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- Three-dimensional solitons and vortices in dipolar Bose-Einstein condensates
- Quantum dynamics of Raman-coupled Bose-Einstein condensates using Laguerre-Gaussian beams
- Multiply quantized vortices in trapped Bose-Einstein condensates
- Bose-Einstein condensates in traps of time-dependent topology
- Vortices and Ring Solitons in Bose-Einstein Condensates
- Bose-Einstein condensates with vortices in rotating traps
- Vortices in attractive Bose-Einstein condensates in two dimensions
- Spontaneous vortices in the formation of Bose-Einstein condensates
- Toroidal optical dipole traps for atomic Bose-Einstein condensates using Laguerre-Gaussian
- Velocity of vortices in inhomogeneous Bose-Einstein condensates

Vortices in Bose-Einstein condensates con?ned in a multiply connected Laguerre-Gaussian optical trap.

J. Tempere and J. T. Devreese

Departement Natuurkunde, Universiteit Antwerpen UIA, Universiteitsplein 1, B-2610 Antwerpen,

arXiv:cond-mat/0101014v1 2 Jan 2001

Belgium

E. R. I. Abraham

Department of Physics and Astronomy, University of Oklahoma, 440 W. Brooks, Norman OK 73019, USA (5/12/2000)

Abstract

The quantized vortex state is investigated in a Bose-Einstein condensate, con?ned in a multiply connected geometry formed by a Laguerre-Gaussian optical trap. Solving the Gross-Pitaevskii equation variationally, we show that the criterium for vortex stability is that the interatomic interaction strength must exceed a critical interaction strength. The time evolution of a freely expanding Laguerre-Gaussian condensate with a vortex is calculated, and used to derive the interference pattern of such a condensate overlapping with a parabolically trapped condensate.

Typeset using REVTEX 1

I. INTRODUCTION

Super?uidity, and its characteristic manifestation as a state with quantized circulation (a vortex), is intimately connected with the phenomenon of Bose-Einstein condensation, yet the precise relation between super?uid persistent currents and Bose-Einstein condensation (BEC) is only beginning to be elucidated [1]. Before 1995, the link between super?uidity and BEC was almost exclusively studied in the context of liquid helium (4 He and 3 He) [2], where the study of the relation between super?uidity and BEC is complicated by the strong interatomic interactions between the atoms in the liquid. In 1995, Bose-Einstein condensation was realized in magnetically trapped clouds of alkali atoms [3]. In these novel condensates, the bosonic atoms are weakly interacting (contrary to the case of liquid helium), and as such these systems have the potential to shed new light on super?uidity. Soon after the initial creation of alkali gas condensates, several experimental groups set out to create a vortex – a quantum of super?uid circulation – in this novel system [4]. The initial attempts to create a vortex in a condensate by stirring the trapped condensate with a blue detuned laser beam [4] were unsuccessful, even though early theoretical work by Dalfovo et al. indicated that persistent super?uid currents can indeed manifest themselves in Bose-Einstein condensates as vortices [5]. Subsequent analysis [6] showed that vortices are unstable in the simply-connected, not-stirred, spin-polarized condensates formed in the original experiments [3]. This can be understood as follows. Along the vortex line, the order parameter of the Bose-Einstein condensate has to vanish. Phrased metaphorically, a “hole” has to be “drilled” in the condensate along the vortex line. In the magnetic trap, the modulus square of the order parameter of a condensate without a vortex is largest in the center of the trap. As a consequence, a vortex line through the center of the trap will perturb the order parameter more than a vortex line at the edge of the condensate (since, using our metaphor again, it will cost more energy to “drill” through the center of the condensate than through its edge). It will be energetically favorable for the vortex line to be at the edge of the condensate. 2

