Structure of vortices in rotating Bose-Einstein condensates

We calculate the structure of individual vortices in rotating Bose-Einstein condensates in a transverse harmonic trap. Making a Wigner-Seitz approximation for the unit cell of the vortex lattice, we derive the Gross-Pitaevskii equation for the condensate wave function in each cell of the lattice, including effects of varying coarse grained density. We calculate the Abrikosov parameter, the fractional core area, and the energy of individual cells.

Figure 1

A Wigner-Seitz cell of the vortex lattice. The position of the center of the cell $j$ relative to the center of the trap is ${\mathbf{R}}_j$, and $\mathbf{\rho }$ is the transverse coordinate measured from ${\mathbf{R}}_j$.

Figure 2

The first five basis functions $f_i^{\left(0\right)}$ of the free Hamiltonian $H_0$, for $\Omega =\omega$ for the central cell, where $\tilde{\omega }=\omega$.

Figure 3

(Color online) Vortex core structure at various $\Omega$ for the central cell obtained from the Gross-Pitaevskii equation in a Wigner-Seitz cell, with ${\kappa }_0=10$ (black lines) and ${\kappa }_0=100$ (blue lines).

Figure 4

(Color online) Comparison of the numerical solution of the local wave function $f$ with the LLL structure $f_{\mathrm{LLL}}$ and with the single vortex form $f_{\mathrm{c}}$, calculated in the central cell.

Figure 5

The amplitudes $c_i$ of the states $f_i^{\left(0\right)}$ in the expansion of $f$, for the cases of ${\kappa }_0=10$ shown in Figs. 3, 4. In all cases here, $c_1>0.995$ and $c_i<0.1$ for $i⩾2$.

Figure 6

Abrikosov parameter $b$ vs ${\Gamma }_{\mathrm{LLL}}^{-1}\equiv 2\hslash \Omega ∕g_{2\mathrm{D}}n\left(0\right)$, for ${\kappa }_0=10$ (for $\Omega ⩾0.425\omega$) and ${\kappa }_0=100$ (for $\Omega ⩾0.5\omega$).

Figure 7

(Color online) Fractional area of the core for the central cell, measured by $r_{\mathrm{c}}^2∕{\ell }^2$ (solid lines) and $r_{\mathrm{c}}^{\prime 2}∕{\ell }^2$ (dashed lines) (see text), as functions of ${\Gamma }_{\mathrm{LLL}}^{-1}$. The lines a and c are for ${\kappa }_0=100$ and the lines b and d for ${\kappa }_0=10$. The dotted lines show the empirical expression for the fractional area, $1.34{\Gamma }_{\mathrm{LLL}}^{-1}$, obtained for low $\Omega$ [9] and the asymptotic value of $r_{\mathrm{c}}^2∕{\ell }^2$ for large ${\Gamma }_{\mathrm{LLL}}^{-1}$ [8].

Figure 8

(Color online) The energy of the central Wigner-Seitz cell $E_0$ (solid line) in units of $n{\hslash }^2∕m$. The kinetic plus trapping potential energy $E_{0,0}$ (dashed line) and interaction energy $E_{0,\mathrm{int}}$ (dash-dotted line) are also plotted. (In the noninteracting case, $E_0=2\pi$.) The “binding” energy $E_{0,\mathrm{b}}$ is the energy gain compared to the energy for the first basis state $f_1^{\left(0\right)}$.

Figure 9

(Color online) Spatial dependence of the vortex core structure for ${\kappa }_0=100$ and $\Omega =0.9\omega$.