Quantized superfluid vortex dynamics on cylindrical surfaces and planar annuli
Superfluid vortex dynamics on an infinite cylinder differs significantly from that on a plane. The requirement that a condensate wave function be single valued upon once encircling the cylinder means that such a single vortex cannot remain stationary. Instead, it acquires one of a series of quantized translational velocities around the circumference, the simplest being , with the mass of the superfluid particles and the radius of the cylinder. A generalization to a finite cylinder automatically includes these quantum-mechanical effects through the pairing of the single vortex and its image in either the top or bottom end of the surface. The dynamics of a single vortex on this surface provides a hydrodynamic analog of Laughlin pumping. The interaction energy for two vortices on an infinite cylinder is proportional to the classical stream function , and it crosses over from logarithmic to linear when the intervortex separation becomes larger than the cylinder radius. An Appendix summarizes the connection to an earlier study of Ho and Huang for one or more vortices on an infinite cylinder. A second Appendix reviews the topologically equivalent planar annulus, where such quantized vortex motion has no offset, but Laughlin pumping may be more accessible to experimental observation.
The dynamics of point vortices in an incompressible nonviscous fluid has been of great interest since the late 19th century (1). For example, given an initial vortex configuration, the subsequent motion obeys first-order equations of motion, which differs greatly from the usual second-order Newtonian equations describing point masses. In addition, the and coordinates of each vortex serve as canonical variables, analogous to and for a Newtonian point particle.
This description has found wide application to superfluid He which acts like an incompressible fluid for vortex motion much slower than the speed of sound m/s (2). Such superfluid systems involve a complex macroscopic condensate wave function , whose phase determines the superfluid velocity , where is the atomic mass. In this way, the quantum-mechanical phase becomes the velocity potential. The creation of dilute ultracold superfluid atomic Bose-Einstein condensates (BECs) in 1995 has subsequently stimulated many new applications of the same formalism (3); (4); (5).
Classical nonviscous irrotational and incompressible hydrodynamics describes well the dynamics of vortices in superfluid He, with the additional condition of quantized circulation (2). Although dilute ultracold superfluid BECs are compressible, local changes in the density become small in the Thomas-Fermi (TF) limit, which typically describes many important experiments (6). In this limit, the condition of current conservation for steady flow reduces to the condition of incompressibility . For such incompressible flow, the stream function provides an important alternative description of the superfluid flow. Specifically, for two-dimensional flow in the plane, the velocity becomes
where is the unit vector normal to the two-dimensional plane, following the right-hand rule.
For such irrotational incompressible flow in two dimensions (), the complex variable provides a natural framework for vortex dynamics. It is helpful to introduce a complex potential , with . For any such analytic function, the Cauchy-Riemann conditions yield the components of velocity:
These conditions give the compact representation of the hydrodynamic flow velocity components
in terms of the first derivative of the complex potential.
Note that the representation of in terms of the velocity potential ensures that the flow is irrotational (namely ), apart from singularities associated with the vortex cores. This applies to all superfluids in both two and three dimensions. In contrast, the representation in terms of the stream function ensures that the flow is incompressible with , for it can be rewritten as . This condition does not apply generally to all superfluids, but it can be very useful in many specific cases.
For flow in a plane, the stream function has the special advantage that takes a constant value along a streamline of the hydrodynamic flow. In addition, as shown below, the interaction energy between two vortices at and is directly proportional to , where is the intervortex separation, as discussed in (7).
All these results are familiar in the case of point vortices in a plane. For example, the complex potential for a positive unit vortex at the origin is , where and . Hence . It is not difficult to verify that and that the vorticity is singular, with . It follows from familiar vector identities that the stream function for a point vortex at the origin obeys an inhomogeneous equation with the vorticity as its source:
and is thus effectively a two-dimensional Coulomb Green’s function. In general, the stream function also satisfies various boundary conditions. Typically, boundaries break translational invariance, and the stream function depends symmetrically on the two variables: .
More recently, the behavior of singularities in various order parameters on curved surfaces has attracted great interest. The simplest such case is a superfluid vortex with a single complex order parameter, although liquid crystals present many more intricate examples (8); (9).
