Dissipative dynamics of a qubit coupled to a nonlinear oscillator

Abstract

We consider the dissipative dynamics of a qubit coupled to a nonlinear oscillator (NO) embedded in an Ohmic environment. By treating the nonlinearity up to first order and applying Van Vleck perturbation theory up to second order in the qubit-NO coupling, we derive an analytical expression for the eigenstates and eigenfunctions of the coupled qubit-NO system beyond the rotating wave approximation. In the regime of weak coupling to the thermal bath, analytical expressions for the time evolution of the qubit's populations are derived: they describe a multiplicity of damped oscillations superposed to a complex relaxation part toward thermal equilibrium. The long time dynamics is characterized by a single relaxation rate, which is maximal when the qubit is tuned to one of the resonances with the nonlinear oscillator.Comment: 24 pages, 7 figures, 1 table; in the text between Eq. (8) and (9) there were misprints in the published version until 3rd Dec 2009: in the second order correction for the nonlinear oscillator and in the corresponding relative error. The correct expressions are given here. The results of the paper are not changed, as we consider the nonlinearity up to first order perturbation theor