Anisotropic and Long-Range Vortex Interactions in Two-Dimensional Dipolar Bose Gases

We perform a theoretical study into how dipole-dipole interactions modify the properties of superfluid vortices within the context of a two-dimensional atomic Bose gas of co-oriented dipoles. The reduced density at a vortex acts like a giant antidipole, changing the density profile and generating an effective dipolar potential centred at the vortex core whose most slowly decaying terms go as $1/{\rho }^2$ and $\ln (\rho )/{\rho }^3$. These effects modify the vortex-vortex interaction which, in particular, becomes anisotropic for dipoles polarized in the plane. Striking modifications to vortex-vortex dynamics are demonstrated, i.e., anisotropic corotation dynamics and the suppression of vortex annihilation.

Figure 1

Vortex density profiles in the presence of DDIs parametrized by ${\epsilon }_{\mathrm{dd}}=g_{\mathrm{dd}}/g$. The left-hand (right-hand) side of each figure shows the profile along $x$ ($y$). (a) Polarization along $z$ ($\alpha =0$). Stable density ripples form close to the onset of the RI (dashed red line). (b) For off-axis dipoles ($\alpha =\pi /4$) the vortex becomes highly anisotropic. Insets: Vortex density $n(x,y)$ for examples of ${\epsilon }_{\mathrm{dd}}$ [area $(40{\xi }_0{)}^2$ for each]. Gray bands indicate unstable regimes where no steady state solutions exist, while the labels $g>0$ and $g<0$ on the individual solution sheets indicate that they are only stable for the specified value of $g$.

Figure 2

(a) The total dipolar potential $\Phi$ (black lines labelled $\Phi$), local component ${\Phi }_{\mathrm{L}}$ (blue lines labelled ${\Phi }_{\mathrm{L}}$), and nonlocal component ${\Phi }_{\mathrm{NL}}$ (red lines labelled ${\Phi }_{\mathrm{NL}}$) for a vortex with ripples (${\epsilon }_{\mathrm{dd}}=-1.2$, $g<0$, dashed lines) and without ripples (${\epsilon }_{\mathrm{dd}}=5$, $g>0$, solid lines). (b) The decay of ${\Phi }_{\mathrm{L}}/{\Phi }_0$, compared with $1/{\rho }^{\prime 2}$ (gray line). (c) The decay of ${\Phi }_{\mathrm{NL}}/{\Phi }_0$, compared with $(A\ln {\rho }^{\prime }+B)\sigma /{\rho }^{\prime 3}$. Parameters are $\alpha =0$, $\sigma =0.5$.

Figure 3

Vortex interaction energy $E_{\mathrm{int}}$ versus separation for (a) VA and (b) VV pairs with various ${\epsilon }_{\mathrm{dd}}$ values. Insets: Energy difference from the nondipolar case, $\Delta E_{\mathrm{int}}=E_{\mathrm{int}}(d,{\epsilon }_{\mathrm{dd}})-E_{\mathrm{int}}(d,{\epsilon }_{\mathrm{dd}}=0)$. Parameters are $\alpha =0$, $\sigma =0.5$.

Figure 4

Angular dependence of the vortex interaction energy $\Delta E_{\mathrm{int}}^{\prime }(\eta )=E_{\mathrm{int}}(\eta )-E_{\mathrm{int}}(\eta =0)$, for (a) VA and (b) VV pairs of differing separation. Parameters are $\alpha =\pi /4$, $\sigma =0.5$, and ${\epsilon }_{\mathrm{dd}}=5$ ($g>0$). Example (c) VA and (d) VV pair density profiles for $d=5{\xi }_0$ [area $(20{\xi }_0{)}^2$ for each].

Figure 5

(a) A VA pair (initial separation $d=1.5\xi$) with no DDIs annihilates while DDIs ($\alpha =0$) stabilize the pair (red lines). (b) A VV pair ($d=2{\xi }_0$) with no DDIs corotates in a circular path. Off-axis ($\alpha =\pi /4$) DDIs lead to anisotropic paths and even suppression of corotation altogether.