Here we focus on the dynamics of point vortices in a thin superfluid film on a cylindrical surface of radius . We start with an infinite cylinder in Sec. II and show that the identification of the velocity potential as the quantum-mechanical phase requires a single vortex to move uniformly around the cylinder with (in the simplest case) one of two specific quantized values.
In Sec. III we study the dynamics of two vortices on a cylinder, which is unexpectedly rich. Vortices with opposite signs move uniformly perpendicular to the relative vector , in the direction of the flow between them. This behavior is closely related to the quantized vortex velocity found in Sec. II. In contrast, two such vortices with the same sign maintain their centroid , displaying both bound orbits and unbounded orbits, in close analogy to the motion of a simple pendulum.
In Sec. IV, we evaluate the interaction energy of two vortices, relating it to the relevant stream function . This result allows us to re-express the dynamics of two or more vortices in terms of forces, including the Magnus force (7).
Section V considers a vortex on a finite cylinder of length , where the method of images provides an exact solution in terms of the first Jacobi function (10); (11). The resulting dynamics under the action of additional external rotation constitutes a direct hydrodynamic analog of the Laughlin pumping.
Ii Point vortex on an infinite cylinder
On the surface of a cylinder of radius , let represent the coordinate around the circumference and the unbounded coordinate along the cylinder’s axis. The unit vector is then the outward normal to the surface of the cylinder. For a thin superfluid film, the problem is apparently equivalent to the infinite plane with periodic repetitions of a strip of width along . As seen below, however, this classical picture violates the quantum-mechanical requirement of a single-valued condensate wave function on once encircling the cylinder. We find that a single vortex on an infinite cylinder must move around the cylinder with a set of quantized velocities.
Throughout this section we consider a single point vortex at the origin of the cylindrical surface (). Note that is just the azimuthal angle in cylindrical polar coordinates. With the usual complex notation and a function of this complex variable, the complex potential corresponds to a positive vortex at each zero of . In particular, Sec. 156 of (1) notes that the choice
represents a one-dimensional periodic array of positive vortices at positions , with .
At first sight, this complex potential should also represent a single superfluid vortex at the origin of an infinite cylinder. Note, however, that changes sign for . Consequently, the present velocity potential is not acceptable as the phase of a single-valued quantum-mechanical condensate wave function, because remains unchanged only for .
ii.1 Velocity potential for one vortex on a cylinder: classical hydrodynamics vs. quantized superfluidity
Here, we explore the source of this discrepancy by evaluating in detail the velocity potential
As expected, this reduces to for .
In the quantum interpretation, the velocity potential is also the phase of the condensate wave function, which leads to the following inconsistency: Note that increases by when , where . Hence the condensate wave function would be antiperiodic on once going around the cylinder. As seen below, the actual fluid velocity itself is indeed continuous and periodic, so that this complex potential is acceptable as a classical solution but not as a quantum mechanical one if the vortex itself remains at rest.
Consider the lines of constant phase. For a single vortex on a plane, these lines extend radially from the center of the vortex, rather like electric field lines from a two-dimensional point charge. On a cylinder, the periodicity means that half the phase lines go upward and half go downward (see Fig. 1 top row). The net change in phase on going once around the circumference of the cylinder is , depending on the sign of .
An illuminating way to think about this question focuses on a vorticity flux of associated with a singly quantized vortex (measured in units of ). For a positive stationary vortex with charge at the origin of the surface, the flux comes from inside the cylinder and emerges radially outward along through the center of the vortex. In the present case of a vortex at rest, the flux comes symmetrically with flowing downward from and flowing upward from , as is clear from Fig. 1 top row. Physically, the vortex on the surface can be considered the end of a vortex line in a superfluid filling the interior of the cylinder. Clearly the vortex line must come wholly from one end or the other, so that such a flux splitting is not possible. We’ll see that a moving vortex indeed satisfies these conditions (for special values of the motion).
More generally, the flow velocity is . Hence adding a uniform flow in the direction will alter the phase gradient and thus alter the phase change in going around the cylinder. Specifically, consider the more general complex potential for a vortex located at on the cylinder
where is a dimensionless real constant. The additional uniform velocity is . As a result, the previous expression for acquires the additional term . The additional net phase change on going once around the cylinder is . Thus merely adjusting the value can yield any desired phase change; for example, the choice would give zero total phase change for and total phase change for . In essence, this behavior is simply the hydrodynamic analog of the familiar Bohm-Aharonov effect. Alternatively, it represents a sort of gauge freedom to alter the phase.
