Lindblad Dissipation, Symmetry, and Criticality in the Open Quantum Rabi Model

Abstract
The effects of dissipation in open quantum systems depend not only on decay rates but also on the operator structure of the system-environment coupling. The open quantum Rabi model provides a minimal setting for examining how these features shape symmetry, criticality, and relaxation. One- and two-photon loss lead to different ​​ Z₂ symmetry structures of the Liouvillian, with one-photon loss retaining a weak symmetry while two-photon loss can preserve a strong symmetry. These symmetry structures constrain the stationary states, low-lying spectra, and relaxation dynamics. Conditional no-jump evolution generated by a non-Hermitian effective Hamiltonian is compared with the full trace-preserving Lindblad dynamics. Preliminary results indicate that mode softening and eigenvalue degeneracy follow distinct trajectories as quantum-jump recycling is restored. The analysis combines Bogoliubov transformations, symmetry-resolved Liouvillian calculations, Jordan-block diagnostics, and direct time evolution. An open Z_N Weyl-Rabi model extends the symmetry analysis to N-photon and lower-order photon loss channels, while an absorbing Baxter-Fendley construction provides an exact benchmark for studying the relation between the conditional non-Hermitian spectrum and the full Liouvillian spectrum.