8.8 Conclusions
289
8.8 Conclusions
In this chapter, we have focused on the scattering properties of open quantum
systems for which there is a fairly sharp spatial separation between the reaction
region and the asymptotic scattering region. As we have seen, for such systems a
formalism exists that allows the comparison between the scattering properties of
systems with a classically chaotic reaction region and the predictions of random
matrix theory. We have limited our discussion to the case of systems that are
rotationally invariant and are invariant under time translation.
In this chapter, we have not attempted to address the question of integrability as
regards scattering processes. This has been discussed in some detail in a review
article by Jung and Seligman (1997), where they ask the question: Given an
integrable Hamiltonian, under what conditions will the S-matrix be integrable?
There has been a considerable amount of work attempting to compare the
scattering properties of systems which are chaotic with the scattering properties
of systems that exhibit random disorder. The formalisms used to examine these two
types of systems are very similar. They both use supersymmetry techniques. There
are some excellent review articles that address these topics and that will lead the
reader to a vast body of literature. These include (Alhassid 1997; Mucciolo et al.
1997; Beenakker 2000; Guhr et al. 1998).
By considering only scattering processes which are time reversal and rotationally
invariant, we have excluded scattering processes that involve magnetic fields. Yet
there has been a considerable amount of experimental and theoretical work devoted
to the effect of magnetic fields on conduction in chaotic and in near-integrable open
quantum systems. This was stimulated by work of Jalabert et al. (1990), who showed
that conduction fluctuations may contain information about the shape of cavities
(Jensen 1991; Marcus et al. 1992, 1993; Chang et al. 1994). Ketzmerick (1996)
showed that the hierarchical structure of nonlinear resonances in the classical phase
space can give rise to fractal structure of conductance fluctuations when plotted as
a function of applied magnetic field. Magnetic fields provide probes that are easily
varied in the laboratory. Accompanying this has been considerable work computing
averages of scattering parameters using supersymmetry techniques based on the
Gaussian unitary ensemble. An excellent discussion of supersymmetry using GUE
can be found in (Haake 2001). Reviews of RMT predictions of scattering properties
of systems with broken time reversal symmetry can be found in (Fyodorov and
Sommers 1997) and (Guhr et al. 1998).
As we have seen in Chap. 7, the electromagnetic fields in flat microwave cavities
can be made to follow a dynamics identical to that of the Schrödinger equation.
Thus it is possible to use microwave cavities to check the predictions of RMT
for quantum systems. Interesting studies of the behavior of poles of the S-matrix
and their relation to cavity dynamics can be found in (Persson et al. 1998, 2000;
Stöckmann et al. 2002), and (Okolowicz et al. 2003). Acoustic cavities have also
been used to look for the effects of container shapes on the spectral properties
of acoustic resonances, and transitions in spectral statistics have been observed in
289
8.8 Conclusions
In this chapter, we have focused on the scattering properties of open quantum
systems for which there is a fairly sharp spatial separation between the reaction
region and the asymptotic scattering region. As we have seen, for such systems a
formalism exists that allows the comparison between the scattering properties of
systems with a classically chaotic reaction region and the predictions of random
matrix theory. We have limited our discussion to the case of systems that are
rotationally invariant and are invariant under time translation.
In this chapter, we have not attempted to address the question of integrability as
regards scattering processes. This has been discussed in some detail in a review
article by Jung and Seligman (1997), where they ask the question: Given an
integrable Hamiltonian, under what conditions will the S-matrix be integrable?
There has been a considerable amount of work attempting to compare the
scattering properties of systems which are chaotic with the scattering properties
of systems that exhibit random disorder. The formalisms used to examine these two
types of systems are very similar. They both use supersymmetry techniques. There
are some excellent review articles that address these topics and that will lead the
reader to a vast body of literature. These include (Alhassid 1997; Mucciolo et al.
1997; Beenakker 2000; Guhr et al. 1998).
By considering only scattering processes which are time reversal and rotationally
invariant, we have excluded scattering processes that involve magnetic fields. Yet
there has been a considerable amount of experimental and theoretical work devoted
to the effect of magnetic fields on conduction in chaotic and in near-integrable open
quantum systems. This was stimulated by work of Jalabert et al. (1990), who showed
that conduction fluctuations may contain information about the shape of cavities
(Jensen 1991; Marcus et al. 1992, 1993; Chang et al. 1994). Ketzmerick (1996)
showed that the hierarchical structure of nonlinear resonances in the classical phase
space can give rise to fractal structure of conductance fluctuations when plotted as
a function of applied magnetic field. Magnetic fields provide probes that are easily
varied in the laboratory. Accompanying this has been considerable work computing
averages of scattering parameters using supersymmetry techniques based on the
Gaussian unitary ensemble. An excellent discussion of supersymmetry using GUE
can be found in (Haake 2001). Reviews of RMT predictions of scattering properties
of systems with broken time reversal symmetry can be found in (Fyodorov and
Sommers 1997) and (Guhr et al. 1998).
As we have seen in Chap. 7, the electromagnetic fields in flat microwave cavities
can be made to follow a dynamics identical to that of the Schrödinger equation.
Thus it is possible to use microwave cavities to check the predictions of RMT
for quantum systems. Interesting studies of the behavior of poles of the S-matrix
and their relation to cavity dynamics can be found in (Persson et al. 1998, 2000;
Stöckmann et al. 2002), and (Okolowicz et al. 2003). Acoustic cavities have also
been used to look for the effects of container shapes on the spectral properties
of acoustic resonances, and transitions in spectral statistics have been observed in