Interpretation of the imaginary parts of the single-term decay-rate solutions

Identify the physical oscillation frequency corresponding to the two distinct imaginary parts of the complex values of the decay rate obtained by setting the determinant of the single-term perturbation ansatz to zero for deviations from a nonzero-acoustic-mode steady state of the driven dissipative oscillator system.

Background

For perturbations around a steady state with cac≠0c_{ac}\neq 0, the paper first tests a single-frequency, single-decay-rate ansatz for the acoustic- and optical-mode deviations. The resulting four-dimensional real homogeneous system depends on an unknown decay rate γ\gamma, and numerical attempts to impose a vanishing determinant produce four complex values of γ\gamma rather than physically admissible real decay rates.

The authors observe that the real parts of these complex values reproduce the two decay rates found using the correct multi-term ansatz, but the associated imaginary parts do not match any frequency observed in numerical simulations. Determining the physical meaning or correspondence of these imaginary parts remains unresolved within the analysis.

References

Interestingly, the four obtained $\gamma$ values take the form of two pairs, each of which are complex conjugates of each other. The two distinct real parts of the different $\gamma$ values obtained in this way correspond to the two decay rates that we find in the solutions described below. The two distinct imaginary parts (ignoring the plus/minus sign differences) are almost equal, but we cannot identify their value with any frequency obtained in our numerical simulations.

— Steady states and oscillation modes of two driven dissipative oscillators with non-Hermitian coupling  (2608.12799 - Ashhab et al., 13 Aug 2026) in Section III.C, “Normal modes near steady-state solutions,” Case 1: $c_{ac}\neq 0$