Velocity of vortices in inhomogeneous Bose–Einstein condensates

Abstract

We derive, from the Gross–Pitaevskii equation, an exact expression for the velocity of any vortex in a Bose–Einstein condensate, in equilibrium or not, in terms of the condensate wave function at the center of the vortex. In general, the vortex velocity is a sum of the local superfluid velocity, plus a correction related to the density gradient near the vortex. A consequence is that in rapidly rotating, harmonically trapped Bose–Einstein condensates, unlike in the usual situation in slowly rotating condensates and in hydrodynamics, vortices do not move with the local fluid velocity. We indicate how Kelvin’s conservation of circulation theorem is compatible with the velocity of the vortex center being different from the local fluid velocity. Finally, we derive an exact wave function for a single vortex near the rotation axis in a weakly interacting system, from which we derive the vortex precession rate.

Vortices in rapidly rotating trapped Bose–Einstein condensates (1–4) do not move with the local fluid flow, despite Lord Kelvin’s circulation theorem, which implies that a vortex in a perfect fluid generally cannot escape from a contour comoving with the fluid (5, 6). To understand how vortices do in fact move in rapidly rotating condensates, we derive here an exact expression for the velocity of a vortex, applicable both to equilibrium and nonequilibrium situations involving arbitrary numbers of vortices, through analyzing the motion of the vortex singularity directly from the Gross–Pitaevskii (GP) equation (Eq. 2 below). This differential equation, a generalization of the Schrödinger equation, determines the time dependence of the order parameter Ψ (the condensate wave function) of the condensed system; its use in studying vortex motion was pioneered by Fetter (7). The great advantage of the GP equation, as opposed to the hydrodynamic equations of a perfect fluid, is that it provides a detailed model for the vortex core and therefore may be used in the regime in which the vortex core size is comparable with or greater than other length scales in the problem.

We elucidate through this approach how the velocity of the center of a vortex and the local fluid velocity differ when the density of the condensate varies sufficiently rapidly in the neighborhood of a vortex: the situation in rapidly rotating condensates. Furthermore, we show how the difference between the vortex velocity and the fluid flow in such cases is compatible with Kelvin’s theorem, a result with more general applications in hydrodynamics.

Velocity of a Vortex

The difference between the local fluid velocity and the velocity of the center of the vortex is brought out very clearly by the simple example of a single off-center vortex at radius b in a two-dimensional (2D) harmonic trap, V(r⃗), of frequency ω, in the limit in which the interparticle interaction is negligible. Such a system is described by a wave function, Ψ, that is a linear superposition of the (s-wave) oscillator ground state and a p-wave eigenstate of the trapping potential. At time t = 0, Ψ(ζ, ζ*) ∼ (ζ − b)e^−|ζ|2/2d2, in the usual complex notation in which ζ = x + iy; here, d = $\sqrt{\hslash /m\omega }$ is the oscillator length, and m is the particle mass. Because the oscillator ground state has energy ℏω and the p-state has energy 2ℏω, the time-dependent wave function is

At time t the vortex is located at ζ = be^iωt, and thus its center precesses in the positive sense about the origin at the trap frequency. On the other hand, the fluid velocity, given by (ℏ/m)∇φ, where φ is the phase of Ψ, equals ℏ/mρ around the vortex, where ρ is the distance from the vortex. There is no background fluid flow; the only flow is that due to the vortex itself, and yet the vortex precesses with frequency ω. A naive application of Kelvin’s theorem would suggest that the vortex should be stationary.

We turn now to a more general analysis of the motion of singly quantized vortices in two dimensions, as described by the time-dependent GP equation

where V(r⃗) is the transverse trapping potential, and g₂ is the 2D coupling constant. We normalize Ψ by ∫d²r|Ψ²| = N, where N is the total number of particles. For a system that is uniform in the axial direction, g₂ = 4πℏ²a_s/mZ, where a_s is the s-wave interatomic scattering length, and Z is the height of the system. Given Ψ, the left side evaluated at the position of the vortex tells us the velocity of the vortex position. Thus, the instantaneous velocity of the vortex center depends only on the value of ∇²Ψ at the vortex position. The simplest case is that of a singly quantized cylindrically symmetric vortex at position ζ_i in a spatially uniform system, which is described by a wave function Ψ(ζ), ∼(ζ − ζ_i) close to the vortex. More generally, the wave function of an asymmetric vortex, e.g., in an elliptic container, can include a term ∼(ζ − ζ_i)* as well. Thus, to evaluate the GP equation at the vortex position, we can write the wave function in the neighborhood of a singly quantized vortex at position ζ_i, without loss of generality, as

where |α| < 1 for a vortex with positive circulation, and Q(ζ, ζ*, t) and φ_b(ζ, ζ*, t) are real, smooth functions at ζ = ζ_i(t). The background phase, φ_b, is that of the superfluid with the singular contribution from the vortex at ζ_i removed, whereas the function Q describes the background density variation at the vortex. The background fluid velocity is v_b ≡ v_bx + iv_by = 2(ℏ/m)∂φ_b/∂ζ*. Then the GP equation implies