In the presence of dissipation, this will cause the vortex line to migrate to the edge of the condensate so that the vortex condensate will decay into a non-vortex state. This argument, sketched here with some roughness, has been worked out with precision by Fetter and coworkers in [7]. Several schemes have been proposed, both theoretically and experimentally, to stabilize vortices: rotating the trapping potential (analogous to rotating a bucket containing 4 He) [6] or stirring the condensate with an o?-resonance laser [8], raising the temperature (to ‘pin’ the vortex in the potential created by the non-condensate fraction at the center of the vortex) [9], phase imprinting methods [10], and various other techniques [11]. Recently, vortices were created experimentally, both with a ‘rotating bucket’ experiment [12] and with the use of a two-component condensate [13]. In the latter experiment, one of the components of the spinor condensate ‘pins’ the vortex present in the other component and a Ramsey type interference between the two components is used to detect the vortex. The long lifetimes of the two-component condensates in ref. [13] is due to similar singlet and triplet scattering lengths of rubidium, resulting in an anomalously low inelastic loss rate [14]. Nevertheless, stable vortices have not yet been realized in a spin-polarized condensate in non-rotating traps, and new methods of stabilization and detection must be developed. A promising scheme, based on an analysis similar to that of Fetter [7], is the use of multiplyconnected condensates. A candidate trap to create a multiply connected condensates consists of a red-detuned laser beam in a Laguerre-Gauss mode [15,16], which we discuss in section II. If the condensate order parameter has a toroidal geometry, a vortex line threaded through the cylindrical symmetry axis of the torus will not perturb the condensate order parameter strongly, and moreover create a metastability barrier for vortex decay. One of the goals of this paper is to verify this statement about vortex stability (in section III). In a condensate with a toroidal order parameter, the vortex line can no longer be detected as a line along which the density of Bose-Einstein condensed atoms vanishes. To detect vorticity, we propose (in section IV) a method based on interference, similar to that proposed in refs. [17,18] for simply connected condensates. 3

II. BOSE-EINSTEIN CONDENSATION IN LAGUERRE GAUSS TRAPS

A. Trapping geometry

Toroidal con?nement for ultracold atoms can be obtained by an optical dipole trap [19] which consists of a laser beam in a Laguerre-Gaussian mode. The Laguerre-Gaussian mode {n, m} is characterized by an intensity pro?le given by [20]: In,l (r, z) ∝

2 2r 2 (2r 2/W0 )l 2 L2 2r 2 /W0 exp ? 2 2 1 + (2z/kW0 )2 n,l W0

,

(1)

where r is the radial distance from the center of the beam, z is the position along the propagation direction of the beam, W0 (referred to as the ‘waist parameter’) is a parameter controlling the minimal width of the beam, k is the wave number of the laser, and Ln,l is the Laguerre polynomial of order {n, l}. Such laser beams have modes which show a node in the center, and trap the atoms in a cylindrical shell around the axis of propagation of the beam. The intensity pro?le of the laser beam in the Laguerre-Gaussian mode is illustrated in Figure 1. Laguerre-Gaussian (LG) laser beams have already been used to successfully trap atoms [16], and subsequent theoretical work has shown that toroidal traps formed by a red-detuned LG beam can be loaded from initial conditions similar to those of conventional magnetic traps [21]. Once the trap has been loaded, one possibility to create a vortex in the trapped gas would be by a phase imprinting method, which already successfully resulted in the creation of solitons in condensates [22]. Alternatively, a rotating perturbing potential can be used to stir the condensate and set up persistent ?ow in the toroidal geometry [23]. In this paper we investigate the properties of the vortex Bose-Einstein condensate, optically trapped by a laser beam in a Laguerre-Gaussian propagation mode {n, l} = {0, 1}. Along the z-axis (the direction of propagation of the laser beam) an additional magnetic trap [15] results in a parabolic z-axis con?nement with frequency ?, which prevents the atoms from escaping along the direction of propagation of the laser beam. This ‘plugging’ of the optical trap was achieved in [16] by using blue-detuned ‘plugging beams’. The condensate in

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the Laguerre-Gauss geometry [15] will be denoted by ‘Laguerre-Gaussian condensate’ (LG condensate), in contrast with the condensate in a parabolic con?nement. The intensity pro?le of such a Laguerre-Gauss beam in the zy-plane (where z is the axis of propagation of the laser beam), given by (1), is shown in Figure 1. In the remainder of this paper, we use units so that = m = ? = 1 (where m is the mass of the atoms). In

these units, the laser beam parameters for Figure 1 are chosen as follows: k = 2, W0 = 5. Figure 2 illustrates a surface of constant intensity of the laser beam; a hollow cylindrical shell. This will also be the shape of the cloud of trapped atoms [21]. The extension of the cloud along the z-axis can be tuned by selecting the ? frequency of the magnetic trap along the z-axis.

