Manipulating Goldstone modes via superradiant light in a bosonic lattice gas inside a cavity

We study the low-energy excitations of a bosonic lattice gas with cavity-mediated interactions. By performing two successive Hubbard-Stratonovich transformations, we derive an effective field theory to study the strongly coupling regime. Taking into account the quantum fluctuation, we report the unusual effect of the superradiant cavity light-induced density imbalance, which has been shown to have a negligible effect on the low-energy excitation spectrum in previous studies. Instead, we show that such infinitesimal fluctuation of density imbalance dramatically changes the behavior of low-energy excitations and results in a parameter-driven switching between two types of Goldstone modes, i.e., type I and type II with odd and even power energy-momentum dispersion, respectively. Our proposal would open a horizon for manipulating Goldstone modes by bridging the cavity light and strongly interacting quantum matters.