Dynamics of quantum vortices in a toroidal trap

Transcription

1 PHYSICAL REVIEW A 79, Dnamics of quantum vortices in a toroidal trap Peter Mason and Natalia G. Berloff Department of Applied Mathematics and Theoretical Phsics, Universit of Cambridge, Wilberforce Road, Cambridge CB3 WA, United Kingdom Received 21 December 28; published 21 April 29 The dnamics of quantum vortices in a two-dimensional annular condensate held under a toroidal trapping potential composed of a harmonic term plus Gaussian term are considered b numericall simulating the Gross-Pitaevskii equation. Families of solitar wave sequences are reported, both without and with a persistent flow, for various values of interaction strength. It is shown that in the toroidal geometr the dispersion curve of solutions is much richer than in the cases of a semi-infinite channel or uniform condensate studied previousl. In particular, the toroidal condensate is found to have states of single vortices at the same position and circulation that move with different velocities. The stabilit of the solitar wave sequences for the annular condensate without a persistent flow is also investigated b numericall evolving the solitar wave solutions in time. In addition, the interaction of vorte-vorte pairs and vorte-antivorte pairs is considered and it is demonstrated that the collisions are either elastic or inelastic depending on the magnitude of the angular velocit. The similarities and differences between numericall simulating the Gross-Pitaevskii equation and using a point vorte model for these collisions are elucidated. DOI: 1.113/PhsRevA PACS number s : 3.7.Lm, 3.7.Kk,.4. a, j I. INTRODUCTION A Bose-Einstein condensate BEC offers an ecellent quantum sstem to investigate both eperimentall and theoreticall. Eperimentall, virtuall all of the phsical parameters can be controlled and thus the dnamics of quantum sstems on a macroscopic scale are readil observed. It is a challenge, theoreticall, to devise relevant and accurate models to accompan the increasing number of phsical phenomena currentl being reported. Quantum vortices are one of such nonlinear phsical phenomena that have been observed in BECs 1 6 and that have been investigated theoreticall see, for instance, 7 9. In a BEC, the vortices have quantized circulation and their interactions on an inhomogeneous densit background have been studied etensivel Recent eperimental interest 1 17 has shifted to focus on BECs in toroidal trapping potentials which are now more eperimentall accessible. A BEC in a toroidal trap will form a ring shaped condensate. The first eperimental observation of spontaneous vorte formation as the alkali gas is cooled through the transition temperature to create a BEC has been reported in 17 for both a toroidal and a harmonic trap. In previous studies vorte formation through the transition temperature could be attributed to a number of different factors such as stirring of the condensate b a laser beam. The conditions under which a ring condensate eists and the manipulation of a condensate in a toroidal trap has recentl been considered in 18 and the generation of solitar waves have been studied in 19. Eperimental interest in the dnamics of superfluids in a ring is not new. Donnell and Fetter 2 noted that, in superfluid 4 He, vortices appear in the ring at a particular angular velocit when vortices can first compensate for the difference in irrotational velocit between the inner and outer boundaries of the ring. Eperimentall, rotation of the condensate will create instabilities around the edge of the condensate. As the rotation velocit is increased the instabilities continue to grow and eventuall form vortices see 21,22. A three-dimensional toroidal trapping potential composed of a harmonic term plus Gaussian term, which creates a ring shaped condensate, might provide the necessar energ requirements to ensure stationar solitar wave solutions eist. In a recent eperiment, the observation of persistent flow in a toroidal trap and the eistence of vortices has been reported in 16. Persistent flow in a BEC is the persistent circulation facilitated b the frictionless flow of a superfluid sstem. A useful wa to view persistent flow is to consider a single vorte pinned to the center of the ring shaped condensate. The energ at the center is a local minimum and as such it will cost too much energ for the vorte to drift awa from the center. The vorte creates a constant circulation around itself: a persistent flow. A persistent flow in a BEC is also referred to as a supercurrent. The properties of a persistent flow in a toroidal trap have been theoreticall considered in 23. It is thought that studies on persistent flows in a BEC held under a toroidal trap could lead to crucial insights into the features of the critical velocit in superfluid 4 He and to the fundamental relationship between superfluidit and Bose-Einstein condensation 16. A two-dimensional condensate with eternal potential trap of the form used in 1,16 is considered, V et,,z = 1 2 m V ep /w 2, 1 for a constant potential V, waist w the aial distance from the laser beam s narrowest point, mass m, and harmonic frequenc components, which will create the required ring shaped condensate. Reduction to a twodimensional condensate is equivalent to a tightl trapped, or confined, condensate in the aial coordinate. Such a confinement is eperimentall achievable and reduction to twodimensional condensates has been theoreticall considered in 24. Previous theoretical investigations 2 28 have con /29/79 4 / The American Phsical Societ

