Dynamical properties of two coupled quantum cavities with single-mode amplification

Coupled optical cavities provide one of the simplest possible schemes to engineer the interaction of bosonic modes. This paper investigates a two-mode model where, in addition to the usual mode coupling, the presence of an amplification term associated to one of the modes triggers an unexpectedly rich dynamical scenario. The resulting nontrivial model is diagonalized by implementing the dynamical-algebra method, a group-theoretic approach which allows one to determine the stability diagram of the model Hamiltonian in terms of the two mode frequencies for given values of the interaction and amplification parameters. The mode interaction significantly modifies the simple amplification effect of the noninteracting model causing the separation of the unstable domain (where the amplification takes place) into two subdomains, one of which is stable, features no amplification effect, and exhibits an extension controlled by the interaction parameter. The analysis of stability properties is corroborated by the fully analytic study of the energy spectrum which exhibits the transition from a discrete to a continuous structure whenever the system undergoes the transition from a stable region to an unstable region where the amplification effect occurs. This scenario is further confirmed by the calculation of the time evolution of the mode populations.