Figure 1(middle and bottom) shows the phase pattern for a single positive point vortex with additional flow velocity . These choices ensure that the lines of constant phase all collect into the lower (upper) part of the cylinder for leaving the fluid asymptotically at rest in the upper (lower) end of the cylinder. In these special cases, the net change in phase upon once encircling the cylinder now will be or , merely by counting the phase lines crossing the path. Note that the two solutions may be mapped onto each other by a rotation of 180, which effectively interchanges the two ends of the cylinder. Evidently, quantum mechanics requires that the phase lines from a single vortex on a cylinder must flow to in multiples of to ensure that the condensate wave function is single valued.
To clarify these questions, consider a single positive vortex on a cylinder at the origin. Integrate in a positive sense the hydrodynamic fluid velocity around a closed rectangular contour (namely the circulation) with vertical sides at . Clearly, the contributions of these vertical sides cancel because of the periodicity of the flow velocity. In addition, the circulation integral will be if the horizontal parts do not enclose the vortex, and if the horizontal parts do enclose the vortex. For (namely no external flow), the top and bottom parts each contribute to the dimensionless circulation. If , the contributions of the top and bottom parts each shift linearly in such a way that the net circulation is unchanged. In particular, for with an integer that specifies the quantum of circulation on the horizontal path above the vortex, each horizontal part contributes a multiple of .
The additional uniform flow means that the vortex now moves uniformly around the cylinder with quantized velocity, required to satisfy the quantum-mechanical condition that the condensate wave function be single valued. For any complex velocity function that contains a vortex at some point , the following limit gives the complex velocity of that vortex as
In the present case, this expression simply reproduces the previous result that the vortex moves with the local uniform flow velocity. For with no applied flow, any particular vortex remains at rest, either from this mathematical treatment or more physically by noting that the induced flow at any particular vortex cancels because of the left-right symmetry of the one-dimensional periodic array.
Focus on the two simplest cases with , in which case the flow vanishes as [see Fig. 1 (middle and bottom)]. The corresponding complex potential becomes
Apart from the additive constant, this complex potential is just that considered by Ho and Huang (12) as the two possible conformal transformations from a plane to a cylinder (corresponding to the choice ). This connection clarifies the special role of the two values . We consider this point in detail in Appendix A.
ii.2 Stream function for one vortex on a cylinder
As noted in Sec. I, the stream function provides a clear picture of the hydrodynamic flow through its contour plots. In the present case, is a little intricate, which illustrates a principal advantage of this complex formalism. Specifically, the stream function for one vortex on the surface of a cylinder of radius is
Familiar complex trigonometric identities give
where each form is useful in different contexts.
This stream function has the proper periodicity in and reduces to the result for a single vortex at the origin when . In contrast, for , the stream function has the very different and asymmetric behavior , independent of . Correspondingly, in this limit, and the hydrodynamic flow velocity reduces to a uniform flow (from ) plus an antisymmetric uniform flow: .
To understand this asymptotic behavior, consider the induced flow of the corresponding infinite one-dimensional array of positive vortices in the plane (for simplicity, take ). Close to each vortex, the flow circulates around that vortex in the positive sense, but for , the combined flow instead resembles that of a “vortex sheet” (see Sec. 151 of (1)). Specifically, a vortex sheet arises when the transverse velocity has a discontinuity. For example, consider the antisymmetric flow field . Here, the vorticity is , which follows either by direct differentiation or with Stokes’s theorem. In particular, the asymptotic flow from a periodic array of positive unit vortices along the axis with spacing approximates a vortex sheet with .
Evidently, the hydrodynamic flow for a single vortex on a cylinder is considerably more complicated than that for a single vortex in the plane. Note that the hydrodynamic flow arises from an analytic function , so that and both satisfy Laplace’s equation (apart from the local singularity associated with the vortex). Such a two-dimensional function cannot be periodic in both directions. Instead, the sum of the curvatures associated with and must vanish, so that the solution necessarily decays exponentially for large (in this case, to a nonzero constant), as seen here from the hyperbolic functions in and .
