Novel techniques to analyze dynamical properties of quantum chaos with peculiar evidence of hybrid systems confinement.

Recent results demonstrate the dynamical peculiarities of the quantum chaos within the hybrid systems by chaotic parameters and probe the pattern formation under the influence of condensation. The complex dynamic behavior of the considered systems was determined with numerical simulation and presented an efficient technique that studied fractional systems comprising chaos-coherence fractions. The findings divulge the peculiar association between the coherence structure and the correlations at finite relative momenta. Thus the present study helps to explore the partially chaos hybrid systems in order to stimulate the experimental applications of nonlinear phenomena. The coherent-chaotic parameters can be measured by examining the chaos peculiarities that possess explicit relations with the condensations to demonstrate the environs of the physical systems. We investigate the influence of the multiplicities, chaos, momentum and temperature of the nonlinear system on the coherent-chaotic normalized correlations. The chaotic parameters are suppressed considerably with the coherence fraction and it appears numerically zero at maximum condensation and one at ideal chaos emissions. We procure that the meaningful parameters decrease significantly with the multiplicity of the nonlinear systems and increase with the momentum in the specified regimes. The identical multiplicity leads to contemplating the coherence and thus the normalized chaotic parameters within its spectacular influences exhibit significance worth contemplating in earnest. The findings underscore the significance of cogitating correlations in deciphering the nonlinear system characteristics and bestowing extraordinary perceptiveness into the convoluted essence of complex systems. The contemplated methodology can be applied to evaluating and analyzing the nonlinear systems and such an innovative approach computes the problems of celestial mechanics, heartbeats and chemical reactions in engineering and medical fields.

Quantum information scrambling and chemical reactions.

The ultimate regularity of quantum mechanics creates a tension with the assumption of classical chaos used in many of our pictures of chemical reaction dynamics. Out-of-time-order correlators (OTOCs) provide a quantum analog to the Lyapunov exponents that characterize classical chaotic motion. Maldacena, Shenker, and Stanford have suggested a fundamental quantum bound for the rate of information scrambling, which resembles a limit suggested by Herzfeld for chemical reaction rates. Here, we use OTOCs to study model reactions based on a double-well reaction coordinate coupled to anharmonic oscillators or to a continuum oscillator bath. Upon cooling, as one enters the tunneling regime where the reaction rate does not strongly depend on temperature, the quantum Lyapunov exponent can approach the scrambling bound and the effective reaction rate obtained from a population correlation function can approach the Herzfeld limit on reaction rates: Tunneling increases scrambling by expanding the state space available to the system. The coupling of a dissipative continuum bath to the reaction coordinate reduces the scrambling rate obtained from the early-time OTOC, thus making the scrambling bound harder to reach, in the same way that friction is known to lower the temperature at which thermally activated barrier crossing goes over to the low-temperature activationless tunneling regime. Thus, chemical reactions entering the tunneling regime can be information scramblers as powerful as the black holes to which the quantum Lyapunov exponent bound has usually been applied.

Quasi-integrability and nonlinear resonances in cold atoms under modulation.

Quantum dynamics of a collection of atoms subjected to phase modulation has been carefully revisited. We present an exact analysis of the evolution of a two-level system (represented by a spinor) under the action of a time-dependent matrix Hamiltonian. The dynamics is shown to evolve on two coupled potential energy surfaces (PESs): one of them is binding, while the other one is scattering type. The dynamics is shown to be quasi-integrable with nonlinear resonances. The bounded dynamics with intermittent scattering at random moments presents a scenario reminiscent of Anderson and dynamical localization. We believe that a careful analytical investigation of a multi-component system that is classically non-integrable is relevant to many other fields, including quantum computation with multi-qubit systems.

Quadratic Growth of Out-of-Time-Ordered Correlators in Quantum Kicked Rotor Model.

We investigate both theoretically and numerically the dynamics of out-of-time-ordered correlators (OTOCs) in quantum resonance conditions for a kicked rotor model. We employ various operators to construct OTOCs in order to thoroughly quantify their commutation relation at different times, therefore unveiling the process of quantum scrambling. With the help of quantum resonance condition, we have deduced the exact expressions of quantum states during both forward evolution and time reversal, which enables us to establish the laws governing OTOCs' time dependence. We find interestingly that the OTOCs of different types increase in a quadratic function of time, breaking the freezing of quantum scrambling induced by the dynamical localization under non-resonance condition. The underlying mechanism is discovered, and the possible applications in quantum entanglement are discussed.

Instantons and the quantum bound to chaos.

