Tricriticalities and Quantum Phases in Spin-Orbit-Coupled Spin-1 Bose Gases

We study the zero-temperature phase diagram of a spin-orbit-coupled Bose-Einstein condensate of spin 1, with equally weighted Rashba and Dresselhaus couplings. Depending on the antiferromagnetic or ferromagnetic nature of the interactions, we find three kinds of striped phases with qualitatively different behaviors in the modulations of the density profiles. Phase transitions to the zero-momentum and the plane-wave phases can be induced in experiments by independently varying the Raman coupling strength and the quadratic Zeeman field. The properties of these transitions are investigated in detail, and the emergence of tricritical points, which are the direct consequence of the spin-dependent interactions, is explicitly discussed.

Figure 1

Phase diagram for antiferromagnetic interactions for $g_0\overline{n}/k_0^2=0.4$ and $g_2\overline{n}/k_0^2=0.012$. The ratio $g_2/g_0=0.03$ is typical of ${}^{23}\mathrm{Na}$. Continuous (dashed) lines represent first-(second-)order transitions. The blue lines define the region of the first polar striped (PS1) phase. The region of the second polar striped (PS2) phase is delimited by the dashed green and blue lines and by the continuous red one. The black line represents the transition between the zero-momentum (ZM) and plane-wave (PW) phases. The inset displays a zoom close to the tricritical point $C_P$.

Figure 2

Phase diagram for ferromagnetic interactions for $g_0\overline{n}/k_0^2=0.4$ and $g_2\overline{n}/k_0^2=-0.2$. The ratio $g_2/g_0=-0.5$ is typical of ${}^7\mathrm{Li}$. Continuous (dashed) lines represent first-(second-)order transitions. The blue lines define the region of the ferromagnetic striped (FS) phase. The black line represents the transition between the zero-momentum (ZM) and plane-wave (PW) phases. At the dotted green (red) line, the ZM (PW) state disappears as a local minimum of the energy.