2 PETER MASON AND NATALIA G. BERLOFF centrated solel on solitar wave solutions in the absence of a persistent flow, in particular vorte solutions, in an idealized two-dimensional annular shaped condensate where the densit is uniform within the annulus and zero outside of the annulus a bolike toroidal trap. Their analsis uses established point vorte model techniques from classical hdrodnamics and has shown that the motion of vortices is similar to the motion of classical vortices in an annular shaped domain 29. Trapping potential 1 creates an inhomogeneous condensate that is markedl different to a bolike condensate. Solitar wave sequences have previousl been considered in both two- and three-dimensional homogeneous condensates 3,31, a three-dimensional cigar-shaped condensate 32 34, and also a two-dimensional channel condensate 1. The toroidal geometr considered here ensures that there are no phase differences in the condensate. As a result, the dispersion curve of the solitar wave sequences is far richer than has previousl been seen. This paper will focus solel on the dnamics of solitar wave solutions in a two-dimensional condensate held under a toroidal trapping potential. Section II will introduce the required mathematical formulation for the annular condensate without a persistent flow. The families of solitar wave sequences that eist are then reported in Sec. III. In Sec. IV, the stabilit of the solitar wave sequences is numericall investigated and in Sec. V the evolution and collisions of vortevorte pairs and vorte-antivorte pairs are considered. Section VI adds a persistent flow to the annular condensate and reports the families of solitar wave sequences. Finall, the paper ends with a conclusion Sec. VII. II. FORMULATION The dnamics of an annular shaped condensate are accuratel described b the time-dependent dimensional Gross- Pitaevskii GP equation in terms of the macroscopic wave function = r,,t, i 2 = t 2m 2 E v V et r, U 2, for two-dimensional eternal potential trap V et r,. The two-dimensional coupling constant is U, the mass of a boson is m, and E v is the chemical potential of the sstem. The eternal potential trap is taken to be solel in the radial direction such that V et r, V et r = V ep 2r 2 /w m 2 r 2 for a constant potential V, waist w, frequenc =, z =, and where r 2 = For small r the eponential term will dominate over the harmonic term and, for a sufficientl tuned V, a circular hole in the center of the condensate, an inner boundar, is created where no atoms will be present. Conversel, at larger values of r the harmonic term will dominate causing the condensate to form an outer boundar. Note that the toroidal trap can also be modeled b an eternal potential trap of the form 2 3 PHYSICAL REVIEW A 79, V et r = 1 r4 m 2 + r 4 2 b for = 1 as has been considered b man authors see, for eample, 22 for oscillator length b and relative strength of the quartic term. Furthermore, an annular geometr can be created b using Eq. 4 with = +1 and placing the condensate into rapid rotation. Above a certain critical angular velocit of rotation, a central hole is created in the condensate resulting in an annular geometr see However onl Eq. 3 will be used throughout this paper. Equation 2 can be nondimensionalized according to n1/2, t 1 t, r a a r, 2 where the transverse oscillator length is a = / 2m 1/2 and the number densit of the ground state is n=g 2 / 2mU where the nondimensional coupling potential g has been introduced. Thus the nondimensional GP equation describing the dnamics in the annulus is 2i t = 2 V r g 2 6 for chemical potential = 2Ev /, coupling constant g= 2nU / a 2, and where the eternal potential trap is V r = A ep l 2 r r2 for potential A= 2V / and inverse waist l= 2 /m w 2 1/2. Equation 6 is a four parameter sstem g,a,l, and is subject to the normalization, V 2 rdrd =2 V 2 rdr =1, 8 where V is the entire spatial domain. A good approimation to the ground state for a particular choice of parameters, g,a,l,, can be garnered from the Thomas-Fermi TF approimation, given b = 1/2 V r TF for V r r g 9 otherwise. The four parameter set g,a,l, can be reduced to three b substitution of TF approimation 9 into normalization condition 8, i.e., A ep l 2 r r2 rdr =1, 1 2 V g such that the value of the chemical potential sa is determined b the choice of g,a,l. An eample of the ground state for g=, A=1, and l=.9 is shown in Fig. 1. Note that since Thomas-Fermi profile 9 is onl an approimation to the ground state, a numerical procedure, described in Sec. III, is required to find the eact ground state for a particular choice of g,a,l. The corresponding chemical potential is found, b the same numerical procedure, to be =

3 DYNAMICS OF QUANTUM VORTICES IN A TOROIDAL TRAP (a) ρ.3 (b) The energ functional and angular momentum for the inhomogeneous condensate are given, respectivel, b 1,3 E f = V r 2 + g 2 V 2 4 rdrd, p = i 2 V drd The energ of the solitar waves can be determined from E=E f E g, where E g is the energ of the ground state in the absence of an solitar waves and is found numericall b evaluating E f for =. Alternativel, an approimate analtical epression for E g is found b evaluating E f for = TF. If solitar wave solutions are found to eist it is entirel plausible that a multitude of possible solutions could eist for various. A restriction to the tpe of solutions sought is therefore taken so that onl solutions with densit minima along = /2 will be considered. Despite this restriction, the essential dnamics of solitar waves in an annulus will still be reflected. III. SOLITARY WAVES r Solitar wave solutions of Eq. 6 that preserve their form and move with a constant angular velocit at constant r subject to eternal potential 7 are sought. A tpical eample of such a solitar wave solution is a single vorte, which will move in the inhomogeneous condensate because of the velocit field created b the surface of the condensate. Other solitar waves that are sought are a vorte pair, dark gra and black solitons, and finite amplitude sound waves rarefaction waves. Thus, Eq. 6 is recast in the frame rotating with the solitar wave using = t so that / t /. Hence FIG. 1. Color online The ground states of a condensate acting under toroidal potential trap 7 for g=, A=1, and l=.9. Frame a is a contour plot of the eact ground state, while b is a slice along constant of the eact ground state solid line and the Thomas-Fermi approimation TF 9 dashed line. Distances are measured in units of a and densit in units of n/a 2. PHYSICAL REVIEW A 79, i = 2 + V r g 2, 13 where is replaced b for convenience and is solved subject to boundar conditions r,,t r as r,. 