Non-monotonic population scaling and re-entrant photon number dynamics in the dissipative-pumped Tavis--Cummings model

Abstract

This paper investigates nonlinear dynamics in the dissipative-pumped Tavis--Cummings model (TCM). Under global excitation constraints, the competition between photon decay and pumping leads to two significant effects. First, the steady-state peak population exhibits a non-monotonic plateau: contrary to the expected monotonic decay trend at high energy levels, intermediate energy levels show comparable peak populations, and this plateau gradually shifts to higher energies and broadens as the number of atoms increases. Second, the average free photon number exhibits re-entrant dynamics. Under strong pumping, the photon number experiences an initial rise and decay, then slowly increases again, exceeding the initial peak when the system is sufficiently large, with clear thresholds in both system size and pumping rate. To reveal these phenomena, we develop a distributed computing framework suitable for large Hilbert spaces to solve the Lindblad master equation. By exploiting the sparsity of the jump operator and combining it with Cannon's algorithm, we reduce the complexity of the non-unitary terms from $\mathcal{O}(MN^3)$ to $\mathcal{O}(MN)$. Furthermore, the dynamic subspace construction method effectively eliminates redundant quantum states and significantly compresses the Hilbert space. Although the scalability of unitary evolution is limited by communication overhead, the framework as a whole still achieves efficient parameter space exploration. This study demonstrates that the dissipative-pumped TCM serves as a rich platform for studying non-equilibrium nonlinear dynamics and provides a practical numerical tool for its investigation.