B. Mean-?eld approach

Con?ned Bose-Einstein condensates are well described by a mean-?eld theory where the properties of the condensate are derived from a complex function Ψ. This function (the order parameter) is interpreted as a macroscopic wave function and obeys the Gross-Pitaevskii equation [24]:

2

?

2m

?Ψ + Vconf (ρ, z)Ψ + U0 |Ψ|2 Ψ = EΨ.

(2)

The term nonlinear in Ψ in (2) arises from the interparticle interaction potential, which is treated as a contact potential with scattering length ascat so that U0 = 4π 2 ascat /m where m is the mass of an atom. The optical con?nement is generated by a red-detuned LaguerreGaussian laser beam in the {0, 1} mode, with waist parameter W0 and wave number k, and propagating along the z-axis [20]. In addition to this, a harmonic con?nement (characterized by a frequency ?) is present which con?nes the atoms along in the z-direction to a region

2 z < kW0 [15]. The potential energy corresponding to this geometry is given, in cylindrical

coordinates {ρ, θ, z}, by: m?2 2 2ρ2 2 2 z . Vconf (ρ, θ, z) = ?A0 2 exp ?2ρ /W0 + W0 2 5 (3)

In the present treatment, the Gross-Pitaevskii equation (2) will be solved variationally [25]. Consider for this purpose the following two trial functions, expressed in cylindrical coordinates {ρ, θ, z}: ψ(ρ, θ, z) = N0 exp{?σ0 (ρ ? ρ0 )2 /2 ? ζ0 z 2 /2}, ψv (ρ, θ, z) = N1 ρ exp{?σ1 (ρ ? ρ1 )2 /2 ? ζ1 z 2 /2}eiθ . (4) (5)

The variational parameters in the trial functions are σ0 , σ1 , controlling the width of the functions in the radial direction; ζ0 ,ζ1 , controlling the widths in the z-direction; and ρ0 , ρ1 , controlling the radial displacement of the maximum of the function away from the z-axis. N0 and N1 are normalization constants, which are determined by demanding that the norm of the trial function equals the number of particles in the condensate. The circulation around a closed loop in the trapped, interacting Bose gas is de?ned by κ =

m

?S, where S represents the phase of the order parameter solving the Gross-

Pitaevskii equation (2). The single-valuedness of the order parameter ensures that the circulation in the trapped Bose gas is quantized: κ = nh/m with n = 0, 1, 2, .... A vortex is present whenever n > 0. The trial function ψ(ρ, θ, z) has a constant phase and hence there is no circulation, no vortex. On the other hand, the phase of ψv (ρ, θ, z) changes by 2π along any closed loop encircling the z-axis. Thus ψv is a trial function for the order parameter of a condensate with one quantum of vorticity: its circulation equals κ = h/m. The trial function ψv must have a node along the z-axis (the factor ρ in expression (5)): if this node were not present, the variational kinetic energy would diverge along the z-axis. We have used ψ and ψv as variational trial functions to solve the Gross-Pitaevskii equation for a condensate without and with a vortex, respectively. The resulting variational energy is shown in Figure 3 as a function of Nascat /aHO where N is the number of atoms and aHO = /(m?). The quantity Nascat /aHO is a dimensionless measure of the strength

of the interaction: upon increasing Nascat /aHO > 0 the interatomic interaction becomes more repulsive. The energy of the LG condensate with a vortex is larger than the energy of the LG condensate without a vortex for all interaction strengths investigated. The di?erence 6

is smallest for the non-interacting gas and increases monotonically as Nascat /aHO increases. This means that for all investigated interactions strengths the vortex LG condensate is either metastable or unstable. In inset (a) of Figure 3, the optimal value of the variational parameters σ, ζ are shown for both trial functions as a function of the interaction strength. Both the variational trial function of the LG condensate with vortex and without vortex are broadened under the in?uence of the repulsive interactions.