14 In the above statement for the boundar conditions, the ground state wave function at r=, is zero, i.e., = =. The solitar wave solutions are found numericall b a Newton-Raphson iteration technique. In order to numericall model the shape of the condensate, a cut in the infinite twodimensional domain is taken along = and the domain is unfurled so that the new two-dimensional rectangular domain in r, occupies the upper half plane. This semiinfinite numerical domain is mapped b the transformation rˆ =tan 1 Dr to a finite grid, /2,2, where D is a constant chosen to lie in the range D.4.8. The boundar conditions at = and =2 are taken to be periodic. The resulting equations are epressed in second-order finitedifference form. Taking 21 2 grid points in the finite domain, the discretized nonlinear equations are solved in polar coordinates on the rectangular grid b a Newton-Raphson iteration procedure using a banded matri linear solver based on the biconjugate gradient stabilized method. The accurac of the obtained solutions is verified b evaluating various integral identities. Indeed, an alternative epression for the energ can be found b taking b for constant b in the epressions for energ and angular momentum 11 and 12 and considering the variational relationship b E p b=1 =. 1 The result is an alternative epression for the energ 1 E = V r 2 drd, 16 which can be used as a check on the numerical results. The accurac of the numerical scheme is found to be ecellent to two decimal places. In Sec. IV, a time evolution of the GP equation Eq. 6 in Cartesian geometr is described and can be used as an additional check on the numerics of this section, particularl the validit of the rectangular grid that is used. An initial ansatz for the numerical procedure for wave function can be obtained from = sol, 17 where is the eact ground state wave function obtained numericall from the TF approimation TF and sol is motivated b the Tsuzuki dark soliton solution in homogeneous 39 and inhomogeneous 1,33 condensates, sol = ic 1 + c 2 tanh c 3 r cos, 18 for constants c 1, c 2, and c 3 to be chosen. Notice that Eq. 18 creates a soliton at cos = to ensure that solitar wave solutions eist onl along = /2. An alternative ansatz

4 PETER MASON AND NATALIA G. BERLOFF PHYSICAL REVIEW A 79, which looks directl for vorte solutions can be found from.2 = v, where the wave function for the vorte at r, is v = r cos + i r sin r r 2 cos 2 + r sin r 2 +2/ Additional vortices can easil be included into Eq. 19 b writing v = N i=1 i, where N is the number of vortices, as required. The position of the vorte is detected numericall b finding zeros of the real and imaginar parts of the wave function. As a vorte moves close to the boundar of the condensate, where, it becomes impossible to distinguish between a rarefaction wave and a vorte solution. In what follows the solitar wave solution is referred to as a rarefaction wave when the numerics cannot resolve whether or not the real imaginar part of changes its sign along the polar angle see also 1 for discussion. An asmptotic epression for the group angular velocit is obtained from the variation + in Eqs. 11 and 12 and allows a comparison with the results of the numerics to be made. On use of the GP equation Eq. 13 one obtains = E p, 21 with the derivatives taken along the direction of propagation of the solitar waves. To obtain an approimation to the angular velocit as a function of r, the position of the vorte Eq. 21 is rewritten as = E p = E/ r 22 p/ r for the energ and angular momentum given as in Eqs. 16 and 12, respectivel. Of the three parameters contained within the sstem g,a,l the value of is determined from the other three, the interaction strength g can be considered to be the dominant factor. The interaction strength directl determines the number of healing lengths that span the condensate and thus in turn directl determines the number of solitar wave solutions, along a constant, which can be present in the condensate. A range of values of g has been simulated 3 g 1 which two tpical eamples will be detailed below. In each eample the energ-angular momentum dispersion curve is smmetric about p=. The effect of this smmetr is to simpl consider solitar wave solutions that move with the opposite circulation. In order to aid the analsis throughout this section different shades of colors will be used. Shades of black will represent a condensate with one solitar wave solution present, shades of dark gra red will represent a condensate with two solitar wave solutions present, and shades of light gra green will represent a condensate with either three or four solitar wave solutions present. E p FIG. 2. Color online The energ-angular momentum dispersion curve for the annular condensate governed b the GP equation Eq. 13 for parameter set g,a,l = 1,4,1. A single vorte solution for = /2 is represented b the thick solid black line while the thick dashed black line represents a rarefaction wave for = /2. The thick dashed dark gra red line corresponds to a rarefaction wave for = /2. The dispersion curve is smmetric about p=. Energ is measured in units of and angular momentum in units of m 1/2. A. Low interaction strength A condensate with a low interaction strength will first be considered with the values of the parameters taken as g,a,l = 1,4,1 with the corresponding value of the chemical potential =8.9. The complete famil of solitar wave solutions in the annulus is then found and the energangular momentum dispersion curve is shown in Fig. 2. Two distinct solutions are found to eist. The first is a single vorte thick solid black line of positive circulation p which eists for angular velocit.29.4 and along = /2 4. The vorte will transcribe circles around the origin for constant angular velocit and for constant radius r.at =.38, the single vorte loses its circulation and becomes a rarefaction wave which is present for increasing angular velocities until the termination angular velocit is reached for the range of parameters chosen here, the termination angular velocit is c=.4. No solutions are present above the termination angular velocit. A plot of the angular velocit of the vorte against distance, r, from the center of the condensate is shown in Fig. 3. The shape of the curve is qualitativel similar in the bulk of the condensate to that observed in a bolike toroidal condensate see 2,27. However, at the boundaries of the condensate there is no qualitative agreement: the curve of Fig. 3 reaches finite values for the angular velocit at the inner and outer boundaries. Notice that the vorte has zero angular velocit when r For.29 the single vorte solution does not eist. Instead a cusp is formed similar to that observed in semiinfinite cigar-shaped condensates 1,33. Over the cusp,

