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) surface modes. For
L/N=1, a vortexlike state is formed and all the atoms occupy the
m=1 state. For small negative values of $L/N\ensuremath{-}1,$ the states with
m=0 and
m=2 become populated, and for small positive values of $L/N\ensuremath{-}1,$ atoms in the states with
m=5 and
m=6 carry the additional angular momentum. In the whole region
0< L/N< 1, we have strong analytic and numerical evidence that the interaction energy drops linearly as a function of
L/N. We have also found that an array of singly quantized vortices is formed as
L/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 study the energy eigenvalue problem and the yrast states for a harmonically trapped two-component weak-interacting Bose-Einstein condensate (BEC) using an analytical method. The yrast states and energy eigenvalues for
L=0,1,2,3 are given in a special case. We illustrate the different eigenstate behaviors between even L and odd L for the two-component BEC. The odd-even effect of the yrast spectrum in the two-component BEC is also pointed out.
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
N. However, the exact yrast states reveal significant structure not captured by the mean-field approximation: Even in the limit of large
N, the corresponding mean-field solution accounts for only a fraction of the total weight of the exact quantum state.
In a k-dimensional system of weakly interacting Bose atoms trapped by a spherically symmetric and harmonic external potential, an exact expression is obtained for the rotating ground states at a fixed angular momentum. The result is valid for arbitrary interactions obeying minimal physical requirements. Depending on the sign of a modified scattering length, it reduces to either a collective rotation or a condensed vortex state, with no alternative. The ground state can undergo a kind of quantum phase transition when the shape of the interaction potential is smoothly varied.
We present an exact diagonalization study of the breathing mode collective excitations for a rotating Bose-Einstein condensate of
N=10 spinless bosons interacting via repulsive finiterange Gaussian potential and harmonically confined in quasi-two-dimension. The yrast state and the low-lying excited states are variationally obtained in given subspaces of the quantized total angular momentum
L employing the beyond lowest Landau level approximation in slowly rotating regime with
0≤L<2N. For a given
L, the low-energy eigenspectra (bands) are obtained in weakly to moderately interacting regime. Further, for a given interaction, the split in low-lying eigenenergies with increasing
L is the precursor to spontaneous symmetry breaking of the axisymmetry associated with the entry of the first vortex. With increase in repulsive interaction, the value of the first breathing mode increases for stable total angular momentum states
L=0 and
N, but decreases for intermediate
0<L<N metastable states. The position of the observed first breathing modes in the eigenspectrum remains unchanged as the interaction is varied over several orders of magnitude.