Quantum droplets and condensates in an optical lattice coupled to a dissipative cavity: Collective excitations and nonequilibrium dynamics

Motivated by recent experiments on light-matter interacting systems, we investigate a dilute Bose gas and self-bound quantum droplets in a one-dimensional optical lattice coupled to a lossy cavity mode. Using a classical-field approach, we determine the stationary states and collective excitations of this non-equilibrium system. Apart from the usual Bogoliubov modes, we identify a polariton-like gapped excitation, the frequency of which softens as a precursor of the density ordering transition. Moreover, its relaxation time diverges as the critical point is approached, signaling the non-equilibrium nature of this transition. Dynamically, this polariton-like mode can be probed by inducing cavity field fluctuations, which in turn generates spatio-temporal oscillations of both the condensate and droplet states. In the droplet regime, we also analyze the bound modes which bear the characteristics of such non-equilibrium self-bound state. In addition, we uncover solitonic non-equilibrium states, including condensate with kink-like configuration and double-droplet state, and investigate their robustness following a sudden quench. Remarkably, although these states become unstable beyond a critical coupling, they continue to manifest in the dynamics, akin to the scarring phenomena. Our results demonstrate that dissipative cavity coupling provides a versatile route for exploring rich non-equilibrium dynamics of condensates and quantum droplets within experimentally accessible settings.