Dissipative dynamics of a biased qubit coupled to a harmonic oscillator: analytical results beyond the rotating wave approximation

We study the dissipative dynamics of a biased two-level system (TLS) coupled to a harmonic oscillator (HO), the latter interacting with an Ohmic environment. Using the Van Vleck perturbation theory and going to second order in the coupling between the TLS and the HO, we show how the Hamiltonian of the TLS-HO system can be diagonalized analytically. Our model represents an improvement on the usually used Jaynes-Cummings Hamiltonian as an initial rotating wave approximation is avoided. By assuming a weak coupling to the thermal bath, analytical expressions for the time evolution of the populations of the TLS are found: the population is characterized by a multiplicity of damped oscillations together with a complex relaxation dynamics towards thermal equilibrium. The long time evolution is characterized by a single relaxation rate, which is largest at resonance and whose expression can be given in a closed analytic form.