The rate at which information scrambles in a quantum system can be quantified using out-of-time-ordered correlators. A remarkable prediction is that the associated Lyapunov exponent [Formula: see text] that quantifies the scrambling rate in chaotic systems obeys a universal bound [Formula: see text]. Previous numerical and analytical studies have indicated that this bound has a quantum-statistical origin. Here, we use path-integral techniques to show that a minimal theory to reproduce this bound involves adding contributions from quantum thermal fluctuations (describing quantum tunneling and zero-point energy) to classical dynamics. By propagating a model quantum-Boltzmann-conserving classical dynamics for a system with a barrier, we show that the bound is controlled by the stability of thermal fluctuations around the barrier instanton (a delocalized structure which dominates the tunneling statistics). This stability requirement appears to be general, implying that there is a close relation between the formation of instantons, or related delocalized structures, and the imposition of the quantum-chaos bound.

Unveiling order from chaos by approximate 2-localization of random matrices.

Quantum many-body systems are typically endowed with a tensor product structure. A structure they inherited from probability theory, where the probability of two independent events is the product of the probabilities. The tensor product structure of a Hamiltonian thus gives a natural decomposition of the system into independent smaller subsystems. It is interesting to understand whether a given Hamiltonian is compatible with some particular tensor product structure. In particular, we ask, is there a basis in which an arbitrary Hamiltonian has a 2-local form, i.e., it contains only pairwise interactions? Here we show, using analytical and numerical calculations, that a generic Hamiltonian (e.g., a large random matrix) can be approximately written as a linear combination of two-body interaction terms with high precision; that is, the Hamiltonian is 2-local in a carefully chosen basis. Moreover, we show that these Hamiltonians are not fine-tuned, meaning that the spectrum is robust against perturbations of the coupling constants. Finally, by analyzing the adjacency structure of the couplings [Formula: see text], we suggest a possible mechanism for the emergence of geometric locality from quantum chaos.

Semi-Poisson Statistics in Relativistic Quantum Billiards with Shapes of Rectangles.

Rectangular billiards have two mirror symmetries with respect to perpendicular axes and a twofold (fourfold) rotational symmetry for differing (equal) side lengths. The eigenstates of rectangular neutrino billiards (NBs), which consist of a spin-1/2 particle confined through boundary conditions to a planar domain, can be classified according to their transformation properties under rotation by π (π/2) but not under reflection at mirror-symmetry axes. We analyze the properties of these symmetry-projected eigenstates and of the corresponding symmetry-reduced NBs which are obtained by cutting them along their diagonal, yielding right-triangle NBs. Independently of the ratio of their side lengths, the spectral properties of the symmetry-projected eigenstates of the rectangular NBs follow semi-Poisson statistics, whereas those of the complete eigenvalue sequence exhibit Poissonian statistics. Thus, in distinction to their nonrelativistic counterpart, they behave like typical quantum systems with an integrable classical limit whose eigenstates are non-degenerate and have alternating symmetry properties with increasing state number. In addition, we found out that for right triangles which exhibit semi-Poisson statistics in the nonrelativistic limit, the spectral properties of the corresponding ultrarelativistic NB follow quarter-Poisson statistics. Furthermore, we analyzed wave-function properties and discovered for the right-triangle NBs the same scarred wave functions as for the nonrelativistic ones.

Spectral Form Factor and Dynamical Localization.

Quantum dynamical localization occurs when quantum interference stops the diffusion of wave packets in momentum space. The expectation is that dynamical localization will occur when the typical transport time of the momentum diffusion is greater than the Heisenberg time. The transport time is typically computed from the corresponding classical dynamics. In this paper, we present an alternative approach based purely on the study of spectral fluctuations of the quantum system. The information about the transport times is encoded in the spectral form factor, which is the Fourier transform of the two-point spectral autocorrelation function. We compute large samples of the energy spectra (of the order of 106 levels) and spectral form factors of 22 stadium billiards with parameter values across the transition between the localized and extended eigenstate regimes. The transport time is obtained from the point when the spectral form factor transitions from the non-universal to the universal regime predicted by random matrix theory. We study the dependence of the transport time on the parameter value and show the level repulsion exponents, which are known to be a good measure of dynamical localization, depend linearly on the transport times obtained in this way.

Quantum Chaos and Level Dynamics.

We review the application of level dynamics to spectra of quantally chaotic systems. We show that the statistical mechanics approach gives us predictions about level statistics intermediate between integrable and chaotic dynamics. Then we discuss in detail different statistical measures involving level dynamics, such as level avoided-crossing distributions, level slope distributions, or level curvature distributions. We show both the aspects of universality in these distributions and their limitations. We concentrate in some detail on measures imported from the quantum information approach such as the fidelity susceptibility, and more generally, geometric tensor matrix elements. The possible open problems are suggested.

