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arXiv:1705.00867 · Physical Review A

Exact lowest-Landau-level solutions for vortex precession in Bose-Einstein condensates

Anxo Biasi, Piotr Bizoń, Ben Craps, Oleg Evnin

Published Nov 10, 2017• 43 citations
arXivHTMLPDF

Abstract

The lowest Landau level (LLL) equation emerges as an accurate approximation for a class of dynamical regimes of Bose-Einstein condensates (BEC) in two-dimensional isotropic harmonic traps in the limit of weak interactions. Building on recent developments in the field of spatially confined extended Hamiltonian systems, we find a fully nonlinear solution of this equation representing periodically modulated precession of a single vortex. Motions of this type have been previously seen in numerical simulations and experiments at moderately weak coupling. Our paper provides a controlled analytic prediction for trajectories of a single vortex, suggests new targets for experiments, and opens up the prospect of finding analytic multivortex solutions.

DOI
10.1103/physreva.96.053615

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The lowest Landau level (LLL) equation emerges as an accurate approximation for a class of dynamical regimes of Bose-Einstein condensates (BEC) in two-dimensional isotropic harmonic traps in the limit of weak interactions. Building on recent developments in the field of spatially confined extended Hamiltonian systems, we find a fully nonlinear solution of this equation representing periodically modulated precession of a single vortex. Motions of this type have been previously seen in numerical simulations and experiments at moderately weak coupling. Our paper provides a controlled analytic prediction for trajectories of a single vortex, suggests new targets for experiments, and opens up the prospect of finding analytic multivortex solutions.
We consider a periodic vortex lattice in a rotating Bose-Einstein-condensed gas, where the centrifugal potential is exactly compensated by the external harmonic trap. By introducing a gauge transformation which makes the Hamiltonian periodic, we numerically solve the two-dimensional (2D) Gross-Pitaevskii equation finding the exact mean field ground state. In particular, we explore the crossover between the Thomas-Fermi regime, holding for large values of the coupling constant, and the lowest Landau level limit, corresponding to the weakly interacting case. Explicit results are given for the equation of state, the vortex core size, as well as the elastic shear modulus, which is crucial for the calculation of the Tkachenko frequencies.
The fast rotation of a Bose Einstein condensate in a harmonic trap leads to a hexagonal lattice of vortices, called the Abrikosov lattice. Vortices are described as the zeros of the condensate wave function, which is a solution to a nonlinear Schrödinger equation type. We study the solutions of this equation on a specific eigenspace of the limiting problem, the Lowest Landau Level. Upper and lower bounds for the energy allow to determine the presence of a vortex lattice or the lack of visible vortices, as a function of the anisotropy and the rotation. The study of the critical regime is open and would need some numerical simulations. This constitutes a project in collaboration with Xavier Blanc and Nicolas Lerner.
Trapped Bose-Einstein condensates (BECs) differ considerably from the standard textbook example of a uniform Bose gas. In an isotropic harmonic potential V(r) = ½Mω2 r 2, the single-particle ground state introduces a new intrinsic scale of length [the ground-state size d = √ℏ/(Mω)] and energy [the ground-state energy E 0 = 32 \frac{3} {2} 23​ ℏω]. When the trap rotates at a low angular velocity, the behavior of a single vortex illustrates the crucial role of discrete quantized vorticity. For more rapid rotation, the condensate contains a vortex array. The resulting centrifugal forces expand the condensate radially and shrink it axially; thus, the condensate becomes effectively two dimensional. If the external rotation speed approaches the frequency of the radial harmonic confining potential, the condensate enters the “lowest-Landau-level” regime, and a simple description again becomes possible. Eventually, the system is predicted to make a quantum phase transition to a highly correlated state analogous to the fractional quantum Hall states of electrons in a strong magnetic field.
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.
We consider the two-dimensional Gross-Pitaevskii equation describing a Bose-Einstein condensate in an isotropic harmonic trap. In the small coupling regime, this equation is accurately approximated over long times by the corresponding nonlinear resonant system whose structure is determined by the fully resonant spectrum of the linearized problem. We focus on two types of consistent truncations of this resonant system: first, to sets of modes of fixed angular momentum, and, second, to excited Landau levels. Each of these truncations admits a set of explicit analytic solutions with initial conditions parametrized by three complex numbers. Viewed in position space, the fixed angular-momentum solutions describe modulated oscillations of dark rings, while the excited Landau level solutions describe modulated precession of small arrays of vortices and antivortices. We place our findings in the context of similar results for other spatially confined nonlinear Hamiltonian systems in recent literature.
In a rapidly rotating Bose-Einstein condensate (BEC) the condensate wave function is analogous to that of a type II superconductor close to the critical magnetic field Hc2{H}_{c2}Hc2​. This wave function is a superposition of the lowest Landau levels (LLL) of a charged particle in a magnetic field and corresponds to a vortex lattice. This leads to the inverted-parabola density profile of the BEC in the parabolic trap similar to slower rotation outside the LLL regime. Then one can apply the continuum theory of the Tkachenko modes (transverse sound modes in the vortex lattice) suggested for slower rotation, but with another shear modulus also known for type II superconductors at Hc2{H}_{c2}Hc2​. The theory yields a simple expression for the Tkachenko eigenfrequencies of the BEC cloud. The comparison with experiment shows that the experimental conditions have not yet reached the LLL regime.