Abstract
A periodically driven quantum system, when coupled to a heat bath, relaxes to a non-equilibrium asymptotic state. In the general situation, the retrieval of this asymptotic state presents a rather non-trivial task. It was recently shown that in the limit of an infinitesimal coupling, using the so-called rotating wave approximation (RWA), and under strict conditions imposed on the time-dependent system Hamiltonian, the asymptotic state can attain the Gibbs form. A Floquet–Gibbs state is characterized by a density matrix which is diagonal in the Floquet basis of the system Hamiltonian with the diagonal elements obeying a Gibbs distribution, being parametrized by the corresponding Floquet quasi-energies. Addressing the non-adiabatic driving regime, upon using the Magnus expansion, we employ the concept of a corresponding effective Floquet Hamiltonian. In doing so we go beyond the conventionally used RWA and demonstrate that the idea of Floquet–Gibbs states can be extended to the realistic case of a weak, although finite system-bath coupling, herein termed effective Floquet–Gibbs states.
Highlights
When coupled to a heat bath, a quantum system with a time-independent Hamiltonian typically relaxes to an overall thermal equilibrium state [1,2,3]
Rotating wave approximation (RWA), and under strict conditions imposed on the time-dependent system Hamiltonian, the asymptotic state can attain the Gibbs form
A Floquet–Gibbs state is characterized by a density matrix which is diagonal in the Floquet basis of the system Hamiltonian with the diagonal elements obeying a Gibbs distribution, being parametrized by the corresponding
Summary
When coupled to a heat bath, a quantum system with a time-independent Hamiltonian typically relaxes to an overall thermal equilibrium state [1,2,3]. In the infinitesimal coupling limit, this thermal state is specified by the canonical Gibbs density matrix ; i.e., μ e-bHS, where HS is the system Hamiltonian and β denotes the inverse temperature of the heat bath [4] This result is quite intuitive, the mechanism behind its universal form and its emergence from the system-specific quantum evolution remains the focus of active studies and debates up to this date [5,6,7,8,9]. In order to have the asymptotic density matrix diagonal in the Floquet basis, it has to be guaranteed that all dissipative effects are relevant on a time scale larger than any intrinsic characteristic timescale of the isolated system (including the period of the driving T).
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