III. VORTEX METASTABILITY BARRIERS

A Laguerre-Gaussian condensate with a vortex can be metastable if there exists an energy barrier separating this state from a state without a vortex. In this section we estimate the height of the energy barrier both by the method proposed by Benakli et al. [26] for 2D traps, and by the method proposed by Fetter and co-workers [7].

A. Hydrodynamic and microscopic instability

The hydrodynamical and the microscopic instabilities studied by Fetter and co-workers [7] involve the displacement of the vortex core relative to the center of the trap. As discussed in the introduction, for a parabolically trapped condensate it is energetically favorable for the distance between the vortex core and the center of the trap to increase. Thus, if dissipation is present, the vortex condensate in the parabolic trap can decay into a non-vortex condensate through a migration of the vortex to the edge of the cloud. This e?ect is di?cult to observe experimentally, since the vortex core can tilt and the image contrast between the vortex core and the cloud of atoms is reduced [27]. To estimate the height of the energy barrier separating the vortex condensate from the condensate without a vortex in our present case, we introduce a third variational function to represent the order parameter of a vortex LG-condensate where the vortex core is at a given distance R from the cylindrical symmetry axis of the Laguerre-Gauss trapping beam:

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ψv (R; ρ, θ, z) = N2 r(ρ, θ) exp{?σR (ρ ? ρR )2 /2 ? ζR z 2 /2}eiΘ(ρ,θ) . The function r(ρ, θ) gives the distance between the point {ρ, θ, 0} and the vortex core: r(ρ, θ) = (ρ cos θ ? R)2 + ρ2 sin2 θ,

(6)

(7)

and the function Θ(ρ, θ) gives the angle between the line connecting the vortex core with the point {ρ, θ, 0} and the x-axis: Θ(ρ, θ) = arctan ρ sin θ ρ cos θ ? R . (8)

The parameters {σR , ρR , ζR } are determined variationally for every studied distance R between the vortex core and the center of the trapping geometry (the axis of propagation of the LG beam), and N2 is a normalization constant. The limit R → 0 retrieves the results for the vortex condensate studied earlier. In the limit R → ∞, the result tends to the result for the condensate without a vortex. Using the variational approach, we calculate the energy E(R) of the displaced vortex state for any intermediate R. If the energy E(R) decreases monotonously with increasing R, the vortex is unstable - in the presence of dissipation the vortex condensate will decay. If there exists a maximum energy E(Rmax ) for an Rmax di?erent from zero, there is an energy barrier Ebarrier = E(Rmax ) ? E(R = 0) which will hinder the vortex decay. In Figure 4, the energy barrier Ebarrier is shown as a function of Nascat /aHO , the dimensionless measure of the interaction strength used in the previous section. The calculations

2 were performed for a con?guration so that A0 = 5 ?, W0 = 20 a2 . We found that a HO

metastability barrier exists (Ebarrier > 0) for Nascat /aHO > 1.6 ± 0.1. This means that vortices are metastable only if the e?ective interaction strength (controlled by the scattering length, the number of atoms and the con?nement strength) is large enough: the metastability is induced by interactions. Upon further increasing the interaction strength above the threshold value, the metastability barrier increases. The variational energy E(R) is shown as a function of R in the insets: once for an interaction strength such that the vortex is not stable, and once for a situation where the vortex is metastable. A reasonable estimate of 8

the experimental parameters required to realize this trapping geometry gives aHO = 4 ?m and ? = 50 Hz. This implies that for

87

Rb, the critical number of trapped atoms neces-

sary to make vortices metastable is of the order of 103 . Since one can reasonably expect to trap more than 105 atoms in the trap, vortices created in a Bose gas in the con?nement potential mentioned above, will be well into the metastable regime (Nascat /aHO ≈ 625 for N = 500000).

