We compare the quantum and classical dynamics of a particle moving in a cosine potential while subject to a time-dependent force. We concentrate here on the behavior of an initially well-localized wave packet at times before the classically chaotic motion is fully developed. We find that the quantum and classical dynamics are indistinguishable well beyond the Ehrenfest time where the wave packet delocalizes. The quantum and classical descriptions first differ precisely when the classical probability density is folded in the vicinity of a hyperbolic fixed point. At this point, the wave function acquires a nodal structure which we show to be the result of a simple beating phenomenon between paths in the semiclassical propagator.
We compare the quantum and classical dynamics of a particle moving in a cosine potential while subject to a time-dependent force. We concentrate here on the behavior of an initially well-localized wave packet at times before the classically chaotic motion is fully developed. We find that the quantum and classical dynamics are indistinguishable well beyond the Ehrenfest time where the wave packet delocalizes. The quantum and classical descriptions first differ precisely when the classical probability density is folded in the vicinity of a hyperbolic fixed point. At this point, the wave function acquires a nodal structure which we show to be the result of a simple beating phenomenon between paths in the semiclassical propagator.
I compare the quantum and classical dynamics of a particle moving in a cosine potential while subject to a time-dependent force. I concentrate here on the behavior of an initially well-localized wave packet at times before the classically chaotic motion is fully developed. I find that the quantum and classical dynamics are indistinguishable well beyond the Ehrenfest time where the wave packet delocalizes. The quantum and classical descriptions first differ precisely when the classical probability density is folded in the vicinity of a hyperbolic fixed point. At this point, the wave function acquires a nodal structure which I show to be the result of a simple beating phenomenon between paths in the semiclassical propagator. When the interaction of the classical manifold with the hyperbolic fixed point leads to escape from the remnant separatrix, representing the onset of classical chaos, the interference associated with the tendril leads to a long-lived accumulation of quantum amplitude on top of the cosine barrier. This effect also has a semiclassical interpretation, meaning that it arises from the interference between classical paths, but one must expand to second order in Planck's constant to describe the behavior correctly. Both the nodal structure and this pinning of amplitude on the barrier are dynamic mechanisms for the quantum inhibition of mixing. I then couple the system to a bath of harmonic oscillators in order to study the effect of the environment on these mechanisms. Although an oscillator bath environment brings dissipation as well as noise to the problem, the noise effect dominates for the high temperature, weak coupling regime I study. When there is sufficient noise to render the interfering classical paths indistinguishable, I find that the quantum interference gets erased. This dephasing occurs at very early times, long before there is appreciable dissipation of energy to the environment. Consequently, one can argue that the presence of an environment, even if its effect would be negligible in a nonchaotic setting, allows for the possibility of quantum mixing.
The relationship between the diverging of classical trajectories in chaotic many-body systems, the spreading of quantum wave packets, and the validity and use of classical molecular dynamics is explored. This analysis, which is based on the semiclassical description of wave function propagation in terms of a weighted integration over a traveling fixed width coherent state basis, suggests that the exponential divergence of nearby classical trajectories in chaotic many-body systems should result in the rapid delocalization of an initially localized quantum wave packet describing the state of the system. Thus the justification for the use of classical molecular dynamics procedures for these supposedly classical systems cannot be based on the picture of the system wave function remaining localized as its center follows a nearly classical trajectory. The quantum evolution of the system density, on the other hand, requires two propagators, and each of these propagators is represented as an integration over trajectories in the semiclassical picture. The interference between the contributions from these two integrations over classical trajectories focuses the analysis on the most important points in this trajectory pair space, which are shown to occur when both trajectories in the pair are the same. Given reasonable assumptions for the initial density for a system that is expected to be well described by classical molecular dynamics, and given an appropriate choice for the width of the coherent state basis which is employed in the semiclassical description, it is shown that the semiclassical expressions for time dependent observables and correlation functions reduce the purely classical expressions, despite the fact that an initially localized wave packet would rapidly delocalize for the same system.
We study the quantum chaotic dynamics of an initially well-localized wave packet in a cosine potential perturbed by an external time-dependent force. For our choice of initial condition and with \ensuremath{\Elzxh} small but finite, we find that the wvae packet behaves classically (meaning that the quantum behavior is indistinguishable from that of the analogous classical system) as long as the motion is confined to the interior of the remnant separatrix of the cosine potential. Once the classical motion becomes unbounded, however, we find that quantum interference effects dominate. This interference leads to a long-lived accumulation of quantum ampliutde on top of the cosine barrier. This pinning of the amplitude on the barrier is a dynamic mechanism for the quantum inhibition of classical mixing.
We explore the onset of delocalization as a function of coupling strength for time-dependent wave packets started on specific phase-space structures in the quantum standard map. Coherent-state wave packets, initially placed on fixed points of the classical map, are propagated for a long time and the delocalization is quantitatively measured. In most cases the delocalization occurs rather abruptly as a function of the kicking strength. The relative quantum stability of various phase-space structures is determined.
The quantum evolution of the Wigner function for Gaussian wave packets generated by a non-Hermitian Hamiltonian is investigated. In the semiclassical limit $\ensuremath{\hbar}\ensuremath{\rightarrow}0$ this yields the non-Hermitian analog of the Ehrenfest theorem for the dynamics of observable expectation values. The lack of Hermiticity reveals the importance of the complex structure on the classical phase space: The resulting equations of motion are coupled to an equation of motion for the phase-space metric---a phenomenon having no analog in Hermitian theories.
In this paper, we investigate the quantum dynamics of wave packets of bound states in the nonlinear Henon–Heiles (HH) system. The time evolution of a wave packet was separated into two processes involving (a) the motion of its first moments and (b) its spreading. On the time scale when wave packet spreading can be neglected, we were able to establish relations between wave packet quantum dynamics expressed in terms of the initial-state population probability P(t) and classical trajectory dynamics. We have demonstrated that the initial states, which are characterized by a periodic time evolution of P(t) over many periods, are related to the five elliptic periodic orbits in the HH system. The decay of P(t) over a few vibrational periods is related to the classical features of quasiperiodic trajectories and can be accounted for in terms of overlap reduction effects. When the center of the initial wave packet is moved further away from the classical periodic orbits, P(t) is characterized by a single initial peak followed by a ’’noisy’’ background. There is no qualitative distinction between the ’’short-time’’ behavior of a wave packet initiated in distant integrable regions and in classical stochastic regions. For long times, wave packet spreading effects set in. Quantum mechanical estimates for the spreading time τs, which depends on the initial conditions, were provided. For wave packets initiated on periodic trajectories, τs is very long, becoming shorter as one moves away from the fixed points. The quantum mechanical long time (t≫τs) behavior of P(t) exhibits small-amplitude noisy time dependence for all initial conditions. This behavior may be described as ’’quantum mechanical stochasticity’’ and is related to the universal feature of irregularity of the spectral resolution of all wave packets in a nonlinear system. From this point of view, all wave packets exhibit ’’quantum stochastic behavior,’’ but there are vast quantitative differences for different initial conditions that are manifested in terms of the values of τs and of the average noisy amplitude. It should be emphasized that this ’’quantum stochasticity’’ is not related to classical stochasticity phenomena.