Parametric correlations of the energy levels of ray-splitting billiards
N. Savytskyy,
1,2
A. Kohler,
3
Sz. Bauch,
1
R. Blu
¨
mel,
4
and L. Sirko
1,2
1
Institute of Physics, Polish Academy of Sciences, Aleja Lotniko ´w 32/46, 02-668 Warszawa, Poland
2
College of Science, Aleja Lotniko ´w 32/46, 02-668 Warszawa, Poland
3
MATFORSK, Osloveien 1, 1430 Ås, Norway
4
Department of Physics, Wesleyan University, Middletown, Connecticut 06459-0155
Received 24 January 2001; published 24 August 2001
Parameter-dependent statistical properties of the spectra of ray-splitting billiards are studied experimentally
and theoretically. The autocorrelation functions c ( x ) and c
˜
( , x ) of level velocities as well as the generalized
conductance C (0) are calculated for two different classically chaotic ray-splitting billiards. Experimentally a
modified Sinai ray-splitting billiard is studied consisting of a thin microwave rectangular cavity with two
quarter-circle-shaped Teflon inserts. The length of the cavity serves as the experimentally adjustable parameter.
For the theoretical estimates of the parametric correlations we compute the quantum spectrum of a scaling
triangular ray-splitting billiard. Our experimental and numerical results are compared with each other and with
the predictions of random matrix theory.
DOI: 10.1103/PhysRevE.64.036211 PACS numbers: 05.45.Mt
A wide class of quantum chaotic systems depends on an
external parameter X, where X may, for instance, represent
the strength of an external field or the shape of the confining
perimeter of the quantum system. Upon the rescaling of X,
the correlation functions of the X-dependent energy levels
are expected to be universal 1–5. Of all possible correla-
tion functions, the velocity autocorrelation function takes
center stage. It has been studied theoretically in great detail
in cases where X is a magnetic field 1–7 or where X char-
acterizes the shape of a quantum billiard 8,9. The velocity
autocorrelation function has also been calculated for random
matrix dynamics 10–13. Given the enormous theoretical
interest in the velocity autocorrelation function, it is surpris-
ing that experiments addressing its measurement are scarce.
We are aware of only four experimental investigations study-
ing the velocity autocorrelation function for i a conven-
tional Sinai billiard 14, ii the Sinai quartz block 15, iii
a vibrating plate 16, and iv a ray-splitting microwave bil-
liard 9.
In order to evaluate the autocorrelation function of level
velocities one should eliminate system-dependent features of
the spectra and, instead of the original energy levels E
i
,
consider the unfolded energies
i
=N
a v
E
i
, 1
where
N
a v
E =
E
a v
E ' dE ' 2
is the integrated average level density
a v
( E ). The paramet-
ric motion of the levels has to be unfolded too. This is
achieved by introducing the dimensionless parameter
x =
X
i
X
C 0 dX , 3
where X
i
, X is the interval of integration,
C 0 =
1
N
j
j
X
2
4
is the generalized conductance, and N is the number of en-
ergy levels under consideration.
The autocorrelation functions of the level velocities c ( x )
1,2,17 and c
˜
( , x ) 3 are defined as follows:
c x =
j
x
¯
x
¯
j
x
¯
x
¯
+x
5
and
c
˜
, x =
i , j
i
x
¯
-
j
x
¯
+x -
i
x ¯
x ¯
j
x ¯
x
¯
+x
i , j
i
x
¯
-
j
x
¯
+x -
. 6
In Eq. 5 the average is over the parameter x
¯
and over all
levels. In Eq. 6 the average is only over the parameter x
¯
. In
contrast to c ( x ), which correlates velocities belonging to the
same level, c
¯
( , x ) measures the averaged autocorrelation of
velocities separated by a distance x in parameter space and
by a distance in energy.
In this paper, we study the velocity autocorrelation func-
tions for two different ray-splitting systems 18–23: the
PHYSICAL REVIEW E, VOLUME 64, 036211
1063-651X/2001/643/0362115/$20.00 ©2001 The American Physical Society 64 036211-1