Completely classical behavior is very different from completely quantum-mechanical behavior, particularly for nonlinear or chaotic systems, even though the transition between the two happens as a function of controllable parameters, such as the size of the system or environmental effects. I report on recent work exploring this multi-parameter transition. The first set of results is on the behavior of calculated quantum Lyapunov exponents Lambda for a Duffing oscillator system as a function of effective action beta as well as the system damping parameter Gamma. In general Lambdas decrease as beta increases (chaos decreases as the system becomes more quantal, as expected). However, we identify anomalous regions where Lambdas increase with beta, including going from negative to positive with increasing beta; and also regions where the quantum results do not tend smoothly to the classical results. All anomalous results correspond to windows of regularity embedded in a larger chaotic parameter regime, which inverts the usual paradigm: The classically regular behavior is the most challenging for quantum-classical correspondence. I also report on progress on various other projects including (a) studying how classical control algorithms work in controlling quantum systems across this parameter regime and (b) on the control of the quantum chaos.
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