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arXiv:1009.6058

Analytical investigation of revival phenomena in periodically driven system

Muhammad Ahsan Shahzad

Published Sep 30, 2010• 0 citations
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Abstract

We present theoretical study of revival phenomena for a wave packet initially well localized in a one-dimensional potential in the presence of an external periodic modulating field. The classical motion, revival, and super-revival time scale are derived exactly for wave packet excited in the one dimension box potential.
DOI
10.48550/arxiv.1009.6058

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•Jan 1, 2012•0 citations
We calculate the quantum revival time for a wave packet initially well localized in a one-dimensional potential in the presence of an external periodic modulating field. The dependence of the revival time on various parameters of the driven system is shown analytically. As an example of an application of our approach, we compare the analytically obtained values of the revival time for various modulation strengths with the numerically computed ones in the case of a driven gravitational cavity. We show that they are in very good agreement.
We present an analytical investigation of revival phenomena in the finite square-well potential. The classical motion, revival, and super-revival time scales are derived exactly for wave packets excited in the finite well. These time scales exhibit a richer dependence on wave-packet energy and on potential-well depth than has been found in other quantum systems: They explain, for example, the difficulties in exciting wave packets with strong classical features at the bottom of a finite well, or with clearly resolved super-revivals in a shallow well. In the proper regions of validity, the time scales predict the instances of wave-packet reformation extremely accurately. Revivals at the bottom of the well are explored as a ``universal'' limit of the general theory, which offers the clearest connection with the series of fractional and full revivals seen in the dynamics of the infinite square-well potential.
We study periodically driven one dimensional box in the presence of weak modulating force and provide analytical calculations for time of quantum revivals and explain via spatiotemporal behavior. Presently introduced analysis can be applied to calculate the time of revivals in the system subject to any periodically changing weak external driving force. Spatiotemporal behavior provides the modified numerical values of revival times which have good agreement with our analytical results. Moreover, spatiotemporal evolution shows absence of symmetry beyond a revival time, which contributes to change of slope in subsequent canals and ridges.
We present a generic treatment of wave-packet revivals for quantum-mechanical systems. This treatment permits a classification of certain ideal revival types. For example, wave packets for a particle in a one-dimensional box are shown to exhibit perfect revivals. We also examine the revival structure of wave packets for quantum systems with energies that depend on two quantum numbers. Wave packets in these systems exhibit quantum beats in the initial motion as well as new types of long-term revivals. As an example, we consider the revival structure of a particle in a two-dimensional box.
The idea of wave packet, that clarified many conceptual difficulties, has more recently been revisited to address interferometric and dynamical issues.Present work focuses on the spatio-temporal dynamics of the quantum wave packets in the Fermi-Ulam Accelerator Model.The specific system considered here is a one-dimensional box under periodic oscillations at one boundary. Such a seemingly simple system can be mapped to non-trivial problems like BEC trapped in an oscillating cavity or dynamics of atomic wave packet in different potential in presence or absence of gravitational field.The system is shown to display rich dynamical features including transitions from regular to stochastic and further to chaotic domain in classical phase space. We have obtained the approximated eigen-value solutions of the problem through time dependent Schrodinger’s equation under rescaling transformations that effectively replace moving boundary with a static one. Consequently the running time becomes a nonlinear function of the continuously changing boundary.Based on these solutions, revivals or the resurrection of the original wave packet, in the driven and un-driven Fermi-Ulam model are presented. Such revivals correspond to the constructive interference whereas the wave packet’s collapse owes to destructive interference. Furthermore, interplay of constructive and destructive interferences also readily mimics the fractional revivals.It is further demonstrated that initial position and momentum play a highly crucial role in the emergent revival and fractional revival patterns.It is shown that the symmetries exhibited by such structures in un-driven systems are usually lost under driven conditions and the revival/fractional revival times can only be predicted through a transcendental equation whose solution mainly depends on the strength of the external driving force.In present thesis we have proposed a comprehensive mathematical technique to calculate these hitherto unpredictable times.In addition, a graphical method is also suggested to observe these asymmetric revival/fractional revival times.This apparent asymmetry, do display an overall periodicity over a comparatively long span covering many times of revivals.It is found that whenever the ratio between the period of wall oscillation and quantum revival time `α/β' is a rational number then the revivals time shows a periodicity after β number of revival times over α wall oscillation. Standard tools of autocorrelation and quantum carpets, a consequence of quantum interference, have been employed for visual manifestation of these revivals.These carpets amply demonstrate the explicit asymmetric shifting of revival times for the driven case.These micro changes exhibited in quantum carpet morphologies consequently yield amended phase velocities associated with the canals and ridges in the driven framework.Furthermore, to grasp the quantum phase space dynamics in its entirety, we have also utilized the more sophisticated techniques of the Wigner function and information entropy.It is demonstrated mathematically that revivals not apparent via autocorrelation become prominent in the Wigner phase space description along with the appearance of smooth mini wave packets in the fractional revivals.These interference patterns emerge due to the state vector formed through the linear combination of Gaussians with different positions and momenta having pure Gaussian as well as oscillatory terms. Moreover, the information entropy, like the Wigner function also predicts the revivals when and where they happen.Thus a rich pattern of fractional revivals, including some altogether new ones, are observed through this tool in both driven and un-driven situations.Finally we have elucidated various scenarios including cold atoms in magneto-optical traps where such a system can be experimentally realized along with the suggestions for future work in this specific domain