Quantum signature of chaos and thermalization in the kicked Dicke model

Classical route to ergodicity and scarring in collective quantum systems

Ergodicity, a fundamental concept in statistical mechanics, is not yet a fully understood phenomena for closed quantum systems, particularly its connection with the underlying chaos. In this review, we consider a few examples of collective quantum systems to unveil the intricate relationship of ergodicity as well as its deviation due to quantum scarring phenomena with their classical counterpart. A comprehensive overview of classical and quantum chaos is provided, along with the tools essential for their detection. Furthermore, we survey recent theoretical and experimental advancements in the domain of ergodicity and its violations. This review aims to illuminate the classical perspective of quantum scarring phenomena in interacting quantum systems.

Analysis of chaos and regularity in the open Dicke model

We present an analysis of chaos and regularity in the open Dicke model, when dissipation is due to cavity losses. Due to the infinite Liouville space of this model, we also introduce a criterion to numerically find a complex spectrum which approximately represents the system spectrum. The isolated Dicke model has a well-defined classical limit with two degrees of freedom. We select two case studies where the classical isolated system shows regularity and where chaos appears. To characterize the open system as regular or chaotic, we study regions of the complex spectrum taking windows over the absolute value of its eigenvalues. Our results for this infinite-dimensional system agree with the Grobe-Haake-Sommers (GHS) conjecture for Markovian dissipative open quantum systems, finding the expected 2D Poisson distribution for regular regimes, and the distribution of the Ginibre unitary ensemble (GinUE) for the chaotic ones, respectively.

Chaos in the three-site Bose-Hubbard model: Classical versus quantum

We consider a quantum many-body system-the Bose-Hubbard system on three sites-which has a classical limit, and which is neither strongly chaotic nor integrable but rather shows a mixture of the two types of behavior. We compare quantum measures of chaos (eigenvalue statistics and eigenvector structure) in the quantum system, with classical measures of chaos (Lyapunov exponents) in the corresponding classical system. As a function of energy and interaction strength, we demonstrate a strong overall correspondence between the two cases. In contrast to both strongly chaotic and integrable systems, the largest Lyapunov exponent is shown to be a multivalued function of energy.

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.

Quantum transitions, ergodicity, and quantum scars in the coupled top model

We consider an interacting collective spin model known as coupled top (CT), exhibiting a rich variety of phenomena related to quantum transitions, ergodicity, and formation of quantum scars, discussed in our previous work [Mondal, Sinha, and Sinha, Phys. Rev. E 102, 020101(R) (2020)2470-004510.1103/PhysRevE.102.020101]. In this work, we present a detailed analysis of the different type of transitions in the CT model, and find their connection with the underlying collective spin dynamics. Apart from the quantum scarring phenomena, we also identify another source of deviation from ergodicity due to the presence of nonergodic multifractal states. The degree of ergodicity of the eigenstates across the energy band is quantified from the relative entanglement entropy as well as multifractal dimensions, which can be probed from nonequilibrium dynamics. Finally, we discuss the detection of nonergodic behavior and different types of quantum scars using "out-of-time-order correlators," which has relevance in the recent experiments.

Dynamical route to ergodicity and quantum scarring in kicked coupled top

Unlike classical systems, understanding ergodicity from phase space mixing remains unclear for interacting quantum systems due to the absence of phase space trajectories. By considering an interacting spin model known as kicked coupled top, we elucidate the manifestation of phase space dynamics on local ergodic behavior of its quantum counterpart and quantum scarring phenomena. A transition to chaos occurs by increasing the kicking strength, and in the mixed phase space, the islands of regular motions within the chaotic sea clearly exhibit deviation from ergodicity, which we quantify from entanglement entropy and survival probability. Interestingly, the reminiscence of unstable orbits and fixed points can be identified as scars in quantum states, exhibiting athermal behavior and violation of Berry's conjecture for ergodic states. We also discuss the detection of quantum scars by a recently developed method of "out-of-time-order correlators," which has experimental relevance.

Fingerprint of chaos and quantum scars in kicked Dicke model: an out-of-time-order correlator study

We investigate the onset of chaos in a periodically kicked Dicke model (KDM), using the out-of-time-order correlator (OTOC) as a diagnostic tool, in both the oscillator and the spin subspaces. In the large spin limit, the classical Hamiltonian map is constructed, which allows us to investigate the corresponding phase space dynamics and to compute the Lyapunov exponent. We show that the growth rate of the OTOC for the canonically conjugate coordinates of the oscillator is able to capture the Lyapunov exponent in the chaotic regime. The onset of chaos is further investigated using the saturation value of the OTOC, that can serve as an alternate indicator of chaos in a generic interacting quantum system. This is also supported by a system independent effective random matrix model. We further identify the quantum scars in KDM and detect their dynamical signature by using the OTOC dynamics. The relevance of the present study in the context of ongoing cold atom experiments is also discussed.

