Superscars in Billiards: A Model for Doorway States in Quantum Spectra

In a unifying way, the doorway mechanism explains spectral properties in a rich variety of open mesoscopic quantum systems, ranging from atoms to nuclei. A distinct state and a background of other states couple to each other which sensitively affects the strength function. The recently measured superscars in the barrier billiard provide an ideal model for an in-depth investigation of this mechanism. We introduce two new statistical observables: the full distribution of the maximum coupling coefficient to the doorway and directed spatial correlators. Using random matrix theory and random plane waves, we obtain a consistent understanding of the experimental data.

Figure 1

Examples for measured superscars induced by the barrier and concentrated along the indicated classical periodic orbits (dashed lines). They are members of four different families. Top row: horizontal bouncing ball $\mathsf{B}\mathsf{B}$ and inverted $\mathsf{V}$ superscars; bottom row: diamond $\mathsf{D}$ and $\mathsf{W}$ superscars. The gray level indicates the value of the wave function (black: highest positive, white: most negative value; see 12).

Figure 2

Doorway strength functions. Left: Overlap between constructed superscar state of the $\mathsf{V}$ family with $m=45$, $n=1$ and the measured wave functions versus rescaled frequency. The solid curve is a BW function. The inset shows the overlap on a logarithmic scale over a large frequency interval. Right: Spectrum of the $(p,p^{\prime })$ reaction at 200 MeV on ${}^{120}\mathrm{Sn}$ in the region of the isoscalar giant quadrupole resonance (ISGQR) as an example for a nuclear doorway state [6]. The solid curve is a BW function fitted to the data.

Figure 3

Left: $c_{\max }^2$ distributions of measured superscars (histogram), and the fit of the RMT model predictions (solid line). Right: Normalized distributions of superscar strength spread over all states on a logarithmic scale. Experimental distributions (dots) are compared with the RMT model predictions.

Figure 4

The wave function correlators. The $J_0(kr)$ prediction is always given as dashed line. The top row shows the correlation function of the constructed $\mathsf{V}$ superscar state as solid lines: the isotropic $C(kr)$ as well as the directed $C^{\perp }(kr)$ and $C^{||}(kr)$. In the middle row, the same observables are depicted as solid lines for the averages over all experimental wave functions in the barrier billiard. In the bottom row, the correlators averaged over all observed $\mathsf{V}$ superscars are displayed as solid lines and the correlators resulting from Eq. (7) with $c_{\max }^2=0.69$ are shown as filled circles.