B. Uniform transition to a non-vortex state

The other method which we used to study the metastability of the vortices in LaguerreGauss condensates, was proposed by Benakli et al. [26] for 2D traps with a an axial hole, punctured by an o?-resonance laser beam. In this method, a trial solution for the GrossPitaevskii equation is constructed as a superposition between the vortex state and the state without a vortex : Ψ(ρ, θ, z) = C0 ψ(ρ, θ, z) + C1 ψv (ρ, θ, z) where |C0 |2 + |C1 |2 = 1. Adapting the coe?cients C0 and C1 of this superposition, the function changes from that of a LG condensate with a vortex to that without a vortex. If the energy of the intermediate states is found to be higher than the energy of the vortex state, this constitutes an energy barrier against uniform transition from vortex state to the state without a vortex, making the former metastable. Using the variationally optimized trial functions for the order parameter of a LG condensate with a vortex and without a vortex, we found with the method of Benakli et al. that such a metastability barrier exists for Nascat /aHO > 1.9 ± 0.1. The dependence of the energy barrier on Nascat /aHO is shown in Figure 5. In the inset of Figure 5, the variational energy of the trial function Ψ is shown as a function of |C0 |2 for an interaction strength below (a) and above (b) the threshold value for metastable vortices. The calculations were

2 performed for a con?guration so that A0 = 5 ?, W0 = 20 aHO . Note that the method of

ref. [26] followed here does not describe the decay of vortices mediated by the excitations, for example by nucleation of vortex loops. However, for a two-dimensional system punctured 9

by an o? resonance laser beam, Benakli et al. [26] could show that excitation-mediated decay of vortices only becomes appreciable for vortices in higher angular momentum states. Furthermore, for a purely one-dimensional toroidal system, Kagan et al. [28] showed that for a weakly interacting system at low temperature, the relaxation time for the decay the super?uid persistent current due to phonons is strongly suppressed. Both the method of Fetter [7] and the method of Benakli et al. [26] give qualitatively the same result, namely that above a critical strength of interaction, expressed by Nascat /aHO , the vortex state of the LG condensate is stabilized by a metastability barrier. Quantitative di?erences remain between the two methods, which is to be expected since they describe di?erent possible mechanisms of vortex decay. The general conclusion – the stabilization of the vortex state due to the toroidal geometry – is further supported by a calculation for a vortex condensate in a mexican hat potential [29].The metastability barrier calculated as by the method of ref. [7] is smaller than the barrier calculated used the method of ref. [26] for Nascat /aHO 3.5, which indicates that the latter method [26] overestimates the real

metastability barrier.

IV. INTERFERENCE AND DETECTION OF VORTICITY

Phase coherence, a property necessary for the existence of vortices, was demonstrated experimentally using interference experiments [30]. Two parabolically trapped condensates, displaced by a given distance, are allowed to expand freely. In the region where the two expanding condensates overlap, an interference pattern is observed. This type of experiment has also been proposed to observe vorticity in parabolically trapped condensates [17,18]: the presence of a vortex leads to an observable edge dislocation in the pattern of otherwise parallel interference fringes. Consider a parabolically trapped condensate positioned on the symmetry axis of the LG condensate. The LG condensate forms a cylindrical shell surrounding the parabolically trapped condensate in the center. As the trapping potentials are switched o?, both 10

condensates will expand and exhibit an interference pattern in the region of overlap. This interference pattern is the subject of the present section. To ?nd the function representing the order parameter at a given time t after the start of the free expansion, the original function ψv (ρ, θ, z; t = 0) is expanded in free particle eigenfunctions. These eigenfunctions acquire a phase factor as time elapses, so that ψv at a time t is given by the resummed expansion with the ‘time evolved’ eigenfunctions. More explicitly, for the LG condensate with a vortex, ?rst the coe?cients of the plane wave expansion are evaluated: ψv (ρ, θ, z; t = 0) = eik.r dk, c(k) (2π)3/2 (9)

c(kρ , φ, kz ) =

ψv (ρ, θ, z)

exp{?ikρ ρ cos(θ ? φ) ? ikz z} dk (2π)3/2

∞

(10) (11)

2 exp {?kz /2ζ} √ × ei(φ+π/2) = 2πζ

0

dρ ρ2 J1 (kρ ρ) exp{?σ(ρ ? ρ0 )2 /2}.