Eigenstate thermalization scaling in approaching the classical limit

According to the eigenstate thermalization hypothesis (ETH), the eigenstate-to-eigenstate fluctuations of expectation values of local observables should decrease with increasing system size. In approaching the thermodynamic limit-the number of sites and the particle number increasing at the same rate-the fluctuations should scale as ∼D^{-1/2} with the Hilbert space dimension D. Here, we study a different limit-the classical or semiclassical limit-by increasing the particle number in fixed lattice topologies. We focus on the paradigmatic Bose-Hubbard system, which is quantum-chaotic for large lattices and shows mixed behavior for small lattices. We derive expressions for the expected scaling, assuming ideal eigenstates having Gaussian-distributed random components. We show numerically that, for larger lattices, ETH scaling of physical midspectrum eigenstates follows the ideal (Gaussian) expectation, but for smaller lattices, the scaling occurs via a different exponent. We examine several plausible mechanisms for this anomalous scaling.

Chaos and Quantum Scars in Bose-Josephson Junction Coupled to a Bosonic Mode

We consider a model describing Bose-Josephson junction (BJJ) coupled to a single bosonic mode exhibiting quantum phase transition (QPT). Onset of chaos above QPT is observed from semiclassical dynamics as well from spectral statistics. Based on entanglement entropy, we analyze the ergodic behavior of eigenstates with increasing energy density which also reveals the influence of dynamical steady state known as π-mode on it. We identify the imprint of unstable π-oscillation as many body quantum scar (MBQS), which leads to the deviation from ergodicity and quantify the degree of scarring. Persistence of phase coherence in nonequilibrium dynamics of such initial state corresponding to the π-mode is an observable signature of MBQS which has relevance in experiments on BJJ.

Chaos and quantum scars in a coupled top model

We consider a coupled top model describing two interacting large spins, which is studied semiclassically as well as quantum mechanically. This model exhibits a variety of interesting phenomena such as a quantum phase transition (QPT), a dynamical transition, and excited-state quantum phase transitions above a critical coupling strength. Both classical dynamics and entanglement entropy reveal ergodic behavior at the center of the energy density band for an intermediate range of coupling strength above QPT, where the level spacing distribution changes from Poissonian to Wigner-Dyson statistics. Interestingly, in this model we identify quantum scars as reminiscent of unstable collective dynamics even in the presence of an interaction. The statistical properties of such scarred states deviate from the ergodic limit corresponding to the random matrix theory and violate Berry's conjecture. In contrast to ergodic evolution, the oscillatory behavior in the dynamics of the unequal time commutator and survival probability is observed as the dynamical signature of a quantum scar, which can be relevant for its detection.

Dynamics of quasiperiodically driven spin systems

We study the stroboscopic dynamics of a spin-S object subjected to δ-function kicks in the transverse magnetic field which is generated following the Fibonacci sequence. The corresponding classical Hamiltonian map is constructed in the large spin limit, S→∞. On evolving such a map for large kicking strength and time period, the phase space appears to be chaotic; interestingly, however, the geodesic distance increases linearly with the stroboscopic time implying that the Lyapunov exponent is zero. We derive the Sutherland invariant for the underlying SO(3) matrix governing the dynamics of classical spin variables and study the orbits for weak kicking strength. For the quantum dynamics, we observe that although the phase coherence of a state is retained throughout the time evolution, the fluctuations in the mean values of the spin operators exhibit fractality which is also present in the Floquet eigenstates. Interestingly, the presence of an interaction with another spin results in an ergodic dynamics leading to infinite temperature thermalization.

Drive-induced delocalization in the Aubry-André model

Motivated by the recent experiment by Bordia et al. [Nat. Phys. 13, 460 (2017)NPAHAX1745-247310.1038/nphys4020], we study the single particle delocalization phenomena of the Aubry-André (AA) model subjected to periodic drives. In two distinct cases we construct an equivalent classical description to illustrate that the drive-induced delocalization phenomena stems from an instability and the onset of chaos in the underlying dynamics. In the first case we analyze the delocalization and the thermalization in a time modulated AA potential with respect to driving frequency and demonstrate that there exists a threshold value of the amplitude of the drive. In the next example, we show that the periodic modulation of the phase of the hopping amplitude induced by a gauge field leads to an unusual effect on delocalization with a nonmonotonic dependence on the driving frequency. Within a window of such a driving frequency a delocalized Floquet band with a mobility edge appears, exhibiting multifractality in the spectrum as well as in the Floquet eigenfunctions. Finally, we explore the effect of interaction and discuss how the results of the present analysis can be tested experimentally.