Motion of vortices in inhomogeneous Bose-Einstein condensates

Abstract

We derive a general and exact equation of motion for a quantised vortex in an inhomogeneous two-dimensional Bose-Einstein condensate. This equation expresses the velocity of a vortex as a sum of local ambient density and phase gradients in the vicinity of the vortex. We perform Gross-Pitaevskii simulations of single vortex dynamics in both harmonic and hard-walled disk-shaped traps, and find excellent agreement in both cases with our analytical prediction. The simulations reveal that, in a harmonic trap, the main contribution to the vortex velocity is an induced ambient phase gradient, a finding that contradicts the commonly quoted result that the local density gradient is the only relevant effect in this scenario. We use our analytical vortex velocity formula to derive a point-vortex model that accounts for both density and phase contributions to the vortex velocity, suitable for use in inhomogeneous condensates. Although good agreement is obtained between Gross-Pitaevskii and point-vortex simulations for specific few-vortex configurations, the effects of nonuniform condensate density are in general highly nontrivial, and are thus difficult to efficiently and accurately model using a simplified point-vortex description.

Vortex mass in a superfluid

“…2 ] is consistent with the solution of the corresponding point-vortex model [26], which can be mapped onto the exactly soluble two-dimensional electrostatic problem of a charge inside a conducting ring. Once M v has been determined, it could be included in simulations of vortex dynamics that use point-vortex and vortex filament models, to include effects due to vortex inertia.…”

“…(20) by e −iθ . Dynamics causes the positions of these phase singularities to be spatially separated and leads to an intrinsic vortex velocity dipole moment (vVDM) of the vortex [26,27] due to the resulting kelvon induced dipolar superflow within the vortex core. The vVDM may be understood from the energetic perspective as the doubly charged phase singularity, e ±i2θ , in one of the quasiparticle components has a tendency to split into two spatially separated singly charged singularities via a critical point explosion [48].…”

“…The influence of the dipolar velocity field due to the kelvon quasiparticle is limited to within the vortex core and yields the vortex velocity dipole moment (vVDM) [26,27]. The equation of motion of the vortex is obtained explicitly by linearising the formal solution of the generalized time-dependent Gross-Pitaevskii equation, Eq.…”

“…The equation of motion of the vortex is obtained explicitly by linearising the formal solution of the generalized time-dependent Gross-Pitaevskii equation, Eq. (2), and finding the position r v (t+δt) of the phase singularity of the vortex wavefunction ψ v (r, t + δt), where δt is an infinitesimal time increment [26]. The resulting velocity of the vortex…”

Einstein–Bose condensation of Onsager vortices

“…of the state (21) defines the superfluid velocity field v s , the incompressible component of which is remarkably well approximated by the velocity field of Onsager's point vortex model [59].…”