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arXiv:quant-ph/0602236 · Physical Review A

Quantum revivals in periodically driven systems close to nonlinear resonances

Farhan Saif, Fortunato

Published Dec 10, 2001• 24 citations
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Abstract

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.

DOI
10.1103/physreva.65.013401

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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 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.
Quantum revivals are investigated for the dynamics of an atom in a driven gravitational cavity. It is demonstrated that the external driving field influences the revival time significantly. Analytical expressions are presented which are based on second-order perturbation theory and semiclassical secular theory. These analytical results explain the dependence of the revival time on the characteristic parameters of the problem quantitatively in a simple way. They are in excellent agreement with numerical results.
Recently it has been shown that interparticle interactions generically destroy dynamical localization in periodically driven systems, resulting in diffusive transport and heating. In this Rapid Communication we rigorously construct a family of interacting driven systems which are dynamically localized and effectively decoupled from the external driving potential. We show that these systems exhibit tunable periodic or quasiperiodic revivals of the many-body wave function and thus of all physical observables. By numerically examining spinless fermions on a one-dimensional lattice we show that the analytically obtained revivals of such systems remain stable for finite systems with open boundary conditions while having a finite lifetime in the presence of static spatial disorder. We find this lifetime to be inversely proportional to the disorder strength.
We have investigated quantum revivals of wave packets in a one-dimensional infinite square well potential containing a $\ensuremath{\delta}$ well in the middle. The time-dependent Schr\"odinger equation for this composite potential admits formally exact solutions. We present analytical results for revival properties in three physically motivated approximations: wave packets containing eigenstates with large numbers in the presence of an arbitrary $\ensuremath{\delta}$ well, ``shallow'' and ``deep'' $\ensuremath{\delta}$ wells. Analytical results in the case of a ``shallow'' $\ensuremath{\delta}$ well have been tested numerically.
In this paper we discuss the quantum dynamics and fractional quantum revivals of an integrable nonlinear system, consisting of an atom bouncing vertically from an evanescent field, for two cases with the simplified infinite-potential and the more practical exponential potential, respectively. We study the two cases separately, then contrast and compare the results and reach the conclusion that provided the starting position of the atoms is not too close to the reflecting surface supporting the evanescent wave (this condition is always satisfied in present experiments in this field), the two cases will produce the same results. This means that the idealized infinite potential is a good approximation to the more realistic exponential potential. Because the quantum analysis of the infinite-potential case is quite simple and straighforward (since its Schr\"odinger equation has analytical solutions), this will greatly simplify the quantum analysis of the more complicated exponential potential case and hence has practical significance.