Quantization of Integrable and Chaotic Three-Particle Fermi-Pasta-Ulam-Tsingou Models.

We study the transition from integrability to chaos for the three-particle Fermi-Pasta-Ulam-Tsingou (FPUT) model. We can show that both the quartic ß-FPUT model (α=0) and the cubic one (ß=0) are integrable by introducing an appropriate Fourier representation to express the nonlinear terms of the Hamiltonian. For generic values of α and ß, the model is non-integrable and displays a mixed phase space with both chaotic and regular trajectories. In the classical case, chaos is diagnosed by the investigation of Poincaré sections. In the quantum case, the level spacing statistics in the energy basis belongs to the Gaussian orthogonal ensemble in the chaotic regime, and crosses over to Poissonian behavior in the quasi-integrable low-energy limit. In the chaotic part of the spectrum, two generic observables obey the eigenstate thermalization hypothesis.

Generalized Survival Probability.

Survival probability measures the probability that a system taken out of equilibrium has not yet transitioned from its initial state. Inspired by the generalized entropies used to analyze nonergodic states, we introduce a generalized version of the survival probability and discuss how it can assist in studies of the structure of eigenstates and ergodicity.

Quantum Bounds on the Generalized Lyapunov Exponents.

We discuss the generalized quantum Lyapunov exponents Lq, defined from the growth rate of the powers of the square commutator. They may be related to an appropriately defined thermodynamic limit of the spectrum of the commutator, which plays the role of a large deviation function, obtained from the exponents Lq via a Legendre transform. We show that such exponents obey a generalized bound to chaos due to the fluctuation-dissipation theorem, as already discussed in the literature. The bounds for larger q are actually stronger, placing a limit on the large deviations of chaotic properties. Our findings at infinite temperature are exemplified by a numerical study of the kicked top, a paradigmatic model of quantum chaos.

A Physical Measure for Characterizing Crossover from Integrable to Chaotic Quantum Systems.

In this paper, a quantity that describes a response of a system's eigenstates to a very small perturbation of physical relevance is studied as a measure for characterizing crossover from integrable to chaotic quantum systems. It is computed from the distribution of very small, rescaled components of perturbed eigenfunctions on the unperturbed basis. Physically, it gives a relative measure to prohibition of level transitions induced by the perturbation. Making use of this measure, numerical simulations in the so-called Lipkin-Meshkov-Glick model show in a clear way that the whole integrability-chaos transition region is divided into three subregions: a nearly integrable regime, a nearly chaotic regime, and a crossover regime.

Universality and beyond in Optical Microcavity Billiards with Source-Induced Dynamics.

Optical microcavity billiards are a paradigm of a mesoscopic model system for quantum chaos. We demonstrate the action and origin of ray-wave correspondence in real and phase space using far-field emission characteristics and Husimi functions. Whereas universality induced by the invariant-measure dominated far-field emission is known to be a feature shaping the properties of many lasing optical microcavities, the situation changes in the presence of sources that we discuss here. We investigate the source-induced dynamics and the resulting limits of universality while we find ray-picture results to remain a useful tool in order to understand the wave behaviour of optical microcavities with sources. We demonstrate the source-induced dynamics in phase space from the source ignition until a stationary regime is reached comparing results from ray, ray-with-phase, and wave simulations and explore ray-wave correspondence.

Canonical Density Matrices from Eigenstates of Mixed Systems.

One key issue of the foundation of statistical mechanics is the emergence of equilibrium ensembles in isolated and closed quantum systems. Recently, it was predicted that in the thermodynamic (N→∞) limit of large quantum many-body systems, canonical density matrices emerge for small subsystems from almost all pure states. This notion of canonical typicality is assumed to originate from the entanglement between subsystem and environment and the resulting intrinsic quantum complexity of the many-body state. For individual eigenstates, it has been shown that local observables show thermal properties provided the eigenstate thermalization hypothesis holds, which requires the system to be quantum-chaotic. In the present paper, we study the emergence of thermal states in the regime of a quantum analog of a mixed phase space. Specifically, we study the emergence of the canonical density matrix of an impurity upon reduction from isolated energy eigenstates of a large but finite quantum system the impurity is embedded in. Our system can be tuned by means of a single parameter from quantum integrability to quantum chaos and corresponds in between to a system with mixed quantum phase space. We show that the probability for finding a canonical density matrix when reducing the ensemble of energy eigenstates of the finite many-body system can be quantitatively controlled and tuned by the degree of quantum chaos present. For the transition from quantum integrability to quantum chaos, we find a continuous and universal (i.e., size-independent) relation between the fraction of canonical eigenstates and the degree of chaoticity as measured by the Brody parameter or the Shannon entropy.

