The energy spectrum for an ultracold rotating Bose gas in a harmonic trap is calculated exactly for small systems, allowing the atoms to occupy several Landau levels. Two vortexlike states and two strongly correlated states (the Pfaffian and Laughlin) are considered in detail. In particular, their critical rotation frequencies and energy gaps are determined as a function of particle number, interaction strength, and the number of Landau levels occupied (up to three). For the vortexlike states, the lowest-Landau-level (LLL) approximation is justified only if the interaction strength decreases with the number of particles; nevertheless, the constant of proportionality increases rapidly with the angular momentum per particle. For the strongly correlated states, however, the interaction strength can increase with particle number without violating the LLL condition. The results suggest that, in large systems, the Pfaffian and Laughlin states might be stabilized at rotation frequencies below the centrifugal limit for sufficiently large interaction strengths, with energy gaps a significant fraction of the trap energy.
The energy spectrum for an ultracold rotating Bose gas in a harmonic trap is calculated exactly for small systems, allowing the atoms to occupy several Landau levels. Two vortexlike states and two strongly correlated states (the Pfaffian and Laughlin) are considered in detail. In particular, their critical rotation frequencies and energy gaps are determined as a function of particle number, interaction strength, and the number of Landau levels occupied (up to three). For the vortexlike states, the lowest-Landau-level (LLL) approximation is justified only if the interaction strength decreases with the number of particles; nevertheless, the constant of proportionality increases rapidly with the angular momentum per particle. For the strongly correlated states, however, the interaction strength can increase with particle number without violating the LLL condition. The results suggest that, in large systems, the Pfaffian and Laughlin states might be stabilized at rotation frequencies below the centrifugal limit for sufficiently large interaction strengths, with energy gaps a significant fraction of the trap energy.
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.
Rapidly rotating atomic gases provide a platform for studying phenomena akin to type-II superconductors and quantum Hall systems. Recently, these systems have attracted renewed interest due to technological advances in the trap anisotropy control, in-situ observation capabilities, and cooling and rotating complex atomic species such as dipolar gases. Understanding the vortex lattice formation and quantum melting is crucial for exploring quantum Hall physics in these systems. In this paper, we theoretically investigate the vortex lattices in anisotropic quantum gases. We formulate the rotating gas Hamiltonian in the Landau gauge, and consider the effects of additional perturbations such as the trap potential in the lowest Landau level (LLL). Focusing on the gases with short-range interactions, we obtain the many-body Hamiltonian projected to lowest Landau level. We consider the limit of full Bose-Einstein condensation and obtain the governing Gross-Pitaevskii equation to identify the possible vortex phases. We numerically solve the Gross-Pitaevskii equation using the imaginary time evolution, and demonstrate the possible vortex lattices as a function of anisotropy, rotation speed and interaction strength. We show that the number of states with a support in the LLL, which determines the number of vortices, follows a Thomas-Fermi type scaling albeit with slightly different coefficients from the usual condensates.
In this dissertation, we study rotational properties of ultracold bosonic quantum gases in two trapped configurations, a quasi-two-dimensional gas in a harmonic trap and a quasi-one-dimensional gas in a toroidal trap. First, we investigate the effects of correlations on the properties of the ground state of the rotating harmonically-trapped Bose gas by adding Bogoliubov fluctuations to the mean-field ground state of an N-particle single-vortex system. We demonstrate that the fluctuation-induced correlations lower the energy compared to that of the mean-field ground state, that the vortex core is pushed slightly away from the center of the trap, and that an unstable mode with negative energy (for rotations slower than a critical frequency) emerges in the energy spectrum, thus, pointing to a better state for slow rotation. We construct mean-field ground states of 0-, 1-, and 2-vortex states as a function of rotation rate and determine the critical frequencies for transitions between these states, as well as the critical frequency for appearance of a metastable state with an off-center vortex and its image vortex in the evanescent tail of the cloud. Then, we show how the configuration-space form of the above-mentioned Bogoliubov ground-state wave function of a bosonic condensate with a single vortex in a harmonic trap can be described in terms of bosonic Jastrow correlations. We then generalize this result to study the effects of such correlations on a mean-field vortex lattice state and show that the included correlations lower the energy below that of the mean-field state. Although the reduction is relatively small, it is a precursor of the more general expected effect of correlations in describing the melting of the vortex lattice at a high angular momentum per particle. Finally, we study the stability and dynamics of an ultracold bosonic gas trapped in a toroidal geometry and driven by rotation in the absence of dissipation. We first delineate, via the Bogoliubov mode expansion, the regions of stability and the nature of instabilities of the system for both repulsive and attractive interaction strengths. To study the response of the system to variations in the rotation rate, we introduce a “disorder” potential, breaking the rotational symmetry. We demonstrate the breakdown of adiabaticity as the rotation rate is slowly varied and find forced tunneling between the eigenstates of the system. The nonadiabaticity is signaled by the appearance of a swallow-tail loop in the lowest-energy level, a general sign of hysteresis. Then, we show that this system is in one-to-one correspondence with a trapped gas in a double-well potential and thus exhibits macroscopic quantum self-trapping. Finally, we show that self-trapping is a direct manifestation of the behavior of the lowest-energy level.
We study the rotational properties of a two-component Bose gas trapped in a rotating harmonic potential, effectively in two dimensions. We first consider a completely homogeneous interaction between particles, and assume the bosons to be in the lowest Landau level. We derive analytical expressions for some wave functions and energies at low angular momenta, and use these results together with exact numerical diagonalization results to test the applicability of composite fermion (CF) trial wave functions to the system. We find that a general procedure based on CF wave functions reproduces the exact ground states and many of the excited states for sufficiently low angular momenta, and give very good approximate results at angular momenta up to the two-component analogy of a single vortex. Additionally, we produce very simple CF states that have remarkably high overlaps with the lowest-lying states. Finally, we briefly discuss the case of inhomogeneous interactions, pointing out the difficulties arising from this generalization.
We consider an ultracold rotating Bose gas in a harmonic trap close to the critical angular velocity, so that the system can be considered to be confined to the lowest Landau level. With this assumption we prove that the Gross-Pitaevskii energy functional accurately describes the ground-state energy of the corresponding N-body Hamiltonian with contact interaction provided the total angular momentum L is much less than N2. While the Gross-Pitaevskii energy is always an obvious variational upper bound to the ground-state energy, a more refined analysis is needed to establish it as an exact lower bound. We also discuss the question of Bose-Einstein condensation in the parameter range considered. Coherent states together with inequalities in spaces of analytic functions are the main technical tools.
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.