Vortex patterns and the critical rotational frequency in rotating dipolar Bose-Einstein condensates

Based on the two-dimensional mean-field equations for pancake-shaped dipolar Bose-Einstein condensates in a rotating frame, for both attractive and repulsive dipole-dipole interaction (DDI) as well as arbitrary polarization direction, we study the profiles of the single vortex state and show how the critical rotational frequency changes with the $s$-wave contact interaction strength, DDI strength, and polarization angle. In addition, we find numerically that at the “magic angle” $\vartheta =arccos(\sqrt{3}/3)$, the critical rotational frequency is almost independent of the DDI strength. By numerically solving the dipolar Gross-Pitaevskii equation at high rotation speed, we identify different patterns of vortex lattices which strongly depend on the polarization direction. As a result, we undergo a study of vortex lattice structures for the whole regime of polarization direction and find evidence that the vortex lattice orientation tends to be aligned to the dipole polarization axis for positive DDI strength and to the perpendicular direction of the dipole axis for negative DDI strength.

Figure 1

Density of a rotating dipolar BEC around the critical rotational frequency ${\mathrm{\Omega }}_c$ for fixed $\gamma =10$, $g=250$, $g_d=-100$, different polarization axis $\mathbf{n}=(sin\vartheta ,0,cos\vartheta )$ [(a),(b) $\vartheta =0$; (c),(d) $\vartheta =\pi /4$; (e),(f) $\vartheta =\pi /2]$. The critical rotational frequency ${\mathrm{\Omega }}_c$ is found to be $0.356<{\mathrm{\Omega }}_c<0.357$ (top panels), $0.275<{\mathrm{\Omega }}_c<0.276$ (middle panels), $0.236<{\mathrm{\Omega }}_c<0.237$ (bottom panels), with the corresponding lower bound of rotational frequency for the nonvortex states and the upper bound for the vortex state.

Figure 2

Density of a rotating dipolar BEC around the critical rotation frequency ${\mathrm{\Omega }}_c$ for fixed $\gamma =10$, $g=250$, $g_d=200$, different polarization axis $\mathbf{n}=(sin\vartheta ,0,cos\vartheta )$ [(a),(b) $\vartheta =0$; (c),(d) $\vartheta =\pi /4$; (e),(f) $\vartheta =\pi /2]$. The critical rotational frequency ${\mathrm{\Omega }}_c$ is found to be $0.195<{\mathrm{\Omega }}_c<0.196$ (top panels), $0.232<{\mathrm{\Omega }}_c<0.233$ (middle panels), $0.357<{\mathrm{\Omega }}_c<0.358$ (bottom panels), with the corresponding lower bound of rotational frequency for the nonvortex states and the upper bound for the vortex state.

Figure 3

The critical rotational frequency ${\mathrm{\Omega }}_c$ of a rotating dipolar BEC vs the $s$-wave contact interaction strength $g=4\pi N{a}_s/a_r$ for fixed $\gamma =10$, dipole axis $\mathbf{n}=(0,0,1)$, and a natural dimensionless parameter ${\varepsilon }_{\mathrm{dd}}:=g_d/g=-0.5$, 0, and 1, respectively.

Figure 4

The critical rotational frequency ${\mathrm{\Omega }}_c$ of a rotating dipolar BEC vs polarization angle $\vartheta$ for fixed $\gamma =10$, $g=250$, and ${\varepsilon }_{\mathrm{dd}}=-0.4$, $-0.2$, 0, 0.4, and 0.8, respectively.

Figure 5

Illustration of the Bravais lattice basis vectors and the lattice parameters.

Figure 6

Density of a rotating dipolar BEC for different dipole polarization direction $\mathbf{n}=(sin\vartheta ,\mathbf{0},cos\vartheta )$, with $\vartheta =0,arcsin\left(0.5\right),arcsin\left(0.8\right),arcsin\left(0.95\right),arcsin\left(0.96\right),\pi /2$. The rotational frequ- ency is $\mathrm{\Omega }=0.95$, $\gamma =10$, $g=250$, and $g_d=250$.

Figure 7

Static structure factor S. Same parameters as in Fig. 6.

Figure 8

Density of a rotating dipolar BEC for different dipole polarization angles with $\phi =0$ and $\vartheta =0,arcsin\left(0.3\right)$, $arcsin\left(0.8\right),arcsin\left(0.98\right),arcsin\left(0.99\right),\pi /2$ (from left to right) and for different DDI strengths with ${\varepsilon }_{{}_{dd}}=-0.5,-0.2,0.3,0.8,0.95,1.0$ (from bottom to top). The rotation frequency is $\mathrm{\Omega }=0.99$, $\gamma =10$, and $g=250$.

Figure 9

Lattice orientation parameter $\eta$ vs $\vartheta$. Same parameters as in Fig. 6.