Emergence of quantum ergodicity in rough billiards

In 1974, Shnirelman proved a theorem according to which quantum eigenstates in chaotic billiards become ergodic for sufficiently high level numbers. Later it was demonstrated by Bohigas, Giannoni and Schmit that in this regime the level spacing statistics $P(s)$ is well described by random matrix theory. However, one can ask the question how this quantum ergodicity emerges with increasing level number $N$? This question becomes especially important for diffusive billiards with weakly rough elastic boundary where the time of classical ergodicity $\tau_D$ due to diffusion on the energy surface in the angular momentum $l$-space is much larger than the collision time with the boundary $\tau_b$. As is shown in [89,92] in such a situation quantum localization on the energy surface may break classical ergodicity eliminating the level repulsion in $P(s)$. The investigation of rough billiards [89,92] showed that this change of $P(s)$ happens when the localization length $\ell$ in $l$-space becomes smaller than the size of the energy surface characterized by the maximal $l=l_{max}$ at given energy ( $\ell < l_{max}$). The localization length $\ell =D \approx 4 l^2_{max}\tilde\kappa^2$ is determined by diffusion rate in $l$ being proportional to the billiard roughness $\tilde\kappa^2=<(d R/d\theta/R_0)^2>$. In the localized regime $P(s)$ has a large Shnirelman peak at small spacings which contains about 30-40% of all spacings. The appearance of quasidegeneracy in integrable billiards was proved by Shnirelman in 1975. Its physical origin was explained in [70] on the basis of tunneling between states rotating in different directions. In some sense the degeneracy between the states connected by time-reversal symmetry is destroyed by tunneling between the future and the past.

For $\ell>l_{max}$ the eigenfunctions are extended over the whole energy surface but surprisingly they are not necessarily ergodic. The usual scenario of ergodicity breaking was based on an image of transition from the quantum eigenstates ergodic on this surface ( Shnirelman ergodicity) to the exponential localized states. In [92] it is shown that this transition between localized and Shnirelman ergodic states can pass through an intermediate phase of Wigner ergodicity. In this Wigner phase the eigenstates are nonergodic and composed of rare strong peaks distributed on the whole energy surface. The description and understanding of this case is based on the mapping of the billiard problem with weakly rough (random) boundary on to a superimposed band random matrix (SBRM). This model is characterized by strongly fluctuating diagonal elements corresponding to a preferential basis of the unperturbed problem. Such type of matrices was studied recently in the context of the problem of particle interaction in disordered systems (see [69,75] and papers of Fyodorov and Mirlin and Frahm and Müller-Groeling). There it was found that the eigenstates can be extended over the whole matrix size while having a very peaked structure. The origin of this behavior is due to the Breit-Wigner form of the local density of states according to which only unperturbed states in a small energy interval $\Gamma_E$ contribute to the final eigenstate. The condition for Wigner ergodicity in rough billiards are determined in [92].

The physical realizations of rough billiards can be quite different. As examples, it is possible to mention surface waves in the droplets which are practically static for the light, nonideal surfaces in microdisk lasers, and capillary waves on a surface of small metallic clusters (see refs. [7, 8, 10] in [89]). Recently, the above theoretical results were confirmed in experiments with microwave rough billiards realized by Stöckmann et al. at Marburg.

The results of this section were obtained with Klaus Frahm who also developed a supersymmetry approach to rough billiards in separate publications. This direction of research is represented in his habilitation.