Superfluid vortex dynamics on a spherical film
Sálvio Jacob Bereta, Mônica A. Caracanhas, Alexander L. Fetter
SSuperfluid vortex dynamics on a spherical film
S´alvio Jacob Bereta ∗ Instituto de F´ısica de S˜ao Carlos, Universidade de S˜ao Paulo, Brazil
Mˆonica A. Caracanhas
Instituto de F´ısica de S˜ao Carlos, Universidade de S˜ao Paulo, Brazil
Alexander L. Fetter † Departments of Physics and Applied Physics, Stanford University, Stanford, CA 94305-4045, USA (Dated: January 19, 2021)Motivated by ongoing experimental efforts to make closed Bose-Einstein condensate (BEC) shellsin microgravity environments, this work studies the energy and dynamics of singly quantized vorticeson a thin spherical superfluid shell, where the overall vortex charge must vanish (as on any compactsurface). For each vortex, stereographic projection yields the corresponding complex potential onthe tangent plane. The resulting stream function then provides both the total energy and thedynamics of a system of overall neutral vortices on a spherical film. Although a single vortex dipolefollows a simple dynamical orbit, four vortices can present a variety of situations. We study a fewsymmetric initial configurations and then focus on the special case of two small vortex dipoles.
I. INTRODUCTION
The remarkable creation (1995) of Bose-Einstein con-densates (BECs) in ultracold dilute atomic gases has re-newed interest in bosonic superfluids (see, for example,Refs [1, 2]). These highly flexible many-body coherentsystems now appear in many different forms, for exam-ple, three-dimensional bulk condensates ranging from flatpancakes to elongated cigars, and various optical lattices.They also provide a basis for tests of fundamental quan-tum matter Ref. [3].Recent proposals to create thin spherical traps inspace (leading to “bubble” BECs) Refs. [4, 5] along withthe related works in Refs. [6, 7] have stimulated re-search on thin-film superfluid dynamics on curved sur-faces Ref. [8, 9]. Here we study thin spherical superfluidsthat we model as ideal classical fluid films with quan-tized vortices. In addition, the compact spherical surfacerequires that the system have zero net vorticity.Most investigations have emphasized the energy ofquantized point vortices on general curved surfaces (see,for example, Ref. [10]), usually based on the phase S ( r )of the superfluid condensate wave function. In contrast,we focus here on the dynamics of these vortices and spe-cialize to the surface of a sphere.To this end, we rely on the stream function χ (Ω) at apoint with spherical polar angular coordinates Ω = ( θ, φ ).In addition, χ also depends parametrically on the loca-tion of the various point vortices at Ω j . Not only does χ determine directly the motion of the set of vortices, butit also immediately gives the interaction energy of thesame vortices with no additional analysis. A combina-tion of these results generalizes the familiar Hamiltonian ∗ [email protected] † [email protected] formulation of vortex dynamics on a plane to the curvedsurface of a sphere, with the angular coordinates of avortex Ω j = ( θ j , φ j ) as canonical Hamiltonian variables(see, for example, Sec. 157 of Ref. [11]).In Sec. II, we summarize vortex dynamics in a plane,where we review the complementary roles of the conden-sate phase S (which is effectively the velocity potential)and of the stream function χ in determining the localflow velocity v ( r ). For the surface of a sphere (Sec. III),we use a stereographic projection to determine the cor-responding stream function χ (Ω) and hence the vortexdynamics on a sphere. As an example, we study the dy-namics of a single vortex dipole. Section IV obtains theenergy of a set of vortices on a sphere in terms of the samestream function, which then provides a Hamiltonian for-malism for the vortex dynamics. Section V studies thedynamics of four vortices in a few simple symmetric con-figurations. The energy and dynamics of two small vortexdipoles are considered in Sec. VI, and Sec. VII providesa summary and conclusions. II. SUPERFLUID VORTEX DYNAMICS ON APLANE
