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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Nonlinear response of a periodically driven damped two-state system

Milena Grifoni, Maura Sassetti, Jürgen T. Stockburger, Ulrich Weiß

Published Nov 1, 1993• 52 citations
Publisher

Abstract

We study the nonlinear response of a periodically driven dissipative two-state system. The exact formal solution in the form of a series in the number of system transitions is derived. The series is summed in analytic form in the parameter region where incoherent transitions prevail. We also present the exact solution for the time evolution of the system in the entire region of temperatures and driving strength for a special value of the Ohmic viscosity. The destruction of quantum coherence by incoherent processes which are induced by the fluctuating and the driving forces is discussed on the basis of these solutions.

DOI
10.1103/physreve.48.3497

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We study the quantum dynamics of a periodically driven biased two-state system which additionally is subject to either Ohmic or to frequency-dependent damping. Within the noninteracting-blip approximation for the stochastic forces, the solution is given in terms of a series which is defined by recursion relations. Avoiding rotating wave and Markovian approximation, we deduce the solution in closed form for the important cases of high- and low-frequency driving. For low-frequency driving, the dynamics is governed by a rate equation with a time-dependent rate and with a time-dependent adiabatic equilibrium state. In the high-frequency case, we find novel cooperative effects, such as resonances in the spectral distribution near fractional values of the static detuning energy.
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The quantum dynamics of a quartic double well, subjected to a harmonically oscillating field, is studied in the framework of the Floquet formalism. The modifications of the familiar tunneling process due to driving and dissipation are investigated numerically and explained in terms of the quasienergy spectrum. In absence of dissipation, there is a onedimensional manifold in the parameter space spanned by amplitude and frequency of the driving force, where tunneling is almost completely suppressed by the coherent driving. The influence of dissipation is described on basis of the reduced density matrix in the Floquet representation. In particular, we consider the effect of weak Ohmic damping. In the classical limit, this system corresponds to a damped bistable Duffing oscillator
The dynamics of periodically driven quantum systems coupled to a thermal environment is investigated. The interaction of the system with the external coherent driving field is taken into account exactly by making use of the Floquet picture. Treating the coupling to the environment within the Born-Markov approximation one finds a Pauli-type master equation for the diagonal elements of the reduced density matrix in the Floquet representation. The stationary solution of the latter yields a quasistationary, time-periodic density matrix which describes the long-time behavior of the system. Taking the example of a periodically driven particle in a box, the stationary solution is determined numerically for a wide range of driving amplitudes and temperatures. It is found that the quasistationary distribution differs substantially from a Boltzmann-type distribution at the temperature of the environment. For large driving fields it exhibits a plateau region describing a nearly constant population of a certain number of Floquet states. This number of Floquet states turns out to be nearly independent of the temperature. The plateau region is sharply separated from an exponential tail of the stationary distribution which expresses a canonical Boltzmann-type distribution over the mean energies of the Floquet states. These results are explained in terms of the structure of the matrix of transition rates for the dissipative quantum system. Investigating the corresponding classical, nonlinear Hamiltonian system, one finds that in the semiclassical range essential features of the quasistationary distribution can be understood from the structure of the underlying classical phase space.