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arXiv:1611.09256 · Physical Review A

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

A. Roussou, J. Smyrnakis, Manolis Magiropoulos, Nikolaos K. Efremidis, G. M. Kavoulakis

Published Mar 3, 2017• 11 citations
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Abstract

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.

DOI
10.1103/physreva.95.033606

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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 NNN 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 NNN as the mean-field state and the corresponding eigenstate of the Hamiltonian, however, it has a lower energy to subleading order in NNN and thus it is energetically favorable.
Abstract We consider a Bose–Einstein condensate, which is confined in a very tight toroidal/annular trap, in the presence of a potential, which breaks the axial symmetry of the Hamiltonian. We investigate the stationary states of the condensate, when its density distribution co-rotates with the symmetry-breaking potential. As the strength of the potential increases, we have a gradual transition from vortex excitation to solid-body-like motion. Of particular importance are states where the system is static and yet it has a nonzero current/circulation, which is a realization of persistent currents/reflectionless potentials.
We investigate the ground and low excited states of a rotating, weakly interacting Bose-Einstein condensed gas in a harmonic trap for a given angular momentum. Analytical results in various limits as well as numerical results are presented, and these are compared with those of previous studies. Within the mean-field approximation and for repulsive interaction between the atoms, we find that for very low values of the total angular momentum per particle, $L/\stackrel{\ensuremath{\rightarrow}}{N}0,$ where $L\ensuremath{\Elzxh}$ is the angular momentum and N is the total number of particles, the angular momentum is carried by quadrupole (∣m∣=2)(|m|=2)(∣m∣=2) surface modes. For L/N=1,L/N=1,L/N=1, a vortexlike state is formed and all the atoms occupy the m=1m=1m=1 state. For small negative values of $L/N\ensuremath{-}1,$ the states with m=0m=0m=0 and m=2m=2m=2 become populated, and for small positive values of $L/N\ensuremath{-}1,$ atoms in the states with m=5m=5m=5 and m=6m=6m=6 carry the additional angular momentum. In the whole region 0< L/N< 1,0<~L/N<~1,0< L/N< 1, we have strong analytic and numerical evidence that the interaction energy drops linearly as a function of L/N.L/N.L/N. We have also found that an array of singly quantized vortices is formed as L/NL/NL/N increases. Finally, we have gone beyond the mean-field approximation and have calculated the energy of the lowest state up to order N for small negative values of $L/N\ensuremath{-}1,$ as well as the energy of the low-lying excited states.
We consider a weakly interacting Bose-Einstein condensate that rotates in either a harmonic, or a weakly anharmonic trapping potential. Performing numerical calculations, we investigate the behavior of the gas in these two cases as the angular momentum, or equivalently as the rotational frequency of the trap increases. While in the case of a purely harmonic potential the gas makes a transition from the mean-field regime to the correlated Laughlin regime, in the case of anharmonic confinement the mean-field approximation remains always valid. We compare our derived results in these two cases, using both the mean-field approximation, as well as the diagonalization of the many-body Hamiltonian considering a small atom number.
For a weakly interacting Bose gas rotating in a harmonic trap we relate the yrast states of small systems (that can be treated exactly) to the thermodynamic limit (derived within the mean-field approximation). For a few dozens of atoms, the yrast line shows distinct quasiperiodic oscillations with increasing angular momentum that originate from the internal structure of the exact many-body states. These finite-size effects disappear in the thermodynamic limit, where the Gross-Pitaevskii approximation provides the exact energy to leading order in the number of particles NNN. However, the exact yrast states reveal significant structure not captured by the mean-field approximation: Even in the limit of large NNN, the corresponding mean-field solution accounts for only a fraction of the total weight of the exact quantum state.
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.