Two-State Bogoliubov Theory Of A Molecular Bose Gas

We present an analytic Bogoliubov description of a Bose-Einstein condensate of polar molecules trapped in a quasi-two-dimensional geometry and interacting via internal state-dependent dipole-dipole interactions. We derive the mean-field ground-state energy functional, and we derive analytic expressions for the dispersion relations, Bogoliubov amplitudes, and static structure factors. This method can be applied to any homogeneous, two-component system with linear coupling and direct, momentum-dependent interactions. The properties of the mean-field ground state, including polarization and stability, are investigated, and we identify three distinct instabilities: a density-wave rotonization that occurs when the gas is fully polarized, a spin-wave rotonization that occurs near zero polarization, and a mixed instability at intermediate fields. The nature of these instabilities is clarified by means of the real-space density-density correlation functions, which characterize the spontaneous fluctuations of the ground state, and the momentum-space structure factors, which characterize the response of the system to external perturbations. We find that the gas is susceptible to both density-wave and spin-wave responses in the polarized limit but only a spin-wave response in the zero-polarization limit. These results are relevant for experiments with rigid rotor molecules such as RbCs, Lambda-doublet molecules such as ThO that have an anomalously small zero-field splitting, and doublet-Sigma molecules such as SrF where two low-lying opposite-parity states can be tuned to zero splitting by an external magnetic field.