Strength Distributions and Symmetry Breaking in Coupled Microwave Billiards
B. Dietz,
1
T. Guhr,
2
H. L. Harney,
3
and A. Richter
1,
*
1
Institut fu ¨r Kernphysik, Technische Universita ¨t Darmstadt, D-64289 Darmstadt, Germany
2
Matematisk Fysik, LTH, Lunds Universitet, S-22100 Lund, Sweden
3
Max-Planck-Institut fu ¨r Kernphysik, D-69029 Heidelberg, Germany
(Received 20 January 2006; published 26 June 2006)
Flat microwave cavities can be used to experimentally simulate quantum mechanical systems. By
coupling two such cavities, we study the equivalent to symmetry breaking in quantum mechanics. As the
coupling is tunable, we can measure resonance strength distributions as a function of the symmetry
breaking. We analyze the data by employing a qualitative model based on random matrix theory and show
that the results derived from the strength distribution are consistent with those previously obtained from
spectral statistics.
DOI: 10.1103/PhysRevLett.96.254101 PACS numbers: 05.45.Mt
The breaking of quantum mechanical symmetries rep-
resents a prominent object of study in a rich variety of
systems, ranging from high energy to condensed matter
physics. Sometimes it is possible to determine the size of
the symmetry breaking by analyzing statistical observ-
ables. We mention parity violation [1], the breaking of
atomic and molecular symmetries [2,3], and isospin mix-
ing [4 – 9] in nuclei. Symmetry breaking influences the
spectral statistics as well as the wave function statistics,
such as the distributions of partial widths and transition
matrix elements. While symmetry breaking cannot be con-
trolled or tuned in nuclei, atoms, and molecules, this was
possible for elastomechanical resonances in quartz crystals
by successively breaking crystal symmetries [10]. An anal-
ogy to the nuclear case of symmetry breaking was given by
two coupled microwave cavities [11] with a tunable cou-
pling. The statistical properties of both systems are fully
equivalent to those of a quantum system [12,13]. Whereas
these two investigations focused on the spectral statistics,
width distributions for two different types of resonances in
elastic aluminum plates were studied in Ref. [14] for a
varying degree of mixing.
Here we present experimental results on the distribution
of products of partial widths in two coupled chaotic mi-
crowave cavities. First, we measure the distribution for
different couplings. It can be normalized such that it de-
pends only on the symmetry breaking. As this is different
from Ref. [14], we can apply a qualitative statistical model
which extends the model of Ref. [8] in order to, second,
extract the size of the symmetry breaking from the data.
We then show, third, that the symmetry breaking thereby
obtained is consistent with the one previously found from
the spectral statistics.
In the experiment, we used two flat cylindrical micro-
wave cavities having the shape of the quarter of a
Bunimovich stadium [11]. In both resonators (see inset in
Fig. 1), the radius of the quarter circle is 0:2m. The ratios
of the length of the rectangular part to the radius are
1
1 and
2
1:8, respectively. Therefore, the level densities
increase with different slopes as functions of frequency
[15,16]. The cavities were put on top of each other and
circular holes, 4 mm in diameter, were drilled through the
walls of both resonators. The coupling was realized by a
niobium pin, 2 mm in diameter, penetrating through the
holes into both resonators. Because of the ring-shaped gap
between the niobium pin and the hole surface, the pin acts
like an antenna, which in the experimental frequency range
supports one TEM mode. The coupling is controlled by the
penetration depth [11].
Information about the partial widths is obtained from
transmission spectra. At a given frequency f, the relative
power transmitted from antenna a to antenna b is propor-
tional to the absolute square of the scattering matrix ele-
ment, P
out;b
=P
in;a
jS
ab
fj
2
. For sufficiently isolated
resonances, one has
S
ab
f
ab
i
a
b
q
f f
i
2
(1)
for f close to the frequency f
of the th resonance. The
quantities
a
and
b
are the partial widths related to the
0.1 0.15 0.2 0.25
λ
0
0.1
0.2
0.3
0.4
0.5
η(λ)
FIG. 1. Size of the tunable coupling between the two cavities
sketched in the inset. The function enters Eq. (6).
PRL 96, 254101 (2006)
PHYSICAL REVIEW LETTERS
week ending
30 JUNE 2006
0031-9007= 06=96(25)=254101(4) 254101-1 © 2006 The American Physical Society