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Journal of Physics B Atomic Molecular and Optical Physics

Controlled vortex states in Bose–Einstein condensates

Tonći Ramljak

Published Mar 9, 2021• 0 citations
Publisher

Abstract

Abstract There is no known experimental technique to create and control shape and orientation of the vortex lattice in Bose–Einstein condensates and the subject of this paper is the numerical study of creation of controlled vortex states with time varying potentials of various geometries formed by laser beams. Gross–Pitaevskii equation in ultra-cold limit models well the behavior of Bose–Einstein condensates and is solved with time-splitting Hermite pseudo-spectral method where we model the creation of vortex states by exciting collective modes. Random behavior of laser beams is introduced to find under what conditions the rotating solutions remain stable.

DOI
10.1088/1361-6455/abece5

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Abstract There is no known experimental technique to create and control shape and orientation of the vortex lattice in Bose–Einstein condensates and the subject of this paper is the numerical study of creation of controlled vortex states with time varying potentials of various geometries formed by laser beams. Gross–Pitaevskii equation in ultra-cold limit models well the behavior of Bose–Einstein condensates and is solved with time-splitting Hermite pseudo-spectral method where we model the creation of vortex states by exciting collective modes. Random behavior of laser beams is introduced to find under what conditions the rotating solutions remain stable.
Realistic methods to create vortices in spin-orbit-coupled Bose-Einstein condensates are discussed. It is shown that, contrary to common intuition, rotation of the trap containing a spin-orbit condensate does not lead to an equilibrium state with static vortex structures but gives rise instead to nonequilibrium behavior described by an intrinsically time-dependent Hamiltonian. We propose here the following alternative methods to induce thermodynamically stable static vortex configurations: (i) to rotate both the lasers and the anisotropic trap and (ii) to impose a synthetic Abelian field on top of synthetic spin-orbit interactions. Effective Hamiltonians for spin-orbit condensates under such perturbations are derived for most currently known realistic laser schemes that induce synthetic spin-orbit couplings. The Gross-Pitaevskii equation is solved for several experimentally relevant regimes. The new interesting effects include spatial separation of left- and right-moving spin-orbit condensates, the appearance of unusual vortex arrangements, and parity effects in vortex nucleation where the topological excitations are predicted to appear in pairs. All these phenomena are shown to be highly nonuniversal and depend strongly on a specific laser scheme and system parameters.
One of the key issues related to superfluidity is the existence of quantized vortices. Following recent experiments on Bose–Einstein condensates exhibiting vortices, we investigate the behavior of the wave function which solves the Gross–Pitaevskii equation. For a rotating Bose–Einstein condensate in a harmonic trap, we give a simplified expression of the Gross–Pitaevskii energy in the Thomas–Fermi regime, which only depends on the number and shape of the vortex lines. This allows us to study in detail the structure of the line of a single quantized vortex, which has a U or S shape. S type vortices exist for all values of the angular velocity Ω but are not minimizers of the energy while U vortices are minimizers and exist only for Ω sufficiently large. Finally, we study the drag created by the movement of a detuned laser beam in a condensate and the nucleation of vortices in the low density region close to the boundary layer of the cloud.
Since the first Bose-Einstein Condensate (BEC) was demonstrated in a dilute atomic gas [1], rotating BECs have been of interest to researchers due to their quantized circulation. We propose a new method of creating controlled vortices in a BEC. Atoms are held in the focus of a scanning beam that rapidly scans to produce a time-averaged potential [2]. We have made theoretical calculations to measure the heating caused by a scanning beam trap and to find the regime where a scanning beam produces a time-averaged potential. The scanning beam splits the BEC into four parts then imprints a relative phase on each part such that, when the parts recombine, a vortex forms. We have simulated such a system using the Gross-Pitaevskii equation and have demonstrated the laser scanning pattern using a two-dimensional acousto-optic modulator (AOM). The optics for the scanning beam trap have been designed and the experimental trap dimensions are used in our simulations. To carry out the experimental part of this project, we are currently building and testing an all-optical BEC apparatus: we have made a magneto-optical trap (MOT) which cools the 87Rb atomsto 30uK. We are currently setting up an optical dipole trap to perform the final stages of cooling.
In this work, we consider vortex lattices in rotating Bose–Einstein condensates composed of two species of bosons having different masses. Previously (Barnett et al 2008 New J. Phys. 10 043030), it was claimed that the vortices of the two species form bound pairs and the two vortex lattices lock. Remarkably, the two condensates and the external drive all rotate at different speeds owing to the disparity of the masses of the constituent bosons. In this paper, we study the system by solving the full two-component Gross–Pitaevskii equations numerically. Using this approach, we verify the stability of the putative locked state that is found to exist within a disc centered on the axis of rotation and that depends on the mass ratio of the two bosons. We also derive a refined estimate for the locking radius tailored to the experimentally relevant case of a harmonic trap and show that this agrees with the numerical results. Finally, we analyze in detail the rotation rates of the different components in the locked and unlocked regimes.
This body of work examines the non-equilibrium dynamics of vortex lattice carrying Bose–Einstein condensates. We solve the mean-feld Gross–Pitaevskii equation for a two-dimensional pancake geometry, in the co-rotating frame within the limit of high rotation frequencies. The condensate responds to this by creating a large periodic lattice of vortices with 6-fold triangular symmetry. By applying two distinct perturbations to this lattice, we examine the resulting effects on the vortices during time evolution. The frst perturbation involves applying an optical potential with matching geometry to the vortex lattice. We observe the appearance of interference fringes, and we show that these can be described by moiré interference theory. This is backed up by a decomposition of the kinetic energy spectra of the condensate. The applied perturbation only modifes the condensate density, with the vortex positions largely unaffected. From this we conclude that the vortex lattice is very stable and robust against phononic disturbances. Next, by removing vortices at predefned positions in the lattice using phase imprinting techniques, we examine the resulting order of the lattice. By performing this we generate stable topological defects in the crystal structure. The resulting lattice remains highly ordered in the presence of low numbers of these defects, where crystal structure and order of the lattice shows to be highly robust. By varying the type of imprinted phases we can create controllable degrees of disorder in the lattice. This disorder is analysed using orientational correlations, Delaunay triangulation, and Voronoi diagrams of the vortex lattice, and demonstrates a method for examining order and generating disorder in vortex lattices in Bose–Einstein condensates. All work described makes extensive use of GPU computing techniques, and allows for the simulation of these systems to be realised in short times. The implementation of the calculations using GPU computing are also discussed, where the software is shown to be the fastest of its kind out of the independently tested software suites.
Vortex nucleation in a Bose-Einstein condensate subject to a stirring potential is studied numerically using the zero-temperature, two-dimensional Gross-Pitaevskii equation. In the case of a rotating, slightly anisotropic harmonic potential, the numerical results reproduce experimental findings, thereby showing that finite temperatures are not necessary for vortex excitation below the quadrupole frequency. In the case of a condensate subject to stirring by a narrow rotating potential, the process of vortex excitation is described by a classical model that treats the multitude of vortices created by the stirrer as a continuously distributed vorticity at the center of the cloud, but retains a potential flow pattern at large distances from the center.