Spectral properties and dynamical tunneling in constant-width billiards

We determine with unprecedented accuracy the lowest 900 eigenvalues of two quantum constant-width billiards from resonance spectra measured with flat, superconducting microwave resonators. While the classical dynamics of the constant-width billiards is unidirectional, a change of the direction of motion is possible in the corresponding quantum system via dynamical tunneling. This becomes manifest in a splitting of the vast majority of resonances into doublets of nearly degenerate ones. The fluctuation properties of the two respective spectra are demonstrated to coincide with those of a random-matrix model for systems with violated time-reversal invariance and a mixed dynamics. Furthermore, we investigate tunneling in terms of the splittings of the doublet partners. On the basis of the random-matrix model we derive an analytical expression for the splitting distribution which is generally applicable to systems exhibiting dynamical tunneling between two regions with (predominantly) chaotic dynamics.

Figure 1

Sketches of the two billiard shapes. That of billiard B1 is shown in red (dark gray) and that of B2 in blue (light gray). They are barely distinguishable.

Figure 2

PSOS for billiard B1. For its construction particles were launched into the billiard in the clockwise direction. Due to unidirectionality only the upper half ($p>0$) of the PSOS is filled. The chaotic sea is bordered by a region of whispering gallery orbits around $p=1$ plotted in red (dark gray) and a region of regular orbits around the BBOs at $p=0$ depicted in blue (light gray). A zoom into these regions is shown in Fig. 3 for B1 and B2. The inset shows an example for a whispering gallery orbit in red (dark gray) and a BBO in blue (light gray).

Figure 3

Regular regions in the PSOS of B1 (left) and B2 (right). Shown are a zoom into the region around the whispering gallery orbits ($p\simeq 1$) and one into that around the BBOs ($p\simeq 0$).

Figure 4

Small regular islands in the chaotic sea of the PSOS (black spots). The left inset illustrates the periodic orbit corresponding to them. Only this one was found. For the billiard used in the experiment it has a length of $2.305$ m. The right inset shows a zoom into one of the small islands.

Figure 5

Photo of the microwave cavity with the shape of B1. It was constructed from copper plates. A hole with the shape of the billiard was milled out of the bottom plate. The top plate has been removed. For the measurements both plates were lead-plated and screwed together tightly through the holes along the cavity boundary.

Figure 6

Resonance spectrum measured with B1. The upper panel shows it in a frequency range from 15.4 to 15.9 GHz and the lower one a zoom into a triplet of nearly degenerate pairs of resonances.

Figure 7

Differences between the zeros of the $J_0$ Bessel function and the measured resonance frequencies of the singlets for B1 (crosses) and B2 (circles). Shown are the absolute values of the differences of the corresponding wave numbers $k$, where $k=2\pi f/c$.

Figure 8

Comparison of the experimentally (black full line) and the numerically (red dashed line) determined fluctuating parts of the integrated resonance density for B1. This was done to identify in the list of possible eigenvalues obtained with the method of Saraceno and Vergini [34] the eigenvalues of the billiards.

Figure 9

Squared modulus of the computed wave functions in the billiard plane (upper panels), the associated Husimi functions (middle panels), and the projection of the Husimi function of the left doublet partner onto the $q$ axis (lower panels) for B1, where the color scale is given to the left of the upper panels. Shown are the results for the chaotic modes with numbers 153 (left) and 154 (right). The wave functions are spread over the whole billiard area and the Husimi functions extend over the whole chaotic part of the PSOS. Likewise, the projection $H_n\left(p\right)$ is nonvanishing for most values of $p$.

Figure 10

Same as Fig. 9. Panels (a) show the wave functions, Husimi functions, and the projection of the Husimi function of the left doublet partner onto the $q$ axis for a singlet state of B1. Panels (b) present these functions for the pair of barrier KAM modes with numbers 575 and 576 and panels (c) those for that with numbers 579 and 580. The wave functions are localized along the BBOs and the Husimi functions around the barrier of the KAM tori. Their projections exhibit a maximum along the $p=0$ line for the former pair, a mimimum that is bordered by two sharp maxima for the latter one.

