210
7 Bounded Quantum Systems
been analyzed in Graf et al. (1992) and Alt et al. (1999). In spite of these special
orbits, the classical stadium billiard is dominated by chaotic behavior.
The first experimental evidence that quantum billiards have spectral statistics
that are Poisson-like for integrable billiards and Wigner-like for chaotic billiards
was done in a microwave resonator by Stöckmann and Stein (1990) and Stöckmann
(1999). They constructed a microwave cavity in the shape of a stadium (actually
one quarter of it). The resonator cavity had brass walls perpendicular to the x − y
plane, which traced out the shape of a quarter stadium in the x − y plane. It
had flat walls parallel to the x − y plane, a distance d apart along the z-axis.
The parameters of the experimental cavity were d = 0.8 cm, a = 36 cm, and
r = 20 cm, where a is one-half the length of the straight side of the stadium
and r is the radius of the circular end of the stadium. For microwave frequencies
below f max = c/2d≈18 GHz, where c is the speed of light, the cavity is effectively
two-dimensional. Only one mode can exist between the flat walls because, as
discussed in Appendix H, the cavity will only support transverse magnetic modes
at these frequencies. For microwave frequencies below f max , the z-component of
the electric field satisfies a Schrödinger-like equation, Eq. (H.12), and therefore, the
resonances of the microwave field will have the same spectral statistics as the energy
eigenvalues of a quantum particle in a similar-shaped two-dimensional cavity.
The experiment of Stöckmann and Stein (1990) was done at room temperature,
where the skin depth for microwave radiation of frequency 10 GHz is of order
1.5 µm. This gives rise to absorption of microwave energy in the walls and
broadens the observed resonance widths. If this broadening exceeds the resonance
spacing, then not all resonances can be observed in the experiment. For the room
temperature experiment, the smallest distance between resonances that could be
resolved was about 3 MHz. In a typical measurement, a total of 1002 eigenvalues of
the microwave cavity were observed in the interval 0 < f < 18 GHz. Stockmann
and Stein estimated that about 15% of the expected resonances were not observed
either because they were too closely spaced and could not be resolved or because
they could not be detected due to the placement of the contacts used to measure the
radiation in the cavity. Nevertheless, they found fairly good qualitative agreement
between the spectral spacing distribution for the stadium billiard and the expected
result of random matrix theory. They also did the measurements for a rectangular
two-dimensional microwave cavity and found that the spectral spacing distribution
of resonances agreed qualitatively with a Poisson distribution, at least for larger
spacings.
In 1992, the group of Richter (Graf et al. 1992) repeated the experiment
using a quasi-two-dimensional superconducting niobium microwave resonator at
a temperature of 2 K. The dimensions of the cavity were the same as those of
Stockmann and Stein. When the walls become superconducting, the skin depth is
reduced by several orders of magnitude, and the absorption by the walls is similarly
reduced. The smallest spacing that could be resolved is of order 10 kHz. In Fig. 7.9,
we show data taken at room temperature and data taken at 2 K, where the walls
are superconducting. Resonance frequencies in the interval 0 < f < 17.5 GHz
were counted. At room temperature, where the walls are normal conductors, 898
7 Bounded Quantum Systems
been analyzed in Graf et al. (1992) and Alt et al. (1999). In spite of these special
orbits, the classical stadium billiard is dominated by chaotic behavior.
The first experimental evidence that quantum billiards have spectral statistics
that are Poisson-like for integrable billiards and Wigner-like for chaotic billiards
was done in a microwave resonator by Stöckmann and Stein (1990) and Stöckmann
(1999). They constructed a microwave cavity in the shape of a stadium (actually
one quarter of it). The resonator cavity had brass walls perpendicular to the x − y
plane, which traced out the shape of a quarter stadium in the x − y plane. It
had flat walls parallel to the x − y plane, a distance d apart along the z-axis.
The parameters of the experimental cavity were d = 0.8 cm, a = 36 cm, and
r = 20 cm, where a is one-half the length of the straight side of the stadium
and r is the radius of the circular end of the stadium. For microwave frequencies
below f max = c/2d≈18 GHz, where c is the speed of light, the cavity is effectively
two-dimensional. Only one mode can exist between the flat walls because, as
discussed in Appendix H, the cavity will only support transverse magnetic modes
at these frequencies. For microwave frequencies below f max , the z-component of
the electric field satisfies a Schrödinger-like equation, Eq. (H.12), and therefore, the
resonances of the microwave field will have the same spectral statistics as the energy
eigenvalues of a quantum particle in a similar-shaped two-dimensional cavity.
The experiment of Stöckmann and Stein (1990) was done at room temperature,
where the skin depth for microwave radiation of frequency 10 GHz is of order
1.5 µm. This gives rise to absorption of microwave energy in the walls and
broadens the observed resonance widths. If this broadening exceeds the resonance
spacing, then not all resonances can be observed in the experiment. For the room
temperature experiment, the smallest distance between resonances that could be
resolved was about 3 MHz. In a typical measurement, a total of 1002 eigenvalues of
the microwave cavity were observed in the interval 0 < f < 18 GHz. Stockmann
and Stein estimated that about 15% of the expected resonances were not observed
either because they were too closely spaced and could not be resolved or because
they could not be detected due to the placement of the contacts used to measure the
radiation in the cavity. Nevertheless, they found fairly good qualitative agreement
between the spectral spacing distribution for the stadium billiard and the expected
result of random matrix theory. They also did the measurements for a rectangular
two-dimensional microwave cavity and found that the spectral spacing distribution
of resonances agreed qualitatively with a Poisson distribution, at least for larger
spacings.
In 1992, the group of Richter (Graf et al. 1992) repeated the experiment
using a quasi-two-dimensional superconducting niobium microwave resonator at
a temperature of 2 K. The dimensions of the cavity were the same as those of
Stockmann and Stein. When the walls become superconducting, the skin depth is
reduced by several orders of magnitude, and the absorption by the walls is similarly
reduced. The smallest spacing that could be resolved is of order 10 kHz. In Fig. 7.9,
we show data taken at room temperature and data taken at 2 K, where the walls
are superconducting. Resonance frequencies in the interval 0 < f < 17.5 GHz
were counted. At room temperature, where the walls are normal conductors, 898