Theoretical and computational investigations of the quantum dynamics of a particle in a square-well potential driven by a sinusoidal force are carried out, with particular attention to the issue of quantum-classical correspondence or noncorrespondence in a classically chaotic square-well system. Central to our analysis is the concept of nonlinear resonance which has been found useful in many past theoretical studies of classical chaos. In particular, the structure and characteristics of nonlinear resonances in a quantum square-well system are compared in detail with those in the corresponding classical system and are shown to reveal clearly the well-known phenomenon of the quantum suppression of classical chaos.
Taking advantage of the fact that the Hamiltonian for our driven square-well system is periodic in time, we employ the Floquet approach and obtain quasienergies and Floquet states by numerical computation. From investigation of the Husimi phase-space plots of the Floquet states, useful information concerning the structure and characteristics of quantum nonlinear resonances is obtained. The information so obtained in turn provides the basic knowledge necessary to understand the quantum dynamical behavior of the system being considered.
As for the potential, both the single and double square-well potentials are considered. For the case of the driven single square-well system, it is found, through theoretical analysis and numerical computation of quasienergies and Floquet states, that the width of quantum resonances is generally smaller than that of the corresponding classical resonances and that high-order high-period resonances play a weaker role in the quantum system than in the corresponding classical system. All these observations are consistent with the notion of the quantum suppression of classical chaos.
For the case of the driven double square-well system, two dynamical aspects absent in the single square-well system are studied; namely the effect of the symmetry of the potential on the quantum dynamical behavior and quantum tunneling of the particle from one well to the other.
The effect of the symmetry of the potential is studied by comparing the time development of the system when the barrier dividing the wells is place on center and off center. Our numerical computation of the Schr$ö$dinger equation indicates that the probability distribution tends to spread less when the double square-well potential is symmetric than when it is not. This is consistent with the finding that the critical amplitude of the driving force at which resonances overlap and the onset of classical chaos occurs takes on a higher value when the potential is symmetric. No indication of chaos is seen, however, from the quantum dynamical behavior in either the symmetric or the nonsymmetric case.
For our study of quantum tunneling in the double square-well system, the attention is focused on the effect of the driving force on the tunneling probability. The potential chosen for this study is a symmetric double square well with a tall and thin barrier placed at the center. It is shown that the tunneling rate can be greatly enhanced or reduced by appropriately selecting the amplitude, frequency and phase of the driving force. A theoretical analysis of such a control of tunneling is given in the framework of the Floquet theory, and the results of analysis are interpreted in terms of the structure and characteristics of quantum nonlinear resonances.
