Quantum nonlinear resonance and quantum chaos in Aharonov-Bohm oscillations in mesoscopic semiconductor rings

We consider Aharonov-Bohm oscillations in a mesoscopic semiconductor ring threaded by both a constant magnetic flux and a time-dependent, resonant magnetic field with one or two frequencies. Working in the ballistic regime, we establish that the theory of ''quantum nonlinear resonance'' applies, and thus that this system represents a possible solid-state realization of ''quantum nonlinear resonance'' and ''quantum chaos.'' In particular, we investigate the behavior of the time-averaged electron energy at zero temperature in the regimes of (i) an isolated quantum nonlinear resonance and (ii) the transition to quantum chaos, when two quantum nonlinear resonances overlap. The time-averaged energy exhibits sharp resonant behavior as a function of the applied constant magnetic flux, and has a staircase dependence on the amplitude of the external time-dependent field. In the chaotic regime, the resonant behavior exhibits complex structure as a function of flux and frequency. We compare and contrast the quantum chaos expected in these mesoscopic ''solid-state atoms'' with that observed in Rydberg atoms in microwave fields, and discuss the prospects for experimental observation of the effects we predict.