Metastable supersolid in spin-orbit coupled Bose-Einstein condensates

Supersolid is a special state of matter with both superfluid properties and spontaneous modulation of particle density. In this paper, we focus on the supersolid stripe phase realized in a spin-orbit coupled Bose-Einstein condensate and explore the properties of a class of metastable supersolids. In particular, we study a one-dimensional supersolid whose characteristic wave number $k$ (magnitude of wave vector) deviates from $k_{m}$, i.e., the one at ground state. In other words, the period of density modulation is shorter or longer than the one at ground state. We find that this class of supersolids can still be stable if their wave numbers fall in the range $k_{c1}<k<k_{c2}$, with two thresholds $k_{c1}$ and $k_{c2}$. Stripes with $k$ outside this range suffer from dynamical instability with complex Bogoliubov excitation spectrum at long wavelength. Experimentally, these stripes with $k$ away from $k_m$ are accessible by exciting the longitudinal spin dipole mode, resulting in temporal oscillation of stripe period as well as $k$. Within the mean-field Gross-Pitaevskii theory, we numerically confirm that for a large enough amplitude of spin dipole oscillation, the stripe states become unstable through breaking periodicity, in qualitative agreement with the existence of thresholds of $k$ for stable stripes. Our work extends the concept of supersolid and uncovers a new class of metastable supersolids to explore.