In Fig. 1(top middle) we show a contour plot of the stream function for a single vortex on the surface of a cylinder with . Lamb (1) has a similar figure in Sec. 156. As expected, the streamlines on the cylindrical surface exhibit both the periodicity in and the exponential decay of the motion in the direction with the characteristic length . Note the occurrence of two topologically different types of trajectories. This phase plot resembles that of a simple pendulum, which reflects the similar canonical roles of for a vortex and for a pendulum. Here, the separatrix is parametrized by , namely by . Inside the closed curves of the separatrix, the flow circulates around the vortex and its periodic images. Outside the separatrix, the flow continues in one direction, like a pendulum with large energy. In the present hydrodynamic context, streamlines inside the separatrix correspond to “libration” of the pendulum and encircle the vortex with zero winding number around the cylinder. Otherwise, streamlines correspond to “rotation” of the pendulum. They do not encircle the vortex but have winding number around the cylinder, depending on the value of .
With standard trigonometric identities, it is not hard to find the hydrodynamic flow velocity induced by the single positive vortex at the origin on the surface of a cylinder (here, for simplicity, we take ):
The resulting flow pattern is shown in Fig. 1(top right). For and , the hydrodynamic flow field reduces to the familiar expression , which falls off inversely with the distance from the vortex in all directions. For large on a cylinder, in contrast, the flow velocity reduces to a constant . In this region the periodicity around the cylinder dominates the flow pattern, rather than the single vortex.
Iii Multiple vortices on a cylinder
It is now straightforward to generalize the previous discussion to the case of vortices on an infinite cylinder, each located at complex position and with charge (). As in electrostatics, the complex potential for multiple vortices on the cylindrical surface is simply the sum of the complex potentials of the individual vortices, always with the option of adding a uniform flow of the form . For an even number of vortices, however, this term is unnecessary.
The corresponding velocity potential and stream function are the imaginary and real parts of and need not be given explicitly.
iii.1 Induced motion of two vortices on a cylinder
As noted at the end of Sec. II.1, in the absence of external flow a single vortex on a cylinder remains stationary. Consequently, the motion of each vortex arises only from the presence of the other vortex. Equation (8) immediately gives the complex velocity of the first vortex
iii.2 Two vortices with opposite signs (vortex dipole)
When and , the vortex dipole moves with no internal rotation (so that and remain constant, simplifying the subsequent dynamics). Furthermore, the centroid moves with uniform translational velocity
at fixed and . Several limits are of interest:
If and , then the translational velocity is the same as that for a vortex dipole on a plane:
Detailed analysis confirms that the vortex dipole moves in the direction of the flow between their centers.
If , then the ratio of hyperbolic functions leaves only the component, with . This value reflects the hydrodynamic flow from a periodic array of vortices, as mentioned near the end of Sec. II.2. In this limit, the vortex dipole will circle the cylinder in a time . Note that this motion is the same as that induced for one vortex with , discussed in Sec. II.1. Hence the additional induced motion of a single vortex on an infinite cylinder can alternatively be thought to arise from a phantom negative vortex placed at , corresponding to .
As an example, Figure 2 shows the hydrodynamic streamlines for various orientations of the relative position . Specifically, we plot the corresponding stream function for three typical cases: with motion along , , with motion along , and with motion along .
iii.3 Two vortices with same signs
In the case of two positive vortices (), it follows immediately that vanishes, so that the centroid of the two vortices remains fixed. In contrast, the relative vector obeys the nontrivial equation of motion
Unlike the case of opposite charges (where remains fixed), the relative vector now becomes time dependent. In fact, the last form given above shows that the motion of the two positive vortices precisely follows the hydrodynamic flow velocity of a single vortex . Hence the streamlines in Fig. 1 (central column) completely characterize the motion. Several cases are of interest.
If , then the two positive vortices simply circle in the positive sense around their common center . The curvature of the surface is irrelevant and the motion is the same as on a flat plane.
If , the two vortices execute closed orbits in the positive sense around their common center , but the general orbits are not circular [by definition, they remain inside the separatrix in Fig. 1 (top row, central column)].
