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

Abstract

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 that enable students and researchers to implement and numerically solve the resulting nonunitary dynamics from first principles. We numerically solve the standard Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation and the dressed-picture master equation (DME), derived by Beaudoin, Gambetta, and Blais, for various initial field states, including coherent states, odd Schrödinger cat states, squeezed vacuum states, squeezed coherent states, and thermal states. The coupling strength is varied in the range $0.05\omega$ to $0.8\omega$. 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 different reservoir spectral densities are considered: white noise and Ohmic noise. The differences between the two master-equation approaches are illustrated through numerical results for several physical observables, including the qubit excited-state population, the mean photon number, the Mandel $Q$-factor, the negativity (used as a measure of entanglement), the subsystem purities, and the photon-number probability distribution at selected times. As expected, for many of these quantities the predictions of the standard GKSL master equation differ substantially from those of the dressed-picture master equation. However, in certain parameter regimes and for specific observables, the discrepancies remain comparatively small.

Chapter 5 of the book of proceedings "Modern Topics in Mathematical, Quantum and Statistical Physics: Proceedings of the 2025 CIF-UnB conferences". ISBN: 978-65-5563-767-0. https://lfeditorial.com.br/produto/modern-topics-in-mathematical-quantum-and-statistical-physics-proceedings-of-the-2025-cif-unb-conferences/