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arXiv:cond-mat/0306402 · Physical Review A

Rotating spin-1 bosons in the lowest Landau level

J. W. Reijnders, F. J. M. van Lankvelt, Kareljan Schoutens, N. Read

Published Feb 19, 2004• 52 citations
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Abstract

We present results for the ground states of a system of spin-1 bosons in a rotating trap. We focus on the dilute, weakly interacting regime, and restrict the bosons to the quantum states in the lowest Landau level (LLL) in the plane (disk), sphere, or torus geometries. We map out parts of the zero-temperature phase diagram, using both exact quantum ground states and LLL mean-field configurations. For the case of a spin-independent interaction we present exact quantum ground states at angular momentum $L<~N.$ For general values of the interaction parameters, we present mean-field studies of general ground states at slow rotation and of lattices of vortices and skyrmions at higher rotation rates. Finally, we discuss quantum Hall liquid states at ultrahigh rotation.

DOI
10.1103/physreva.69.023612

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We present results for the ground states of a system of spin-1 bosons in a rotating trap. We focus on the dilute, weakly interacting regime, and restrict the bosons to the quantum states in the lowest Landau level (LLL) in the plane (disk), sphere, or torus geometries. We map out parts of the zero-temperature phase diagram, using both exact quantum ground states and LLL mean-field configurations. For the case of a spin-independent interaction we present exact quantum ground states at angular momentum L< N.L<~N.L< N. For general values of the interaction parameters, we present mean-field studies of general ground states at slow rotation and of lattices of vortices and skyrmions at higher rotation rates. Finally, we discuss quantum Hall liquid states at ultrahigh rotation.
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 a model of bosons in the lowest Landau level in a rotating trap where the confinement potential is a sum of a quadratic and a quartic term. The quartic term improves the stability of the system against centrifugal deconfinement and allows consideration of rotation frequencies beyond the frequency of the quadratic part. The interactions between particles are modeled by a Dirac $\ensuremath{\delta}$ potential. We derive rigorously conditions for ground states of the system to be strongly correlated in the sense that they are confined to the kernel of the interaction operator, and thus contain the correlations of the Bose-Laughlin state. Rigorous angular momentum estimates and trial state arguments indicate a transition from a pure Laughlin state to a state containing in addition a giant vortex at the center of the trap (Laughlin quasihole). There are also indications of a second transition where the density changes from a flat profile in a disk or an annulus to a radial Gaussian confined to a thin annulus
Abstract We show that ultracold fermions in an artificial magnetic field open up a new window to the physics of the spinful fractional quantum Hall (FQH) effect. We numerically study the lowest energy states of strongly interacting few-fermion systems in rapidly rotating optical microtraps. We find that skyrmion-like ground states with locally ferromagnetic, long-range spin textures emerge. To realize such states experimentally, rotating microtraps with higher-order angular momentum components may be used to prepare fermionic particles in a lowest Landau level. We find parameter regimes in which skyrmion-like ground states should be accessible in current experiments and demonstrate an adiabatic pathway for their preparation in a rapidly rotating harmonic trap. The addition of long range interactions will lead to an even richer interplay between spin textures and FQH physics.
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.
The fractional quantum Hall effect (FQHE) is theoretically investigated, with numerical and algebraic approaches, in assemblies of a few spinful ultracold neutral fermionic atoms, interacting via repulsive contact potentials and confined in a single rapidly rotating two-dimensional harmonic trap. Going beyond the commonly used second-order correlations in the real configuration space, the methodology in this paper will assist the analysis of experimental observations by providing benchmark results for NNN-body spin-unresolved, as well as spin-resolved, momentum correlations measurable in time-of-flight experiments with individual particle detection. Our analysis shows that the few-body lowest-Landau-level (LLL) states with good magic angular momenta exhibit inherent ordered quantum structures in the NNN-body correlations, similar to those associated with rotating Wigner molecules (WMs), familiar from the field of semiconductor quantum dots under high magnetic fields. The application of a small perturbing stirring potential induces, at the ensuing avoided crossings, formation of symmetry-broken states exhibiting ordered polygonal-ring structures, explicitly manifest in the single-particle density profile of the trapped particles. Away from the crossings, an LLL state obtained from exact diagonalization of the microscopic Hamiltonian, found to be well described by a (1,1,1) Halperin two-component variational wave function, represents also a spinful rotating WM. Analysis of the calculated LLL wave function enables a two-dimensional generalization of the Girardeau one-dimensional ``fermionization'' scheme, originally invoked for mapping of bosonic-type wave functions to those of spinless fermions.