If , the two vortices move in opposite directions, executing periodic closed orbits around the cylinder with unit winding number. The upper vortex moves monotonically to the left and the lower vortex moves monotonically to the right, as seen in Fig. 1 (top row, central column) (they remain outside the separatrix).
For relatively large , an expansion of the above equation yields the approximate form
Asymptotically for , the variable varies linearly in time. The leading correction to this uniform horizontal motion is a small periodic modulation for both and components.
Iv Energy of two vortices
The stream function provides the hydrodynamic flow velocity through Eq. (1), which is its usual role. As shown below, however, the stream function also determines the interaction energy between two point vortices through Eq. (23). The analogous electrostatic situation is familiar in that the electrostatic potential gives both the electric field from a single point charge and the interaction energy of two point charges. For electrostatics, this connection follows directly as the work done to bring the second charge in from infinity. For vortices, however, such an argument is less clear, since vortices do not act like Newtonian particles and obey first-order equations of motion. Hence we present a straightforward analysis that gives the interaction energy of two vortices by integrating the kinetic-energy density, which is proportional to the squared velocity field. This approach is clearly analogous to finding the electrostatic interaction energy of two charges by integrating the electrostatic-energy density, which is proportional to the squared electrostatic field.
In the present model, the total energy of two vortices at () with unit charge is the spatial integral of the kinetic-energy density
over the surface of the cylinder. Here, is the hydrodynamic velocity field of a single positive unit vortex at the origin, is the two-dimensional number density, and is the atomic mass.
iv.1 Interaction energy
As noted in Secs. II.1 and II.2, for large , the asymptotic velocity field of a single vortex on a cylinder is uniform. Hence the kinetic energy of any single vortex diverges linearly as the upper and lower integration boundaries on the cylinder become large (namely ). As a result, each term in the above kinetic energy of two vortices on a cylinder separately diverges. The only case with a finite total kinetic energy is the vortex dipole with (say) and , since the two asymptotic hydrodynamic velocity flow fields then cancel.
It is convenient to use the stream function to characterize the local fluid velocity of the th vortex: , where [compare Eq. (1)]. The operation simply rotates the following vector through and we find
We follow de Gennes’s argument for type-II superconductors (13), but the analysis is also familiar from classical electrostatics. Here, the two-dimensional surface integral runs over the region and , where .
The first term above involves the divergence of the total derivative , and the divergence theorem reduces it to an integral on the boundary with outward unit normals. The contributions from the vertical parts at cancel because the integrand is periodic with period . In general, the contributions from the horizontal parts at separately diverge linearly, except for the special case of a vortex dipole with . The relevant quantity is for large . Equation (II.2) gives (here we take since the system is neutral)
where the corrections are exponentially small for large . It is now clear that each horizontal contribution vanishes for the present case of a vortex dipole, reflecting the overall charge neutrality.
It remains to evaluate the second line of Eq. (21). We already noted that , and the interaction energy (the terms involving the cross product of and ) thus becomes
for general choice of . The dynamics of two vortices involves gradient operations like , so that any divergent constant becomes irrelevant (alternatively, we can redefine the zero of the energy). As in Eq. (7) of (7), it is convenient to take out a factor , writing
which properly reduces to for small intervortex separation.
iv.2 Self energy of one vortex
Equation (21) also contains two self energy terms, one for each vortex. Consider a single vortex at the origin with self energy
A heuristic approach for the self energy terms (those involving ) in Eq. (21) is to cut off the singularity at the small core radius , which gives
The finite total energy of a vortex dipole is simply the sum of the interaction energy and the two self energies
Note that this total vortex energy reduces to the familiar for small . Otherwise it has a very different form and grows linearly for (see Fig. 3). This interaction energy was already discussed in previous studies of Berezinskii-Kosterlitz-Thouless behavior for a thin cylindrical film (14), and of vortex dipoles on capped cylinders (9).
This analysis holds whenever a stream function describes the flow, even in the presence of boundaries when involves two separate variables. Here, it yields the general result
augmented by the cutoff at when .
iv.3 Vortex motion as response to applied force
We can interpret the quantity as the force that vortex 2 exerts on vortex 1, and similarly with , where the last relation follows because depends only on the difference of the coordinates.
