The Interplay of Chaos and Dissipation in a Driven Double-Well Potential

Abstract

As a characteristic feature of the spectrum of a driven bistable potential, there exist crossings of chaotic quasienergy singlets with regular tunnel doublets. As the two partners of the tunnel doublet are have opposite symmetry, the crossing with one of them is avoided, the other is exact. Close to the avoided crossing, the tunnel splittings are drastically increased, resulting in a correspondingly higher tunnel rate (chaos-assisted tunneling) [1].

In the presence of dissipation, the tunneling becomes a transient phenomenon within the relaxation towards a quantum-mechanical attractor. The transient dynamics near a singlet-doublet crossing involves at least all three states in the crossing. Depending on temperature and parameters of the crossing, it can be qualitatively different from the familiar fading out of two-state tunneling [2].

As a model for dissipation, we couple the driven system to an ensemble of harmonic oscillators (Caldeira-Leggett model) and introduce a Markov approximation to the driven system, integrated within the Floquet formalism. This results in a master equation which describes the dissipative dynamics in the basis of the Floquet states of the conservative system.

We discuss both the coherent and the incoherent dynamics near singlet-doublet crossings in terms of a simple three-state model [1]. By comparison with exact numerical results, we identify its limitations. In particular, we investigate the duration of the coherent tunneling and the type of quantum-mechanical attractor, depending on temperatur and distance to the avoided crossing.