Vortex patterns in a fast rotating Bose-Einstein condensate

For a fast rotating condensate in a harmonic trap, we investigate the structure of the vortex lattice using wave functions minimizing the Gross-Pitaevskii energy in the lowest Landau level. We find that the minimizer of the energy in the rotating frame has a distorted vortex lattice for which we plot the typical distribution. We compute analytically the energy of an infinite regular lattice and of a class of distorted lattices. We find the optimal distortion and relate it to the decay of the wave function. Finally, we generalize our method to other trapping potentials.

Figure 1

The structure of the ground state of a rotating Bose-Einstein condensate described by an LLL wave function $(\Lambda =3000)$: (a) vortex location and (b) atomic density profile (with a larger scale). The reduced energy defined in Eq. (2.6) is $ϵ=31.4107101$. The unit for the positions $x$ and $y$ is ${[\hslash ∕\left(m\omega \right)]}^{1∕2}$.

Figure 2

The single particle energy spectrum for $\Omega =0.9$. The index $k$ labels the Landau levels. The energy is expressed in units of $\hslash \omega$.

Figure 3

The minimum reduced energy ${ϵ}_n$ as a function of the number of vortices in the trial wave function $(\Lambda =3000)$.

Figure 4

Vortex distribution minimizing the reduced energy for $\Lambda =3000$ and $n=70$ vortices.

Figure 5

The regular lattice with $\mathcal{A}=\pi$ and distorted lattice minimizing the energy for $\Lambda =3000$ and $n=52$ vortices.

Figure 6

The radial density distribution ${\rho }_{\mathrm{rad}}\left(r\right)$ for $\Lambda =3000$. The unit along the vertical direction is arbitrary. The dotted line is a fit using the inverted parabola with the radius $R_1={(2b\Lambda ∕\pi )}^{1∕4}$, with an adjustable amplitude.

Figure 7

An example of a distorted lattice generated by the transformation Eq. (A.1). The radius of the circle is ${\lambda }_{\alpha }\alpha R_0$. In the regular part outside the circle, the cell area is ${\mathcal{A}}_{\alpha }$.