Signatures of classical structures in the leading eigenstates of quantum dissipative systems

Abstract

By analyzing a paradigmatic example of the theory of dissipative systems-the classical and quantum dissipative standard map-we are able to explain the main features of the decay to the quantum equilibrium state. The classical isoperiodic stable structures typically present in the parameter space of these kinds of systems play a fundamental role. In fact, we have found that the period of stable structures that are near in this space determines the phase of the leading eigenstates of the corresponding quantum superoperator. Moreover, the eigenvectors show a strong localization on the corresponding periodic orbits (limit cycles). We show that this sort of scarring phenomenon (an established property of Hamiltonian and projectively open systems) is present in the dissipative case and it is of extreme simplicity.

Three-dimensional classical and quantum stable structures of dissipative systems

“…The corresponding eigenvalues are no more real and the spectral gap is large. This is a generic behaviour similar to what we have found in the 2D case [16], and most importantly, it is a clear indication that the morphology of the qISS is the same for both β.…”

“…5 both show a leading real eigenvalue extremely close to the invariant one. This is a typical feature of large qISS whose invariant and leading eigenstates are localized around the corresponding classical limit cycle, within quantum uncertainty [16]. This is clearly noticed by looking at Figs.…”

“…1. The first thing we notice is that this system has the same richness as the 2 parameter dissipative kicked rotor [16], i.e. we find a big regular region together with large ISSs intertwined with smaller shrimp-like ones, and all of them embedded in a chaotic background.…”

“…Then, why is there such a striking difference between the quantum behaviour at these two β values for which the classical ISS is essentially the same? In the following we will answer this question by using some tools of our previously developed theory for quantum 2D dissipative systems [13,16].…”

“…Also, the leading eigenstates which rule the transitory behaviour have a phase space structure dominated by limit cycles of neighbouring ISSs, and their eigenvalues have the same periodicity. This leads to scarring (localization) [15] on the corresponding unstable periodic orbits [16].…”