In this expression J1 (x) is the Bessel function of ?rst order of the ?rst kind and the wave number k is expressed in cylindrical coordinates {kρ , φ, kz }. As time elapses, the free particles eigenfunctions acquire a phase factor exp{i k 2 t/(2m)}. The function at time t after the start of the free expansion is found by resumming the eigenfunctions at time t : ψv (ρ, θ, z; t) = = ×

0

eik.r+i k t/(2m) dk c(k) (2π)3/2 1 ζz exp ? 1 + iζt 2(1 + iζt)

∞

2

(12) × eiθ . (13)

dkρ (ρ′ )2

?J1 (ρ′ ρ/t) ?σ(ρ′ ?ρ0 )2 /2 ρ2 + (ρ′ )2 e exp i t 2t

The time evolution for the free expansion of a condensate prepared in a parabolic con?nement is derived analogously [17]. The total measured density generated by the two condensates is then given by |ψv (ρ, θ, z; t) + ψ(ρ, θ, z; t)|2 - remember that each condensate function is normalized to the number of particles in the given condensate. Several time frames of the resulting evolution of the density of the expanding condensates are shown in Figure 3. The frames in Figure 3 show a cross-section of the density along the xy-plane. 11

Immediately after the traps are switched o?, the density is that of a cylindrical, LaguerreGaussian condensate with a parabolically trapped condensate in the middle. As time goes by, both condensates expand: the parabolically trapped condensate expands radially, and the cylindrical shell of the LG condensate broadens. As the expanding condensates start to overlap the fringe pattern appears. If the LG condensate does not contain a quantum of super?uid circulation (i.e. no vortex), the interference pattern consists of a series of concentric circles with linearly increasing radius. If however the LG condensate does contain a vortex, the interference pattern is an Archimedean spiral. At higher vorticity, the number of arms in the Archimedean spiral equals the number of vortex quanta in the LG condensate. Figure 3 shows that, as a function of time, the spiral interference pattern rotates around the cylindrical symmetry axis of the trapped condensates, with a frequency of the order of the frequencies characterizing the parabolic approximation to the trapping potentials. Furthermore, as can be seen from Figure 3, the distance between two successive windings of the spiral increases as time increases. Hence, to detect the spiral interference pattern, the density has to be measured on a time scale shorter than that given by the inverse of the characteristic trapping frequencies.

V. CONCLUSIONS

In conclusion, we have studied the properties of vortices in a condensate in an optical trap generated by a laser beam in a Laguerre-Gaussian mode and proposed a method to detect these vortices. The energy of the vortex state was calculated variationally in a mean?eld framework, and a threshold interaction strength was found beyond which there exists a metastability barrier stabilizing the LG vortex state against a transition to a non-vortex state. Both the hydrodynamic instability [7] and the uniform transition to a non-vortex state [26] were considered in the investigation of the vortex stability. Below the critical interaction strength the metastability barrier vanishes, irrespective of the fact that the present con?ned geometry allows for the vortex core to be positioned in a region where the condensate is 12

nonzero. The time evolution under free expansion of a LG condensate was derived, and subsequently used to calculate the interference pattern which arises from the overlap of an expanding LG condensate with an expanding parabolically trapped condensate in its center. When super?uid circulation is present, the interference pattern consists of an Archimedean spiral, clearly distinct from the series of concentric cylindrical fringes which arise in the interference pattern when no vorticity is present. The observation of such a spiral interference pattern would constitute clear evidence for vorticity in a spin-polarized Bose-Einstein condensates.