Typical superfluids like He-II and atomic BECs havescalar complex order parameters Ψ = | Ψ | e iS . The asso-ciated superfluid velocity is v = ( (cid:126) /M ) ∇ S, (1)where 2 π (cid:126) = h is Planck’s constant and M is the rel-evant atomic mass. Apart from an overall factor, thephase S ( r ) is the velocity potential. Equation (1) im-plies that the superfluid velocity field is irrotational with ∇ × v = 0 except at singular points that represent quan-tized vortices. Each vortex has quantized circulation (cid:73) C d l · v = (cid:126) M (cid:73) C d l · ∇ S = 2 π (cid:126) M ν, (2) a r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n where ν is the winding number of the phase around theclosed contour C . The fundamental unit of circulation is2 π (cid:126) /M , which has the dimension of length squared overtime, like a diffusivity or a kinematic viscosity.In many cases, the fluid is also incompressible with ∇ · v = 0. This property allows an alternative repre-sentation of the superfluid velocity field v = ( (cid:126) /M ) ˆ n × ∇ χ, (3)where χ ( r ) is the stream function and ˆ n is the unit nor-mal vector to the plane. For a single quantized vortexwith integer vortex charge q at r , the vorticity is sin-gular, with ∇ × v = 2 π (cid:126) M ˆ n q δ (2) ( r − r ) . (4)It is not hard to show from Eqs. (3) and (4) that thescalar function χ satisfies Poisson’s equation ∇ χ = 2 π N v (cid:88) j =1 q j δ (2) ( r − r j ) , (5)with the N v vortices at r j and charges q j serving assources for the stream function. Note the clear analogy tothe two-dimensional electrostatic potential arising froma set of two-dimensional point charges. Like the electro-static potential, the total stream function χ is definedonly up to an additive constant that is generally irrele-vant.Equations (1) and (3) together give the following equa-tions for the Cartesian components of the velocity field M v x (cid:126) = ∂S∂x = − ∂χ∂y ; M v y (cid:126) = ∂S∂y = ∂χ∂x . (6)By inspection, we see that χ and S satisfy the Cauchy-Riemann equations and can be linked to form a singleanalytic function of a complex variable F ( z ) = χ + iS, (7)where z = x + iy ; its real part immediately provides thedesired stream function with χ = Re F . It also followsdirectly that v y + iv x = (cid:126) M dFdz = (cid:126) M F (cid:48) ( z ) . (8)The complex potential for a single vortex with positivecharge q = 1 at z is F ( z ) = ln( z − z ) . (9)Any complex function Z ( z ), such as z − z , can be writtenin polar form: Z ( z ) = | Z ( z ) | exp[ i arg Z ( z )], and notethat | Z | = ( ZZ ∗ ) / , where Z ∗ is the complex conjugate.Correspondingly, the logarithm becomes ln Z = ln | Z | + i arg Z and the real part is ln | Z | = ln | Z | . For a single positive vortex [compare Eq. (9)], thestream function χ ( r ) = Re F is χ ( r ) = ln | z − z | = ln | r − r | . (10)The resulting superfluid velocity v ( r ) follows fromEq. (8) v y + iv x = (cid:126) M z − z = (cid:126) M z ∗ − z ∗ | z − z | . (11)If the vortex is at the origin ( z = 0), we readily recoverthe familiar axisymmetric circulating irrotational flow v ( r ) = (cid:126) M r ˆ φ , (12)where ( r, φ ) are plane polar coordinates and ˆ φ = ˆ n × ˆ r is the unit azimuthal vector in the polar direction.More generally, for a system of N v vortices at r j withvortex charge q j ( j = 1 , · · · , N v ), the total hydrody-namic flow field v ( r ) is the sum of contributions fromeach vortex. To make this connection more precise, de-fine χ j ( r ) = ln | z − z j | = ln | r − r j | . (13)We here consider only singly charged vortices with q j = ±
1, although most of our analysis applies more generally.The total stream function χ ( r ) is the linear combination χ ( r ) = N v (cid:88) j =1 q j χ j ( r ) (14)giving the total hydrodynamic flow field v ( r ) = (cid:126) M ˆ n × ∇ χ ( r ) = (cid:126) M ˆ n × ∇ (cid:88) j q j χ j ( r ) . (15)In an ideal fluid, a given vortex moves with the localflow velocity at its position, typically arising from all the other vortices. In the present context of an unboundedplane, it follows directly that the k th vortex has the ve-locity ˙ r k = (cid:126) M ˆ n × (cid:88) j (cid:48) q j [ ∇ χ j ( r )] r → r k , (16)where the primed sum omits the single term j = k . Foreach term of the sum, the j th coordinate is fixed, and wecan simplify this result to˙ r k = (cid:126) M ˆ n × ∇ k (cid:88) j (cid:48) q j χ j ( r k ) , (17)which now depends only on the coordinates of the vor-tices since χ j ( r k ) = ln | r k − r j | . In this way, the in-dividual stream functions χ j determine both the totalhydrodynamic superfluid flow through Eq. (15) and thedynamical motion of each vortex through Eq. (17).The quantity ∇ k χ j ( r k ) = ( r k − r j ) / | r k − r j | is oddunder the interchange j ↔ k . It follows immediatelyfrom Eq. (17) that N v (cid:88) k =1 q k ˙ r k = 0 , or, equivalently, N v (cid:88) k =1 q k r k = const . (18)Hence the combined dynamical motion of the vortices ona plane conserves this vector quantity (cid:80) N v k =1 q k r k . Thisresult is well known for two vortices, but its generaliza-tion will be useful in Sec. VI. III. STEREOGRAPHIC PROJECTION FORVORTICES ON A SPHERE
