Spectral properties of noisy classical and quantum propagators

Abstract

We study classical and quantum maps on the torus phase space, in the presence of noise. We focus on the spectral properties of the noisy evolution operator, and prove that for any amount of noise, the quantum spectrum converges to the classical one in the semiclassical limit. The small-noise behaviour of the classical spectrum highly depends on the dynamics generated by the map. For a chaotic dynamics, the outer spectrum consists of isolated eigenvalues ('resonances') inside the unit circle, leading to an exponential damping of correlations. In contrast, in the case of a regular map, part of the spectrum accumulates along a one-dimensional 'string' connecting the origin with unity, yielding a diffusive behaviour. We finally study the non-commutativity between the semiclassical and small-noise limits, and illustrate this phenomenon by computing (analytically and numerically) the classical and quantum spectra for some maps.

Basis Markov Partitions and Transition Matrices for Stochastic Systems

“…Making use of this crucial feature of the noise profile we may expand the kernel of the Frobenius-Perron operator (19),…”

Spectral properties and classical decays in quantum open systems

“…By limiting the resolution of the functional space, one can effectively truncate the PF to a nonunitary operator of finite size (say N × N ) with a spectrum lying entirely inside the unit circle, except for the simply degenerate eigenvalue 1. In the, prop- * Electronic address: garciama@tandar.cnea.gov.ar † Electronic address: saraceno@tandar.cnea.gov.ar erly taken, limit of infinite size and no coarse graining, the isolated eigenvalues turn out to be the RP resonances [17,18]. As shown in [8], the linear entropy and the Loschmidt echo, for asymptotic times much longer than the Ehrenfest time 1 , also show characteristic decay rates governed by the classical RP resonances [9].…”

“…We model the unitary dynamics by means of a quantum map and implement a diffusive superoperator represented by a Kraus sum. Two recent works by Blank, et al [17] and Nonnenmacher [18] provide a rigorous theoretical underpinning to our calculations for quantum and classical maps on the torus.…”

Phase-space contraction and quantum operations

“…Following recent works [12,13,14,16,17,28,29] we study the effect of the dissipative noise channel described in Sec. V when composed with a unitary map.…”

“…In this work we follow an alternative approach to the description of dissipative quantum noise which consists in directly modeling the superoperator in its Kraus represen- * Electronic address: garciama@tandar.cnea.gov.ar tation. The formalism of quantum operations to describe open systems, which is especially well adapted in the context of quantum information and quantum computing [10,11], has also been used for open quantum maps [12,13,14,15,16,17], which are quantum maps in the usual sense with some nonunitary noise that can be considered the effect of an interaction with some environment.…”