This equation shows how the instantaneous velocity of the vortex center depends only on the slope and curvature of the wave function in the neighborhood of the vortex core; the velocity depends on the trapping potential, V, and the interaction strength, g₂, only insofar as they affect the gradients of Q and φ_b at the vortex position.

After straightforward algebra we find that the velocity, u⃗_i, of the center of the vortex at ζ_i is, in Cartesian coordinates,

where λ = (1 − α)/(1 + α), and all quantities on the right are evaluated at x_i, y_i. For α = 0

The vortex velocity is the sum of the local fluid velocity, v⃗_b, plus a correction, ∼∇Q, of order (ℏ/m)∇n_b, where n_b ∼ e^−2Q describes the background density in the vortex core region. This correction is nonnegligible if the background density varies significantly over the vortex core. In the simple example above, v_b = 0 and the entire vortex velocity arises from the density gradient. As this calculation illustrates, vortices do not in general move with the local fluid velocity generated, e.g., by other vortices in the system. For the simple example in Eq. 1, φ_b = 0, whereas ∇Q = r⃗/d², and u⃗_i = ω(ẑ × r⃗_i). We note that these results hold for generalizations of the GP equation in which the interaction energy per particle is given by a local function of the density, as in hydrodynamics.

How Kelvin’s Theorem Is Satisfied

How can Eq. 5 be reconciled with Kelvin’s theorem, that the circulation, ∮_𝒞v⃗·ds⃗ around a contour, 𝒞, moving with the local fluid velocity, v⃗(r), is conserved in time? (Here ds⃗ is the line element.) Kelvin’s theorem applies to the GP equation, as well as the Schrödinger equation, because the flow described by these equations is potential. [The proof of Kelvin’s theorem for the GP equation is given by Damski and Sacha (8). As stressed in ref. 8 and earlier in ref. 9, the conditions for the proof of the theorem do not apply should the velocity be singular on the contour. This situation does not arise in the cases we consider here.] Because the theorem implies that a vortex cannot escape from a comoving contour, no matter how small, the vortex would appear to be constrained to move with the background local fluid velocity. However, as is illustrated in Fig. 1 for an off-center vortex in the harmonic oscillator potential, comoving contours do not necessarily remain regular in time, but rather, because of the difference between the fluid motion about the vortex and the motion of the vortex center, can become highly distorted. Although in this example, the flow velocity calculated from Eq. 1 is always circular about the vortex center, the density gradient term in the vortex velocity (Eq. 6), which makes the vortex move with respect to the background flow, causes a mismatch between the motion of the contour and the motion |v| = bω of the vortex center. An initially circular contour around the vortex begins to develop a lobe toward larger radii from the center. With time the lobe becomes extended and begins to wrap around. In Fig. 1, we show the fate, calculated numerically, of an initially circular contour centered on the vortex, at successive times within a quarter period, π/2ω. The standard argument that the vortex moves with the local fluid velocity assumes that the average velocity on the contour equals the fluid velocity at the center of the vortex. As this calculation indicates, it is not possible in general to deduce the motion of the vortex from Kelvin’s theorem when the average velocity on the contour is no longer simply related to the vortex motion.

Fig. 1.

Contour comoving with the fluid, surrounding a single off-center vortex in a harmonic trap. The contour is shown in the stationary lab frame at successive equally spaced times, a–d, within one-quarter of an oscillation period, π/2ω. Distances are measured in units of the oscillator length, d. The vortex is at radius d, and the initial radius of the contour is 0.3d. Note the sharp fold of the contour in d.