Most of the previous summary about vortices on aplane applies also to vortices on the surface of a sphere,but there is one important difference. On a plane, a sim-ple closed curve C divides the plane into an interior andan exterior, with a clear distinction between them. Forexample, a point in the interior can serve as a local originthat defines a positive sense of rotation for C . The phasewinding around C is the net vortex charge enclosed by C ,and the resulting lines of constant phase crossing C canextend unimpeded through the exterior region.The situation is very different on the surface of a spherebecause of the compact topology. In particular, a closedcurve C divides the spherical surface into two regions, butthe choice of interior region appears arbitrary. Choose apoint in one of the two regions and use it to define thepositive sense for C , with a resulting phase winding num-ber. From the perspective of the other region, however, C has a negative sense with the negative of the same phasewinding number. This symmetry means that any col-lection of point vortices on the surface of a sphere musthave zero total vortex charge. Sections 80 and 160 of[11] quote this property in treating a vortex dipole on asphere as the simplest case with zero net vortex charge,using without proof the method of stereographic projec-tion to give the translational velocity of such a vortexdipole. Here we provide a more complete derivation.Consider a point vortex in a superfluid film on thesurface of a sphere of radius R . To determine the velocityfield on the sphere, we use a stereographic projection [12]onto a complex plane. Familiar methods from Sec. IIprovide the velocity field on the complex plane, and theappropriate coordinate transformation then determinesthe corresponding solution for the sphere.In the stereographic projection, we consider a tangentplane at the north pole. Each point on the sphere has aone-to-one correspondence with a point on tangent plane,except for the point at the south pole. As seen fromFig. 1, a point on the sphere with spherical polar coor-dinates ( θ, φ ) has the corresponding complex coordinateon the plane z = ρe iφ with ρ = 2 R tan ( θ/
2) (19) and the same azimuthal angle φ . θ/2 P P’R ρ θ FIG. 1. Stereographic projection of a sphere of radius R ontoa tangent plane at the north pole. The figure shows the pro-jection of a generic point P on the sphere to the point P (cid:48) onthe complex plane, with radial distance ρ = 2 R tan( θ/
2) andthe same azimuthal angle φ . Section II shows that the complex potential F ( z ) of avortex dipole with complex coordinates z ± and charges q ± = ± F dip ( z ) = ln( z − z + ) − ln( z − z − ) = ln (cid:18) z − z + z − z − (cid:19) , (20)where z is on the tangent plane. The stereographicprojection in Eq. (19) gives the transformation z ± =2 R tan( θ ± / e iφ ± , where ( θ ± , φ ± ) are the spherical coor-dinates of the members of the vortex dipole on the sphere,leading to the associated complex function on the sphere F dip = ln (cid:18) tan( θ/ e iφ − tan( θ + / e iφ + tan( θ/ e iφ − tan( θ − / e iφ − (cid:19) , (21)where the overall scale factors 2 R cancel.It is helpful to introduce the abbreviation u = tan( θ/ u ± . Application of Eq. (10) for eachvortex yields the stream function for a vortex dipole ona sphere χ dip (Ω) = 12 ln (cid:18) u + u − uu + cos( φ − φ + ) u + u − − uu − cos( φ − φ − ) (cid:19) , (22)where Ω = ( θ, φ ) is a convenient notation. Use of thetrigonometric identitytan( θ/
2) = sin θ θ = (cid:18) − cos θ θ (cid:19) / (23)simplifies this result, which can be written as the differ-ence of two separate quantities, each involving a singlevortex [compare Eq. (14)] χ dip (Ω) = χ + (Ω) − χ − (Ω) . (24)Here [compare Eq. (13)] χ j (Ω) = ln [2 − θ cos θ j − θ sin θ j cos( φ − φ j )] (25)depends on the variable angular coordinate Ω and thefixed angular coordinate Ω j of the j th vortex. Note thatthe factor 2 merely adds a constant term to χ j and ischosen for convenience. Also, we omit a constant factor1 + cos θ j that depends only on the vortex coordinateand does not affect the fluid velocity. Since Eq. (25) issymmetric in its variables, we can also write it as χ j (Ω) = χ (Ω , Ω j ), which will be useful below.To interpret Eq. (25), recall the unit radial vector ˆ r =sin θ cos φ ˆ x +sin θ sin φ ˆ y +cos θ ˆ z . The dot product withthe corresponding ˆ r j for the j th vortex isˆ r · ˆ r j = cos γ j = sin θ sin θ j cos( φ − φ j )+cos θ cos θ j , (26)where γ j is the angle between the two vectors. Evidently,Eq. (25) has the equivalent and simpler form χ j (Ω) = χ (Ω , Ω j ) = ln(2 − γ j ) = ln | ˆ r − ˆ r j | = ln (cid:2) ( γ j / (cid:3) = ln [2 sin( | γ j | / , (27)where the square root requires the absolute value | γ j | [compare Eq. (13) for the stream function of a vortex ona plane].To provide a geometric interpretation, Fig. 1 showsthat 2 R sin( θ/
