Rotating Bose-Einstein condensates with a finite number of atoms confined in a ring potential: Spontaneous symmetry breaking beyond the mean-field approximation

Motivated by recent experiments on Bose-Einstein condensed atoms which rotate in annular and/or toroidal traps, we study the effect of the finiteness of the atom number $N$ on the states of lowest energy for a fixed expectation value of the angular momentum, under periodic boundary conditions. To attack this problem, we develop a general strategy, considering a linear superposition of the eigenstates of the many-body Hamiltonian, with amplitudes that we extract from the mean-field approximation. This many-body state breaks the symmetry of the Hamiltonian; it has the same energy to leading order in $N$ as the mean-field state and the corresponding eigenstate of the Hamiltonian, however, it has a lower energy to subleading order in $N$ and thus it is energetically favorable.

Figure 1

Solid lines: The single-particle density distribution $n\left(\theta \right)$ corresponding to $\left|\mathrm{\Phi }\right(\ell )\rangle$ of a finite system of atoms, within the space of states with $m=-2,\cdots ,3$, for $N=4$ atoms, $\gamma =(N-1)U/\epsilon =0.9$, and $\ell =L/N=0.50$, 0.75, and 1.00, from top to bottom. Dashed lines: The single-particle density distribution $n_{\mathrm{MF}}\left(\theta \right)$ corresponding to $|{\mathrm{\Phi }}_{\mathrm{MF}}\left(\ell \right)\rangle$, derived within the mean-field approximation, via the minimization of the energy, for the same set of parameters.

Figure 2

Same as Fig.1, with $N=8$ and $\gamma =0.9$. The difference between the two curves is hardly visible.

Figure 3

The energy per particle as a function of the angular momentum per particle for $N=4$ atoms, $\gamma =(N-1)U/\epsilon =0.9$, and $\ell =L/N=0.50$, 0.75, and 1.00, within the space of states with $m=-2,\cdots ,3$. The lowest curve corresponds to the expectation value of the energy of $\left|\mathrm{\Phi }\right(\ell )\rangle$; the middle curve, to the eigenvalues of the yrast states $|{\mathrm{\Phi }}_{\mathrm{ex}}\left(L\right)\rangle$; and the top curve, to the energy of the mean-field state $|{\mathrm{\Phi }}_{\mathrm{MF}}\left(\ell \right)\rangle$. We stress that in the middle curve $L$ on the $x$ axis is the eigenvalue of $\widehat{L}$, while in the other two curves $L$ is the expectation value of $\widehat{L}$.