  The interplay of three light-matter interaction processes - rotating and
counter-rotating interactions and two-photon relaxation of the light field - is
a topic of interest in quantum optics and quantum information processing. In
this work, we theoretically investigate the three light-matter interaction
processes using the anisotropic Rabi model, which accounts for different
strengths of rotating and counter-rotating interactions and the unique
occurrence of photon escape exclusively in pairs. By numerically solving the
Lindblad master equation, we analyze the excitation-relaxation dynamics and
derive a non-Hermitian effective Hamiltonian to gain further physical insights.
To explore the individual effects of these interactions, we examine three
analytically tractable limits of the effective Hamiltonian. Our analysis
reveals that the three competitive light-matter interaction processes exhibit
sensitivity to parity, leading to intriguing phenomena in both transient and
steady states. Particularly interesting dynamical patterns resembling quantum
phase transitions emerge when these three interaction terms compete. This work
deepens the understanding of ultrastrong light-matter interaction in open
quantum systems and offers valuable insights into cavity-based quantum
computations.
