Opening a gap in the dispersion of the collective excitations of a driven-dissipative condensate subject to an external coherent drive

Abstract

We build a minimal theoretical model to describe the opening of a gap in the dispersion of the collective excitations of a driven-dissipative condensate when the condensate phase is fixed by an additional coherent phase-locking drive. We map out the phase diagram as a function of the amplitude and frequency of the coherent drive, identifying distinct regions corresponding to different steady-state regimes. For each region, we analyze the dispersion of the collective excitations and determine whether the spectrum is gapped, with either a purely imaginary gap or a finite real part. When the coherent drive is unable to lock the condensate phase, a gapless Goldstone mode is recovered. Within the same phase diagram, we further identify regions of finite-wavevector dynamical instability, where the condensate tends to develop a supersolid-like spatial modulation. While our theoretical framework is directly related to recent experiments with exciton-polariton condensates, it can be applied to describe the effect of external injection also in a variety of spatially extended optical parametric oscillators or laser devices.