  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 fulfills 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.
