Manipulating Goldstone modes via the 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 single particle excitation in the previous studies. Instead, we show that such negligible fluctuation of density imbalance dramatically changes the behavior of the low-energy excitation and results in a free switching between two types of Goldstone modes in its single particle excitation, i.e., type I and type II with odd and even power energy-momentum dispersion, respectively. Our proposal would open a new horizon for manipulating Goldstone modes from bridging the cavity light and strongly interacting quantum matters.