Classical transients and the support of open quantum maps

Abstract

The basic ingredients in a semiclassical theory are the classical invariant objects serving as a support for quantization. Recent studies, mainly obtained on quantum maps, have led to the commonly accepted belief that the classical repeller-the set of nonescaping orbits in the future and past evolution-is the object that suitably plays this role in open scattering systems. In this paper we present numerical evidence warning that this may not always be the case. For this purpose we study recently introduced families of tribaker maps [L. Ermann, G. G. Carlo, J. M. Pedrosa, and M. Saraceno, Phys. Rev. E 85, 066204 (2012)], which share the same asymptotic properties but differ in their short-time behavior. We have found that although the eigenvalue distribution of the evolution operator of these maps follows the fractal Weyl law prediction, the theory of short periodic orbits for open maps fails to describe the resonance eigenfunctions of some of them. This is a strong indication that new elements must be included in the semiclassical description of open quantum systems. We provide an interpretation of the results in order to have hints about them.

Resonance Eigenfunction Hypothesis for Chaotic Systems

“…For the structure of resonance eigenfunctions some aspects have been studied, e.g. for open billiards [38][39][40][41][42], optical microcavities [43][44][45][46][47][48], potential systems [35], and maps [32,[49][50][51][52][53][54][55][56]. However, there exists no analogue to the semiclassical eigenfunction hypothesis for scattering systems.…”

Theory of short periodic orbits for partially open quantum maps

“…In order to quantify the behaviour of the short POs method we define its performance P [11] as the fraction of long-lived eigenvalues that it is able to reproduce within an error given by given by the semiclassical theory, respectively. We restrict our analysis to the number of exact eigenvalues with modulus greater than ν c , which is a critical value that depends on R. In our calculations we have tried to keep the number constant at around n c = 60 since for low values of R this represents the outer ring of eigenvalues that is a typical feature of the open quantum baker maps.…”

“…III now associated to the right |Ψ R j and left Ψ L j | eigenstates, which are related to the eigenvalue z j . We calculate the sum of the first j of these projectors [11], ordered by decreasing modulus of the corresponding eigenvalues (|z j | |z j | with j ≤ j ) up to completing the set of long-lived resonances.…”

“…In this approach the main ingredients * E-mail address: carlo@tandar.cnea.gov.ar are the shortest POs contained in the repeller, they provide all the necessary information to construct a basis set of scar functions in which the quantum non unitary operators can be written. The number of trajectories needed to reproduce the quantum repeller [10,11] is related to the fractal Weyl law [12].…”

Semiclassical theory of chaotic quantum resonances

“…Our result, thus, yields a characteristic value for the decay rate, where the important role of fluctuations is left unaccounted for. Other effects not captured by our analysis include transient features of the classical dynamics outside the repeller's area which, as recent work shows [23,24], may have important influence on the resonances of open quantum system.…”