Two-state Bogoliubov theory of a molecular Bose gas

Abstract

We present an analytic Bogoliubov description of a BEC of polar molecules trapped in a quasi-2D 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 dynamic 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. These instabilities are 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 response 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.Comment: 21 pages, 18 figures, comments welcom

Bogoliubov theory of a Bose-Einstein condensate of rigid rotor molecules

“…(For further justification of this approximation, see Ref. [33].) In addition, since we treat the system as being free in-plane, we can make the approximation that the ground state has a uniform density.…”

“…where l = ̵ h mω is the oscillator length for the harmonic trapping. These approximations have been shown to be only qualitatively accurate [33].…”

“…It is well-known that a fully-polarized dipolar gas in quasi-2D will go unstable at a critical interaction strength due to localized density fluctuations [27][28][29][30]39] caused by local sampling of the attractive part of the dipole-dipole interaction. In addition, in the weak external-field limit, a BEC of polarizable molecules can go unstable due to localized fluctuations in the polarization [32,33] caused by attractive interactions between oppositely-aligned dipole moments. In contrast, here, attractive interactions between tilted hω.…”

“…This instability is associated with the onset of a polarization wave (see Ref. [32,33]). While not relevant for a gas of bare rigid rotor molecules, it is possible that such an approximation is relevant for microwave-dressed molecules.…”

“…The signatures of this density wave * Contact author: brandon.peden@wwu.edu roton mode have recently been observed in a quasi-1D gas of erbium atoms interacting via strong magnetic ddi [31]. Later, researchers predicted that when the internal structure of molecules is incorporated at the level of a two-state approximation, a different instability, dominated by polarization fluctuations, emerges at low external field and high interaction strengths [32,33]. Here we extend this work to include multiple internal molecular configurations in the form of quantized rotational states.…”