable as well as nonintegrable models of interacting spins and hard-core bosons on a lattice. These simulations strongly suggest that "stiffness" and "smoothness" are indeed of vital importance for the validity of the IFT for microcanonical and pure states. Our work contrasts with recent approaches to the IFT based on Lieb-Robinson speeds and the eigenstate thermalization hypothesis.

Eigenstate Thermalization Hypothesis and Its Deviations from Random-Matrix Theory beyond the Thermalization Time.

The eigenstate thermalization hypothesis explains the emergence of the thermodynamic equilibrium in isolated quantum many-body systems by assuming a particular structure of the observable's matrix elements in the energy eigenbasis. Schematically, it postulates that off-diagonal matrix elements are random numbers and the observables can be described by random matrix theory (RMT). To what extent a RMT description applies, more precisely at which energy scale matrix elements of physical operators become truly uncorrelated, is, however, not fully understood. We study this issue by introducing a novel numerical approach to probe correlations between matrix elements for Hilbert-space dimensions beyond those accessible by exact diagonalization. Our analysis is based on the evaluation of higher moments of operator submatrices, defined within energy windows of varying width. Considering nonintegrable quantum spin chains, we observe that matrix elements remain correlated even for narrow energy windows corresponding to timescales of the order of thermalization time of the respective observables. We also demonstrate that such residual correlations between matrix elements are reflected in the dynamics of out-of-time-ordered correlation functions.

Quantum chaos and the correspondence principle.

The correspondence principle is a cornerstone in the entire construction of quantum mechanics. This principle has been recently challenged by the observation of an early-time exponential increase of the out-of-time-ordered correlator (OTOC) in classically nonchaotic systems [E. B. Rozenbaum et al., Phys. Rev. Lett. 125, 014101 (2020)PRLTAO0031-900710.1103/PhysRevLett.125.014101]. Here, we show that the correspondence principle is restored after a proper treatment of the singular points. Furthermore, our results show that the OTOC maintains its role as a diagnostic of chaotic dynamics.

Chaotic dynamics of a non-Hermitian kicked particle.

We investigate both the classical and quantum dynamics of a kicked particle withPTsymmetry. In chaotic situation, the mean energy of the real parts of momentum linearly increases with time, and that of the imaginary momentum exponentially increases. There exists a breakdown time for chaotic diffusion, which is obtained both analytically and numerically. The quantum diffusion of this non-Hermitian system follows the classically chaotic diffusion of Hermitian case during the Ehrenfest time, after which it is completely suppressed. Interestingly, the Ehrenfest time decreases with the decrease of effective Planck constant or the increase of the strength of the non-Hermitian kicking potential. The exponential growth of the quantum out-of-time-order correlators (OTOC) during the initially short time interval characterizes the feature of the exponential diffusion of imaginary trajectories. The long time behavior of OTOC reflects the dynamical localization of quantum diffusion. The dynamical behavior of inverse participation ratio can quantify thePTsymmetry breaking, for which the rule of the phase transition points is numerically obtained.

Directed momentum current induced by the PT-symmetric driving.

We investigate the directed momentum current in the quantum kicked rotor model with PT-symmetric deriving potential. For the quantum nonresonance case, the values of quasienergy become complex when the strength of the imaginary part of the kicking potential exceeds a threshold value, which demonstrates the appearance of the spontaneous PT symmetry breaking. In the vicinity of the transition point, the momentum current exhibits a staircase growth with time. Each platform of the momentum current corresponds to the mean momentum of some eigenstates of the Floquet operator whose imaginary parts of the quasienergy are significantly large. Above the transition point, the momentum current increases linearly with time. Interestingly, its acceleration rate exhibits a kind of "quantized" increment with the kicking strength. We propose a modified classical acceleration mode of the kicked rotor model to explain such an intriguing phenomenon. Our theoretical prediction is in good agreement with numerical results.

Characterization of random features of chaotic eigenfunctions in unperturbed basis.

In this paper we study random features manifested in components of energy eigenfunctions of quantum chaotic systems, given in the basis of unperturbed, integrable systems. Based on semiclassical analysis, particularly on Berry's conjecture, it is shown that the components in classically allowed regions can be regarded as Gaussian random numbers in a certain sense, when appropriately rescaled with respect to the average shape of the eigenfunctions. This suggests that when a perturbed system changes from integrable to chaotic, deviation of the distribution of rescaled components in classically allowed regions from the Gaussian distribution may be employed as a measure for the "distance" to quantum chaos. Numerical simulations performed in the Lipkin-Meshkov-Glick model and the Dicke model show that this deviation coincides with the deviation of the nearest-level-spacing distribution from the prediction of random-matrix theory. Similar numerical results are also obtained in two models without classical counterpart.

Internal temperature of quantum chaotic systems at the nanoscale.

The extent to which a temperature can be appropriately assigned to a small quantum system, as an internal property but not as a property of any large environment, is still an open problem. In this paper, a method is proposed for solving this problem, by which a studied small system is coupled to a two-level system as a probe, the latter of which can be measured by measurement devices. A main difficulty in the determination of possible temperature of the studied system comes from the back-action of the probe-system coupling to the system. For small quantum chaotic systems, we show that a temperature can be determined, the value of which is sensitive to neither the form, location, and strength of the probe-system coupling, nor the Hamiltonian and initial state of the probe. The temperature thus obtained turns out to have the form of Boltzmann temperature.

Correlations in eigenfunctions of quantum chaotic systems with sparse Hamiltonian matrices.

In most realistic models for quantum chaotic systems, the Hamiltonian matrices in unperturbed bases have a sparse structure. We study correlations in eigenfunctions of such systems and derive explicit expressions for some of the correlation functions with respect to energy. The analytical results are tested in several models by numerical simulations. Some applications are discussed for a relation between transition probabilities and for expectation values of some local observables.