Applications

We now apply the result Eq. 5 to discuss more general vortex motion in detail in two limiting cases. The first is when the density scale height R is large compared with the vortex core size, determined in this case by the healing length, ξ; such a situation arises, e.g., in a system slowly rotating at angular frequency Ω, when the interaction energy, ∼g₂n (where n is the mean 2D density), dominates the rotational energy, ℏΩ, a strongly interacting regime in this sense. The second is the weakly interacting rapidly rotating lowest Landau level (LLL) limit in a harmonic trapping potential, where g₂n ≪ ℏΩ. In the former case a vortex does move with the local fluid velocity, but as the harmonic oscillator example indicates, it does not in the latter case. The motion of vortices in the limit of large g₂n may be explicitly calculated by the method of asymptotic matching (10–14), which we recall here. In this approach, one solves the GP equation near the vortex by expanding the wave function about the solution for a uniform system to first order in the gradient of the potential. In the far field, at distances large compared with ξ, the velocity is determined by solving the continuity equation with the density taken to be that in the absence of the vortex. Matching of the inner and outer solutions in the region where the radial coordinate ρ, measured with respect to the position of the vortex, is in the range ξ ≪ ρ ≪ R, gives the vortex velocity and the parameters of the core wave function. The solution near the vortex core, which enters Eq. 5, has the form (10–13)

where Ψ₀ is the vortex solution in a uniform medium, η and χ are real functions of ρ, and the azimuthal angle θ about the center of the vortex is zero in the outward direction (opposite to the background density gradient near the vortex). Comparing this structure with the general form (Eq. 3), we see that α vanishes in this limit, and the corrections due to asymmetry of the vortex core are negligible; the background phase is φ_b = η(ρ)sinθ. By using the analysis of refs. 10–13, we find χ ∼ ρ⁴ at small ρ, and η ∼ ρ. In addition, for small ρ, Q = −χ cos θ/|Ψ₀| ∼ ρ³ cos θ. Thus, as we see from Eq. 6, the only contribution to the vortex velocity comes from the background phase of the wave function; the vortex moves with the background flow velocity. The background fluid velocity, found from the far-field solution (12, 13, 15, 16), is given to logarithmic accuracy by

in the region ρ ≫ ξ, whereas in the core region, ρ/R should be replaced by ξ/R. Here, n is the smooth (Thomas–Fermi) density outside the vortex core, given by ∇n = −∇V(r)/g₂. The flow velocity is induced by the vortex as a consequence of the density gradient in the system.

In the rapidly rotating limit, on the other hand, in which the size of the vortex core is comparable with other lengths in the problem, vortices do not move with the background flow. In a harmonic trap, the wave function in this limit is, to a first approximation, a linear superposition of LLL in the Coriolis force (17, 18), Ψ ∼ ∏_j(ζ − ζ_j)e^−|ζ|2/2d2. Thus, α = 0, and the velocity of the individual vortices, Eq. 4, is given by

i.e., u⃗_i = ωẑ × r⃗_i. In other words, each vortex precesses at the trap frequency, independent of the flow from other vortices! This result implies immediately that in any motion of the vortices other than precession at the trap frequency, as for example in Tkachenko modes in the rapidly rotating limit (1–4), higher Landau levels play a crucial role in the wave function.

As a first step in describing the effects of interactions on vortex motion, we consider the problem of steady-state precession, in which there exists a frame rotating at angular velocity Ω in which the system is stationary. The problem then is to determine Ω as a function of the interaction strength. In general, Ω = ∂〈H〉/∂〈L〉, where L is the z component of the angular momentum. In a system composed primarily of LLLs, 〈H〉 = ω(〈L〉 + N) + 〈H_int〉, so that Ω − ω = ∂〈H_int〉/∂〈L〉. As the angular momentum of the system increases, the particles spread out, lowering the average density; thus, the derivative of the interaction energy with respect to the angular momentum is negative, implying that Ω < ω. The first deviation of Ω from ω is of order g₂.

Instead of calculating Ω from ∂〈H〉/∂〈L〉, we discuss the precession of a single vortex [an effect measured, in the small core regime, by Anderson et al. (19)] directly in terms of Eq. 5. Explicitly, we calculate the exact wave function of a single vortex to order b and g₂, and from this calculation, we show how the known vortex precession rate emerges for a vortex at small distance, b, from the origin, to first order in the coupling strength (20). (We set ℏ = 1 in this section). In the frame rotating at Ω, the GP equation assumes the form

where H₀ is the 2D oscillator Hamiltonian, l̂ = (ζ∂/∂ζ − ζ*∂/∂ζ*) is the angular momentum operator, and μ is the chemical potential in the rotating frame (equal to ω in the absence of interactions). Because Ω − ω is first order in g₂, as is μ − ω, it follows from Eq. 10 that the interaction term mixes LLL components of order g₂⁰ in Ψ for all angular momenta ν. In the sense of quantum-mechanical perturbation theory, whereas the interaction is of order g₂, the splitting of the LLL and thus energy denominators are also of order g₂, leading to g₂⁰ corrections.

