240
8 Manifestations of Chaos in Quantum Scattering Processes
using random matrix theory (Wigner 1955, 1959). As we show in Sect. 8.2, this
explanation for the lack of close spacings was found to agree very well with data
from some nuclear scattering experiments.
In the late 1970s, the field of “quantum chaos” underwent a transformation
when two seemingly different branches of physics, random matrix theory and chaos
theory, merged. Contact between random matrix theory and chaos theory occurred
when numerical studies of the statistical properties of the quantized chaotic billiards
showed agreement with some nuclear scattering data (see, for example, McDonald
and Kaufman 1979). This led to the realization that one might see signatures of
chaos in nuclear scattering data that involved moderately high-energy nuclear states,
and it opened new directions for the application of quantum chaos theory in open
quantum systems.
The theory that has been most widely used to study the statistical properties of
scattering processes, and their relation to the underlying Hamiltonian, is the reaction
matrix theory of scattering developed by Wigner and Eisenbud (1947). The original
version of this theory was used to describe nuclear scattering processes. The basic
idea was to separate the spatial region occupied by the nucleus (the reaction region
in which the dynamics is largely unknown) from the asymptotic scattering region,
for which the dynamics is completely known. This formalism then allows one to
write the scattering matrix (S-matrix) explicitly in terms of a Hamiltonian for the
reaction region and as a function of the incident particle energy. In Sect. 8.3, we use
projection operators to derive the Wigner–Eisenbud expression for the scattering
matrix, and then apply it to the scattering of quantum particles in waveguides.
One of the most important quantities that can be derived from the S-matrix is
the Wigner–Smith delay time, and we discuss its properties in detail in Sect. 8.4.
The Wigner–Smith delay time is a measure of the time that a particle is delayed
in the reaction region (relative to the time it would take to traverse the reaction
region if no scattering processes were present). If we are given a scattering matrix,
which is a unitary matrix, its eigenvalues will be complex numbers that lie on the
unit circle. To each eigenvalue we can associate an eigenphase. The slopes of these
eigenphases, when plotted as a function of energy, are called the partial delay times.
Their average value, at any given energy, is called the Wigner–Smith delay time. In
the neighborhood of energies for which the S-matrix has complex energy poles, the
delay times can become very large.
In Sect. 8.5, we use the Wigner–Eisenbud theory to obtain the scattering matrix
and delay times for particles propagating in waveguides with reaction regions that
contain ripple cavities. The ripple cavities can exhibit a variety of behaviors, ranging
from nonlinear resonance structures and KAM orbits to fully developed chaos, and
these different behaviors affect the scattering properties of the system.
The reaction matrix theory of Wigner and Eisenbud allows a connection between
the Hamiltonian describing the reaction region and the S-matrix for the scattering
process. One of the goals of random matrix theory has been to determine what
scattering properties result if the reaction region is described by Hamiltonians
from the Gaussian orthogonal ensemble (GOE). Can one use the ensemble of
Hamiltonians obtained from the GOE to build an ensemble of S-matrices that belong
8 Manifestations of Chaos in Quantum Scattering Processes
using random matrix theory (Wigner 1955, 1959). As we show in Sect. 8.2, this
explanation for the lack of close spacings was found to agree very well with data
from some nuclear scattering experiments.
In the late 1970s, the field of “quantum chaos” underwent a transformation
when two seemingly different branches of physics, random matrix theory and chaos
theory, merged. Contact between random matrix theory and chaos theory occurred
when numerical studies of the statistical properties of the quantized chaotic billiards
showed agreement with some nuclear scattering data (see, for example, McDonald
and Kaufman 1979). This led to the realization that one might see signatures of
chaos in nuclear scattering data that involved moderately high-energy nuclear states,
and it opened new directions for the application of quantum chaos theory in open
quantum systems.
The theory that has been most widely used to study the statistical properties of
scattering processes, and their relation to the underlying Hamiltonian, is the reaction
matrix theory of scattering developed by Wigner and Eisenbud (1947). The original
version of this theory was used to describe nuclear scattering processes. The basic
idea was to separate the spatial region occupied by the nucleus (the reaction region
in which the dynamics is largely unknown) from the asymptotic scattering region,
for which the dynamics is completely known. This formalism then allows one to
write the scattering matrix (S-matrix) explicitly in terms of a Hamiltonian for the
reaction region and as a function of the incident particle energy. In Sect. 8.3, we use
projection operators to derive the Wigner–Eisenbud expression for the scattering
matrix, and then apply it to the scattering of quantum particles in waveguides.
One of the most important quantities that can be derived from the S-matrix is
the Wigner–Smith delay time, and we discuss its properties in detail in Sect. 8.4.
The Wigner–Smith delay time is a measure of the time that a particle is delayed
in the reaction region (relative to the time it would take to traverse the reaction
region if no scattering processes were present). If we are given a scattering matrix,
which is a unitary matrix, its eigenvalues will be complex numbers that lie on the
unit circle. To each eigenvalue we can associate an eigenphase. The slopes of these
eigenphases, when plotted as a function of energy, are called the partial delay times.
Their average value, at any given energy, is called the Wigner–Smith delay time. In
the neighborhood of energies for which the S-matrix has complex energy poles, the
delay times can become very large.
In Sect. 8.5, we use the Wigner–Eisenbud theory to obtain the scattering matrix
and delay times for particles propagating in waveguides with reaction regions that
contain ripple cavities. The ripple cavities can exhibit a variety of behaviors, ranging
from nonlinear resonance structures and KAM orbits to fully developed chaos, and
these different behaviors affect the scattering properties of the system.
The reaction matrix theory of Wigner and Eisenbud allows a connection between
the Hamiltonian describing the reaction region and the S-matrix for the scattering
process. One of the goals of random matrix theory has been to determine what
scattering properties result if the reaction region is described by Hamiltonians
from the Gaussian orthogonal ensemble (GOE). Can one use the ensemble of
Hamiltonians obtained from the GOE to build an ensemble of S-matrices that belong