5 DYNAMICS OF QUANTUM VORTICES IN A TOROIDAL TRAP PHYSICAL REVIEW A 79, (vi) (v) Ω (vii) (iv) E.1 (ii) (iii).1.1 (i) r. FIG. 3. The angular velocit plotted against the distance r of the single vorte of positive circulation depicted in Fig. 2 for p. The inner edge of the condensate is at r=r 1 =.78 and the outer edge of the condensate is at r=r 2 =4.6. Angular velocit is measured in units of /m 1/2 and distance in units of a. where now the angular velocit is increased, a second solution is found to eist as two rarefaction waves along = /2 thick dashed dark gra red line in Fig. 2. The two rarefaction wave solution eists all the wa to p = where a maimum on the dispersion curve is attained and the solution corresponds to a black soliton. A sample of the two distinct solutions on the dispersion curve is provided b the contour plots in Fig. 4. The single vorte of positive circulation at =.1 frame a and the two rarefaction waves at =.2 frame b are depicted. Note that b Eq. 21, the angular velocit is the gradient of the energ-angular momentum dispersion curve. B. Midrange interaction strength As the interaction strength is increased new features are seen to develop in the energ-angular momentum dispersion curve that are unepected. An eample of the features observed to occur can be seen b taking the parameter set g,a,l =,1,.9 with the corresponding value of chemical potential = The dispersion curve is given in (a) (b) FIG. 4. Color online Two contour plots depicting two solutions of the GP equation Eq. 13 for parameter set g,a,l = 1,4,1. Frame a contains a single vorte at =.1 r =2.49 and frame b pair of rarefaction waves along = /2 for =.2. Distances are measured in units of a p FIG.. Color online The energ-angular momentum dispersion curve for the annular condensate governed b the GP equation Eq. 13 for parameter set g,a,l =,1,.9. There are seven distinct branches i vii the others can be obtained b smmetr about p=. Branch i corresponds to a single vorte of positive circulation thick solid black line, branch ii corresponds to two vortices on = /2 thin solid dark gra red line, branch iii corresponds to a single vorte of negative circulation thin solid black line, branch iv corresponds to two vortices at smmetric distances on = /2 thin dashed-dotted dark gra red line, branch v corresponds to the four-vorte solution thick solid light gra green line, branch vi to a three-vorte solution thin dashed-dotted light gra green line, and branch vii corresponds to two vortices at antismmetric distances on = /2 thick solid dark gra red line. Units of energ and angular momentum are as in Fig. 2. Fig. and selections of contour plots for the different solutions of the dispersion curve are given in Figs. 6 and 7. Note that the dnamics within this particular sstem are much more diverse than for the low interaction strength g=1 detailed in Sec. III A. The reason for the increase in the number and tpe of solutions realized is the increase in the condensates spatial etent. It is now possible for the condensate to contain at least two solitar wave solutions for an = /2. The dispersion curve contains seven distinct branches with each branch allocated labels i vii. At zero energ and zero angular momentum there is no solitar wave solution; however an increase in energ and positive angular momentum will create a single solitar wave solution moving near the termination angular velocit at the far edge of the condensate along = /2. The solitar wave solution is a single vorte of positive circulation. As the angular velocit is decreased, branch i in the dispersion curve is transcribed and the position of the vorte is seen to move toward the center of the condensate. At =.11 a kink develops in the single vorte solution branch with energ and angular momentum given b E, p =.13,.7. The same single vorte of positive circulation solution is found to be continuous over the kink. The solution is present until =.12 at 4362-

6 PHYSICAL REVIEW A 79, 4362 共29兲 PETER MASON AND NATALIA G. BERLOFF (i) (iv) (v) (vi) (vii).2 (ii) FIG. 7. 共Color online兲 Contour plots of the four different configurations 共iv兲 共vii兲 given on the dispersion curve 共Fig. 兲 for angular velocities =.2, =.1, =.1, and =.1, respectivel. Distances are measured in units of a. (iii) FIG. 6. 共Color online兲 Contour plots of the three different configurations 共i兲 共iii兲 given on the dispersion curve 共Fig. 兲 for angular velocities =.1, =, and =.1, respectivel. Distance is measured in units of a. which point the branch terminates with the vorte at r = 2.3. A new solution is found to eist that joins with branch 共i兲. The new branch, 共ii兲, is characterized b the creation of a new vorte of negative circulation at the far edge of the condensate on = / 2 关see Fig. 6, frame 共ii兲 for a contour plot of the new solution兴. Branch 共ii兲 eists for.12ⱕ ⱕ.13 and includes a solution where =. As the angular velocit increases both the vorte of positive circulation 共near the inner edge兲 and the vorte of negative circulation 共near the outer edge兲 move toward the origin. At =.13, where the branch terminates, the positive vorte reaches the inner edge of the condensate and is lost while the negative vorte is at r = A new cusp is formed on the dispersion curve at 共E, p兲 = 共.13,.34兲. As the angular velocit is decreased a new branch is traced out 关branch 共iii兲兴 which is characterized b the eistence of a single vorte of negative circulation. Further decreasing the angular velocit shifts the single vorte toward the outer edge of the condensate until at =.3 the vorte is lost and the branch terminates. A further solution can be found b increasing the angular velocit and creating a new branch, 共iv兲. Branch 共iv兲 is characterized b the appearance of two identical vortices at the far edge of the condensate on = / 2 关see Fig. 7, frame 共iv兲兴. The cusp formed b branches 共iii兲 and 共iv兲 is unique to all the other cusps in the dispersion curve. At the cusp the dnamics contain no solitar wave solutions, similar to the termination point of branch 共i兲 at the termination speed 共p = 兲. In the eperimental procedures it is possible that disturbances caused outside the far edge of the condensate could theoreticall manifest into forming solutions ehibited b branch 共i兲, 共iii兲, or 共iv兲. The angular velocit range of eistence of branch 共iv兲 is.3ⱕ ⱕ.13 during which the energ and angular momentum change considerabl. As the angular velocit is increased the two vortices move toward the inner edge of the condensate each keeping the same fied distance from the origin. The solution can be viewed as a smmetric pair of vortices on = / 2 关there is smmetr about = 兴. The solution is present to negative angular momentum and terminates when 共E, p兲 = 共.23,.38兲 with the vortices at r = 2.4. At this point the dispersion curve forms a three-wa cusp. Choosing to follow the thick solid light gra 共green兲 curve in Fig. 