Anisotropic dynamics of a spin-orbit-coupled Bose-Einstein condensate

By calculating the density response function we identify the excitation spectrum of a Bose-Einstein condensate with equal Rashba and Dresselhaus spin-orbit coupling. We find that the velocity of sound along the direction of spin-orbit coupling is deeply quenched and vanishes when one approaches the second-order phase transition between the plane-wave and the zero momentum quantum phases. We also point out the emergence of a roton minimum in the excitation spectrum for small values of the Raman coupling, providing the onset of the transition to the stripe phase. Our findings point out the occurrence of a strong anisotropy in the dynamic behavior of the gas. A hydrodynamic description accounting for the collective oscillations in both uniform and harmonically trapped gases is also derived.

Figure 1

Phase diagram corresponding to the spin-orbit-coupled Hamiltonian (4). The lines corresponding to the I-II [solid blue (dark gray)], II-III (dashed red) and I-III [solid green (light gray)] phase transitions are shown. The parameters are $g=4\pi \times 100a_B$, where $a_B$ is the Bohr radius, $\gamma =0.0012$, and $k_0^2=2\pi \times 80$ Hz, corresponding to a critical density $n^{\left(c\right)}=k_0^2/\left(2\gamma g\right)=4.37\times {10}^{15}$ cm${}^{-3}$.

Figure 2

Sound velocity as a function of the Raman coupling for the following choice of parameters: $G_1/k_0^2=0.2$, $G_2/k_0^2=0.05$. The two sound velocities in phase II correspond to phonons propagating in the directions parallel ($c_{\text{II}}^{+}$) and antiparallel ($c_{\text{II}}^{-}$) to $k_1$. The horizontal dashed line corresponds to the value $\sqrt{2G_1}=0.63k_0$ of the sound velocity in the absence of spin-orbit and Raman coupling. The vertical dashed lines indicate the Raman frequencies at which the I-II and II-III phase transitions take place.

Figure 3

Excitation spectrum (a) in phase II ($\Omega /k_0^2=0.85$) and (b) in phase III ($\Omega /k_0^2=2.25$) as a function of $q_x$ ($q_y=q_z=0$). The blue (dark gray) and green (light gray) lines represent the lower and upper branches, respectively. In phase II the spectrum is not symmetric and exhibits a roton minimum for negative $q_x$, whose energy becomes smaller and smaller as one approaches the transition to the stripe phase at $\Omega /k_0^2=0.09$. The other parameters are $G_1/k_0^2=0.12$ and $\gamma =G_2/G_1={10}^{-3}$.

Figure 4

Static response in phase II as a function of $q_x$ ($q_y=q_z=0$). The curve is symmetric and exhibits a typical peak near the roton momentum. The parameters are $\Omega /k_0^2=0.85$, $G_1/k_0^2=0.12$, and $\gamma =G_2/G_1={10}^{-3}$.

Figure 5

Contribution of the lower branch to the static structure factor in phase II as a function of $q_x$ (blue solid line) compared with the total $S\left(q_x\right)$ (red dashed line). The parameters are $\Omega /k_0^2=0.85$, $G_1/k_0^2=0.12$, and $\gamma =G_2/G_1={10}^{-3}$.