Scarring in Rough Rectangular Billiards.

We study the mechanism of scarring of eigenstates in rectangular billiards with slightly corrugated surfaces and show that it is very different from that known in Sinai and Bunimovich billiards. We demonstrate that there are two sets of scar states. One set is related to the bouncing ball trajectories in the configuration space of the corresponding classical billiard. A second set of scar-like states emerges in the momentum space, which originated from the plane-wave states of the unperturbed flat billiard. In the case of billiards with one rough surface, the numerical data demonstrate the repulsion of eigenstates from this surface. When two horizontal rough surfaces are considered, the repulsion effect is either enhanced or canceled depending on whether the rough profiles are symmetric or antisymmetric. The effect of repulsion is quite strong and influences the structure of all eigenstates, indicating that the symmetric properties of the rough profiles are important for the problem of scattering of electromagnetic (or electron) waves through quasi-one-dimensional waveguides. Our approach is based on the reduction of the model of one particle in the billiard with corrugated surfaces to a model of two artificial particles in the billiard with flat surfaces, however, with an effective interaction between these particles. As a result, the analysis is conducted in terms of a two-particle basis, and the roughness of the billiard boundaries is absorbed by a quite complicated potential.

Spectrum statistics in the integrable Lieb-Liniger model.

We address the old and widely debated question of the spectrum statistics of integrable quantum systems, through the analysis of the paradigmatic Lieb-Liniger model. This quantum many-body model of one-dimensional interacting bosons allows for the rigorous determination of energy spectra via the Bethe ansatz approach and our interest is to reveal the characteristic properties of energy levels in dependence of the model parameters. Using both analytical and numerical studies we show that the properties of spectra strongly depend on whether the analysis is done for a full energy spectrum or for a single subset with fixed total momentum. We show that the Poisson distribution of spacing between nearest-neighbor energies can occur only for a set of energy levels with fixed total momentum, for neither too large nor too weak interaction strength, and for sufficiently high energy. By studying long-range correlations between energy levels, we found strong deviations from the predictions based on the assumption of pseudorandom character of the distribution of energy levels.

Emergence of correlations in the process of thermalization of interacting bosons.

We address the question of the relevance of thermalization and scrambling to the increase of correlations in the quench dynamics of an isolated system with a finite number of interacting bosons. Specifically, we study how, in the process of thermalization, the correlations between occupation numbers increase in time, resulting in the emergence of the Bose-Einstein distribution. Despite the exponential increase of the number of principal components of the wave function, we show, both analytically and numerically, that the two-point correlation function before saturation increases quadratically in time according to perturbation theory. In contrast, we find that the out-of-time-order correlator increases algebraically and not exponentially in time after the perturbative regime and before the saturation. Our results can be confirmed experimentally in traps with interacting bosons and they may be relevant to the problem of black hole scrambling.

Exponentially fast dynamics of chaotic many-body systems.

We demonstrate analytically and numerically that in isolated quantum systems of many interacting particles, the number of many-body states participating in the evolution after a quench increases exponentially in time, provided the eigenstates are delocalized in the energy shell. The rate of the exponential growth is defined by the width Γ of the local density of states and is associated with the Kolmogorov-Sinai entropy for systems with a well-defined classical limit. In a finite system, the exponential growth eventually saturates due to the finite volume of the energy shell. We estimate the timescale for the saturation and show that it is much larger than ℏ/Γ. Numerical data obtained for a two-body random interaction model of bosons and for a dynamical model of interacting spin-1/2 particles show excellent agreement with the analytical predictions.

Timescales in the quench dynamics of many-body quantum systems: Participation ratio versus out-of-time ordered correlator.

We study quench dynamics in the many-body Hilbert space using two isolated systems with a finite number of interacting particles: a paradigmatic model of randomly interacting bosons and a dynamical (clean) model of interacting spins-1/2. For both systems in the region of strong quantum chaos, the number of components of the evolving wave function, defined through the number of principal components N_{pc} (or participation ratio), was recently found to increase exponentially fast in time [Phys. Rev. E 99, 010101(R) (2019)2470-004510.1103/PhysRevE.99.010101]. Here, we ask whether the out-of-time ordered correlator (OTOC), which is nowadays widely used to quantify instability in quantum systems, can manifest analogous time dependence. We show that N_{pc} can be formally expressed as the inverse of the sum of all OTOCs for projection operators. While none of the individual projection OTOCs show an exponential behavior, their sum decreases exponentially fast in time. The comparison between the behavior of the OTOC with that of the N_{pc} helps us better understand wave packet dynamics in the many-body Hilbert space, in close connection with the problems of thermalization and information scrambling.

Temperature of a single chaotic eigenstate.

The onset of thermalization in a closed system of randomly interacting bosons at the level of a single eigenstate is discussed. We focus on the emergence of Bose-Einstein distribution of single-particle occupation numbers, and we give a local criterion for thermalization dependent on the eigenstate energy. We show how to define the temperature of an eigenstate, provided that it has a chaotic structure in the basis defined by the single-particle states. The analytical expression for the eigenstate temperature as a function of both interparticle interaction and energy is complemented by numerical data.