2) is the chordal distance between the point P and the north pole. Hence 2 R sin( | γ j | /
2) is the chordaldistance between ˆ r and ˆ r j , which is less than the corre-sponding great-circle distance between them.For a general set of N v point vortices with (cid:80) j q j = 0,the corresponding total stream function is χ (Ω) = (cid:88) j q j χ (Ω , Ω j ) , (28)which depends on the variable observation location Ωand the given locations of all the vortices Ω j . The totalhydrodynamic velocity field is v (Ω) = (cid:126) M ˆ r × ∇ χ (Ω) = (cid:126) M ˆ r × (cid:88) j q j ∇ χ (Ω , Ω j ) , (29)where ˆ r is the unit normal vector to the surface of thesphere. Note that the operator ˆ r × ∇ is effectively the an-gular momentum operator and involves only the angularparts of the gradient operator. To be very specific,ˆ r × ∇ = − ˆ θ R sin θ ∂∂φ + ˆ φ R ∂∂θ , (30)where ( ˆ θ , ˆ φ , ˆ r ) form an orthonormal triad on the surfaceof the sphere, analogous to ( ˆ x , ˆ y , ˆ n ) on the plane.Just as for a plane, a given vortex on a spherical surfacemoves with the local velocity generated by all the other vortices [see Eq. (16)]. The argument leading to Eq. (17)readily yields˙ r k = (cid:126) M ˆ r k × ∇ k (cid:88) j (cid:48) q j χ (Ω k , Ω j ) . (31)The dynamics of the k th vortex now depends only on theangular coordinates of the N v vortices through the singlesum of stream functions (cid:88) j (cid:48) q j χ (Ω k , Ω j ) = (cid:88) j (cid:48) q j ln [2 sin( | γ kj | / , (32)where cos γ kj = ˆ r k · ˆ r j .A single vortex dipole provides a simple example ofvortex dynamics on a sphere. The rotational symmetryallows us to place the two vortices at the same azimuthangle φ + = φ − , with θ + in the upper hemisphere and θ − in the lower hemisphere (we eventually choose themto be symmetrical around the equator, but we first needto differentiate the stream function with respect to onepolar angle, keeping the other polar angle fixed). Equa-tion (27) gives the relevant stream function χ (Ω + , Ω − ) = ln (cid:20) (cid:18) θ − − θ + (cid:19)(cid:21) . (33)Use of Eq. (31) gives (compare Sec. 160 of [11])˙ r + = ˙ r − = (cid:126) M R cot (cid:18) θ − − θ + (cid:19) ˆ φ = (cid:126) M R tan θ ˆ φ , (34)where we set θ + = θ and θ − = π − θ after apply-ing the gradient operator. Each member of the vor-tex dipole moves uniformly in the positive ˆ φ directionwith fixed separation and speed v dip = (cid:126) tan θ/ (2 M R ).For small θ (cid:28)
1, the dipole is near the poles and themotion is slow. If the pair is near the equator, how-ever, we have θ = π/ − ∆ θ with ∆ θ (cid:28)
1, and v dip = (cid:126) cot ∆ θ/ (2 M R ) ≈ (cid:126) / (2 M R ∆ θ ). This result agrees withthe corresponding planar dipole because 2 R ∆ θ is the lin-ear separation between the two members of the dipole. IV. ENERGY OF VORTICES ON A SPHERE
In the present hydrodynamic model, the total energy issimply the kinetic energy E = M n (cid:82) d r | v ( r ) | , where n is the two-dimensional uniform number density andthe logarithmic divergence near the center of each vortexmust be cut off at some small vortex core radius ξ . Thispicture applies equally to a plane and to the surface of asphere. A combination with Eq. (29) leads to E = (cid:126) n M N v (cid:88) i,j =1 q i q j (cid:90) d r ∇ χ i · ∇ χ j = (cid:126) n M N v (cid:88) i,j =1 q i q j I ij , (35)which defines the dimensionless integrals I ij = (cid:82) d r ∇ χ i · ∇ χ j . The sum includes both the diagonalterms with i = j and the off-diagonal terms with i (cid:54) = j .It is convenient to use the vector identity ∇ A · ∇ B = ∇ · ( A ∇ B ) − A ∇ B, where A and B are scalar functions. For the diagonalterms, we exclude a small circle of radius ξ around r j ,where the relative angle γ j is small and constant. Inthis vicinity, Eq. (27) shows that χ j ≈ ln( γ j ), and adetailed analysis gives I ii = 2 π ln( R/ξ ), where R is theradius of the sphere. Note that the resulting self-energy E self = ( π (cid:126) n/M ) ln( R/ξ ) is independent of position.For the off-diagonal integrals with i (cid:54) = j , the abovevector identity and Eq. (5) immediately gives the result I ij = − πχ ij , where we use the simplified notation χ ij = χ (Ω i , Ω j ). We can now combine our results to find thetotal energy E = (cid:126) nπM N v ln (cid:18) Rξ (cid:19) − (cid:126) nπM (cid:88) i,j (cid:48) q i q j χ ij , (36)where the primed double sum omits terms with i = j .The first term (the