Rotating Bose-Einstein condensates: Closing the gap between exact and mean-field solutions

When a Bose-Einstein-condensed cloud of atoms is given some angular momentum, it forms vortices arranged in structures with a discrete rotational symmetry. For these vortex states, the Hilbert space of the exact solution separates into a “primary” space related to the mean-field Gross-Pitaevskii solution and a “complementary” space including the corrections beyond mean field. Considering a weakly interacting Bose-Einstein condensate of harmonically trapped atoms, we demonstrate how this separation can be used to close the conceptual gap between exact solutions for systems with only a few atoms and the thermodynamic limit for which the mean field is the correct leading-order approximation. Although we illustrate this approach for the case of weak interactions, it is expected to be more generally valid.

Figure 1

Isosurfaces of the density distribution of the bosonic cloud within the mean-field approximation (taken at about one-third of the maximum value) for different ratios of angular momentum to particle number, $L/N=1,1.8$, and 2.3 from left to right (first discussed by Butts and Rokshar [20]). The vortices appear as holes, and the phase of the order parameter [in color scale from light to dark blue (gray)] jumps by $2\pi$ when encircling each vortex.

Figure 2

Interaction energies (in units of $v_0$; see text) for $L=N=100$ evaluated within the model [left, red (light gray) circles] as a function of the highest occupancy $N_c$ of the complementary space for a single-particle basis of orbitals with $0\le m\le 6$. The blue (dark gray) circles to the right show the result from the “exact” approach in the same subspace of single-particle states (see text). The relative error between the model and the full diagonalization in the same subspace is given in Fig. 3 for the yrast state (“Y”) and the Goldstone mode (“G”; see text), converging to numbers as small as ${10}^{-7}$ to ${10}^{-10}$ for values of $N_c\approx N/2$. (The state labeled “CM” is a center-of-mass excitation.)

Figure 3

(a) The relative error in the interaction energy and (b) the deviation from unity overlap of the model yrast state with the exact yrast state, $1-|\langle {\Psi }_{\mathrm{A}}^0|{\Psi }_{\mathrm{ex}}\rangle {|}^2$, on a logarithmic scale as a function of the highest occupancy of the complementary space $N_c$ for the data of Fig. 2. (c) The dimension of the submatrices as a function of $N_c$.

Figure 4

The quantity $1-|\langle {\Psi }_A^0|{\Psi }_{\mathrm{ex}}\rangle {|}^2$ as a function of $N$ for a fixed angular momentum per particle, $L/N=9/5$. Here, at each point the number $N_c$ of particles in the complementary space was increased until numerical convergence of the result was obtained. The (red) squares refer to the space with $0\le m\le 4$ (open squares, $K=1$, and solid squares, $K=4$, where $K$ is the number of lowest-energy states included in the reduced matrix), and the (blue) circles refer to the space with $0\le m\le 6$ (open circles, $K=1$, and solid circles, $K=4$).

Figure 5

Occupancies of the exact yrast states at $L/N=9/5$ as a function of the particle number $N$ on a double-logarithmic scale here for a basis with $0\le m\le 7$. The saturation of the single-particle basis is reflected by the significant reduction in occupancy for higher values of $m$. The circles (connected by solid lines to guide the eye) indicate single-particle states that belong to the primary space; the squares (dashed lines) indicate those that belong to the complementary space. Clearly, with increasing $N$, there is a convergence towards the occupancies obtained from mean field (see Table 1).