Nonexponential Dissipation in a Lossy Elastodynamic Billiard: Comparison with Porter-Thomas and Random Matrix Predictions

We study the dissipation of diffuse ultrasonic energy in a reverberant body coupled to a waveguide, an analog for a mesoscopic electron in a quantum dot. A simple model predicts a Porter-Thomas distribution of level widths and corresponding nonexponential dissipation, a behavior largely confirmed by measurements. For the case of fully open channels, however, measurements deviate from this model to a statistically significant degree. A random matrix supersymmetric calculation is found to accurately model the observed behaviors at all coupling strengths.

Figure 1

A diffuse wave field radiates into a waveguide.

Figure 2

A comparison of the predictions of the Porter-Thomas model [dashed line, Eq. (1)] for transient decay with that of a full supersymmetric calculation [solid line, Eq. (3)].

Figure 3

Sketch of measurement system.

Figure 4

The natural logarithm of mean energy $E(t)$ averaged over 16 distinct positions of source and receiver in a band between 225 to 275 kHz. The Heisenberg time is 34 ms.

Figure 5

Decay rates $\sigma$ as recovered from a fit of the difference $\ln E(t)$ to Eq. (2) in each of several narrow frequency bands (data points) are compared to the prediction: $\sigma =M_{\mathrm{P}\mathrm{o}\mathrm{c}\mathrm{h}\mathrm{h}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{e}\mathrm{r}}/t_H$ (solid line).

Figure 6

$M$ values as recovered from fits to the difference $\ln E(t)$ in each of several narrow frequency bands are compared to expectations based on the cutoff frequencies of the Pochhammer dispersion relation.

Figure 7

Values for $t_H\sum _{}^{}{\sigma }_i$ (dashed line) and for $\sum _{}^{}2/(1+g_i)$ (solid line) from fits of Eqs. (1, 3) to the difference $\ln E(t)$ profiles. The chi-squares of the fits indicate the inability of Eq. (1) to fit the data in those places where coupling is strong.

Figure 8

Fits of the supersymmetric profile [Eq. (3)] (smooth solid lines) and the Porter-Thomas model [Eq. (1)] (dashed lines) to the difference $\ln E(t)$ (irregular solid lines) in representative frequency bins. At 150 kHz, each model fits well: Eq. (1) calls for ${\sigma }_i=\{1,1,0.07686,0.07686\}/t_H$ (the flexural modes are especially weakly coupled at such long wavelength, hence their small $\sigma$.) Eq. (3) calls for $2/(1+g_i)=\{0.5954,1,0.0614,0.0614\}$. At 475 kHz, the fits call for ${\sigma }_i=1/t_H$ and $2/(1+g_i)=0.9$ for all $i$.