To calculate the vortex velocity from Eq. 4, or equivalently, Eq. 5, one also must include the higher Landau level contributions to Ψ explicitly; in general, the exact wave function describing a vortex at position ζ₀ in terms of Landau levels is

where the normalized Landau level wave functions are χ_νσ(ζ) = P_νσ(ζ, ζ*)e^−|ζ|2/2d2, with P_νσ a polynomial, ν = 0, ±1, ±2, …, the angular momentum, and σ = 0, 1, 2, …, the radial quantum number. The state χ_νσ(ζ) is an eigenstate of H₀ with energy (2σ + |ν| + 1)ω.

The GP equation for a wave function stationary in the rotating frame, evaluated at the vortex position ζ = b, is

With Ψ given by Eq. 11, this equation reduces to

with all terms evaluated at b. The ω terms arise from ∂²e^{−|ζ|2/2d2/∂ζ∂ζ*}. Because Ω − ω is of order g₂, we need to keep only LLL terms, σ = 0 in the sum on the left, and of these only the ν = 1 term is of order b⁰. On the right side, only higher Landau levels enter, because the P_ν0 are independent of ζ*.

The degree of mixing of higher ν in the wave function of a single vortex depends on its distance b from the origin. To leading order in b, C_ν0 ∼ b^ν−1 for ν ≥ 1, whereas C₀₀ ∼ b. This relation is readily proven recursively: the interaction mixes terms χ_ν0, χ₁₀, and χ*₀₀ to give a ν + 1 component; because C₀₀ begins at order b, then if C_ν0 is of order b^ν−1, C_ν+1,0 begins at order b^ν; other mixings that give a ν + 1 component lead to terms at least of order b^ν. An immediate consequence of this mixing is that even if one starts in the absence of interactions with a single vortex at b, the wave function in the presence of interactions describes further vortices at large distances, ∼d²/b; in general, the LLL part of Ψ is proportional to a polynomial in ζ of the form ζf(bζ/d²) − b, to leading order in b, where f(0) = 1, so that in addition to the zero at ζ = b, the polynomial has further zeroes at |ζ| ∼ d²/b. However, the particle density is negligible at such distances.

As an explicit illustration, let us start with a single vortex at position b described in the absence of interactions by

then with interactions the LLL wave function, even to order g₂⁰, includes levels (ν, σ = 0) with amplitude ∼b^ν−1 for ν ≥ 1. To first order in b, we must include as well the ν = 2, σ = 0 LLL. Furthermore, the interactions to first order in g₂ mix in higher Landau levels with ν = 0, 1, 2. However, the ν = 2 term contributes to the equation for the motion of the vortex only in order b² and thus can be neglected. Computing the coefficients C_νσ by expansion of the GP equation (10), one readily finds the exact wave function of a single vortex at position b, to first order in b and g₂

where C ∼ $\sqrt{N/\pi }$ is a normalization constant. The term ∼ζ(|ζ/d|² − 2) is the ν = 1, σ = 1 contribution. [We include the higher-order b² term only to indicate that Ψ(ζ) has a zero at ζ = b.]

Noting that α = 0 here, we substitute this wave function into Eq. 4 (or, equivalently, into Eq. 6) to derive the precession rate to first order in g₂ and lowest order in b (20)

This result illustrates the role of higher Landau levels in producing deviations of the precession frequency from the trap frequency. Although the background velocity, v⃗_b, vanishes in the noninteracting case, we see in the presence of interactions, from Eq. 6, that v⃗_b = bωŷ/2, for a vortex centered at (x = b, y = 0), whereas the second term in Eq. 6 equals (1 − mg₂N/4πℏ²)bωŷ/2. The precessional motion arises in part from the background flow and in part from the density gradient term.

In addition, the chemical potential shift is given by μ − ℏω = 3g₂N/8πd², to first order in g₂ and lowest order in b. To derive the b² corrections to the precession rate and chemical potential would require including the ν = 3 components in Ψ.

Summary

In summary, we have derived a general expression for the velocity of the center of a vortex in an atomic Bose–Einstein condensate in terms of the background fluid velocity and the density gradients at the vortex center. The differences of the two velocities become important when the vortex core size becomes comparable with the scale height of the density variation. In particular, in the case of a rapidly rotating condensate in a harmonic trap, vortices precess at the transverse trap frequency, ω, minus corrections of order Na_s/Z.

Abbreviations

GP

Gross–Pitaevskii

LLL

lowest Landau level.