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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Wave-packet initial motion, spreading, and energy in the periodically kicked pendulum

Boon Leong Lan

Published Aug 1, 1994• 11 citations
Publisher

Abstract

The initial quantum dynamics of an initially well-localized, minimum-uncertainty Gaussian wave packet is studied analytically for the periodically kicked pendulum. The general results presented here include our earlier analytic result [Phys. Rev. A 41, 2952 (1990)] as a special case. We find that as long as the packet remains localized, it remains approximately Gaussian, while its center and widths evolve in time approximately according to, respectively, the classical map and a variance map which utilize the instantaneous values of the classical Jacobi matrix. When the classical map is very strongly chaotic, we predict that the average kinetic energy of the well-localized wave packet does not initially grow diffusively (linearly with time). We show that, instead, in the presence of chaos, the deviations initially grow, on the average, exponentially at a rate equal to the corresponding classical Liapunov exponent under the appropriate conditions. The time scale (Ehrenfest time) on which the Ehrenfest-Hamilton correspondence holds in the presence of chaos is estimated and is slightly different from a previous estimate made by Chirikov, Israilev, and Shepelyansky [Sov. Sci. Rev. C 2, 209 (1981)]. Numerical evidence in support of these analytical results is presented.

DOI
10.1103/physreve.50.764

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arXiv:quant-ph/0111044•Journal of Experimental and Theoretical Physics•Jan 1, 1998•9 citations
The initial quantum dynamics of an initially well-localized, minimum-uncertainty Gaussian wave packet is studied analytically for the periodically kicked pendulum. The general results presented here include our earlier analytic result [Phys. Rev. A 41, 2952 (1990)] as a special case. We find that as long as the packet remains localized, it remains approximately Gaussian, while its center and widths evolve in time approximately according to, respectively, the classical map and a variance map which utilize the instantaneous values of the classical Jacobi matrix. When the classical map is very strongly chaotic, we predict that the average kinetic energy of the well-localized wave packet does not initially grow diffusively (linearly with time). We show that, instead, in the presence of chaos, the deviations initially grow, on the average, exponentially at a rate equal to the corresponding classical Liapunov exponent under the appropriate conditions. The time scale (Ehrenfest time) on which the Ehrenfest-Hamilton correspondence holds in the presence of chaos is estimated and is slightly different from a previous estimate made by Chirikov, Israilev, and Shepelyansky [Sov. Sci. Rev. C 2, 209 (1981)]. Numerical evidence in support of these analytical results is presented.
The behavior of the periodically kicked pendulum is determined classically and quantum mechanically. Our earlier analytic study [Phys. Rev. A 41, 2952 (1990)] is confirmed and extended by the numerical results presented here. We are able to construct minimum uncertainty, Gaussian wave packets that maintain their form, and shadow the corresponding classical motion for long times when the classical motion is nonchaotic or only weakly chaotic. However, when the classical motion is robustly chaotic, the quantum-mechanical probability distribution grows macroscopically broad exponentially fast and the quantum-classical correspondence breaks down.
The correspondence limit is illustrated for the periodically kicked pendulum. The classical dynamics of this system can be represented by a discrete map. A corresponding quantum map is derived and then rendered in the Ehrenfest formulation for expectation values. The Ehrenfest representation is studied for a minimum uncertainty Gaussian wave packet. It is shown that as \ensuremath{\Elzxh} gets very small, followed by a decrease in the variance of the wave packet, the quantum map shadows the classical map with an error approaching zero for a length of time approaching infinity. The Gaussian form of the wave packet is preserved by the time evolution provided ${\mathrm{\ensuremath{\omega}}}_{0}$T\ensuremath{\ll}1. This constraint implies that the classical map is predominantly not chaotic with very small regions of very weak chaos. As ${\mathrm{\ensuremath{\omega}}}_{0}$T approaches 1, where the classical map becomes strongly chaotic, the propagation in time of the Gaussian wave packet completely breaks down. The possible significance of this breakdown is discussed.
The problem ``what is the quantum signature of a classically chaotic system'' is studied for the periodically kicked top. We find that the quantum variances initially grow exponentially if the corresponding classical description is chaotic. The rate of growth is connected to the corresponding classical Jacobi matrix and, thereby, to the local, classical, transient expansion rate. These connections were recently established in an analysis of the kicked pendulum for the correspondence between quantum Husimi-O'Connell-Wigner distributions and classical Gaussian ensembles. Here, we present closely related results for the kicked top by using generalized coherent states. An explanation is given for why this quantum signature of classical chaos was missed in earlier studies of the kicked top.
The trajectory predicted by Bohm's quantum equation of motion, using an initially angle-localized minimum-uncertainty Gaussian wave packet, is compared with the trajectory predicted by Newton's classical equation of motion for a macroscopic periodically kicked pendulum system. We find that, in violation of the strong form of the correspondence principle, the Bohmian trajectory does not, in general, agree or correspond with the Newtonian trajectory for all times. The Bohm-Newton trajectory correspondence holds as long as the wave packet remains localized in angle. Delocalization of the wave packet in angle triggers the breakdown of the correspondence. The breakdown occurs exponentially quickly if either the Newtonian trajectory that initially coincides with the Bohmian trajectory or one of its classical neighbors is chaotic. In contrast, the breakdown occurs much slower if neither the Newtonian trajectory nor its neighbors are chaotic. These behaviors of the Bohm-Newton trajectory correspondence are generic, or universal, for the class of macroscopic Hamiltonian dynamical systems with bound coordinate space since the kick pendulum is a prototypical member. We predict that neither Bohm's equation of motion nor Newton's correctly predicts the experimentally observed and measured trajectory of a macroscopic system after the breakdown of the correspondence, implying that the all-times agreement required by the strong form of the correspondence principle is unnecessary.
We describe the quantum-mechanical spreading of a Gaussian wave packet by means of the semiclassical WKB approximation of Berry and Balazs [J. Phys. A 2,2,2, 625 (1979)]. We find that the time scale $\ensuremath{\tau}$ on which this approximation breaks down in a chaotic system is larger than the Ehrenfest times considered previously. In one dimension $\ensuremath{\tau}=\frac{7}{6}{\ensuremath{\lambda}}^{\ensuremath{-}1}\mathrm{ln}(A/\ensuremath{\Elzxh}),$ with $\ensuremath{\lambda}$ the Lyapunov exponent and A a typical classical action.
We examine the dynamics of a wave packet that initially corresponds to a coherent state in the model of a quantum rotator excited by a periodic sequence of kicks. This model is the main model of quantum chaos and allows for a transition from regular behavior to chaotic in the classical limit. By doing a numerical experiment we study the generation of squeezed states in quasiclassical conditions and in a time interval when quantum-classical correspondence is well-defined. We find that the degree of squeezing depends on the degree of local instability in the system and increases with the Chirikov classical stochasticity parameter. We also discuss the dependence of the degree of squeezing on the initial width of the packet, the problem of stability and observability of squeezed states in the transition to quantum chaos, and the dynamics of disintegration of wave packets in quantum chaos.