Observation of an Alice ring in a Bose–Einstein condensate

Physical Review A, 2004

We construct three-dimensional structures of topological defects hosted in trapped wave fields, in the form of vortex stars, vortex cages, parallel vortex lines, perpendicular vortex rings, and parallel vortex rings, and we show that the latter exist as robust stationary, collective states of nonrotating Bose-Einstein condensates. We discuss the stability properties of excited states containing several parallel vortex rings hosted

Physical Review A, 2000

We study the feasibility of preparing a Bose-Einstein condensed sample of atoms in a 2D spin monopole. In this state, the atomic internal spins lie in the x-y plane, and point in the radial direction. PACS: 03.75.Fi,05.30.Jp,32.80.Pj, After the successful generation of Bose-Einstein condensation of alkali atoms, the creation and manipulation of certain macroscopic quantum states remains as one of the fundamental goals in the field of Atomic Physics [1]. During the last year, several ways to create vortices and solitons have been proposed . Under certain circumstances these states are stable , which has motivated several experimental groups to try corresponding experiments. In this letter we study a new kind of macroscopic quantum state for these atomic samples, what we call a 2D spin monopole. It is a state in which the atomic spin (i.e., the magnetization vector) points in the radial direction in the x-y plane [ ]. We show that this state is stable under realistic conditions, and analyze a method to generate it which only requires current experimental technology. We will first study a one dimensional situation, in which the condensate is confined in a ring, and then we will generalize it to the 3D case.

Dynamics of vortex clusters is essential for understanding diverse superfluid phenomena. In this paper, we examine the dynamics of vortex quadrupoles in a trapped two-dimensional (2D) Bose-Einstein condensate. We find that the movement of these vortex-clusters fall into three distinct regimes which are fully described by the radial positions of the vortices in a 2D isotropic harmonic trap, or by the major radius (minor radius) of the elliptical equipotential lines decided by the vortex positions in a 2D anisotropic harmonic trap. In the " recombination " and " exchange " regimes the quadrupole structure maintains, while the vortices annihilate each other permanently in the " annihilation " regime. We find that the mechanism of the charge flipping in the " exchange " regime and the disappearance of the quadrupole structure in the " annihilation " regime are both through an intermediate state where two vortex dipoles connected through a soliton ring. We give the parameter ranges for these three regimes in coordinate space for a specific initial configuration and phase diagram of the vortex positions with respect to the Thomas-Fermi radius of the condensate. We show that the results are also applicable to systems with quantum fluctuations for the short-time evolution. Vortices could be observed in most realm of physics such as hydrodynamics, superfluids, optical fields and even cosmology. The dynamics of quantized vortices is essential for understanding diverse superfluid phenomena such as critical-current densities in superconductors 1 , quantum turbulence 2–6 and novel quantum phases 7–11 in superfluids. Vortices are also topological defects that play key roles in transport, dissipative and coherent properties 12–16 of superfluid systems. The pioneering work of Yarmchuk et al. in 1979 successfully located the ends of parallel vortex lines in superfluid Helium 17. The real-time dynamics of vortex lattice in type II super conductors was observed in 1992 18. Until 2006, direct observation of quantized vortex lines in superfluid Helium in arbitrary three-dimensional configurations has been achieved 19. The realization of Bose-Einstein condensates (BECs) provides an accessible and highly controllable platform for fundamental studies of superfluid vortex dynamics, and has been followed by various theoretical investigations and numerical analyses. It is remarkable that, comparing with other systems ruled by nonlinear Schrödinger equations , BECs are the ideal laboratory for finding these nonlinear excitations due to larger interaction strengths and easier tunable parameters. The size of vortex cores in a BEC is proportional to the healing length of the condensate,  ζ ρ = mg / 2 , where m is the mass of atoms, ρ is the density of the condensate in the absence of vortex, and g is the interatomic interaction strength. For given trap frequencies, g is proportional to the scattering length between atoms, which can be easily adjusted by using magnetic or optical Feshbach resonances 20–22. Due to the matter wave nature of condensates 23 , vortices can be detected in atomic interference. Matthews et al. 24 firstly demonstrated vortex production through an interference measurement, and later, Inouye et al. observed vortex phase singular-ities as dislocations in the interference fringes in BECs 25. As the size of the vortex cores in a trapped condensate is ordinarily several times smaller than the wavelength of light used for imaging, in experiments the condensates are usually allowed to expand to a point at which the vortex cores are large compared to the imaging resolution 24–30. Many works have been done to develop the vortex detection techniques and to understand vortex dynamics during the interference of BECs 25,31–36 , which is important for the applications of matter-wave interferometry. However, the direct, in situ observation of vortices in a trapped condensate without expansion was wachieved