ACKNOWLEDGMENTS

We thank F. Brosens and L. F. Lemmens for intensive discussions. Also discussions with T. Kuga are gratefully acknowledged. This work is performed within the framework of the FWO projects No. 1.5.545.98, G.0287.95, 9.0193.97, G.0071.98 and WO.073.94N (Wetenschappelijke Onderzoeksgemeenschap, Scienti?c Research Community of the FWO on “Low Dimensional Systems”), the “Interuniversitaire Attractiepolen – Belgische Staat, Diensten van de Eerste Minister – Wetenschappelijke, Technische en Culturele Aangelegenheden”, and in the framework of the GOA BOF UA 2000 projects of the Universiteit Antwerpen. One of the authors (J.T., aspirant bij het Fonds voor Wetenschappelijk Onderzoek - Vlaanderen) acknowledges the FWO for ?nancial support.

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Rokhsar, preprint available as cond-mat/9709212. [24] L. P. Pitaevskii, Zh. Eksp. Teor. Fiz. 40, 646 (1961). [25] J. Tempere, F. Brosens, J. T. Devreese and L. F. Lemmens, Solid State Commun. 107, 51 (1998); A. Parola, L. Salasnich and L. Reatto, Phys. Rev. A 57, 3180 (1998); J. Tempere, F. Brosens, L. F. Lemmens, J. T. Devreese, Phys. Rev. A 60, 043605 (2000). [26] M. Benakli, S. Raghavan, A. Smerzi, S. Fantoni and S. R. Shenoy, Europhys. Lett. 46, 275 (1999). [27] B. P. Anderson , P. C. Haljan, C. E.Wieman, and E. A.Cornell, Phys. Rev. Lett. 85, 2857 (2000). [28] Yu. Kagan, N. Prokof’ev, and B. V. Svistunov, Phys. Rev. A 61, 045601 (2000). [29] L. Salasnich, A. Parola, and L. Reatto, Phys. Rev. A 59, 2990 (1999). [30] M. R. Andrews, C. G. Townsend, H. J. Miesner, D. S. Durfee, D. M. Kurn, and W. Ketterle, Science 275, 637 (1997).

16

FIGURE CAPTIONS

Figure 1 : The Laguerre-Gauss (LG) trap is an optical dipole trap consisting of a reddetuned laser beam. A typical intensity pro?le of the laser beam in the LG-mode {0, 1} is shown in this ?gure, in a cross section through the axis of propagation of the beam (the z-axis). The atoms will feel the optical dipole force attracting them to the (toroidal) region √ of highest intensity of the laser beam. 20 = 4. 472 14 Figure 2 :A typical surface of constant intensity of the laser beam in the Laguerre-Gauss (LG) mode {0, 1} is shown. The atoms in the red-detuned LG beam will be attracted to a toroidal or cylindrical region such as that within the surface of Figure 2. An additional parabolic magnetic con?nement potential can be added in the z-direction [15,16]. Figure 3 : The variational result for the energy of the condensate in the LaguerreGaussian optical trap is shown as a function of the interaction strength. The full curve shows the energy of the condensate without a vortex, the dashed curve shows the energy with a vortex. The inset depicts the results for the variation parameters σ0 , σ1 in the trial function for the condensate with vortex (dashed line) and without (full line), as a function of interaction strength. The trapping (beam) parameters were chosen as follows: A0 = 5 ?,

2 W0 = 20 a2 . In all graphs of this Figure, energies are expressed in units ? and lengths HO

in units aHO =

/(m?).

