In this thesis, we study one of the simplest systems that exhibit resonance overlap and classical chaos and that can be quantally realized: a nonrelativistic one-dimensional simple harmonic oscillator driven by a space-time varying force. Theoretical study of nonlinear resonance and quantum chaos in the driven oscillator is carried out by using both classical and quantum second-order perturbation theories, with particular attention to quantum-classical correspondence in generation of nonlinear resonance and classical chaos due to resonance overlap in a driven oscillator.
Chaos signifies irregular or aperiodic behavior that arises from nonlinearity in equations of motion governing a deterministic system. It is essentially distinguishable from noisy behavior caused by stochastic forces such as thermal fluctuations. In the sense that a harmonic oscillator presents a standard linear system, it cannot be allowed to behave chaotically when it is subject to a time-varying force. In this thesis, we show that even the simple harmonic oscillator can exhibit nonlinear resonance and chaos if it is driven by an external force that varies in both time and space. A special type of space-time dependent force that varies periodically both in time and in space applied to a simple harmonic oscillator is encountered frequently in plasma physics in relation to the cyclotron motion of a charged particle interacting with an electromagnetic wave. Here we consider a more general class of a space-time varying force and show, using canonical perturbation theory, that the force generates a series of nonlinear resonances in the phase space of oscillator. As is well known, when the neighboring resonances overlap, chaos occurs.
In order for an oscillator to generate nonlinear resonance and to exhibit chaos in phase space, it is required that the frequency of oscillation varies with respect to energy. The reason why the time-driven simple harmonic oscillator can behave chaotically when it moves at relativisitic velocities is because its frequency is no longer constant in the relativistic region of energy. In the nonrelativistic region, the oscillation frequency of a simple harmonic oscillator remains constant even if a time-varying force is applied. What we have shown in this thesis is that the frequency of the nonrelativistic simple harmonic oscillator is shifted and becomes energy dependent in the presence of a spatially varying time-dependent external force. The frequency shift of the same nature occurs when a time-varying force is applied to the relativistic oscillator of constant period.
We also investigate quantum-classical correspondence in the driven harmonic oscillator, by using quantum unitary perturbation theory. In order for an oscillator to exhibit quantum nonlinear resonance, it needs to have a non-equidistant spectrum. Second-order quantum perturbation calculation shows that the eigenenergies of a simple harmonic oscillator, which are equally spaced in the absence of the external driving field, are shifted by a spatially varying time-dependent external field, in analogy to the classical results obtained by perturbation calculation. Eventually, quantum nonlinear resonances are generated and Floquet states appear to be delocalized in quantum phase space when the neighboring quantum nonlinear resonances overlap. The difference of the quantum and classical results is that the critical value at which the overlap of quantum nonlinear resonances occurs is larger than that of the classical limit. The fundamental reason why the difference occurs is because any structure less than $\bar{h}$ in phase space area is not recognized in the quantum world.