total self-energy of the N v individualvortices) is an irrelevant additive constant, whereas thesecond term (the interaction energy) depends explicitlyon the position and sign of all the vortices because χ ij = ln (cid:2) ( γ ij / (cid:3) involves the relative angle γ ij betweenthe two vortices.A combination with Eq. (31) leads to the importantresult2 π (cid:126) nq k ˙ r k = 2 π (cid:126) nM ˆ r k × ∇ k (cid:88) j (cid:48) q k q j χ (Ω k , Ω j )= − ˆ r k × ∇ k E, (37)where we note that ∇ k (cid:88) i,j (cid:48) q i q j χ (Ω i , Ω j ) = 2 ∇ k (cid:88) j (cid:48) q k q j χ (Ω k , Ω j ) . Also, the total self-energy is constant and hence does notcontribute to the induced velocity. The quantity − ∇ k E can be considered a force on the k th vortex. The hy-drodynamics of ideal fluids then ensures that the vortexmoves perpendicular to this force, in contrast to the usualNewtonian situation for point masses.Although Eq. (37) is correct and instructive as writ-ten, it can be helpful to separate it into spherical polarcomponents on the sphere’s surface. The velocity vectoron the surface has the corresponding components˙ r = R ˙ θ ˆ θ + R sin θ ˙ φ ˆ φ . (38)Also, ∇ k E = 1 R ∂E∂θ k ˆ θ k + 1 R sin θ k ∂E∂φ k ˆ φ k (39) A straightforward comparison of terms shows thatEq. (37) has the equivalent form2 π (cid:126) nq k R ˙ θ k = 1 R sin θ k ∂E∂φ k , (40)2 π (cid:126) nq k R sin θ k ˙ φ k = − R ∂E∂θ k . (41)As noted by Kirchhoff for point vortices on a plane(see [11], Sec. 157), these equations have a Hamilto-nian structure with N v pairs of conjugate canonical vari-ables ( θ k , φ k ). For a sphere, N v must be even with (cid:80) k q k = 0. In the present case, the polar motion ˙ θ k depends on ∂E/∂φ k , and similarly the azimuthal mo-tion ˙ φ k depends on − ∂E/∂θ k , as seen for a single vortexdipole in Eq. (34). V. DYNAMICS OF FOUR VORTICESARRANGED SYMMETRICALLY
In Sec. III, we determined the velocity field of a singlevortex dipole on a sphere. Equation (34) shows that thedipole moves at constant angular separation parallel tothe great circle bisecting them in the direction of the fluidflow between them. The speed of the dipole depends onlyon the cotangent of the half the angular aperture betweenthem.
A. Some simple configurations of four vortices
The next simplest case is four vortices with overallcharge neutrality. Among many initial configurations,some have particularly simple dynamical trajectories.Here we explore two coplanar initial configurations (sym-metric and antisymmetric or exchanged) along with thethree-dimensional tetrahedral configuration.
1. symmetric configuration + + −− Figure 2 illustrates what we call the symmetriccoplanar vortex configuration, where we choose thecoordinate axes with all four vortices initially in the xz plane. Vortices 1 (positive) and 2 (negative) have thesame azimuthal coordinates φ , = 0, and symmetricpolar coordinates θ = θ and θ = π − θ around theequator. The other two vortices 3 (positive) and 4 (neg-ative) are mirror images of vortices 1 and 2, respectively,reflected in the yz plane, with polar angles θ = θ and θ = π − θ , and azimuthal coordinates φ , = π .These four vortices behave intuitively in two simplelimits: When the vortices are near the poles ( θ (cid:28) (cid:126) / (2 M Rθ ), just like two same-sign
FIG. 2. Coplanar symmetric configuration for four vorticesnumbered 1 to 4 and charged as shown. We choose initialazimuthal angles φ = 0 (vortices 1 and 2) and φ = π (vortices3 and 4) and initial polar angles θ (vortices 1 and 3) and π − θ (vortices 2 and 4) measured from the z axis. The figure showsthe angular aperture 2∆ θ = π − θ for vortex dipoles (12) and(34), where ∆ θ = π/ − θ . vortices on a plane. Near the equator ( θ → π/ θ = π/ − θ (cid:28)
1) vortices (1 ,
2) and (3 ,
4) act liketwo independent vortex dipoles and move uniformly withspeed (cid:126) / (2 M R ∆ θ ), straddling the equator. For inter-mediate configurations, all four vortices influence eachother, requiring a more detailed analysis.Use of Eq. (31) shows that˙ r k = (cid:88) j (cid:54) = k (cid:126) M R q j sin ( γ kj / (cid:26) − [sin θ j sin ( φ k − φ j )] ˆ θ k + [sin θ k cos θ j − cos θ k sin θ j cos( φ k − φ j )] ˆ φ k (cid:27) . (42)For each k = 1 , · · · ,
4, a detailed analysis yields the sim-ple result˙ r k = (cid:126) M R θ ˆ φ k = (cid:126) M R θ ˆ φ k , (43)where ∆ θ = π/ − θ . Since these four translational ve-locities are equal and in the positive azimuthal direction,the vortices all rotate together in a positive (counter-clockwise) sense around ˆ z , remaining coplanar. In appro-priate limits, Eq. (43) confirms our previous qualitativediscussion.