Figure 11

Same as Fig. 9. Panels (a) show the wave functions, Husimi functions, and the projection of the Husimi function of the left doublet partner onto the $q$ axis for the doublet with mode numbers 113 (left) and 114 (right) of B1. The wave functions are localized close to the boundary and the Husimi functions near $p=\pm 1$. Likewise, the projection ${\mathrm{H}}_n\left(p\right)$ is nonvanishing only there. Hence, these modes correspond to whispering gallery modes. In panels (b) these functions are depicted for the island modes with numbers 123 and 124. The wave functions are localized around the regular orbit shown in Fig. 4 and the Husimi functions around the islands in the chaotic sea also shown there. Panels (c) present those for the hybrid modes with numbers 259 and 260. The Husimi functions exhibit a high intensity at $p=0$ and at some value $\left|p\right|\lesssim 1$.

Figure 12

Comparison of the spectral properties of billiards B1 and B2 (black) with those of random matrices from the GOE shown in blue (light gray) and the GUE depicted in red (dark gray). The upper left (right) panel shows the nearest-neighbor spacing distribution for billiard B1 (B2). Similarly, the lower panels show the ${\Delta }_3$ statistic.

Figure 13

Comparison of the spectral statistic of the measured eigenvalues shown in black with those of the RMT model Eq. (5) depicted in red (dark gray). The coupling parameter ${\tau }_1=0.135$ for B1 and ${\tau }_1=0.205$ for B2, respectively, was obtained from a fit of the ${\Delta }_3$ statistic deduced from the RMT model (5) to the experimental result.

Figure 14

Experimentally determined splittings, that is, differences $\delta =k_n^r-k_n^l$ for B1 (upper panel) and B2 (lower panel) versus $k=k_n^r$. Here $k_n^r$ and $k_n^l$ denote the wave numbers of the right and the left doublet partners, respectively. The black circles mark the chaotic modes and the orange crosses whispering gallery modes [see Fig. 11]. Furthermore, the red downward and the blue upward triangles mark the splittings of the KAM modes with a maximum and a minimum of the Husimi functions at $p=0$, respectively [see Figs. 10 and 10]. The green diamonds mark hybrid modes [see Fig. 11].

Figure 15

Experimentally determined splitting-weighted density of states $f\left(k\right)$ (upper panel) [see Eq. (7)] and the result for its convolution $f_G\left(k\right)$ with a Gaussian of width ${\sigma }_G$ [see Eq. (8)] with ${\sigma }_G=1.0$ (lower panel) for B1. Large splittings seem to occur periodically.

Figure 16

Same as Fig. 15 but for B2.

Figure 17

Comparison of the Fourier transforms of the splitting-weighted density of states $f\left(k\right)$ (upper panel) with that of $f_G\left(k\right)$ for ${\sigma }_G=1.0$ (lower panel) for B1 where $f\left(k\right)$ and $f_G\left(k\right)$ are shown in Fig. 15. Both exhibit a dominant peak at the length of the diameter.

Figure 18

Same as Fig. 17 but for B2.

Figure 19

Comparison of the experimental splitting distributions shown in black with those obtained from the RMT model Eq. (9) depicted in red (dark gray), with $H^{\mathrm{mixed}}$ replaced by a $N\times N$-dimensional random matrix from the GUE with $N=300$. That for B1 is shown in the upper panel and that for B2 in the lower one. The splittings $\tilde{\delta }$ were obtained by first unfolding the sequences of left and right doublet partners, respectively, to the mean spacing one.

Figure 20

Comparison of the experimentally determined splitting distributions shown as black histogram with that obtained from the RMT model Eq. (9) depicted as a red (dark gray) histogram, with $H^{\mathrm{mixed}}$ replaced by a $2\times 2$ random matrix from the GUE. The green (gray) full curve shows the respective analytical result Eq. (22). The dashed (cyan) curve results from a fit of the truncated Cauchy distribution given in Eq. (23) to the experimental result. The experimental and the numerical splittings $\tilde{\delta }$ were obtained by first unfolding the sequences of left and right doublet partners, respectively, to the mean spacing one.

Figure 21

Comparison of the distribution of the ratios ${\mathcal{W}}_l/E_l$ obtained for random matrices of the form Eq. (9) with $H^{\mathrm{mixed}}$ replaced by a random $2\times 2$ matrix from the GUE (black histogram) with the analytical results Eq. (A8) depicted as the green (gray) full line and Eq. (A10) shown as the red (dark gray) histogram.

Figure 22

Comparison of the distribution of the splittings of the eigenvalues of random matrices of the form Eq. (9) with $H^{\mathrm{mixed}}$ replaced by a random $300\times 300$ matrix from the GUE (black histogram) with the analytical result Eq. (A10) (full red line).