关branch 共v兲兴 and thus decreasing the angular velocit result into two new vortices being created at the edge of the condensate on = / 2 to produce a four-vorte configuration 关see Fig. 7, frame 共v兲兴. The two new vortices are of opposite circulation compared to the two eisting vortices. As the angular velocit is decreased toward zero the four vortices all move toward the inner edge of the condensate

7 DYNAMICS OF QUANTUM VORTICES IN A TOROIDAL TRAP PHYSICAL REVIEW A 79, (v) (vi) condensate and is thus the crucial parameter determining the tpe of dnamics that can be ehibited in the condensate. As a result it is thus epected that a change in A and l will not change the number of solitar waves that can span the condensate at an one time since changing A and l is epected to onl have a minor effect on the width of the condensate, and thus the number of healing lengths that span the condensate. Thus consider potential trap 7. Taking the derivative with respect to r and searching for the minimum one obtains FIG. 8. Color online Contour plots for = for branches v and vi on the dispersion curve Fig. showing the occurrence of a black soliton. Distances are measured in units of a. the positions of the vortices on = /2 alwas remain smmetric to those on = /2. When the angular velocit becomes zero, two black solitons occur see Fig. 8, frame v. For negative velocities the dnamics are smmetric to those for positive velocities such that when the branch terminates at =.13 for E, p =.23,.38, the configuration is identical to that at E, p =.23,.38. Starting at E, p =.23,.38 and instead choosing to follow the thin dashed-dotted light gra green curve in Fig., and thus tracing out branch vi, the penultimate configuration is realized. Here the dnamics consists of three vortices, with the new vorte appearing at the edge of the condensate for = /2 sa. Thus the configuration consists of two vortices on = /2 and a single vorte on = /2 see the contour plot of Fig. 7, frame vi. Increasing the angular velocit will cause all the vortices to graduall move toward the center of the condensate until at zero angular velocit when the two vortices on = /2 deca to form a black soliton while the single vorte on = /2 remains see Fig. 8, frame vi. For positive velocities the single vorte continues to move toward the center of the condensate, but the other two vortices now move toward the outer edge of the condensate. As the branch terminates at =.13, the furthest of these two vortices reaches the outer edge of the condensate and disappears. Meanwhile the two remaining vortices on opposite sides of the condensate are at different radii. The final realizable configuration is mapped out b branch vii see Fig. 7, frame vii. The two vortices can be described as an antismmetric pair since the are at different radii r 1 =2.61, r 2 =3.38 for the vortices on =+ /2 and = /2, respectivel, for =.1 and their dnamics behave differentl. As the angular velocit is decreased, the vorte at r 1 will move toward the inner edge of the condensate while the vorte at r 2 will move toward the outer edge of the condensate. At zero angular velocit r 1 =r 2 note that there is no black soliton here. C. Effect of A and l This section will consider in more detail the tpical effect that changing the values of the parameters A and l has on the dnamics. As eplained before, the interaction strength g is the parameter that is most responsible for the width of the V r = 2Al2 r ep l 2 r 2 + r = r =+l 1 ln 2Al 2 1/2, 23 where the positive root is chosen and 2Al 2 1, both to ensure r. Thus the value of A and l effectivel shift the position of the minimum of the potential and hence shift the position of the maimum of the densit of the condensate. The effect will be to alter the range of angular velocities which a particular solitar wave solution eists; however it is epected that A and l will have onl a minor effect on the width of the condensate, which is dominated b the harmonic term in Eq. 7. A comparison of the energ-angular momentum dispersion curve for two different parameter sets is shown in Fig. 9 for g= and parameter sets: a A,l =,.9 and b A,l =,1.. The dispersion curves can then be compared to Fig. where A,l = 1,.9. The value of g for all parameter sets is the same. Parameter sets a and b var, respectivel, one of the parameters A,l of Fig. while holding the other fied. Thus a direct comparison into how A and l affect the dnamics of the condensate can be achieved. As can be seen in Fig. 9 the effect of changing A is almost negligible compare Figs. and 9 a. Perhaps the onl noticeable difference is a slight increase in the maimum energ, E ma,asa is decreased. The increase in E ma is to be epected since decreasing A will increase the width of the condensate slightl. The effect of changing l is again fairl minimal compare frames a and b of Fig. 9. However there are a couple of points that can be made. First, increasing l has the effect of increasing E ma fairl considerabl from E ma =.28 on frame a to E ma =.3 on frame b. Second new features have developed in the dispersion curve for the three thin dashed-dotted light gra green curves and four thick solid light gra green curves vorte solutions. Ignoring mirror smmetr about p =, the dispersion curve in frame a contains a single three-vorte configuration and a single four-vorte configuration. In contrast the dispersion curve in frame b now contains two distinct three-vorte configurations and four distinct four-vorte configurations see the blowup of the region near E ma on Fig. 9 b. Again the new configurations are to be epected; changing l will shift the position of the maimum densit of the condensate without altering the width of the condensate much. The additional configurations that occur b changing l are important, but the essential dnamics of the problem are unaltered; merel a few new configurations to alread eisting