Figure 4 : The energy barrier for removing a vortex from a condensate in a LaguerreGaussian optical trap, is depicted as a function of interaction strength. This metastability barrier Ebarrier was calculated using the formalism of Fetter and co-workers [7]. In this formalism, Ebarrier is found by deriving the energy E(R) as a function of the distance R between the vortex core and the center of the trapping potential, here the axis of propagation of the laser beam. This is illustrated in the insets. For interaction strengths lower than a critical value Nascat /aHO < 1.6 ± 0.1 the barrier vanishes and vortices in Laguerre-Gaussian 17

condensates are unstable with respect to the non-vortex state. For interaction strengths above this critical value, a metastability barrier exists. Figure 5 : The energy barrier for a uniform transition from a vortex condensate to a condensate without a vortex, in a Laguerre-Gaussian optical trap, is depicted as a function of interaction strength. The energy barrier for this mechanism was calculated by the method of Benakli et al. [26]. In the inset, the energy per particle of the variational function Ψ = C0 ψ + C1 ψv is given as a function of |C0 |2 , for two di?erent interaction strengths: one below (a) and one above (b) the threshold for metastability of vortices, and the relation to the energy barrier for a uniform transition is shown. Figure 6 : Several time frames in the evolution of freely expanding, overlapping condensates are shown. The gray scale represents the density in a cross-section orthogonal to the symmetry axis of the trap (the direction of propagation of the Laguerre-Gaussian laser beam), with black being the maximum density and white the minimum density. The initial situation is depicted in the top left panel: a parabolically trapped condensate in the center is surrounded by a cylindrical shaped Laguerre-Gaussian condensate containing a vortex. The time evolution of the density in the region within the rectangle in the top left panel is shown in more detail in the subsequent panels. When the con?nement of both condensates is switched o?, they expand and overlap - the density in the gap between both condensates increases - and a spiral interference pattern is formed. Units are chosen so that lengths are expressed in aHO = /(m?) and the time is in units of 1/?.

18

This figure "figuur1.jpg" is available in "jpg" format from: http://arXiv.org/ps/cond-mat/0101014v1

This figure "figuur2.jpg" is available in "jpg" format from: http://arXiv.org/ps/cond-mat/0101014v1

Tempere,Devreese,Abraham; Figure 3

0.34 σ

-1.12

Beam parameters: A0=5 ; W0=4.47 aHO

0.32 0.3

? E/(Nh?)

-1.13

0

3 6 Nascat/aHO

9

-1.14

-1.15

-1.16

0

2

4

6

8

10

Nascat/aHO

Tempere, Devreese, Abraham; Figure 4

10

-0.9

8

? Ebarrier/(Nh?) (×10 )

3

? E/(Nh?)

-0.92 -0.94 0

Beam parameters: W0 = 4.47 aHO A0 = 5 ? ? h

Nasc/aHO=1.0

1 2 3 4 5 6 -0.84

? E/(Nh?)

6

R/aHO

4

Ebarrier

-0.85 -0.86 -0.87

Nasc/aHO=7.5

2

vortices are unstable

vortices are metastable

0

1

2

3

4

5

R/aHO

0

0

2

4

6

8

10

Nasc/aHO

Tempere, Devreese, Abraham; Figure 5

10

-0.9

8

? Ebarrier/(Nh?) (×10 )

3

? E/(Nh?)

-0.92 -0.94 -0.96

Beam parameters: W0 = 4.47 aHO A0 = 5 ? ? h

Nasc/aHO=1.0

0 0.2 0.4 0.6

2

6

0.8

1 -0.82

? E/(Nh?)

|C0|

Ebarrier

4

vortices are metastable

-0.84 -0.86 -0.88 -0.9 0

Nasc/aHO=5.0

2

vortices are unstable

0.2

0.4

0.6

2

0.8

1

|C0|

0

0

1

2

3

4

5

6

7

8

Nasc/aHO

10 t0 5 2 4

t0.3 4

t0.9

2

0

0

0

-5

-2

-2

-4 -10 -10 -5 0 5 10 -4 -2 0 t1.5 4 2 4

-4 -4 -2 0 t1.8 4 2 4

t1.2 4

2

2

2

0

0

0

-2

-2

-2

-4 -4 -2 0 2 4

-4 -4 -2 0 2 4

-4 -4 -2 0 2 4

J. Tempere, J.T. Devreese, E.R.I. Abraham, Figure 6