Physical Review A, 2004

Topological phase imprinting is a unique technique for vortex formation in a Bose-Einstein condensate (BEC) of an alkali-metal gas, in that it does not involve rotation: the BEC is trapped in a quadrupole field with a uniform bias field which is reversed adiabatically leading to vortex formation at the center of the magnetic trap. The scenario has been experimentally verified by Leanhardt et al. employing 23 Na atoms. Recently similar experiments have been conducted by Hirotani et al. in which a BEC of 87 Rb atoms was used. In the latter experiments the authors found that fine-tuning of the field reverse time T rev is required to achieve stable vortex formation. Otherwise, they often observed vortex fragmentation or a condensate without a vortex. It is shown in this paper that this behavior can be attributed to the heavy mass of the Rb atom. The confining potential, which depends on the eigenvalue m B of the hyperfine spin F along the magnetic field, is now shifted by the gravitational field perpendicular to the vortex line. Then the positions of two weak-field-seeking states with m B = 1 and 2 deviate from each other. This effect is more prominent for BECs with a heavy atomic mass, for which the deviation is greater and, moreover, the Thomas-Fermi radius is smaller. We found, by solving the Gross-Pitaevskii equation numerically, that two condensates interact in a very complicated way leading to fragmentation of vortices, unless T rev is properly tuned.

We report on the observation of vortex formation in a Bose-Einstein condensate of 87Rb atoms. Vortices are generated by superimposing an oscillating excitation to the trapping potential introduced by an external magnetic field. For small amplitudes of the external excitation field we observe a bending of the cloud axis. Increasing the amplitude we observe formation of a growing number of vortices in the sample. Shot-to-shot variations in both vortex number and position within the condensed cloud are observed, probably due to the intrinsic vortex nucleation dynamics. We discuss the possible formation of vortices and antivortices in the sample as well as possible mechanisms for vortex nucleation.

Physical Review Letters, 2002

We present a calculation of a solitary wave propagating along a cylindrical Bose-Einstein trap, which is found to be a hybrid of a one-dimensional (1D) soliton and a three-dimensional (3D) vortex ring. The calculated energy-momentum dispersion exhibits characteristics similar to those of a mode proposed sometime ago by Lieb within a 1D model, as well as some rotonlike features.

Physical Review E, 2012

We present a method for numerically building a vortex knot state in the superfluid wave-function of a Bose-Einstein condensate. We integrate in time the governing Gross-Pitaevskii equation to determine evolution and stability of the two (topologically) simplest vortex knots which can be wrapped over a torus. We find that the velocity of a vortex knot depends on the ratio of poloidal and toroidal radius: for smaller ratio, the knot travels faster. Finally, we show how unstable vortex knots break up into vortex rings.

The European Physical Journal Special Topics, 2007

We discuss nonlinear excitations in an atomic Bose-Einstein condensate which is trapped in a harmonic potential. We focus on axially symmetric solitary waves propagating along a cylindrical condensate. A quasi one-dimensional dark soliton is the only nonlinear mode for a condensate with weak interactions. For sufficiently strong interactions of experimental interest solitary waves are hybrids of one-dimensional dark solitons and three-dimensional vortex rings. The energymomentum dispersion of these solitary waves exhibits characteristics similar to a mode proposed sometime ago by Lieb in a strictly 1D model, as well as some rotonlike features. We subsequently discuss interactions between solitary waves. Head-on collisions between dark solitons are elastic. Slow vortex rings collide elastically but faster ones form intermediate structures during collisions before they lose energy to the background fluid. Solitary waves and their interactions have been observed in experiments. However, some of their intriguing features still remain to be experimentally identified.

Physical Review A, 2008

We numerically obtain stationary rings of vortices in pancake-shaped Bose-Einstein condensates confined in a three-dimensional nonrotating trap. For this purpose we use a static axisymmetric trapping potential that can sustain locally stable off-axis vortices. We analyze the maximum number of vortices the system can host as a function of the number of particles. We also show that this system provides a very suitable scenario for predicting vortex dynamics in inhomogeneous media. Specifically, on the one hand, we derive an exact and simple analytical expression for the velocity field in a particular vortex position due to the presence of the other vortices of the array. On the other hand, using the fact that in stationary conditions this field should balance all other contributions to the velocity, we investigate the applicability of approximated expressions for the precession velocity obtained in previous works for condensates with a single vortex.