8 PETER MASON AND NATALIA G. BERLOFF PHYSICAL REVIEW A 79, (a) Ω.1 E r E p (b) p FIG. 9. Color online A comparison of the energ-angular momentum dispersion curve for two different parameter sets with g=. Frame a has A,l =,.9 and frame b has A,l =,1.. The colors are as in Fig.. Units of energ and angular momentum are as in Fig. 2. solutions are found for instance, the effect of increasing g might be to create new solutions where greater than four vortices in the condensate are present. Indeed as g is raised to high interaction strengths man more distinct solutions will be present. D. Discussion The dnamics of a condensate held under a toroidal trapping potential have been described for two particular cases: g,a,l = 1,4,1 and g,a,l =,1,.9. For g,a,l =,1,.9 the solutions on branch i and branch iii have both been labeled vorte solutions see Fig. with the vorte on branch i possessing positive circulation and the vorte on branch iii possessing negative circulation. The contour plots of the two branches see Fig. 6,.348 FIG. 1. The angular velocit plotted against the distance r of the solutions of branches i thick line and iii thin line depicted in Fig. for p. The inner edge of the condensate is at r=r 1 =1.22 and the outer edge of the condensate is at r=r 2 =.78. Units of angular velocit and distance are as in Fig. 3. branches i and iii indicate that the solutions are vortices. However this raises an interesting anomal. Figure 1 shows the angular velocit against distance plot for branches i and iii. There is one point on the figure in which the curves intersect. At this intersection, the values of the angular velocit and the distances from the center of the condensate are equal for both branches, however the vortices possess different circulations. A plot of the real and imaginar components of the wave function =u+iv, Fig. 11 for the two branches, shows that the do indeed contain vorte solutions with the plots of the wave function across the direction of propagation confirming that the vortices have opposite circulation. However, closer inspection of the plots around r=7 shows that the solution of branch i depletes to the ground state at a larger value of r than the solution of branch iii. This indicates that branch iii possibl contains both a vorte solution and another embedded solution. Recall that branch iii was created when one of the vortices of branch ii reached the inner edge of (a) (b) tpe I tpe II π.2 2π π 2π θ θ FIG. 11. Plots of the real solid line and imaginar dashed line components of the wave function a across the direction of propagation and b along the direction of propagation of the vorte for g,a,l =,1,.9 and for the two single vorte cases tpe I and tpe II branches i and iii in Fig., respectivel for =.1. Distance is measured in units of a

9 DYNAMICS OF QUANTUM VORTICES IN A TOROIDAL TRAP PHYSICAL REVIEW A 79, the condensate and decaed into sound waves in fact, the vorte that decas into sound waves is the vorte of branch i. Therefore, the further solitar wave solution contained in branch iii is the remnants of the rarefaction waves created when this vorte decaed. Thus the solution of branch i can indeed be called a true vorte. However the solution of branch iii is actuall a vorte with rarefaction waves embedded in the condensate. To highlight their differences but at the same time to stress their similarities, the vorte of branch i will be denoted tpe I and the vorte and rarefaction waves of branch iii will be denoted tpe II. Tpe II solutions will often be referred to as vorte solutions in order to emphasize the similarities with tpe I solutions. For an annular shaped condensate the method of images will produce two infinite sequences of image vortices. To see this consider a single vorte of positive circulation in the condensate at position r=r with the inner edge of the condensate at R 1 and the outer edge at R 2 so that R 1 r R 2. First consider the image of the vorte with the inner edge of the condensate. B the Milne-Thomson circle theorem 41 the image vorte, named I i1, will be of negative circulation and will be at r=r 2 1 /r. The image vorte I i1 will itself have an image vorte, of positive circulation, associated with the outer edge of the condensate. The new image will be at r=r 2 2 r /R 2 1 and is named I o2. The process can be repeated with I o2 now producing an image vorte with the inner edge of the condensate and so on. Second consider the image of the vorte with the outer edge of the condensate. The image vorte, named I o1, will be of negative circulation and be at position r=r 2 2 /r. The image vorte I o1 will have an associated image vorte with the inner edge of the condensate to produce a vorte of positive circulation, named I i2,atr=r 2 1 r /R 2 2. Again the process can be repeated. The upshot is that two infinite sequences of image vortices are produced. For r R 1, the first two terms of the sequence of image vortices are R 1 2 r + 2 R 1 2 R 1, 24 R 2 r where a sign has been added to indicate the circulation of the vorte. For R 2 r, the first two terms of the sequence of image vortices are R 2 2 r + 2 R 2 2 R 2. 