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.
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.
In this paper we study the time evolution of a quasiperiodically kicked spin system whose Hamiltonian is linear in the SU(2) generators. Since this Hamiltonian preserves SU(2) coherent states under time evolution, there is a close correspondence between the quantum evolution and the classical evolution of the system on the Bloch sphere. Such a system has previously been studied by Milonni, Ackerhalt, and Goggin [Phys. Rev. A 35, 1714 (1987)] and, with two incommensurate driving frequencies, the quantum evolution was characterized by (a) decay of the autocorrelation function of the state vector, (b) broadband power spectrum of observables, (c) ergodicity on the Bloch sphere. We expand upon their work to study the corresponding behavior of the motion on a symplectic phase space and the decay of the autocorrelation function of the state vector. When the forcing strength is such that the motion is localized in phase space, the autocorrelation function shows revivals. When the motion is ergodic the autocorrelation function rapidly decays without revivals. The classical motion however is not chaotic, there is no sensitive dependence on initial conditions. For three incommensurate frequencies, the motion is more delocalized than for two frequencies for a fixed forcing strength.
We study the quantum dynamics of the kicked Dicke model (KDM) in terms of the Floquet operator, and we analyze the connection between chaos and thermalization in this context. The Hamiltonian map is constructed by suitably taking the classical limit of the Heisenberg equation of motion to study the corresponding phase-space dynamics, which shows a crossover from regular to chaotic motion by tuning the kicking strength. The fixed-point analysis and calculation of the Lyapunov exponent (LE) provide us with a complete picture of the onset of chaos in phase-space dynamics. We carry out a spectral analysis of the Floquet operator, which includes a calculation of the quasienergy spacing distribution and structural entropy to show the correspondence to the random matrix theory in the chaotic regime. Finally, we analyze the thermodynamics and statistical properties of the bosonic sector as well as the spin sector, and we discuss how such a periodically kicked system relaxes to a thermalized state in accordance with the laws of statistical mechanics.
Using an efficient one and two qubit gate simulator operating on graphical processing units, we investigate ergodic properties of a quantum Ising spin model on a two-dimensional lattice, which is periodically driven by a δ -pulsed transverse magnetic field. We consider three different dynamical properties: (i) level density, (ii) level spacing distribution of the Floquet quasienergy spectrum, and (iii) time-averaged autocorrelation function of magnetization components. Varying the parameters of the model, we found transitions between ordered (non-ergodic) and quantum chaotic (ergodic) phases, but the transitions between flat and non-flat spectral density do not correspond to transitions between ergodic and non-ergodic local observables. Even more surprisingly, we found good agreement of level spacing distribution with the Wigner surmise of random matrix theory for almost all values of parameters except where the model is essentially non-interacting, even in regions where local observables are not ergodic or where spectral density is non-flat. These findings question the versatility of the interpretation of level spacing distribution in many-body systems and stress the importance of the concept of locality.
Motivated by the search for non-equilibrium phases of matter beyond the Floquet paradigm, we propose a route to realize perfect revivals and time quasicrystals (TQCs) in a Fibonacci-driven central-spin model with Ising interactions between the central and satellite spins in the absence of disorder. We analytically demonstrate that when the interaction strength, λ is tuned to 2mπ, no entanglement is generated between the central and satellite spins at stroboscopic times, independent of the strength of the transverse driving field and the drive sequence. This leads to indefinitely long-lived quasiperiodic oscillations in satellite spin magnetization. Furthermore, for initial states with a definite bath magnetization, no entanglement is generated when the transverse driving field, θ is tuned to π for any value of $ λ$, and the satellite spin magnetization oscillations exhibit period-3 revivals at Fibonacci times. Examining the robustness of these oscillations away from this fine-tuned limit, we identify a parameter regime where the quasiperiodic response does not decay over the timescales probed, giving rise to a long-lived π-pulse-proximate TQC regime. We further show that at special Clifford points, the system exhibits perfect quasiperiodic oscillations of the satellite spin magnetization, with a parity-dependent Fourier response. This response persists over the timescales probed, even away from the Clifford points for an even number of satellite spins, leading to the formation of a Clifford-proximate TQC. Finally, we demonstrate that at this Clifford points, the system is steered along an entangled trajectory, leading to the dynamical generation of Bell-cat and spin-cat states. These states exhibit magnetization of collective spin fluctuates quadratically with the number of satellite spins, thereby enabling Heisenberg-limited metrology.
We map the infinite-range coupled quantum kicked rotors over an infinite-range coupled interacting bosonic model. In this way we can apply exact diagonalization up to quite large system sizes and confirm that the system tends to ergodicity in the large-size limit. In the thermodynamic limit the system is described by a set of coupled Gross-Pitaevskii equations equivalent to an effective nonlinear single-rotor Hamiltonian. These equations give rise to a power-law increase in time of the energy with exponent $\ensuremath{\gamma}\ensuremath{\sim}2/3$ in a wide range of parameters. We explain this finding by means of a master-equation approach based on the noisy behavior of the effective nonlinear single-rotor Hamiltonian and on the Anderson localization of the single-rotor Floquet states. Furthermore, we study chaos by means of the largest Lyapunov exponent and find that it decreases towards zero for portions of the phase space with increasing momentum. Finally, we show that some stroboscopic Floquet integrals of motion of the noninteracting dynamics deviate from their initial values over a timescale related to the interaction strength according to the Nekhoroshev theorem.