Vortex lattices in the lowest Landau level for confined Bose-Einstein condensates

We present the results of numerical calculations of the ground states of weakly interacting Bose-Einstein condensates containing large numbers of vortices. Our calculations show that these ground states appear to be close to uniform triangular vortex lattices. However, slight deviations from a uniform triangular lattice have dramatic consequences on the overall particle distribution. In particular, we demonstrate that the overall particle distribution averaged on a length scale large compared to the vortex lattice constant is well approximated by a Thomas-Fermi profile.

Figure 1

The particle density for two vortex lattices which are calculated as the minimum of the interaction energy (1). In the upper entry we have angular momentum $\ell =31$ and in the lower entry $\ell =91$. The gray-scale code is set to black for vanishing particle density and to white for maximum density, which occurs at the central region of the trap.

Figure 2

The particle density for two wave functions obtained under the assumpion of a perfect triangular vortex lattice. In the upper entry we use a lattice constant $a=1.9351a_{\perp }$ which corresponds to angular momentum $\ell =31$ and in the lower entry we have $a=1.9151a_{\perp }$ which gives $\ell =91$. The gray-scale code is set to black for vanishing particle density and to white for maximum density, which occurs at the central region of the trap. A qualitative comparison with Fig. 1 is easy, but note that the white color in Fig. 1 does not correspond to the same value in the present figure.

Figure 3

The angular-averaged particle density as a function of the radial coordinate for (a) $\ell =31$ and (b) $\ell =91$. (The particle density is in units of $a_{\perp }^{-2}$ and $\rho$ in units of $a_{\perp }$.) In both figures, the solid line corresponds to the full variational result, and the dashed line corresponds to the triangular lattice ansatz. The dotted lines are the Thomas-Fermi profiles, Eq. (6), for (a) $\ell =31$ and (b) $\ell =91$, while the dot-dashed curves are the Gaussian profiles (9) for the lattice constant (a) $a=1.9351a_{\perp }$ and (b) $a=1.9151a_{\perp }$.

Figure 4

The interaction energy (1) as a function of the angular momentum per particle, for the full variational ground state (solid line) and for the triangular lattice ansatz (dashed line). [The interaction energy is plotted in units of $2\pi {\hslash }^2{a}_s{N}^2∕\left(M{a}_{\perp }^2{a}_{\Vert }\right)$.]