2 R 1 r It was seen in 42 that the effect of a boundar in an inhomogeneous condensate was to alter the position of the image vorte b an amount equal to the depletion in densit. The result was used in 1 for a channel condensate to find an estimate for the velocit of the vorte as a function of distance from the center of a channel. In the annular shaped condensate there is an asmmetr of the image vortices since the inner edge of the condensate is depleted b an eponential term and the outer edge of the condensate is depleted b a quadratic term. The geometr of the problem unfortunatel does not lend itself to eas calculation of an analtical epression for the angular velocit of a vorte as a function of distance r from the center of the annulus. Various simplifications can be made in the hope of attaining an epression, among them being to consider onl the two closest image vortices I i1 and I o1. However it has been out of reach to find an analtical epression for the angular velocit. IV. STABILITY The stabilit of the solitar wave sequences raises an intriguing question. Because quantized vorticit is an important feature of condensate sstems, the manipulation and observation of vortices are crucial features of an eperiment involving vorticit. Here, a numerical approach is taken to investigate the stabilit of the solitar wave sequences in Sec. III. Families of solitar wave sequences were found in Sec. III for different values of the parameters g,a,l. The stabilit analsis in this section will concentrate on the parameter set g,a,l =,1,.9 which was described in detail in Sec. III B. This parameter set is a tpical eample of the dnamics found in an annular condensate and the energangular momentum dispersion curve is displaed in Fig.. To elucidate the stabilit of the seven distinct branches, a time evolution of the found solutions is performed. Specificall, the time-dependent nondimensional Gross-Pitaevskii equation describing the dnamics in annular condensate 6 with toroidal trap 7 is solved using fourth-order finite difference and fourth-order Runge-Kutta scheme. The stationar solutions found in Sec. III B were found on an irregular polar grid. To translate the solutions to a regular Cartesian grid, cubic splines are used. This different numerical scheme can additionall be used to check the validit of the rectangular grid of Sec. III and therefore the validit of the timeindependent solutions of Sec. III. The solutions in each branch, for various angular velocities, are evolved in time until t 1, a long enough time period to ensure a number of full revolutions of the solutions. It is epected that, over the course of a few revolutions, the unstable solutions will show one of two features: either the will deca into sound waves or in the case of vortices will drift toward the inner or outer edge of the condensate. The stable solutions are epected to be unchanged after the time evolution. It is emphasized that the approach taken here does not prove the stabilit, or instabilit, of the respective branches of the energ-angular momentum dispersion curve, but this approach gives a strong indication to the respective stabilit of the different solutions. Of the seven branches of Fig., it is observed that four branches are stable branches i, iii, iv, and vii while three branches are unstable branches ii, v, and vi. The relative stabilit of the different branches is summarized in Fig. 12. The motions of the single vorte of tpes I and II branches i and iii of Fig., respectivel are both stable. A plot of the position of the vorte in each case is shown in Fig. 13 for angular velocit =.1. One of the other stable branches, branch iv, will be considered in more detail in Sec. VA. Turning now to the unstable solutions, branch v, where there are four vortices of alternate circulation along

10 PETER MASON AND NATALIA G. BERLOFF PHYSICAL REVIEW A 79, t= t= E.1.1 t=12 t= p FIG. 12. The energ-angular momentum dispersion curve for the annular condensate for parameter set g,a,l =,1,.9 of Fig.. The relative stabilities of the different branches are indicated: stable solid line ; unstable dashed line. Units of energ and angular momentum are as in Fig. 2. = /2, is considered as an eample of the dnamics observed. Figure 14 shows snapshots of the contour profile at different intervals of time for angular velocit =.1. As is seen, slight disturbances to the velocit profile cause the four-vorte solution to deca. Initiall, instabilities occur t=11, causing the four vortices to deca into two single vortices and sound waves t=12. After a short time interval, two vortices are observed to remain in the condensate which is now filled with sound waves t=26. The deca of the vortices in branches ii and vi is similar. The relative stabilities of the seven branches are to be epected. Vorte rings and pairs in an infinite homogeneous condensate were found to be stable 31,43 as well as vorte rings and pairs in a semi-infinite channel condensate 33. However multiple vorte rings and pairs are unstable in these geometries. For the case considered in this paper, the unstable solutions are ones which there eist two i.e., multiple vortices along the same polar angle. The two vortices will I II FIG. 13. The path gra lines of the single vorte solutions of tpe I and tpe II for =.1 for integration forward in time. The vortices are at r=3.11 and r=2.87, respectivel, and are epected to be stable. The edges of the condensate are the black lines and are at r=r 1 =1.22 and r=r 2 =.78. Distance is measured in units of a FIG. 14. Color online Contour profiles of the stationar solution for =.1 of branch v for four vortices placed along = /2 at different intervals in time. The deca of the solution over time indicates instabilit. Distance is measured in units of a. deca into a single vorte and sound waves see Fig. 14. In contrast, solutions that contain onl a single vorte along a polar angle are stable. Indeed, for branch vi where there are two vortices along = /2 and one vorte along = /2, the instabilit develops along = /2 while the single vorte along = /2 remains stable. However, over time the sound waves from the deca of the two vortices along = /2 propagate through the condensate and render the single vorte unstable. V. EVOLUTION AND COLLISIONS OF SOLITARY WAVES The evolution of vortices in multipl connected domains has received much attention over the ears. The majorit of the focus has been on modeling classical fluid dnamical situations of interest, for eample, the motion of a vorte around islands 44 or through a gap in a wall 4,46. Traditional hdrodnamical methods for solving such situations revolve around a point vorte model which enables a reduction in the dimension of the sstem while capturing all the essential dnamics. Point vorte models can also be successfull applied to annular condensates see However the point vorte model relies on incompressible dnamics so the densit in the bulk of the condensate is constant and is zero outside of the condensate. Thus, interactions between the background condensate and the vortices are forbidden and so the transfer of energ between the vortices and the background condensate does not occur. Instead, vortices initiall in the condensate will remain in the condensate. Deca of vortices is therefore not possible using a bolike trap and a point vorte model. It has alread been seen Sec. IV that interactions with the background condensate can have a significant effect on the dnamics. Figure 14 shows that the

