Quantum Recurrences in Periodically Driven Systems

The phenomena of recurrence or revival is a beautiful combination of classical mechanics, wave mechanics, and quantum laws. A wave packet evolves over a short period of time in a potential, initially following classical mechanics. It spreads while moving along its classical trajectory, however rebuilds itself after a classical period. It follows wave mechanics in its long time evolution and gradually observes a collapse. However, the discreteness of quantum mechanics leads to the restoration and restructuring of the wave packet. In one degree of freedom systems the phenomena of quantum revivals are well studied both theoretically and experimentally. The quantum revivals were first studied in cavity quantum electrodynamics [1–3]. Recently, the existence of revivals has been investigated in atomic [4–11] and molecular [12–15] wave packet evolution. The periodically driven quantum systems [16–18], and two-degree-of-freedom systems such as stadium billiard [19] indicate the presence of quantum revivals in higher dimensional systems. Latter, it is proved that the recurrence or revival phenomenon is a generic property of the one degree of freedom periodically driven quantum systems [20]. In the present contribution we calculate the classical period and quantum revival time for the driven systems. Moreover, we calculate their interdependence for different dynamical regimes. Latter we explain these interdependences for the power law potentials. The layout of the paper is as follows: In Sec. II, we write general Hamiltonian for the periodically driven time dependent systems. In Sec. III, we calculate the quasi-energy eigen functions and quasi-energies for these systems. In Sec. IV we calculate the classical period and quantum recurrence time and write their interdependence in Sec. V. We dedicate section VI for a discussion of this interdependence in power law potentials.