In this way, the dynamical equations take the intuitive form (see Sec. III of (7))
Hence a vortex moves perpendicular to the applied force, which is often called the Magnus effect. Equivalently, we can introduce the “Magnus force” , and the dynamical equation then becomes . These equations concisely express two-dimensional vortex dynamics in a general form, applying not only to motion on a plane but also on a cylinder.
iv.4 Energy of multiple vortex dipoles
As seen in Sec. III, the stream function for a set of vortices on an infinite cylinder is the sum of individual terms , where we assume is even. The total kinetic energy of the vortices is proportional to over the area of the cylinder. This behavior is completely analogous to the electrostatic energy for two-dimensional point charges on a cylindrical surface, since the electrostatic energy is proportional to the integral , and is the (negative) gradient of the electrostatic potential . Furthermore, the total electrostatic potential is a sum of contributions from each charge, like the similar structure of the total . Finally, Eq. (4) shows that the stream function obeys Poisson’s equation with each vortex as a source, in complete analogy to the electrostatic potential which also obeys Poisson’s equation with the point charges as sources.
Thus, by analogy with two-dimensional electrostatics, the energy of multiple pairs of point vortices on the infinite cylinder follows immediately as the sum over all pairs plus the sum over all self energies
If the system is overall neutral, then the total energy is finite; otherwise, there are divergent constant terms that do not affect the dynamics of individual vortices. Similar divergences appear in two-dimensional electrostatics unless the total electric charge vanishes.
V Single vortex on a cylinder of finite length
As seen in the previous sections, the complex potential generated by a single positive vortex located at the origin of a cylinder with radius and of infinite length, is
The corresponding solution on a cylinder with finite length (with ) follows with the method of images. Consider a physical vortex located at with . Reflect the potential of the unbounded solution along the planes and and reverse the charge of successive image vortices. This procedure creates an infinite set of positive vortices at positions and negative vortices at . We find
where , , and .
Examine the infinite product in Eq. (V) in detail:
where . Here, denotes the first Jacobi function, defined by either its product form or its series form (10)
This function has simple zeros at the complex points , where and is a complex number with positive imaginary part. In addition, the parameter obeys the condition . Here, and hence , as noted above.
The final complex potential for a vortex located at on a cylinder of length and radius has the relatively simple analytic form
The first Jacobi theta function changes sign when , immediately proving that the phase of the wave function changes by integer multiples of when . In particular, the integral , computed at fixed , equals 0 above the vortex , and below (, as is clear from Fig. 4 (left). Hence, the complex potential always generates “quantum-mechanically acceptable” solutions that move uniformly around the cylinder.
By construction, the fluid is basically at rest above the vortex, and in motion below it. This result may be understood by noting that the original vortex and its image below the bottom end of the cylinder replicate a vortex dipole located at . In this basic “building block”, the fluid flow is largely confined to the region between the two vortices and vanishes at large distances from the line (or domain wall) joining the two vortices.
Note furthermore that is manifestly anti-symmetric around a vertical axis passing through the vortex core [namely, ], so that its line integral along a circumference (at fixed ) vanishes. As a consequence, the angular momentum per particle on the cylinder is simply . If angular momentum were to be “pumped” at a constant (slow) rate into the system (namely, if the cylinder were to be spun with a linearly increasing rotation frequency, or if an increasing synthetic flux pierced the surface of the cylinder, as discussed in Ref. (15)), a vortex would enter the lower rim of the cylinder and progressively spiral up the cylinder. Once the vortex reaches the upper rim and leaves the cylinder, the angular momentum per particle would increase by exactly . This mechanism is a direct hydrodynamic analog of the Laughlin pumping (16).
v.1 Velocity of the vortex core
The velocity of the vortex core follows from Eq. (8),
where indicates the derivative of with respect to its variable . This function is purely imaginary, indicating that the vortex moves solely along the direction, and its velocity diverges as it approaches either end of the cylinder (namely, one of the image charges), as shown in Fig. 5. If the vortex is located at the middle of the cylinder at , one may use the property (valid for generic real ) to show that it moves uniformly around the cylinder with speed
as seen in Fig. 5. Here the rightward motion arises because we chose to pair the vortex with its image in the lower boundary of the cylinder. Had we instead used the image in the upper boundary at