Qubit-oscillator relationships in the open quantum Rabi model: the role of dissipation

Using a dissipative quantum Rabi model, we study the dynamics of a slow qubit coupled to a fast quantum harmonic oscillator interacting with a bosonic bath from weak to strong and ultra-strong coupling regimes. Solving the quantum Heisenberg equations of motion, perturbative in the internal coupling between qubit and oscillator, we derive functional relationships directly linking the qubit coordinates in the Bloch sphere to oscillator observables. We then perform accurate time-dependent Matrix Product State simulations and compare our results both with the analytical solutions of the Heisenberg equations of motion, and with numerical solutions of a Lindblad master equation, perturbative in the external coupling between oscillator and environment. Indeed, we show that, up to the strong coupling regime, the qubit state accurately fulfils the derived functional relationships. We analyse in detail the case of a qubit starting with generic coordinates on the Bloch sphere of which we evaluate the three components of the Bloch vector through the averages of oscillator observables. Interestingly, a weak to intermediate oscillator coupling to the bath is able to simplify the Bloch vector evaluation since qubit-oscillator relationships are more immediate. Moreover, by monitoring the qubit fidelity with respect to free limit, we find the parameter regime where the combined effect of internal and external couplings is able to hinder the reliable evaluation of the qubit Bloch vector. Finally, in the ultra-strong coupling regime, non-Markovian effects become robust and the dynamics of qubit and oscillator are inextricably entangled making the qubit Bloch vector evaluation difficult.