2. exchanged (antisymmetric) configuration + − + − Consider now a second coplanar case, the exchangedconfiguration, where we interchange the charge of thevortices 1 and 2 in Fig. 2, with q = − q = +1, asshown in Fig. 3. FIG. 3. Coplanar exchanged (antisymmetric) configurationfor θ = π/
4, including the small deviations δθ that providethe orbital motion around the equilibrium position. For simplicity, we start with θ (cid:28)
1, when the motionis intuitively clear. Vortices (13) form a vortex dipolenear the north pole, with vortices (24) similarly forminga vortex dipole near the south pole. Both dipoles movein the yz plane at fixed φ = π/ θ (cid:28) (cid:126) / (2 M Rθ ) and travels a distance 2 πR in onecomplete cycle. Hence the period is T = 4 πM R θ/ (cid:126) andthe corresponding angular frequency is ω = 2 πT = (cid:126) M R θ . (44)The other simple initial configuration is θ = π/ r k all vanish, yield-ing a static configuration of the four vortices. For smalldeviations of the initial positions with θ = π/ δθ , nu-merical studies (see Supplemental Material [13] ) showthat each vortex executes a closed elliptical orbit aroundits equilibrium static configuration, with positive vorticesmoving in the positive sense and the negative ones in thenegative sense, similar to the motion for the initial con-figuration θ (cid:28)
1. For δθ (cid:28)
1, the elliptical orbit hassemiminor and semimajor axes ≈ R δθ and ≈ R δθ along ˆ θ and ˆ φ directions, respectively. Using the nu-merical data for various values of δθ , we found that theanalytical formula ω ≈ (cid:126) M R δθ ) (45)provides a best fit to the numerical values of orbital fre-quency. For small δθ , this frequency reduces to ω ≈ (cid:126) / ( M R ), whereas for small θ (cid:28) δθ → − π/ ω ≈ (cid:126) / (2 M R θ ), as expected for the or-bital frequency of the dipoles in Eq. (44). The expressionin Eq. (45) may well be exact, but we have not found adetailed proof.For more general asymmetric θ configurations of thefour vortices, they will rapidly lose their initial copla-nar condition, and we are unable to predict their orbitsanalytically.Finally we briefly discuss a three-dimensional tetrahe-dral configuration, with the vortices at four symmetricsites on the sphere. In contrast to four coplanar vor-tices, the tetrahedron has only one possible configura-tion: each positive site will have one positive neighborand two negative neighbors, all at the same separations.Equation (42) shows that the tetrahedron is also a staticvortex configuration. Unlike the planar configurations,however, it is an unstable equilibrium, with complicateddynamics even for small perturbations. VI. ENERGY AND DYNAMICS OF TWOVORTEX DIPOLES
Section IV considered the energy of a general neutralsystem of vortices on the surface of a sphere, as given inEq. (36). The next Sec. V then studied the dynamics offour vortices with zero net vortex charge in some simplesymmetric configurations. Here, we examine the case oftwo small vortex dipoles, which is a different special con-figuration of four vortices with zero net vortex charge.For definiteness, we consider one dipole with charges q = − q = 1 and another dipole with q = − q = 1.Although Eq. (36) gives the total interaction energy ofthese two dipoles as a sum over six pairs of vortices, weinstead provide a more physical picture.As a first step, we focus on the dipole with vortices atangular positions Ω and Ω on the sphere. The totalstream function is χ (Ω) = χ (Ω , Ω ) − χ (Ω , Ω ) , (46)where Eq. (25) gives the stream function for each individ-ual vortex. For the present case of a small vortex dipolecentered at Ω , we write δ Ω = ( δθ , δφ ) = Ω − Ω and expand Eq. (46) to first order in the small angularseparations. Define the vortex dipole moment (a vector) p = Rδθ ˆ θ + R sin θ δφ ˆ φ , (47) which has the dimension of a length, with ˆ θ and ˆ φ theunit spherical-polar vectors at Ω on the surface of thesphere. It is not difficult to find χ dip (Ω) ≈ − p · R (ˆ r − ˆ r ) R (ˆ r − ˆ r ) = − p · ( r − r )( r − r ) , (48)where r = R ˆ r is the coordinate vector normal to thesurface of the sphere and we note that p · ˆ r = 0. Thisexpression is completely analogous to the electrostaticpotential arising from an electric dipole moment p , heresuitably modified for the two-dimensional character of avortex dipole.The interaction energy of this single vortex dipole fol-lows directly from Eq. (36) and the subsequent discussion E = 2 (cid:126) nπM χ = 2 (cid:126) nπM ln (cid:20) (cid:18) | γ | (cid:19)(cid:21) . (49)For a small dipole with | γ | (cid:28)