11 DYNAMICS OF QUANTUM VORTICES IN A TOROIDAL TRAP interactions between the background condensate and the four vortices result in an instabilit developing and the deca of two of the vortices into sound waves. Eperimentall, collisions of three-dimensional solitar waves have been observed in 47 and a recent theoretical stud 48 considered the head-on collisions of vorte rings and solitons in a cigar-shaped condensate. It was observed in 48 that, depending on the speed of approach, the interactions of the solitar waves can be elastic or inelastic, with the solitar waves either repelling, passing through, or annihilating each other. Collisions where the solitar waves pass through each other elastic collision are also forbidden b point vorte models. In a recent paper, Li et al. 14 considered vorteantivorte pairs in a two-dimensional harmonicall trapped condensate. In such a condensate it becomes difficult to simulate vorte-vorte or vorte-antivorte collisions. Instead a toroidal geometr eplicitl allows this possibilit. This section will consider the evolution of vorte-vorte pairs and vorte-antivorte pairs in the annular condensate b numericall simulating the GP equation Eq. 6 b the same method emploed in Sec. IV. It is noted that, b solving Eq. 6 with toroidal trap 7, the bulk condensate densit is not constant and thus the simulations in this section will provide a comparison with the point vorte models emploed in previous papers The parameters of the condensate are again chosen to be g,a,l =,1,.9. The evolution and collisional dnamics of two vortices are the main focus and thus the initial states for the condensate will contain two vortices, one along = /2 and one along = /2. As detailed before, there eist two distinct single vorte solutions denoted tpe I and tpe II branch i and branch iii on Fig., respectivel. The solutions on these branches are isolated note that it is not the vorte solution that is isolated, but the complete solution in the absence of the ground state. The initial state can then be formed from = v+ v for ground state and vorte states v Eq. 2 for vortices along = /2. Different combinations and different circulations of the single vorte solutions of tpe I and tpe II are considered. A. Vorte-vorte pair Consider first the evolution of two vortices of tpe I of the same circulation and placed at equal radii on opposite sides of the condensate. As epected, the vortices traverse paths of constant radius and move with the same angular velocit. The paths of the vortices are shown in Fig. 1 a for vortices placed at r =3.82 where the angular velocit is =.2. Also shown in Fig. 1 b is the identical situation but now for vortices of tpe II placed at radii r =4.38 where the angular velocit is =.2. Note that the construction of these models forms condensates that are identical to branch iv of the dispersion curve in Fig. which was shown to be numericall stable. However, for a given radii of the two vortices, the angular velocit is different in all cases. B. Vorte-antivorte pair Net, the interactions of tpe I and tpe II vortices are considered. A vorte of tpe I at = /2 and a vorte of (a) PHYSICAL REVIEW A 79, (b) FIG. 1. The path gra lines of vortices placed initiall on opposite sides of the condensate with the same circulation. Frame a contains two vortices of tpe I of Fig. at r =3.82 at = /2 and frame b contains two vortices of tpe II at r =4.38. The edges of the condensate are the black lines and are at r=r 1 =1.22 and r=r 2 =.78. Distance is measured in units of a. tpe II at = /2 are placed at opposite radial position r =3.62. The vortices have opposite circulation, however at r =3.62, the vorte of tpe I has positive angular velocit =.18, whereas the vorte of tpe II has negative angular velocit =.1. Thus the vortices are epected to move toward one another but with different magnitudes of angular velocit. Indeed, as time evolves, the vortices begin to approach each other along the same radii with their initial angular velocities. However as the distance between the vortices decreases, the vortices begin to lose angular velocit and move toward the center of the condensate. The continue to move toward the center of the condensate until the reach the inner boundar where the both deca into sound waves. A selection of snap shots of the contour profile in Fig. 16 at different time intervals shows the process. The collision of the two vortices of tpe I and tpe II, has opened up the question as to whether there is an analog with the collisional properties outlined b Komineas and Brand 48. For solitar waves in a cigar-shaped trap, the head-on collisions were elastic for low and high velocities of approach but were inelastic for intermediate values of velocit. To investigate the elasticit of the collisions in an annular condensate, a series of simulations for different angular velocities was considered for two vortices both taken to be of tpe I and placed on opposite sides of the condensate at the same radius but with opposite circulations. The three angular velocities chosen are =.1 with corresponding r =3.11, =.2 r =3.82, and =.3 r =.1. The three simulations are outlined in Figs , respectivel. When the angular velocit is small =.1, initiall the vortices approach one another with constant angular velocit. However as the distance between the vortices begins to reduce, their influence on each other begins to take effect and dominate the flow in the condensate. At such time, the angular velocit is reduced and the vortices move to a new radius where the begin to separate again at a new constant angular velocit. The process is repeated when the vortices net meet at the opposite side of the condensate; however now the vortices move back to the original radius and angular velocit. The process continues forever and is therefore elastic: the vortices upon collision merel transfer to a new energ and angular momentum configuration. In Fig. 17, a plot of the paths of the vortices is provided