Quasistationary distributions of dissipative nonlinear quantum oscillators in strong periodic driving fields Heinz-Peter Breuer, 1 Wolfgang Huber, 2 and Francesco Petruccione 1 1 Fakulta ¨t fu ¨r Physik, Albert-Ludwigs-Universita ¨t, D-79104 Freiburg im Breisgau, Germany 2 IBM, Almaden Research Center, San Jose, California 95120 Received 14 October 1999; revised manuscript received 17 December 1999 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 inde- pendent 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. PACS numbers: 05.70.Ln, 03.65.Sq, 02.50.Ga I. INTRODUCTION Within the Born-Markov approximation, autonomous open quantum systems are described by quantum dynamical semigroups with a time-independent Lindblad generator 1. Under quite general physical conditions such systems relax in the long-time limit to a unique stationary state, which is given by the principles of equilibrium statistical mechanics 2. For example, requiring the condition of detailed balance for the transition rates and some kind of ergodic property regarding the operators which describe the coupling of the system to its environment, one finds an equilibrium station- ary state which is given by the Boltzmann distribution over the energy eigenvalues of the system. For open quantum systems in time-varying external fields, the quantum dynamics must be described, in general, by a time-dependent generator. In the case in which the external driving field is strong, one expects that the long-time dynam- ics differs significantly from the equilibrium stationary state. In this paper, we shall investigate the question of the exis- tence and the basic properties of a certain quasistationary state which governs the long-time behavior for systems in strong, time-periodic driving fields. In our study, the interaction with the external field will be treated exactly using the Floquet representation for time- periodic quantum systems 3,4, whereas the coupling to en- vironment will be taken into account in the Born-Markov approximation. It is known that for this case the diagonal elements of the reduced density matrix in the Floquet repre- sentation obey a closed equation of motion which is formally equivalent to a Pauli-type master equation 5–7. We shall perform analytical and numerical investigations of the stationary solution of the Pauli master equation for a general class of periodically driven, nonlinear oscillators coupled to an environment at finite temperature. Our results reveal that a large class of Hamiltonian systems leads to a unique, quasistationary density matrix which is diagonal in the Floquet representation. The structure of the quasistation- ary distribution will be discussed in detail. We shall also study the connections to the phase flow of the corresponding classical Hamiltonian system, which shows a sharp di- chotomy of quasiregular and chaotic motion. II. MASTER EQUATION FOR OPEN QUANTUM SYSTEMS IN STRONG DRIVING FIELDS A. The density matrix in the Floquet representation We consider in the following a periodically driven quan- tum system coupled to an environment at temperature T for a review, see Ref. 7. The coherent part of the dynamics is generated by some Hamiltonian H ( t ) which is periodic in time with frequency  L , that is, we have H ( t +T L ) =H ( t ), where T L =2  /  L denotes the period. Usually, H ( t ) takes the form H ( t ) =H 0 +H I ( t ), where H 0 is the unperturbed system Hamiltonian and H I ( t ) represents a time-periodic in- teraction with an external driving field. According to the Floquet theorem 3,4, there exists a basis of T L -periodic wave functions u j ( t ) =u j ( t +T L ), the Floquet states, such that any solution  ( t ) of the time- dependent Schro ¨ dinger equation pertaining to the Hamil- tonian H ( t ) can be represented in the form we choose units such that  =1) PHYSICAL REVIEW E MAY 2000 VOLUME 61, NUMBER 5 PRE 61 1063-651X/2000/615/48837/$15.00 4883 ©2000 The American Physical Society