1, we find the simpleexpression E ≈ (cid:126) nπM ln ( | γ | ) = − (cid:126) nπM ln (cid:18) Rp (cid:19) , (50)where p = R | γ | is the separation between the twovortices (it is also the dipole moment). As a result, theenergy of the two separate dipoles is E d = 2 (cid:126) nπM ( χ + χ ) . (51)The interaction energy of the two dipoles is E dd = M n (cid:82) d r v · v , where v and v are the velocityfields of the two dipoles. Formally, we can use Eq. (46)and write E dd = (cid:126) nM (cid:90) d r ∇ χ · ∇ χ = (cid:126) nM (cid:90) d r ∇ ( χ − χ ) · ∇ ( χ − χ )= − π (cid:126) nM ( χ + χ − χ − χ ) . (52)The sum of Eqs. (51) and (52) precisely reproduces thesix terms inferred from Eq. (36).More physically, we can find χ and χ directly fromEq. (48). In this way, the familiar integration by partswith the first line of Eq. (52) gives an explicit expressionfor the interaction energy of two small vortex dipoles p and p (cid:48) at positions r and r (cid:48) : E dd = 2 π (cid:126) nM ( p · p (cid:48) ) ( r − r (cid:48) ) − p · ( r − r (cid:48) ) p (cid:48) · ( r − r (cid:48) )( r − r (cid:48) ) . (53)This rather complicated form is familiar from electro-statics and magnetostatics, suitably modified for thetwo dimensions relevant here. It varies inversely withthe squared separation of the dipoles and is highlyanisotropic through the dependence on each dipole’s ori-entation.We emphasize that Eq. (53) only applies for large sep-arations | r − r (cid:48) | (cid:29) pp (cid:48) and does not hold for two nearbydipoles. In such a case, we must rely on the generalexpression from Eq. (36), involving a sum over all sixdistinct pairs of vortices.It is interesting to consider the dynamics of two smallvortex dipoles on a sphere, say p i and p (cid:48) i . Whenthe two dipoles are well separated, the total energyis the sum of their individual dipole energies E tot = − ( π (cid:126) n/M ) ln( R /p i p (cid:48) i ), and the magnitudes p i and p (cid:48) i remain fixed as each moves along a great circle at speed (cid:126) /M p i and (cid:126) /M p (cid:48) i , respectivelyIf they approach each other, the dynamics becomescomplicated, requiring a detailed analysis. Neverthe-less, some general conclusions are possible. For twosmall nearby dipoles, they interact locally on the tan-gent plane. We can then invoke the conservation lawfrom Eq. (18), which now implies that the total dipolemoment is conserved along with the total energy. In anobvious notation for initial and final dipole moments, wehave p i + p (cid:48) i = p f + p (cid:48) f and the conservation of energyimplies that p i p (cid:48) i = p f p (cid:48) f . FIG. 4. Two vortex dipoles p i and p (cid:48) i move to the right onnearly parallel converging trajectories. Once they are suffi-ciently close, the individual vortices rearrange to form twonew vortex dipoles, with p f moving rapidly to the left and p (cid:48) f moving slowly to the right. The solid lines show calculatedorbits for the initial conditions. We already discussed a specific example of two dipoleson a sphere in connection with Fig. 3, when one dipolestarts at the north pole and the other starts at the southpole as shown for small dipoles with θ (cid:28)
1. Here weassume that p (cid:48) i = p i = 2 Rθ . After the interaction, thereconstructed dipoles move around the equator in oppo-site directions, and the conservation laws then show that p f = p (cid:48) f = 2 Rθ during this part of their orbits.A more unusual situation arises for two nearly paralleldipoles that gradually approach each other (see Fig. 4) in the tangent plane, labeled by axes x and y . Herethe initial dipole moments are p i ≈ p (cid:48) i ≈ p ˆ y , and theirmotion is principally along ˆ x (converging slowly towardthe x axis), with each having a small velocity compo-nent along ˆ y . After the interaction, the two inner vor-tices form a new smaller dipole p f = − p f ˆ y that movesto the left. In contrast, the two outer vortices forma new larger dipole p (cid:48) f = p (cid:48) f ˆ y that continues movingto the right. The conservation of dipole moment re-quires 2 p = p (cid:48) f − p f , and the conservation of energyshows that p = p f p (cid:48) f . Simple algebra yields the fi-nal values p (cid:48) f /p = √ p f /p = √ −
1. Asa result, we have a very asymmetric configuration with p (cid:48) f /p f = 3 + 2 √ ≈ .
83. These two new dipoles movein opposite directions around a single great circle at dif-ferent speeds, with v f /v (cid:48) f = 3 + 2 √ ≈ .
83, but theyeventually meet. After interacting a second time, theyseparate and move on nearby great circles toward theirinitial configuration (see Supplemental Material [13] fora video of the early evolution of two colliding dipoles).
