Vortex lattice formation in dipolar Bose-Einstein condensates via rotation of the polarization

The behavior of a harmonically trapped dipolar Bose-Einstein condensate with its dipole moments rotating at angular frequencies lower than the transverse harmonic trapping frequency is explored in the co-rotating frame. We obtain semi-analytical solutions for the stationary states in the Thomas-Fermi limit of the corresponding dipolar Gross-Pitaevskii equation and utilize linear stability analysis to elucidate a phase diagram for the dynamical stability of these stationary solutions with respect to collective modes. These results are verified via direct numerical simulations of the dipolar Gross-Pitaevskii equation, which demonstrate that dynamical instabilities of the co-rotating stationary solutions lead to the seeding of vortices that eventually relax into a triangular lattice configuration. Our results illustrate that rotation of the dipole polarization represents a new route to vortex formation in dipolar Bose-Einstein condensates.

Figure 1

Stationary solutions, as characterized by $\alpha$, as a function of $\mathrm{\Omega }$: (a) $\gamma =1$ and various ${\epsilon }_{\text{dd}}$; (b) ${\epsilon }_{\text{dd}}=0.5$ and various $\gamma$. Branch I obeys $\alpha \ge 0$ while Branches II and III obey $\alpha \le 0$, and $\mathrm{\Omega }={\mathrm{\Omega }}_{\text{b}}$ when Branches II and III coincide.

Figure 2

Cross sections at $z=0$ of density of a dipolar BEC during an quasi-adiabatic ramp-up of $\mathrm{\Omega }$ up to ${\mathrm{\Omega }}_{\text{f}}=(0.55,0.85,0.95){\omega }_{\perp }$. Snapshots taken at $t=0$ in (a)–(c) and (g)–(i) and $t=500{\omega }_{\perp }^{-1}$ in (d)–(f) and (j)–(l). In (a)–(f) ${\epsilon }_{\text{dd}}=0.1$ and (g)–(l) ${\epsilon }_{\text{dd}}=0.5$. Lengths and density are scaled by $l_{\perp }=\sqrt{\hslash /\left(m{\omega }_{\perp }\right)}$. The black ellipses depict the predicted Branch I TF profiles from Eqs. (20), (21), and (24).

Figure 3

Comparison of the TF solutions to the numerical integration of the dGPE. Black lines show the TF predictions for Branch I, underlying the numerically extracted $\alpha$ for ${\epsilon }_{\text{dd}}=0.1$ (blue triangles) and ${\epsilon }_{\text{dd}}=0.5$ (green circles). Red crosses show solutions that are unstable after a long-time evolution with $\mathrm{\Omega }={\mathrm{\Omega }}_{\text{f}}$ fixed. Dotted lines are the columns of Fig. 2.

Figure 4

Long-time evolution of a dipolar BEC after a ramp up to $\mathrm{\Omega }={\mathrm{\Omega }}_{\text{f}}$, with ${\mathrm{\Omega }}_{\text{f}}=0.850{\omega }_{\perp }$ and ${\epsilon }_{\text{dd}}=0.5$, into a triangular vortex lattice.

Figure 5

Dynamical instability of the TF stationary states: With the truncation parameter $N_{\text{max}}=10$, the shaded region indicates where there exists at least one linearized eigenvalue with a positive real component greater than (a) ${10}^{-4}$ or (b) ${10}^{-2}$. Overlain in red are the trajectories of the dGPE simulations for ${\epsilon }_{\text{dd}}\in \{0.1,0.2,0.3,0.4,0.5\}$ wi9ith filled circles at ${\mathrm{\Omega }}_{\text{v}}$.