Approach for making visible and stable stripes in a spin-orbit-coupled Bose-Einstein superfluid

The striped phase exhibited by a spin-$1/2$ Bose-Einstein condensate with spin-orbit coupling is characterized by the spontaneous breaking of two continuous symmetries: gauge and translational symmetry. This is a peculiar feature of supersolids and is the consequence of interaction effects. We propose an approach to produce striped configurations with high-contrast fringes, making their experimental detection in atomic gases a realistic perspective. Our approach, whose efficiency is directly confirmed by three-dimensional Gross-Pitaevskii simulations, is based on the space separation of the two spin components into a two-dimensional bilayer configuration, causing the reduction of the effective interspecies interaction and the increase of the stability of the striped phase. We also explore the effect of a $\pi /2$ Bragg pulse, causing the increase of the fringe wavelength, and of a $\pi /2$ rf pulse, revealing the coherent nature of the order parameter in the spin channel.

Figure 1

Integrated density profile $\int dydzn$ in the striped phase. (a) displays the situation without separation of the traps for the two spin components. (b) corresponds instead to traps separated along $z$ by a distance $d=a_z$, which increases the contrast of the fringes.

Figure 2

Integrated density profiles $\int dzn$ (top) and $\int dydzn$ (bottom) in the stripe phase, in the same conditions as Fig. 1, after the application of a $\pi /2$ Bragg pulse transferring momentum $\pm 1.8\hslash k_1$.

Figure 3

3D Gross-Pitaevskii simulation for the integrated spin density profiles $\int dydz(n_{\uparrow }-n_{\downarrow })$ in the stripe phase, in the same conditions as Fig. 1, after the application of a fast $\pi /2$ rf pulse.