Universal Single-Mode Lasing in Fully Chaotic Billiard Lasers.

By numerical simulations and experiments of fully chaotic billiard lasers, we show that single-mode lasing states are stable, whereas multi-mode lasing states are unstable when the size of the billiard is much larger than the wavelength and the external pumping power is sufficiently large. On the other hand, for integrable billiard lasers, it is shown that multi-mode lasing states are stable, whereas single-mode lasing states are unstable. These phenomena arise from the combination of two different nonlinear effects of mode-interaction due to the active lasing medium and deformation of the billiard shape. Investigations of billiard lasers with various shapes revealed that single-mode lasing is a universal phenomenon for fully chaotic billiard lasers.

Ray-Wave Correspondence in Microstar Cavities.

In a previous work by the authors (Phys. Rev. Research 2, 012072(R) (2020)) a novel concept of light confinement in a microcavity was introduced which is based on successive perfect transmissions at Brewster's angle. Hence, a new class of open billiards was designed with star-shaped microcavities where rays propagate on orbits that leave and re-enter the cavity. In this article, we investigate the ray-wave correspondence in microstar cavities. An unintuitive difference between clockwise and counterclockwise propagation is revealed which is traced back to nonlinear resonance chains in phase space.

Effective Field Theory of Random Quantum Circuits.

Quantum circuits have been widely used as a platform to simulate generic quantum many-body systems. In particular, random quantum circuits provide a means to probe universal features of many-body quantum chaos and ergodicity. Some such features have already been experimentally demonstrated in noisy intermediate-scale quantum (NISQ) devices. On the theory side, properties of random quantum circuits have been studied on a case-by-case basis and for certain specific systems, and a hallmark of quantum chaos-universal Wigner-Dyson level statistics-has been derived. This work develops an effective field theory for a large class of random quantum circuits. The theory has the form of a replica sigma model and is similar to the low-energy approach to diffusion in disordered systems. The method is used to explicitly derive the universal random matrix behavior of a large family of random circuits. In particular, we rederive the Wigner-Dyson spectral statistics of the brickwork circuit model by Chan, De Luca, and Chalker [Phys. Rev. X 8, 041019 (2018)] and show within the same calculation that its various permutations and higher-dimensional generalizations preserve the universal level statistics. Finally, we use the replica sigma model framework to rederive the Weingarten calculus, which is a method of evaluating integrals of polynomials of matrix elements with respect to the Haar measure over compact groups and has many applications in the study of quantum circuits. The effective field theory derived here provides both a method to quantitatively characterize the quantum dynamics of random Floquet systems (e.g., calculating operator and entanglement spreading) and a path to understanding the general fundamental mechanism behind quantum chaos and thermalization in these systems.

Quantum Chaos in the Extended Dicke Model.

We systematically study the chaotic signatures in a quantum many-body system consisting of an ensemble of interacting two-level atoms coupled to a single-mode bosonic field, the so-called extended Dicke model. The presence of the atom-atom interaction also leads us to explore how the atomic interaction affects the chaotic characters of the model. By analyzing the energy spectral statistics and the structure of eigenstates, we reveal the quantum signatures of chaos in the model and discuss the effect of the atomic interaction. We also investigate the dependence of the boundary of chaos extracted from both eigenvalue-based and eigenstate-based indicators on the atomic interaction. We show that the impact of the atomic interaction on the spectral statistics is stronger than on the structure of eigenstates. Qualitatively, the integrablity-to-chaos transition found in the Dicke model is amplified when the interatomic interaction in the extended Dicke model is switched on.

Chaos and Thermalization in the Spin-Boson Dicke Model.

We present a detailed analysis of the connection between chaos and the onset of thermalization in the spin-boson Dicke model. This system has a well-defined classical limit with two degrees of freedom, and it presents both regular and chaotic regions. Our studies of the eigenstate expectation values and the distributions of the off-diagonal elements of the number of photons and the number of excited atoms validate the diagonal and off-diagonal eigenstate thermalization hypothesis (ETH) in the chaotic region, thus ensuring thermalization. The validity of the ETH reflects the chaotic structure of the eigenstates, which we corroborate using the von Neumann entanglement entropy and the Shannon entropy. Our results for the Shannon entropy also make evident the advantages of the so-called "efficient basis" over the widespread employed Fock basis when investigating the unbounded spectrum of the Dicke model. The efficient basis gives us access to a larger number of converged states than what can be reached with the Fock basis.