Optical-lattice-assisted magnetic phase transition in a spin-orbit-coupled Bose-Einstein condensate

We investigate the effect of a periodic potential generated by a one-dimensional optical lattice on the magnetic properties of an $S=1/2$ spin-orbit-coupled Bose gas. By increasing the lattice strength one can achieve a magnetic phase transition between a polarized and an unpolarized Bloch wave phase, characterized by a significant enhancement of the contrast of the density fringes. If the wave vector of the periodic potential is chosen close to the roton momentum, the transition could take place at very small lattice intensities, revealing the strong enhancement of the response of the system to a weak density perturbation. By solving the Gross-Pitaevskii equation in the presence of a three-dimensional trapping potential, we shed light on the possibility of observing the magnetic phase transition in currently available experimental conditions.

Figure 1

Static density response ${\chi }_{\rho }\left(q\right)$ of a spin-orbit-coupled BEC in the plane-wave phase as a function of the Raman coupling $\mathrm{\Omega }$, at three different values $q=1.5k_1^0$ (green dash-dotted line), $q=2k_1^0$ (red solid line), and $q=3k_1^0$ (blue dashed line) of the momentum transfer. The red dotted curve corresponds to the response at $q=2k_1^0$ of a standard Bose gas without spin-orbit coupling. The vertical black dash-dotted line identifies the critical Raman coupling ${\mathrm{\Omega }}_{\mathrm{tr}}^0$ at which the transition to the striped phase takes place. The other parameters are $\overline{n}{g}_1/E_r=0.4$ and $g_2/g_1=0.0012$.

Figure 2

(a) Condensation quasimomentum, (b) magnetic polarization, and (c) contrast of the fringes as a function of the dimensionless lattice strength $s$ for $\mathrm{\Omega }/E_r=1.0$. In each panel we show the results for three different values $q=1.5k_1^0$ (green dash-dotted line), $q=2k_1^0$ (red solid line), and $q=3k_1^0$ (blue dashed line) of the momentum transfer. The other parameters are $\overline{n}{g}_1/E_r=0.4$ and $g_2/g_1=0.0012$.

Figure 3

Phase diagram as a function of the Raman coupling $\mathrm{\Omega }$ and of the dimensionless lattice strength $s$ for (a) $q=1.5k_1^0$, (b) $q=2k_1^0$, and (c) $q=3k_1^0$. The blue solid line represents the first-order transition from the mixed to the polarized Bloch wave phase. The red dashed line corresponds to the second-order transition from the polarized to the unpolarized Bloch wave phase. For the latter, the prediction of the ideal gas model is also given (red dashed line). The other parameters are $\overline{n}{g}_1/E_r=0.4$ and $g_2/g_1=0.0012$.

Figure 4

Magnetic susceptibility ${\chi }_M^{}$ as a function of the dimensionless lattice strength $s$. The parameters are the same as in Fig. 2, i.e., $\mathrm{\Omega }/E_r=1.0$; $q=1.5k_1^0$ (green dash-dotted line), $q=2k_1^0$ (red solid line), and $q=3k_1^0$ (blue dashed line); $\overline{n}{g}_1/E_r=0.4$; and $g_2/g_1=0.0012$.

Figure 5

(a) Condensation quasimomentum, (b) magnetic polarization, and (c) contrast of the fringes as a function of the dimensionless lattice strength $s$ for $\mathrm{\Omega }/E_r=3.0$ and $q=3k_1^0=1.98k_0$. The dashed line corresponds to the case of an infinite system with $\overline{n}{g}_1/E_r=0.4, g_2/g_1=0.0012$, and $g_3/g_2=2.0$. The squares are the results for a three-dimensional trapped gas in the conditions described in the text, which yield a density at the center of the trap equal to the average density $\overline{n}$ of the infinite system.

Figure 6

(a) and (b) Lowest-lying energy band of the single-particle dispersion of a spin-orbit-coupled BEC in a one-dimensional optical lattice and (c) and (d) the corresponding magnetic polarization as a function of the quasimomentum $k$, for $\mathrm{\Omega }/E_r=1.0$ and $q=3k_1^0=2.9k_0$. The left and right panels show the results for $s=0.1$ and $s=10.0$, respectively. The region between the vertical dashed lines corresponds to the first Brillouin zone.