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.
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 consider a two-state system kicked quasiperiodically by an external force. When the two kicking frequencies assumed for the force are incommensurate, there can be quantum chaos in the sense that (a) the autocorrelation function of the state vector decays, (b) the power spectrum of the state vector is broadband, and (c) the motion on the Bloch sphere is ergodic. The time evolution of the state vector is nevertheless dynamically stable in the sense that memory of the initial state is retained. We also consider briefly the kicked quantum rotator and find, in agreement with Shepelyansky [Physica 8D, 208 (1983)], that the quantum localization effect is greatly weakened by the presence of two incommensurate driving frequencies.
In this paper we consider the time evolution of SU(1,1) coherent states driven by a coherence-preserving Hamiltonian containing periodic or quasiperiodic pulsing terms. This is a generalization of a system consisting of a two-level atom subjected to quasiperiodic pulsing that was recently studied by Milonni, Ackerhalt, and Goggin [Phys. Rev. A 35, 1714 (1987)]. The time-evolution operator in our case is given by a product of two finite group transformations of SU(1,1). Assuming an initial SU(1,1) coherent state, we determine the equivalent classical motion generated by a Poincar\'e map that is a M\"obius transformation on the Lobachevski plane, the interior of the unit circle in the complex plane. The quantum-mechanical evolution of the state vector is calculated exactly and in closed form even though the Hilbert space is infinite dimensional. We also study the autocorrelation function which, as in the work of Milonni, Ackerhalt, and Goggin, is found to decay in the case of quasiperiodic pulsing that may possibly be associated with a manifestation of chaos in a quantum-mechanical system.
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.
A two-level system subject to quasiperiodically modulated kicking is investigated under the aspects of ergodic theory. For weak kicking strength \ensuremath{\kappa} the correlation functions exhibit quasiperiodic behavior, whose character changes with increasing \ensuremath{\kappa} as recurrences become infrequent. A quantum instability with a transition to mixing behavior that was suggested for similar models does not arise. Analytic results are obtained for the case where two of three characteristic frequencies are incommensurate.
A general relation is derived, which expresses the fidelity of quantum dynamics, measuring the stability of time evolution to small static variation in the Hamiltonian, in terms of ergodicity of an observable generating the perturbation as defined by its time correlation function. Fidelity for ergodic dynamics is predicted to decay exponentially on time scale proportional to delta(-2), delta approximately strength of perturbation, whereas faster, typically Gaussian decay on shorter time scale proportional delta(-1) is predicted for integrable, or generally nonergodic dynamics. This result needs the perturbation delta to be sufficiently small such that the fidelity decay time scale is larger than any (quantum) relaxation time, e.g., mixing time for mixing dynamics, or averaging time for nonergodic dynamics (or Ehrenfest time for wave packets in systems with chaotic classical limit). Our surprising predictions are demonstrated in a quantum Ising spin-(1/2) chain periodically kicked with a tilted magnetic field where we find finite parameter-space regions of nonergodic and nonintegrable motion in the thermodynamic limit.