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.
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.
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.
In this thesis, I explore the behavior of rotating ultra-cold Bose gases, by diagonalizing the Hamiltonian. This method has the advantage of being exact in the limit of weak interactions, and thus is a useful complement to the more commonly used Gross-Pitaevskii (GP) equation, which relies on a mean-field approximation. It is generally known that an ultra-cold Bose gas shows some remarkable properties under rotation, such as quantized vortices and persistent currents, and these are one main reason why we choose to study rotation. Abstract A repulsively interacting Bose-Einstein condensate (BEC) in a harmonic trapping potential will form singly quantized vortices when rotated. Papers I-V are examples of different kinds of generalizations to this basic system, each showing its particular non-trivial new features. Abstract In Paper I we make the trapping potential weakly anharmonic, and confirm the existence of multiply quantized vortices in both repulsively and attractively interacting condensates, which were previously predicted in Gross-Pitaevskii calculations. Abstract In Papers II and III, we generalize to a two-component system in a harmonic trap. The vortices that form under rotation will in general be coreless, i.e. one component will form a vortex and rotate around the other component. In Paper II we give some analytical expressions for the dispersion relation and occupation numbers at low angular momenta, and in Paper III we show how multiply quantized vortices can be stable in these systems. Abstract Finally we investigate bosons in potentials which can support persistent currents, i.e. rotating states that do not decay in a finite amount of time. These are possible if there is an energy barrier separating the rotating state from the non-rotating ground state. In Paper IV, we have a two-component condensate in a one-dimensional potential and we calculate among other things the minimum interaction strength where persistent currents can occur. In Paper V, we study an experimentally relevant two-dimensional annulus. Again we find that persistent currents are possible above a certain interaction strength, and we find this strength as a function of the width of the annulus, as well as how it depends on the relative population of the two species.
Bose-Einstein condensation, first predicted by Bose and Einstein in 1924, was finally realized in dilute, weakly interacting gases of alkali atoms in 1995. A few years later, quantized vortices - intimately related to the phenomenon of superfluidity - were created and observed in these novel quantum many-body systems.\n\nIn this Thesis, vortices in dilute atomic Bose-Einstein condensates are studied theoretically and computationally using microscopic mean-field theory formalisms. Special emphasis is on the structure and the stability of vortices at finite temperatures. The formation of quantized circulation in Bose-Einstein condensates is investigated, as well as multiply quantized vorticity and the adiabaticity of the precessing motion of a vortex.\n\nThis research strongly relies on numerically solving the Gross-Pitaevskii equation for the macroscopic condensate wavefunction and the coupled Bogoliubov equations and their variants for the collective quasiparticle excitation eigenmodes. The computed elementary excitation spectra are used, for instance, in determining the local stability of various condensate states. The vortex formation process is studied at finite temperatures in terms of the surface modes of the condensate providing the Landau critical velocities for the breakdown of superfluidity. Free energy calculations are used for determining the thermodynamic equilibrium states.\n\nSingly quantized, axisymmetric vortex states are found to be locally energetically stable even in the zero-temperature limit within the self-consistent finite-temperature approximations. This would imply for the vortex a precession direction opposite to that observed in the experiment. Adiabatic approximation of quantum mechanics is formulated for Bose-Einstein condensed systems and is applied to a case of a moving vortex line. The experimentally observed precessing motion of the vortex is found to be nonadiabatic. A method to create stable multiply quantized vortices using an additional pinning potential is presented. The computed critical trap rotation frequencies for vortex nucleation at finite temperatures are in agreement with experiments and the predictions of zero-temperature field theories.