CATS Acoustic Chaology Group
Parametric level motion
Introduction
Recently, several authors have found universality for various quantities
that relate to the motion of energy levels when a parameter of the system
varies. For example, one can study the level velocity (first derivative with
respect to the parameter) and the curvature (second derivative).
B. D. Simons and coworkers studied the hydrogen atom in a strong
magnetic field, using the field strength as the parameter. They found nice
agreement
between the experimental velocity autocorrelation and the universal prediction
from Random Matrix Theory. From studying the kicked top, Zakrzewski and
Delande suggested a simple form for the curvature distribution, and this was
proven correct by F. von Oppen.
Generally, it is hard to perform experiments on real quantum systems. Instead,
one can study a model system like an acoustic resonator, where the resonance
frequencies play the role of the energy levels of a quantum system. In the
Acoustic
Chaology Group at CATS, parametric experiments have recently been conducted
and highly
significant results have been obtained for the level velocity and curvature.
For an acoustic resonator, one can use the shape of the resonator as the
parameter.
The aluminium stadium plate
This was the first of three parametric experiments conducted during the
winter 1996/97 by K. Schaadt and P. Bertelsen.
We used a stadiumshaped aluminium plate of thickness 2mm.
In order to make the plate fully chaotic, we had to make a 'cut'
on one side of the plate (as discussed on the
Spectral statistics of vibrating plates page). The plate is shown on the
figure below.
'Raw' spaghetti

In this experiment, we used the length of the plate as the parameter.
We cut 0.05 mm off one end of the plate, measured 171 resonance frequencies,
and repeated this procedure 61 times. This enabled us to track the level motion
motion with respect to the length of the plate. However, we used the mass
we had cut off as the parameter ('cut off mass'), since we could
measure this with good accuracy. After doing the usual unfolding of the spectra
(dividing by the mean level spacing), we obtain a 'spaghetti plot' showing the
unfolded resonance frequencies as a function of the cut off mass.

Fitted spaghetti

As shown above, the raw data were very noisy. We therefore applied a fitting
proceduce, which removed most of the noise, without smoothing the structure
away. The result is seen above to the left. From the fit, we could find the
velocity and curvature of the spaghetti in each point, and the statistics
of these quantities were compared to predictions from Random Matrix Theory.
Our main result is the velocity autocorrelation c(x), which measures the
correlation between the velocity of a certain level at two different values
of the parameter (which are a distance x apart).
where v is the velocity, and the average is over the level number, n, and the
parameter, x'. To the right we plot our experimental result with error bars,
compared to the prediction from Random Matrix models. A good agreement is
found, although our experimental data goes a bit below the theoretical curve.


The quartz block
Papers in preparation

Experimental study of the velocity autocorrelation function
using an elastic plate,
P. Bertelsen, C. Ellegaard, T. Guhr, K. Lindemann, M. Oxborrow
and K. Schaadt
In preparation

Experimental study of parametric correlations using a
three dimensional acoustic resonator,
P. Bertelsen, C. Ellegaard, T. Guhr, K. Lindemann, M. Oxborrow
and K. Schaadt
In preparation
Back To Acoustic Group
June 26 , 1997
Mail me: schaadt@nbi.dk
