Abstract We consider time-dependent relaxation of observables in quantum systems of chaotic and regular type. Using statistical arguments and exact numerical solutions we show that the spread of the initial wave function in the Hilbert space and the main characteristics of evolution of observables have certain generic features. The study compares examples of regular dynamics, a completely chaotic case of the Gaussian orthogonal ensemble, a bosonic system with random interactions, and a fully realistic case of the time evolution of various initial non-stationary states in the nuclear shell model. In the case of the Gaussian orthogonal ensemble we show that the survival probability obtained analytically also fully defines the relaxation timescale of observables. This is not the case in general. Using the realistic nuclear shell model and the quadrupole moment as an observable we demonstrate that the relaxation time is significantly longer than defined by the survival probability of the initial state. The full analysis does not show the presence of an analog of the Lyapunov exponent characteristic for examples of classical chaos.
Abstract We consider time-dependent relaxation of observables in quantum systems of chaotic and regular type. Using statistical arguments and exact numerical solutions we show that the spread of the initial wave function in the Hilbert space and the main characteristics of evolution of observables have certain generic features. The study compares examples of regular dynamics, a completely chaotic case of the Gaussian orthogonal ensemble, a bosonic system with random interactions, and a fully realistic case of the time evolution of various initial non-stationary states in the nuclear shell model. In the case of the Gaussian orthogonal ensemble we show that the survival probability obtained analytically also fully defines the relaxation timescale of observables. This is not the case in general. Using the realistic nuclear shell model and the quadrupole moment as an observable we demonstrate that the relaxation time is significantly longer than defined by the survival probability of the initial state. The full analysis does not show the presence of an analog of the Lyapunov exponent characteristic for examples of classical chaos.
A major open question in studies of nonequilibrium quantum dynamics is the identification of the time scales involved in the relaxation process of isolated quantum systems that have many interacting particles. We demonstrate that long time scales can be analytically found by analyzing dynamical manifestations of spectral correlations. Using this approach, we show that the Thouless time tTh and the relaxation time tR increase exponentially with system size. We define tTh as the time at which the spread of the initial state in the many-body Hilbert space is complete and verify that it agrees with the inverse of the Thouless energy. tTh marks the point beyond which the dynamics acquire universal features, while relaxation happens later when the evolution reaches a stationary state. In chaotic systems, ${t}_{\text{Th}}\ensuremath{\ll}{t}_{\text{R}}$, while for systems approaching a many-body localized phase, ${t}_{\text{Th}}\ensuremath{\rightarrow}{t}_{\text{R}}$. Our analytical results for tTh and tR are obtained for the survival probability, which is a global quantity. We show numerically that the same time scales appear also in the evolution of the spin autocorrelation function, which is an experimental local observable. Our studies are carried out for realistic many-body quantum models. The results are compared with those for random matrices.
We investigate quench dynamics in a one-dimensional spin model, comparing both quantum and classical descriptions. Our primary focus is on the different timescales involved in the evolution of the observables as they approach statistical relaxation. Numerical simulations, supported by semianalytical analysis, reveal that the relaxation of single-particle energies (global quantity) and on-site magnetization (local observable) occurs on a timescale independent of the system size L. This relaxation process is equally well-described by classical equations of motion and quantum solutions, demonstrating excellent quantum-classical correspondence, provided the system is strongly chaotic. The correspondence persists even for small quantum spin values (S=1), where a semiclassical approximation is not applicable. Conversely, for the participation ratio, which characterizes the initial state spread in the many-body Hilbert space and which lacks a classical analog, the relaxation timescale is system-size dependent.
Quantum systems whose classical counterparts are chaotic typically have highly correlated eigenvalues and level statistics that coincide with those from ensembles of full random matrices. A dynamical manifestation of these correlations comes in the form of the so-called correlation hole, which is a dip below the saturation point of the survival probability's time evolution. In this work, we study the correlation hole in the spin-boson (Dicke) model, which presents a chaotic regime and can be realized in experiments with ultracold atoms and ion traps. We derive an analytical expression that describes the entire evolution of the survival probability and allows us to determine the time scales of its relaxation to equilibrium. This expression shows remarkable agreement with our numerical results. While the initial decay and the time to reach the minimum of the correlation hole depend on the initial state, the dynamics beyond the hole up to equilibration is universal. We find that the relaxation time of the survival probability for the Dicke model increases linearly with system size.
The quantum dynamics of initial coherent states is studied in the Dicke model and correlated with the dynamics, regular or chaotic, of their classical limit. Analytical expressions for the survival probability, i.e. the probability of finding the system in its initial state at time t, are provided in the regular regions of the model. The results for regular regimes are compared with those of the chaotic ones. It is found that initial coherent states in regular regions have a much longer equilibration time than those located in chaotic regions. The properties of the distributions for the initial coherent states in the Hamiltonian eigenbasis are also studied. It is found that for regular states the components with no negligible contribution are organized in sequences of energy levels distributed according to Gaussian functions. In the case of chaotic coherent states, the energy components do not have a simple structure and the number of participating energy levels is larger than in the regular cases.
We analyze time evolution of spread complexity (SC) in an isolated interacting quantum many-body system when it is subjected to a sudden quench. Characteristics features of the time evolution of the SC after the quench are analyzed for different timescales, both in integrable and chaotic models. For a short time after the quench, the SC shows universal quadratic growth, irrespective of the initial state or the nature of the Hamiltonian, with the timescale of this growth being determined by the local density of states. The characteristics of the SC in the next phase depend on the nature of the system, and we show that, depending on whether the survival probability of an initial state is Gaussian or exponential, the SC can continue to grow quadratically, or it can show linear growth. To understand the behavior of the SC at late times, we consider sudden quenches in two models, a full random matrix in the Gaussian orthogonal ensemble, and a spin-1/2 system with disorder. We observe that, for the full random matrix model and the chaotic phase of the spin-1/2 system, the SC shows linear growth at early times and saturation at late times. The full random matrix case shows a peak in the intermediate-time region, whereas this feature is less prominent in the spin-1/2 system, as we explain.
Randomness generation through quantum-chaotic evolution underpins foundational questions in statistical mechanics and applications across quantum information science, including benchmarking, tomography, metrology, and demonstrations of quantum computational advantage. While statistical mechanics successfully captures the temporal averages of local observables, understanding randomness at the level of higher statistical moments remains a daunting challenge, with analytic progress largely confined to random quantum circuit models or fine-tuned systems exhibiting space-time duality. Here we study how much randomness can be dynamically generated by generic quantum-chaotic evolution under physical, non-random Hamiltonians. Combining theoretical insights with numerical simulations, we show that for broad classes of initially unentangled states, the dynamics become effectively Haar-random well before the system can ergodically explore the physically accessible Hilbert space. Both local and highly nonlocal observables, including entanglement measures, equilibrate to their Haar expectation values and fluctuations on polynomial timescales with remarkably high numerical precision, and with the fastest randomization occurring in regions of parameter space previously identified as maximally chaotic. Interestingly, this effective randomization can occur on timescales linear in system size, suggesting that the sub-ballistic growth of Renyi entropies typically observed in systems with conservation laws can be bypassed in non-random Hamiltonians with an appropriate choice of initial conditions.