Comparison of the standard and dressed-picture master equations for the quantum Rabi model in the ultrastrong coupling regime

The goal of this chapter is to investigate the effects of relaxation and dephasing on the quantum Rabi model in the ultrastrong coupling regime, and to provide explicit formulas to implement and numerically solve the resulting nonunitary dynamics from first principles. The quantum Rabi model constitutes the most fundamental description of light-matter interaction, describing a single two-level system coupled to a single mode of a quantized cavity field. The ultrastrong coupling regime is typically defined by g gtrsim 0.1ω, where ω denotes the cavity-mode frequency. In this regime, the standard master equation of quantum optics - commonly referred to as the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation - becomes inaccurate. The reason is that strong light-matter interaction hybridizes the bare atom and field states, so that dissipation cannot be consistently described in the uncoupled basis. A consistent treatment must therefore incorporate this hybridization directly into the dissipative terms. One such approach is the dressed-picture Markovian master equation derived by Beaudoin, Gambetta, and Blais, in which the qubit-field interaction is explicitly included in the construction of the system-bath coupling operators. In this chapter, we numerically solve both the GKSL master equation and the dressed master equation (DME) for various initial field states, including coherent, odd Schrödinger cat, squeezed vacuum, squeezed coherent, and thermal states. We also examine photon generation from the vacuum induced by external time-dependent modulation of the qubit parameters, as well as multiphoton Rabi oscillations for an initially excited qubit. Two reservoir spectral densities are considered: white and Ohmic noise. The differences between the two approaches are illustrated through numerical results for several physical observables.