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 co-workers 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 stadium-shaped 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.
Stadium plate with cut
Spaghetti plot
'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.
Spaghetti plot
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).
Velocity autocorrelation
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.
Velocity autocorrelation

The quartz block

Spaghetti plot Velocity autocorrelation

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