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

TL;DR

This paper provides an analytical study of the dissipative dynamics of a biased qubit coupled to a harmonic oscillator, extending beyond the rotating wave approximation and including environmental effects.

Contribution

It introduces an analytical diagonalization method for the biased TLS-HO system beyond the rotating wave approximation using Van-Vleck perturbation theory.

Findings

Derived explicit expressions for the time evolution of TLS populations.

Identified a single relaxation rate dominating long-term dynamics.

Showed the relaxation rate peaks at resonance, with a closed-form expression.

Abstract

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 Van-Vleck perturbation theory and going to second order in the coupling between TLS and HO, we show how the Hamiltonian of the TLS-HO system can be diagonalized analytically. Our model represents an improvement to 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 closed analytic form.