Emergence of Quantum Ergodicity in Rough Billiards

Abstract

By analytical mapping of the eigenvalue problem in rough billiards on to a band random matrix model a new regime of Wigner ergodicity is found. There the eigenstates are extended over the whole energy surface but have a strongly peaked structure. The results of numerical simulations and implications for level statistics are also discussed. [5,6] for diffusive billiards where the time of classical ergodicity τ D due to diffusion on the energy surface is much larger than the collision time with the boundary τ b . 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 [6] showed that this change of p(s) happens when the localization length ℓ in the angular momentum l-space becomes smaller than the size of the energy surface characterized by the maximal l = l max at given energy (ℓ < l max ). For ℓ > l max the eigenfunctions are extended over the whole surface but as we will see they are not necessarily ergodic (Fig. 1).In this situation which we will furthermore call Wigner ergodicity the eigenstates are composed of rare strong peaks distributed on the whole energy surface. Such a case is very different from the Shnirelman ergodicity where the eigenstates are uniformly distributed. The usual scenario of ergodicity breaking was based on the image of quantum localization along the energy surface [7]. Here we show that the transition between localized and Shnirelman ergodic states can pass trough an intermediate phase of Wigner ergodicity. Our description and understanding of this phase 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. Recently such type of matrices was studied in the context of the problem of particle interaction in disordered systems [8][9][10][11]. 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 [12] of the local density of states according to which only unperturbed states in a small energy interval Γ E contribute to the final eigenstate.

Quantum chaos and thermalization in isolated systems of interacting particles

“…Therefore, in this case only the left part of the inequality, D ≪ Γ, comes into play, and the Breit-Wigner form of the SF emerges for any strong perturbation. In realistic models with one-body chaos, such as quantum billiards, the dependence of the width of the SF on the strength of interaction with rigid walls may be tricky [78,79].…”

Quantum Andreev Map: A Paradigm of Quantum Chaos in Superconductivity

“…The 1D models get around this constraint through a periodically timedependent external force (''kick''), which eliminates the energy as a constant of motion-but still conserves the quasienergy (analogously to quasimomentum conservation in a periodic lattice). The two paradigms share a common set of phenomena in the fields of quantum chaos and localization [5][6][7][8].The combination of chaos and superconductivity produces an entirely new phenomenology, notably the appearance of an excitation gap as a signature of quantum chaos [9]. The paradigm common to most of the literature is the 2D billiard connected to a superconductor [10], introduced under the name ''Andreev billiard'' in Ref.…”

Nodal Domains in Chaotic Microwave Rough Billiards with and without Ray-Splitting Properties

“…The border of BreitWigner regime is given by N W = M 2 /48k 2 . It means that between N e < N < N W Breit−Wigner ergodicity [17] ought to be observed and for N > N W Shnirelman ergodicity should emerge. In the regime of Shnirelman ergodicity wave functions have to be uniformly spread out in the billiard [18].…”