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Communications on Pure &amp Applied Analysis

Vortex interaction dynamics in trapped Bose-Einstein condensates

Pedro J. Torres, R. Carretero-González, Stephan Middelkamp, Peter Schmelcher, D. J. Frantzeskakis, P. G. Kevrekidis

Published Jan 1, 2011• 28 citations
Publisher

Abstract

Motivated by recent experiments studyingthe dynamics ofconfigurations bearing a small number of vortices inatomic Bose-Einstein condensates (BECs), weillustrate that such systems can be accurately described byordinary differential equations (ODEs) incorporating the precessionand interaction dynamics ofvortices inharmonic traps.This dynamics is tackled in detail at the ODE level, both forthe simpler case of equal charge vortices,and for the more complicated(yet also experimentally relevant) case of opposite charge vortices.In the former case, we identify the dynamics as being chieflyquasi-periodic (although potentially periodic), while in thelatter, irregular dynamics may ensuewhen suitableexternal drive of the BEC cloud is also considered. Our analytical findings arecorroborated by numerical computations of the reduced ODE system.

DOI
10.3934/cpaa.2011.10.1589

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Motivated by recent experiments studyingthe dynamics ofconfigurations bearing a small number of vortices inatomic Bose-Einstein condensates (BECs), weillustrate that such systems can be accurately described byordinary differential equations (ODEs) incorporating the precessionand interaction dynamics ofvortices inharmonic traps.This dynamics is tackled in detail at the ODE level, both forthe simpler case of equal charge vortices,and for the more complicated(yet also experimentally relevant) case of opposite charge vortices.In the former case, we identify the dynamics as being chieflyquasi-periodic (although potentially periodic), while in thelatter, irregular dynamics may ensuewhen suitableexternal drive of the BEC cloud is also considered. Our analytical findings arecorroborated by numerical computations of the reduced ODE system.
In this brief review we summarize a number of recent developments in the study of vortices in Bose-Einstein condensates, a topic of considerable theoretical and experimental interest in the past few years. We examine the generation of vortices by means of phase imprinting, as well as via dynamical instabilities. Their stability is subsequently examined in the presence of purely magnetic trapping, and in the combined presence of magnetic and optical trapping. We then study pairs of vortices and their interactions, illustrating a reduced description in terms of ordinary differential equations for the vortex centers. In the realm of two vortices we also consider the existence of stable dipole clusters for two-component condensates. Last but not least, we discuss mesoscopic patterns formed by vortices, the so-called vortex lattices and analyze some of their intriguing dynamical features. A number of interesting future directions are highlighted.
In this brief review we summarize a number of recent developments in the study of vortices in Bose–Einstein condensates, a topic of considerable theoretical and experimental interest in the past few years. We examine the generation of vortices by means of phase imprinting, as well as via dynamical instabilities. Their stability is subsequently examined in the presence of purely magnetic trapping, and in the combined presence of magnetic and optical trapping. We then study pairs of vortices and their interactions, illustrating a reduced description in terms of ordinary differential equations for the vortex centers. In the realm of two vortices we also consider the existence of stable dipole clusters for two-component condensates. Last but not least, we discuss mesoscopic patterns formed by vortices, the so-called vortex lattices and analyze some of their intriguing dynamical features. A number of interesting future directions are highlighted.
Motivated by experiments in atomic Bose–Einstein condensates (BECs), we compare predictions of a system of ordinary differential equations (ODEs) for dynamics of one and two individual vortices in the rotating BECs with those of the Gross–Pitaevskii mean-field model written as a partial differential equation (PDE). In particular, we characterize orbitally stable vortex configurations in a symmetric harmonic trap due to a cubic repulsive interaction and a steady rotation. The ODE system is analysed in detail and the PDE model is approximated numerically. Good agreement between the two models is established in the semi-classical (Thomas–Fermi) limit that corresponds to the BECs at large values of the chemical potential.
The dynamics and interaction of quantized vortices in Bose-Einstein condensates (BECs) are investigated by using the two-dimensional Gross-Pitaevskii equation (GPE) with/without an angular momentum rotation term. If all vortices have the same winding number, they would rotate around the trap center but never collide. In contrast, if the winding numbers are different, their interaction highly depends on the initial distance between vortex centers. The analytical results are presented to describe the dynamics of the vortex centers when β=0. While if β≠0, there is no analytical result but some conclusive numerical findings are provided for the further understanding of vortex interaction in BECs. Finally, the dynamic laws describing the relation of vortex interaction in nonrotating and rotating BECs are presented.
We study the dynamics of vortices in ideal and weakly interacting Bose-Einstein condensates using a Ritz minimization method to solve the two-dimensional Gross-Pitaevskii equation. For different initial vortex configurations we calculate the trajectories of the vortices. We find conditions under which a vortex-antivortex pair annihilates and is created again. For the case of three vortices we show that at certain times two additional vortices may be created, which move through the condensate and annihilate each other again. For a noninteracting condensate this process is periodic, whereas for small interactions the essential features persist, but the periodicity is lost. The results are compared to exact numerical solutions of the Gross-Pitaevskii equation confirming our analytical findings.
Quantum vortices in superfluid Bose Einstein condensates (BECs) are excellent candidates\nfor studying hydrodynamic phenomena, such as turbulence and chaos, due to the\nquantization of vortex circulation and the absence of dissipation in BECs. Since the early\n2000s, many experimental groups have successfully observed vortices in dilute atomic\nBECs.\n\nThis work mainly focuses on creating quantum chaos with a few vortices in a twodimensional\nBEC of Rubidium􀀀87. We use graphics processing unit (GPU) computing methods to simulate our BEC trapped inside a harmonic potential with tight trapping in one direction. The quantum vortices are created by rotating the condensate. After generating the ground state of the BEC with a small number of singly charged vortices with positive phase winding, we imprint a 4\\pi phase winding on top of one vortex in the opposite orientation to that of the original vortex. This phase imprinting annihilates the preexisting vortex and creates a new singly-charged vortex with negative phase winding. Due to the high resolution of our system and an excellent vortex tracking algorithm, the location of phase imprinting can be delicately controlled.\n\nBoth the individual vortex dynamics and the dynamics of the entire system of vortices\nafter flipping a vortex have shown strong signs of chaos. This chaotic nature is\nfurther confirmed both qualitatively and quantitatively by employing two chaos indicators\non our numerical results. These chaos indicators are the Lyapunov exponent spectrum\nand the Smaller ALignment Index (SALI). The Lyapunov exponent remains positive and\nconverges to a positive value during an experimentally realistic time period, which is the\nsmoking gun for chaotic behaviour.