VII. SUMMARY AND CONCLUSIONS
We presented the dynamics and stability of singlycharged vortices in a thin spherical “bubble” BEC. Thetopology of this system constrains the configuration ofvortices to an even number of vortices with total chargezero. We calculated the velocity fields and energies ofvarious configurations of vortices and dipoles, which al-low us to determine their dynamics on the surface of thesphere. In some cases, two initial small vortex dipolescan interact and exchange partners. The final reconfig-ured dipoles then separate and follow new trajectories.Proposed new experiments in microgravity conditions[14] means that our model of a thin “bubble” trappedBEC may be studied experimentally in the near future.In addition, recent experiments have studied a “bubble”trap potential in the presence of gravity [15, 16], and itmay soon be feasible to manipulate the trap rotation andthe atomic parameters to fix various initial conditions.We anticipate that our theoretical studies will be relevantfor any further studies on the evolution of vortex pairs ona sphere. We plan to provide theoretical explanations forthese experimental results, adapting our model to includethe effect of the gravitational potential [17].
VIII. ACKNOWLEDGMENTS
We are grateful to Pietro Massignan for helpfuldiscussions. This work was supported by the S˜aoPaulo Research Foundation (FAPESP) under Grant No.2013/07276-1. The authors also thank Coordena¸c˜ao deAperfei¸coamento de Pessoal de N´ıvel Superior (CAPES)for their financial support. [1] C. Pethick and H. Smith,
Bose-Einstein condensation inDilute Gases , 2nd ed. (Cambridge University Press, NewYork, 2008).[2] L.P. Pitaevskii and S. Stringari,
Bose-Einstein Con-densation and Superfluidity , 2nd ed. (Oxford UniversityPress, New York, 2016).[3] I. Bloch, J. Dalibard, and W. Zwerger, “Many-bodyphysics with ultracold gases,” Rev. Mod. Phys. , 885–964 (2008).[4] E.R. Elliott, M.C. Krutzik, J.R. Williams, R.J. Thomp-son, and D.C. Aveline, “Nasa (cid:48) s cold atom lab (cal): sys-tem development and ground test status,” npj Micro-gravity , 1–7 (2018).[5] N. Lundblad, R.A. Carollo, C. Lannert, M.J. Gold,X. Jiang, D. Paseltiner, N. Sergay, and D.C. Aveline,“Shell potentials for microgravity Bose-Einstein conden-sates,” npj Microgravity , 1–6 (2019).[6] S.J. Bereta, L. Madeira, V.S. Bagnato, and M.A. Cara-canhas, “Bose-Einstein condensation in spherically sym-metric traps,” American Journal of Physics , 924(2019).[7] A. Tononi and L. Salasnich, “Bose-Einstein condensationon the surface of a sphere,” Phys. Rev. Lett. , 160403(2019).[8] N. Guenther, P. Massignan, and A.L. Fetter, “Quantizedsuperfluid vortex dynamics on cylindrical surfaces andplanar annuli,” Phys, Rev. A , 063608 (2017).[9] P. Massignan and A.L. Fetter, “Superfluid vortex dynam-ics on planar sectors and cones,” Phys. Rev. A , 063602(2019). [10] A.M. Turner, V. Vitelli, and D.R. Nelson, “Vortices oncurved surfaces,” Rev. Mod. Phys. , 1301–1348 (2010).[11] H. Lamb, Hydrodynamics , 6th ed. (Dover Publications,New York, 1945).[12] E. Butkov,
Mathematical Physics (Addison-Wesley,1973).[13] See Supplemental Material at [URL will be inserted bypublisher] for complete simulations of (i) of the dynamicsof two vortex-dipoles; (ii) for a video displaying a simu-lation of the vortices’ orbits for small deviations of theirinitial positions; and (iii) for a video of the collision oftwo asymmetric dipoles (different speeds) with parame-ters set as in Fig. 4 of the main text.[14] D.C. Aveline, J.R. Williams, E.R Elliott, C. Dutenhoffer,J.R. Kellogg, J.M. Kohel, N.E. Lay, K. Oudrhiri, R.F.Shotwell, N. Yu, and R.J. Thompson, “Observation ofBose-Einstein condensates in an earth orbiting researchlab,” Nature , 193–197 (2020).[15] C. De Rossi, R. Dubessy, K. Merloti, M.G. de Herve,T. Badr, A. Perrin, L. Longchambon, and H. Perrin,“Probing superfluidity in a quasi two-dimensional Bosegas through its local dynamics,” New Journal of Physics , 062001 (2016).[16] Y. Guo, R. Dubessy, M.G. de Herve, A. Kumar, T. Badr,A. Perrin, L. Longchambon, and H. Perrin, “Supersonicrotation of a superfluid: A long-lived dynamical ring,”Phys. Rev. Lett. , 025301 (2020).[17] K. Sun, K. Padavi´c, F. Yang, S. Vishveshwara, andC. Lannert, “Static and dynamic properties of shell-shaped condensates,” Phys. Rev. A , 013609 (2018). SUPPLEMENTAL MATERIAL: SUPERFLUID VORTEX DYNAMICS ON A SPHERICAL FILM