For a system of N bosons in a two-dimensional harmonic trap with frequency $\ensuremath{\omega},$ interacting via forces $V\ensuremath{\ll}\ensuremath{\Elzxh}\ensuremath{\omega},$ we develop a method to find the ground states of rotating Bose condensate as a function of two quantum numbers, the total angular momentum and the angular momentum of internal excitations (generalized yrast states). The energies of these condensed vortex states are expressed through the single two-body matrix element of interaction,
V. A broad universality class of predominantly repulsive interactions for which these results hold is described by a simple integral condition on
V. It includes Gaussian, $\ensuremath{\delta}$ function, and log-Coulomb forces.
In this thesis we develop an orthogonalised Hartree-Fock-Bogoliubov (HFB) formalism that has a zero-energy excitation (in contrast with standard HFB). We demonstrate that this formalism satisfies the number and linear/angular momentum conservation laws (as does standard HFB). This formalism is applied to vortices in 2D Bose-Einstein condensates (BECs) in axially-symmetric harmonic traps, where we initially find solutions for on-axis vortices, determining the energy spectrum and hence the lowest core localised state (LCLS) energy. In the T=0 case we identify this with the anomalous mode which gives a zero excitation energy in the frame rotating at this frequency. For this reason the anomalous mode frequency was identified in the earlier literature with the precessional frequency for an off-axis vortex. However the LCLS energy is positive in the finite temperature case. Hence, associating this LCLS energy with the precessional frequency leads to the erroneous conclusion that the vortex precesses in the opposite direction to the T=0 case, which is clearly physically unreasonable. In order to address this problem, we derive an equation for the prediction of vortex precessional frequencies from the continuity equation, and use this equation solved self-consisently with the orthogonal HFB equations in the frame rotating at this predicted frequency to create off-axis vortices at pre-specified positions. Hence we are able to predict the precessional frequencies and show that these are consistent with the T=0 case, and are entirely unrelated to the LCLS energy. We also consider a two-state model and demonstrate that this model is insufficient for the description of single off-axis precessing vortices. We formulate a generalised multi-state model, using the normalisation conditions for the model to derive an equation predicting the precessional frequency of the vortex, and demonstrate equivalence with the continuity equation prediction. We use the time-dependent HFB equations to simulate creation of vortices by stirring the BEC by means of a Gaussian optical potential, finding very good agreement of the measured precessional frequencies of the vortices in stirred BECs with the predicted values. We find the existence of a critical stirring frequency for the creation of vortices in regions of appreciable superfluid density, in qualitative agreement with experiment. We then investigate the consequences of breaking rotational symmetry and find that breaking the axial symmetry of the harmonic trapping potential leads to loss of angular momentum, and hence to the decay of vortices. Finally we develop a finite temperature treatment of the Bose-Hubbard model based upon the Hartree-Fock-Bogoliubov formalism in the Popov approximation to study the effect of temperature upon the superfluid phase of ultracold, weakly interacting bosons in a one-dimensional optical lattice.
We consider a gas of
N(< 15) Bose particles with hard-core repulsion, contained in a quasi-two-dimensional harmonic trap and subjected to an overall angular velocity $\ensuremath{\Omega}$ about the z axis. Exact diagonalization of the $n\ifmmode\times\else\texttimes\fi{}n$ many-body Hamiltonian matrix in given subspaces of the total (quantized) angular momentum
Lz, with $n\ensuremath{\sim}{10}^{5}$ (e.g., for
Lz=N=15,n=240782) was carried out using Davidson's algorithm. The many-body variational ground-state wave function, as also the corresponding energy and the reduced one-particle density-matrix $\ensuremath{\rho}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}})={\ensuremath{\sum}}_{\ensuremath{\mu}}{\ensuremath{\lambda}}_{\ensuremath{\mu}}{\ensuremath{\chi}}_{\ensuremath{\mu}}^{*}(\mathbf{r}){\ensuremath{\chi}}_{\ensuremath{\mu}}({\mathbf{r}}^{\ensuremath{'}})$ were determined. With the usual identification of $\ensuremath{\Omega}$ as the Lagrange multiplier associated with
Lz for a rotating system, the ${L}_{z}\ensuremath{-}\ensuremath{\Omega}$ phase diagram (or the stability line) was determined that gave a number of critical angular velocities ${\ensuremath{\Omega}}_{\mathrm{ci}}, i=1,2,3,\dots{},$ at which the ground-state angular momentum and the associated condensate fraction, given by the largest eigenvalue of the reduced one-particle density matrix, undergo abrupt jumps. For a given N, a number of (total) angular momentum states were found to be stable at successively higher critical angular velocities ${\ensuremath{\Omega}}_{\mathrm{ci}}, i=1,2,3,\dots{}.$ All the states in the regime
N>Lz>0 are metastable. For
Lz>N, the
Lz values for the stable ground states generally increased with increasing critical angular velocities ${\ensuremath{\Omega}}_{\mathrm{ci}},$ and the condensate was strongly depleted. The critical ${\ensuremath{\Omega}}_{\mathrm{ci}}$ values, however, decreased with increasing interaction strength as well as the particle number, and were systematically greater than the nonvariational yrast-state values for the
Lz=N single vortex state. We have also observed that the condensate fraction for the single vortex state (as also for the higher vortex states) did not change significantly even as the two-body interaction strength was varied over several $(\ensuremath{\sim}4)$ orders of magnitude in the moderately to the weakly interacting regime.
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.
By numerically solving the single-particle stationary Schrödinger equation and the Gross-Pitaevskii equation with mean-field interactions at zero temperature, the ground state properties of the rotating spin-orbital-angular-momentum coupled Bose-Einstein condensates in a harmonic trapping potential are investigated in this work. The results show that the rotation lifts the double degeneracy of the single-particle energy spectrum in the angular momentum space, and leads to the vortex state. The angular momentum of the vortex depends on the rotating frequency, the intensity of the laser beam, and the spin-orbital-angular-momentum coupling. In particular, if the rotating frequency is below a critical value, the angular momentum of the ground state vortex remains unaffected by the rotating frequency. When the rotating frequency exceeds the critical value, the angular momentum of the ground state vortex will increase with the rotating frequency increasing. By assuming that the system is confined in a ring trap, the expression of the single-particle energy spectrum in the angular momentum space can be obtained, which clarifies how the rotation frequency affects the angular momentum of the ground state. In the presence of atomic interactions, similar phenomena can also be observed in the mean-field ground state at zero temperature.
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.