Physical Review A, 2012

We theoretically show that the topology of a non-simply-connected annular atomic Bose-Einstein condensate enforces the inner surface waves to be always excited with outer surface excitations and that the inner surface modes are associated with induced vortex dipoles unlike the surface waves of a simply-connected one with vortex monopoles. Consequently, under stirring to drive an inner surface wave, a peculiar population oscillation between the inner and outer surface is generated regardless of annulus thickness. Moreover, a new vortex nucleation process by stirring is observed that can merge the inner vortex dipoles and outer vortex into a single vortex inside the annulus. The energy spectrum for a rotating annular condensate with a vortex at the center also reveals the distinct connection of the Tkachenko modes of a vortex lattice to its inner surface excitations.

Physical Review A, 2002

We present a numerical method for generating vortex rings in Bose-Einstein condensates confined in axially symmetric traps. The vortex ring is generated using the line-source approximation for the vorticity, i.e., the curl of the superfluid velocity field is different from zero only on a circumference of a given radius located on a plane perpendicular to the symmetry axis and coaxial with it. The particle density is obtained by solving a modified Gross-Pitaevskii equation that incorporates the effect of the velocity field. We discuss the appearance of density profiles, the vortex core structure, and the vortex nucleation energy, i.e., the energy difference between vortical and ground-state configurations. This is used to present a qualitative description of the vortex dynamics.

Physical Review E, 2013

In the present work, we consider the problem of a system of few vortices N ≤ 5 as it emerges from its experimental realization in the field of atomic Bose-Einstein condensates. Starting from the corresponding equations of motion for an axially symmetric trapped condensate, we use a twopronged approach in order to reveal the configuration space of the system's preferred dynamical states. On the one hand, we use a Monte-Carlo method parametrizing the vortex "particles" by means of hyperspherical coordinates and identifying the minimal energy ground states thereof for N = 2, . . . , 5 and different vortex particle angular momenta. We then complement this picture with a dynamical systems analysis of the possible rigidly rotating states. The latter reveals supercritical and subcritical pitchfork, as well as saddle-center bifurcations that arise exposing the full wealth of the problem even at such low dimensional cases. By corroborating the results of the two methods, it becomes fairly transparent which branch the Monte-Carlo approach selects for different values of the angular momentum which is used as a bifurcation parameter.

Physical Review A, 2003

The stability of doubly quantized vortices in dilute Bose-Einstein condensates of 23 Na is examined at zero temperature. The eigenmode spectrum of the Bogoliubov equations for a harmonically trapped cigar-shaped condensate is computed and it is found that the doubly quantized vortex is spectrally unstable towards dissection into two singly quantized vortices. By numerically solving the full three-dimensional time-dependent Gross-Pitaevskii equation, it is found that the two singly quantized vortices intertwine before decaying. This work provides an interpretation of recent experiments [A. E. Leanhardt et al. Phys. Rev. Lett. 89, 190403 (2002)].

Physical Review A, 2003

Quasi-one-dimensional solitons that may occur in an elongated Bose-Einstein condensate become unstable at high particle density. We study two basic modes of instability and the corresponding bifurcations to genuinely three-dimensional solitary waves such as axisymmetric vortex rings and non-axisymmetric solitonic vortices. We calculate the profiles of the above structures and examine their dependence on the velocity of propagation along a cylindrical trap. At sufficiently high velocity, both the vortex ring and the solitonic vortex transform into an axisymmetric soliton. We also calculate the energy-momentum dispersions and show that a Lieb-type mode appears in the excitation spectrum for all particle densities.

Physical Review A, 2011

We study the changes in the spatial distribution of vortices in a rotating Bose-Einstein condensate due to an increasing anisotropy of the trapping potential. Once the rotational symmetry is broken, we find that the vortex system undergoes a rich variety of structural changes, including the formation of zig-zag and linear configurations. These spatial re-arrangements are well signaled by the change in the behavior of the vortex-pattern eigenmodes against the anisotropy parameter. The existence of such structural changes opens up possibilities for the coherent exploitation of effective many-body systems based on vortex patterns.