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We study the dissipative dynamics of a biased two-level system (TLS) coupledto a harmonic oscillator (HO), the latter interacting with an Ohmicenvironment. Using Van-Vleck perturbation theory and going to second order inthe coupling between TLS and HO, we show how the Hamiltonian of the TLS-HOsystem can be diagonalized analytically. Our model represents an improvement tothe usually used Jaynes-Cummings Hamiltonian as an initial rotating waveapproximation is avoided. By assuming a weak coupling to the thermal bath,analytical expressions for the time evolution of the populations of the TLS arefound: the population is characterized by a multiplicity of damped oscillationstogether with a complex relaxation dynamics towards thermal equilibrium. Thelong time evolution is characterized by a single relaxation rate, which islargest at resonance and whose expression can be given in closed analytic form.

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