Superstripes and the Excitation Spectrum of a Spin-Orbit-Coupled Bose-Einstein Condensate

Using Bogoliubov theory we calculate the excitation spectrum of a spinor Bose-Einstein condensed gas with an equal Rashba and Dresselhaus spin-orbit coupling in the stripe phase. The emergence of a double gapless band structure is pointed out as a key signature of Bose-Einstein condensation and of the spontaneous breaking of translational invariance symmetry. In the long wavelength limit the lower and upper branches exhibit, respectively, a clear spin and density nature. For wave vectors close to the first Brillouin zone, the lower branch acquires an important density character responsible for the divergent behavior of the structure factor and of the static response function, reflecting the occurrence of crystalline order. The sound velocities are calculated as functions of the Raman coupling for excitations propagating orthogonal and parallel to the stripes. Our predictions provide new perspectives for the identification of supersolid phenomena in ultracold atomic gases.

Figure 1

Density profile along the $x$ direction. The parameters are $\Omega /k_0^2=1.0$, $G_1/k_0^2=0.3$, and $G_2/k_0^2=0.08$.

Figure 2

Lowest four excitation bands along the $x$ direction. The parameters are the same as in Fig. 1. The thin dotted line corresponds to the Feynman relation $\omega =q_x^2/2S(q_x)$.

Figure 3

Density (a) and spin-density (b) static structure factor as a function of $q_x$ (blue solid line). The contributions of the first (red dashed line) and second (black dash-dotted line) bands are also shown. The parameters are the same as in Fig. 1.

Figure 4

Static response as a function of $q_x$ (blue solid line). The contributions of the first (red dashed line) and second (black dash-dotted line) bands are also shown. The parameters are the same as in Fig. 1.

Figure 5

Sound velocities in the first (red) and second (black) bands along the $x$ ($c_x$, solid lines) and transverse ($c_{\perp }$, dashed lines) directions as a function of $\Omega$. The blue dash-dotted line represents the transition from the stripe phase to the plane-wave phase. The values of the parameters $G_1/k_0^2$ and $G_2/k_0^2$ are the same as in Fig. 1.