Vortex Lattices In Bose-Einstein Condensates Research Articles

Topological defect dynamics of vortex lattices in Bose-Einstein condensates

Vortex lattices in rapidly rotating Bose--Einstein condensates are systems of topological excitations that arrange themselves into periodic patterns. Here we show how phase-imprinting techniques can be used to create a controllable number of defects in these lattices and examine the resulting dynamics. Even though we describe our system using the mean-field Gross--Pitaevskii theory, the full range of many particle effects among the vortices can be studied. In particular we find the existence of localized vacancies that are quasi-stable over long periods of time, and characterize the effects on the background lattice through use of the orientational correlation function, and Delaunay triangulation.

Precession of vortices in dilute Bose-Einstein condensates at finite temperature

We demonstrate that the precessional frequencies of vortices in Bose-Einstein condensates (BECs) are determined by a conservation law and not by the lowest-lying excitation energy mode. We determine the precessional frequency for a single off-axis vortex and vortex lattices in BECs using the continuity equation and solve this self-consistently with the time-independent Hartree-Fock-Bogoliubov equations in the rotating frame. We find agreement with zero-temperature calculations (Bogoliubov approximation) and a smooth variation in the precession frequency as the temperature is increased. Time-dependent solutions confirm the validity of these predictions.

Vortex Lattice Theory: A Particle Interaction Perspective

Recent experiments on the formation of vortex lattices in Bose-Einstein condensates has produced the need for a mathematical theory that is capable of predicting a broader class of lattice patterns, ones that are free of discrete symmetries and can form in a random environment. We give an overview of an $N$-particle based Hamiltonian theory which, if formulated in terms of the $O(N^2)$ interparticle distances, leads to the analysis of a nonnormal “configuration” matrix whose nullspace structure determines the existence or nonexistence of a lattice. The singular value decomposition of this matrix leads to a method in which all lattice patterns and the associated particle strengths, in principle, can be classified and calculated by a random-walk scheme which systematically uses the $m$ smallest singular values as a ratchet mechanism to home in on lattices with $m$-dimensional nullspaces, where $0 < m \le N$. The resulting distribution of singular values encodes detailed geometric properties of the lattice and allows us to identify and calculate important quantitative measures associated with each lattice, including its size (as measured by the Frobenius or 2-norms), distance between the lattices (hence lattice density), robustness, and Shannon entropy as a quantitative measure of its level of disorder. This article gives an overview of vortex lattice theory from 1957 to the present, highlighting recent experiments in Bose-Einstein condensate systems and formulating questions that can be addressed by understanding the singular value decomposition of the configuration matrix. We discuss some of the computational challenges associated with producing large $N$ lattices, the subtleties associated with understanding and exploiting complicated Hamiltonian energy surfaces in high dimensions, and we highlight ten important directions for future research in this area.

Vortex lattices in Bose-Einstein condensates with dipolar interactions beyond the weak-interaction limit

We study the ground states of rotating atomic Bose-Einstein condensates with dipolar interactions. We present the results of numerical studies on a periodic geometry which show vortex lattice ground states of various symmetries: triangular and square vortex lattices, ``stripe crystal,'' and ``bubble crystal.'' We present the phase diagram (for systems with a large number of vortices) as a function of the ratio of dipolar to contact interactions and of the chemical potential. We discuss the experimental requirements for observing transitions between vortex lattice ground states via dipolar interactions. We finally investigate the stability of mean-field supersolid phases of a quasi-two-dimensional nonrotating Bose gas with dipolar interactions.

Vortex lattices in Bose-Einstein condensates: From the Thomas-Fermi regime to the lowest-Landau-level regime

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 solve numerically the 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.

Nonequilibrium effects of anisotropic compression applied to vortex lattices in bose-einstein condensates.

We have studied the dynamics of large vortex lattices in a dilute-gas Bose-Einstein condensate. While undisturbed lattices have a regular hexagonal structure, large-amplitude quadrupolar shape oscillations of the condensate are shown to induce a wealth of nonequilibrium lattice dynamics. When exciting an m=-2 mode, we observe shifting of lattice planes, changes of lattice structure, and sheetlike structures in which individual vortices appear to have merged. Excitation of an m=+2 mode dissolves the regular lattice, leading to randomly arranged but still strictly parallel vortex lines.

Formation and Decay of Vortex Lattices in Bose-Einstein Condensates at Finite Temperatures

The dynamics of vortex lattices in stirred Bose-Einstein condensates have been studied at finite temperatures. The decay of the vortex lattice was observed nondestructively by monitoring the centrifugal distortions of the rotating condensate. The formation of the vortex lattice could be deduced from the increasing contrast of the vortex cores observed in ballistic expansion. In contrast to the decay, the formation of the vortex lattice is insensitive to temperature change.

Vortex bending and tightly packed vortex lattices in Bose-Einstein condensates

We study in detail the counterintuitive result that in elongated rotating Bose--Einstein condensates the ground state is composed of one or more vortex lines which bend even in completely symmetric setups. This symmetry breaking allows the condensate to smoothly adapt to rotation and to produce tightly packed arrays of vortex lines.