394
10 Time-Periodic Quantum Systems
In recent years, studies have been done on kicked Rydberg-type atoms that
further demonstrate some of the types of phenomena discussed in this chapter.
For example, Rydberg atoms that are kicked by short periodic pulses have been
found to exhibit dynamical stabilization of the electron (Reinhold et al. 1997)
and transient localization of the electron (Stokely et al. 2002). Also quasi onedimensional Rydberg atoms have been created with very high quantum number
(n ∼ 350) (Stokely et al. 2003).
The Fermi accelerator model (see Sect. 3.9), which consists of a ball bouncing
between a fixed wall and an oscillating wall, is a periodic time-dependent system,
although of a slightly different type than considered so far in this chapter. Jose and
Cordery (1986) have quantized the Fermi accelerator model for a particular form of
wall oscillation and have studied its Floquet spectral statistics. They find a transition
from Poisson-like to Wigner-like spectral statistics as they vary a parameter of the
system.
As we have seen in this chapter, Floquet theory provides a powerful tool for
describing the photon structures that can form when time-periodic fields interact
with nonlinear systems. As was shown in Li and Reichl (1999, 2000); Martinez
and Reichl (2001); Emmanouilidou and Reichl (2002), a Floquet scattering matrix
can be constructed to study scattering processes in the presence of strong timeperiodic fields. The construction of a Floquet scattering matrix allows one to use
the concepts of scattering theory to analyze scattering processes that involve the
emission and absorption of large numbers of photons. This theory allows one
to analyze transmission probabilities, formation of quasibound states, effects of
evanescent modes, and delay times. It has been applied to electron transport in
heterostructures and the scattering of electrons from atomic systems in the presence
of radiation fields.
In this chapter, we have focused on the behavior of quantum systems driven by
time-periodic forces. An introduction to the problem of control of quantum systems
with nonperiodic external forces can be found in Bayfield (1999).
In Sect. 10.3.5, we showed numerical evidence of KAM behavior in quantum
systems and derived a renormalization map that had a stable manifold, again
indicating quantum KAM behavior. We should note that several years ago Hose and
Taylor (1983); Hose et al. (1984) proposed a criterion, based on perturbation theory,
to determine if states of a perturbed nonintegrable Hamiltonian can be assigned a
full set of quantum numbers (equal to the number of degrees of freedom) given
that the unperturbed Hamiltonian is integrable. Such a state would be a quantum
KAM state. However, the Hose-Taylor criterion is limited by the fact that it is a
perturbation scheme and cannot fully account for the effect of quantum nonlinear
resonances (see Ramaswamy (1984) for further discussion and Zheng and Reichl
(1987) for application to the Floquet states of microwave-driven hydrogen).
10 Time-Periodic Quantum Systems
In recent years, studies have been done on kicked Rydberg-type atoms that
further demonstrate some of the types of phenomena discussed in this chapter.
For example, Rydberg atoms that are kicked by short periodic pulses have been
found to exhibit dynamical stabilization of the electron (Reinhold et al. 1997)
and transient localization of the electron (Stokely et al. 2002). Also quasi onedimensional Rydberg atoms have been created with very high quantum number
(n ∼ 350) (Stokely et al. 2003).
The Fermi accelerator model (see Sect. 3.9), which consists of a ball bouncing
between a fixed wall and an oscillating wall, is a periodic time-dependent system,
although of a slightly different type than considered so far in this chapter. Jose and
Cordery (1986) have quantized the Fermi accelerator model for a particular form of
wall oscillation and have studied its Floquet spectral statistics. They find a transition
from Poisson-like to Wigner-like spectral statistics as they vary a parameter of the
system.
As we have seen in this chapter, Floquet theory provides a powerful tool for
describing the photon structures that can form when time-periodic fields interact
with nonlinear systems. As was shown in Li and Reichl (1999, 2000); Martinez
and Reichl (2001); Emmanouilidou and Reichl (2002), a Floquet scattering matrix
can be constructed to study scattering processes in the presence of strong timeperiodic fields. The construction of a Floquet scattering matrix allows one to use
the concepts of scattering theory to analyze scattering processes that involve the
emission and absorption of large numbers of photons. This theory allows one
to analyze transmission probabilities, formation of quasibound states, effects of
evanescent modes, and delay times. It has been applied to electron transport in
heterostructures and the scattering of electrons from atomic systems in the presence
of radiation fields.
In this chapter, we have focused on the behavior of quantum systems driven by
time-periodic forces. An introduction to the problem of control of quantum systems
with nonperiodic external forces can be found in Bayfield (1999).
In Sect. 10.3.5, we showed numerical evidence of KAM behavior in quantum
systems and derived a renormalization map that had a stable manifold, again
indicating quantum KAM behavior. We should note that several years ago Hose and
Taylor (1983); Hose et al. (1984) proposed a criterion, based on perturbation theory,
to determine if states of a perturbed nonintegrable Hamiltonian can be assigned a
full set of quantum numbers (equal to the number of degrees of freedom) given
that the unperturbed Hamiltonian is integrable. Such a state would be a quantum
KAM state. However, the Hose-Taylor criterion is limited by the fact that it is a
perturbation scheme and cannot fully account for the effect of quantum nonlinear
resonances (see Ramaswamy (1984) for further discussion and Zheng and Reichl
(1987) for application to the Floquet states of microwave-driven hydrogen).