Nonlinear dynamical systems with only one space dimension can undergo a transition to chaos if they are driven by a time-periodic force. In this chapter, we focus on the dynamics of quantum systems with time-periodic Hamiltonians as they undergo a transition from a regime in which they exhibit integrable-like behavior to a regime where they exhibit the manifestations of chaos. Time-periodic systems have discrete time-translation invariance, and their dynamics is constrained by conservation laws. For such systems, the Floquet energy (also called quasienergy) is a constant of the motion even in the presence of a transition to chaos in the underlying classical phase space. For intense time-periodic fields, the Floquet states appear to describe coherent photon structures that result from the interaction between the driving field and the nonlinear forces of the driven system.
KeywordsStable Manifold Primary Resonance Microwave Field Localization Length Principal Quantum Number
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