Dynamics of Stripe Patterns in Supersolid Spin-Orbit-Coupled Bose Gases

Despite ground-breaking observations of supersolidity in spin-orbit-coupled Bose-Einstein condensates, until now the dynamics of the emerging spatially periodic density modulations has been vastly unexplored. Here, we demonstrate the nonrigidity of the density stripes in such a supersolid condensate and explore their dynamic behavior subject to spin perturbations. We show both analytically in infinite systems and numerically in the presence of a harmonic trap how spin waves affect the supersolid’s density profile in the form of crystal waves, inducing oscillations of the periodicity as well as the orientation of the fringes. Both these features are well within reach of present-day experiments. Our results show that this system is a paradigmatic supersolid, featuring superfluidity in conjunction with a fully dynamic crystalline structure.

Figure 1

Illustration of the interference effects that lead to the appearance and dynamics of stripe patterns. The Raman process responsible for spin-orbit coupling turns the two-component Bose-Einstein condensate (two big circles) into a system with a four-component wave function. Components with the same spin form a spatial interference pattern.

Figure 2

Dynamics of the stripe pattern in a harmonically trapped system for $\hslash \mathrm{\Omega }/E_r=1.75$. (a),(b) Snapshots of the density profile at different times, showing the compression and dilatation of the fringe spacing (a) as well as the rotation of the stripes (b) after suddenly releasing the longitudinal and transversal spin perturbations $H_{x{\sigma }_z}=-m{\omega }_x^2{x}_{0}x{\sigma }_z$ with $x_0/a_x=0.1$ and $H_{y{\sigma }_z}=-m{\omega }_y^2{y}_{0}y{\sigma }_z$ with $y_0/a_y=0.15$, respectively. (c) Evolution of the magnitude of the stripe wave vector $|\mathit{K}|$ and of the longitudinal spin-dipole moment $⟨x{\sigma }_z⟩$ for the scenario in (a). (d) Time trace of the rotation angle $\theta$ of the stripes and of the transversal spin-dipole moment $⟨y{\sigma }_z⟩$ for the scenario in (b). The oscillation frequencies of the stripe pattern coincide with those of the corresponding spin-dipole moments.