Quantum Tricriticality and Phase Transitions in Spin-Orbit Coupled Bose-Einstein Condensates

We consider a spin-orbit coupled configuration of spin-$1/2$ interacting bosons with equal Rashba and Dresselhaus couplings. The phase diagram of the system at $T=0$ is discussed with special emphasis on the role of the interaction treated in the mean-field approximation. For a critical value of the density and of the Raman coupling we predict the occurrence of a characteristic tricritical point separating the spin mixed, the phase separated, and the zero momentum states of the Bose gas. The corresponding quantum phases are investigated analyzing the momentum distribution, the longitudinal and transverse spin polarization, and the emergence of density fringes. The effect of harmonic trapping as well as the role of the breaking of spin symmetry in the interaction Hamiltonian are also discussed.

Figure 1

$k_1$, energy per particle $E/N$, transverse and longitudinal spin polarization $⟨{\sigma }_x⟩$ and $|⟨{\sigma }_z⟩|$ as a function of $\Omega$. Red dashed lines: stripe phase $k_1\ne 0$ and $\beta =1/4$; blue dotted lines: separated phase $k_1\ne 0$ and $\beta =0$; green solid lines: zero momentum phase $k_1=0$; open circles: ground state. The parameters: $G_1/k_0^2=0.2$, $G_2/k_0^2=0.05$ (a)–(d), $G_2/k_0^2=0.16$ (e)–(h).

Figure 2

Spin polarization $|⟨{\sigma }_z⟩|$ (a) and $k_1/k_0$ (b) as a function of $\Omega$ and density $n/n^{(c)}$ in three different phases with $G_2>0$. The white solid lines represent the phase transition (I–II), (II–III) and (I–III). The parameters: $g=100a_B$, where $a_B$ is the Bohr radius, $\gamma =0.0012$, $k_0^2=2\pi \times 80\mathrm{Hz}$, corresponding to $n^{(c)}=4.37\times {10}^{15}\mathrm{c}{\mathrm{m}}^{-3}$.

Figure 3

Spin polarization $|⟨{\sigma }_z⟩|$ as a function of $\Omega$ for the trapped case (red solid line), and for the uniform case using the density in the center of the trap (blue dashed line). The parameters are chosen as follows: ${\omega }_0=2\pi \times 20\mathrm{Hz}$, $k_0^2/{\omega }_0=4$, $g_{aa}=g_{bb}=101.20a_B$, $g_{ab}=100.99a_B$, where $a_B$ is the Bohr radius. The density in the center of trap corresponds to $n\simeq 3.9\times {10}^{13}\mathrm{c}{\mathrm{m}}^{-3}$.