Supersolid phase of a spin-orbit-coupled Bose-Einstein condensate: a perturbation approach

TL;DR

This paper develops a perturbation method to analyze the supersolid phase of a spin-orbit-coupled Bose-Einstein condensate, providing analytical insights into its properties and confirming results with numerical simulations.

Contribution

The work introduces a perturbation approach for the supersolid phase, offering analytical predictions for key observables and clarifying the nature of gapless modes.

Findings

Analytical predictions match numerical results for large coupling.

Identification of the Goldstone mode as a translation of density fringes.

Superfluid density can be measured via sound velocity ratios.

Abstract

The phase diagram of a Bose-Einstein condensate with Raman-induced spin-orbit coupling includes a stripe phase with supersolid features. In this work we develop a perturbation approach to study the ground state and the Bogoliubov modes of this phase, holding for small values of the Raman coupling. We obtain analytical predictions for the most relevant observables (including the periodicity of stripes, sound velocities, compressibility, and magnetic susceptibility) which are in excellent agreement with the exact (non perturbative) numerical results, obtained for significantly large values of the coupling. We further unveil the nature of the two gapless Bogoliubov modes in the long-wavelength limit. We find that the spin branch of the spectrum, corresponding in this limit to the dynamics of the relative phase between the two spin components, describes a translation of the fringes of the equilibrium density profile, thereby providing the crystal Goldstone mode typical of a supersolid configuration. Finally, using sum-rule arguments, we show that the superfluid density can be experimentally accessed by measuring the ratio of the sound velocities parallel and perpendicular to the direction of the spin-orbit coupling.