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10 Time-Periodic Quantum Systems
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.
We begin in Sect. 10.2 with a discussion of Floquet theory as applied to
Schrödinger equations with time-periodic coefficients. We shall show that it is
possible to describe the time evolution of these quantum systems in terms of a map
(Floquet map) that connects the state at time t to the state at time t + T , where T is
the period of the driving force. The Floquet map can be constructed analytically for
the quantum delta-kicked rotor, but for most other systems it must be constructed
numerically.
In quantum systems, as in classical systems, the transition from integrablelike to nonintegrable behavior occurs because nonlinear resonances overlap and
destroy, locally, some good quantum numbers. Nonlinear resonances also exist in
quantum systems and play a similar role. In Sect. 10.3, we give examples of quantum
nonlinear resonances and show that their overlap causes a change in the Floquet
spectral statistics from Poisson-like to Wigner-like behavior.
In Sect. 10.4, we focus on the effect of time periodic forces on a particle in a
bounded system. The time periodic force induces chaos in the classical phase space
and, as that happens, the Floquet eigenvalues (quasienergies) begin to show level
repulsion. We also show how the structure of the underlying classical phase space
affects the radiation spectrum emitted from the driven particle.
Experimental confirmation of the existence of Floquet states in time-periodic
systems can be seen in recent cold atom optics tunneling experiments, in which
an optical lattice undergoes time-periodic modulation. It is shown that only a few
Floquet states govern large-scale oscillations in the momenta of the atoms that are
observed in the experiment. These results are described in Sect. 10.5.
In Sect. 10.6, we study the dynamical properties of the quantum delta-kicked
rotor, and derive an analytic expression for the Floquet map that governs the
dynamics. This system shows some of the rich variety of phenomena that can occur
in the quantum domain when the underlying classical system undergoes a transition
to chaos. The delta-kicked rotor is also of interest because the equation for Floquet
eigenstates is similar to the tight-binding model of solid state physics and the deltakicked rotor exhibits dynamic Anderson localization.
In Sect. 10.7 we describe some of the experimental results that were obtained
by Bayfield, Koch, and others for microwave-driven hydrogen, and we discuss the
theory that has been developed to explain their results. These experiments provide
evidence for the existence of quantum nonlinear resonances and, indeed, for the
existence of higher-order quantum resonances. Microwave-driven hydrogen provided the first system in which a new multi-photon nonlinear ionization mechanism
was observed.
10 Time-Periodic Quantum Systems
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.
We begin in Sect. 10.2 with a discussion of Floquet theory as applied to
Schrödinger equations with time-periodic coefficients. We shall show that it is
possible to describe the time evolution of these quantum systems in terms of a map
(Floquet map) that connects the state at time t to the state at time t + T , where T is
the period of the driving force. The Floquet map can be constructed analytically for
the quantum delta-kicked rotor, but for most other systems it must be constructed
numerically.
In quantum systems, as in classical systems, the transition from integrablelike to nonintegrable behavior occurs because nonlinear resonances overlap and
destroy, locally, some good quantum numbers. Nonlinear resonances also exist in
quantum systems and play a similar role. In Sect. 10.3, we give examples of quantum
nonlinear resonances and show that their overlap causes a change in the Floquet
spectral statistics from Poisson-like to Wigner-like behavior.
In Sect. 10.4, we focus on the effect of time periodic forces on a particle in a
bounded system. The time periodic force induces chaos in the classical phase space
and, as that happens, the Floquet eigenvalues (quasienergies) begin to show level
repulsion. We also show how the structure of the underlying classical phase space
affects the radiation spectrum emitted from the driven particle.
Experimental confirmation of the existence of Floquet states in time-periodic
systems can be seen in recent cold atom optics tunneling experiments, in which
an optical lattice undergoes time-periodic modulation. It is shown that only a few
Floquet states govern large-scale oscillations in the momenta of the atoms that are
observed in the experiment. These results are described in Sect. 10.5.
In Sect. 10.6, we study the dynamical properties of the quantum delta-kicked
rotor, and derive an analytic expression for the Floquet map that governs the
dynamics. This system shows some of the rich variety of phenomena that can occur
in the quantum domain when the underlying classical system undergoes a transition
to chaos. The delta-kicked rotor is also of interest because the equation for Floquet
eigenstates is similar to the tight-binding model of solid state physics and the deltakicked rotor exhibits dynamic Anderson localization.
In Sect. 10.7 we describe some of the experimental results that were obtained
by Bayfield, Koch, and others for microwave-driven hydrogen, and we discuss the
theory that has been developed to explain their results. These experiments provide
evidence for the existence of quantum nonlinear resonances and, indeed, for the
existence of higher-order quantum resonances. Microwave-driven hydrogen provided the first system in which a new multi-photon nonlinear ionization mechanism
was observed.