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Structural Permeability of Quantum Vortices in Bose-Einstein Condensates and Screening of Tkachenko Waves

Vakhtang Mchedlishvili

Published Jun 18, 2026• 0 citations

Abstract

This paper presents a mathematical formalism for the generation of emergent electrodynamic properties from the internal structure of quantum vortices in Bose-Einstein condensates (BECs) and the screening of transverse Tkachenko waves. From the local energy balance of the condensate, the physical boundary of the vortex core is determined as R = σ, where the kinetic energy of the flow per particle reaches the background chemical potential. Analytical integration of the Gross-Pitaevskii density profile yields the structural redistribution of the vortex mass deficit between the core (47.30%) and the outer halo (52.70%). By analyzing the local displacement of the vortex, wherein the Magnus force is balanced by the Ginzburg-Landau elastic restorative force, we derive the screening of the effective mass density of the Bose-condensate and the emergent permeability coefficient κ_BEC. Sensitivity analysis conducted for various variational coefficients σ_core = γξ demonstrates that the microscopic permeability of the vortex core lies within the interval 4.20 ≤ κ_individual ≤ 7.01, while the macroscopic permeability lies within 1.25 ≤ κ_BEC ≤ 1.48. Utilizing coarse-grained wave equations and the Lagrangian formalism, we derive the frequency screening formula for Tkachenko waves. The theoretical results are compared with JILA experimental data and Simula's numerical solutions, demonstrating high quantitative agreement.

DOI: 10.5281/zenodo.20744342

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This paper presents a mathematical formalism for the generation of emergent electrodynamic properties from the internal structure of quantum vortices in Bose-Einstein condensates (BECs) and the screening of transverse Tkachenko waves. From the local energy balance of the condensate, the physical boundary of the vortex core is determined as R = σ, where the kinetic energy of the flow per particle reaches the background chemical potential. Analytical integration of the Gross-Pitaevskii density profile yields the structural redistribution of the vortex mass deficit between the core (47.30%) and the outer halo (52.70%). By analyzing the local displacement of the vortex, wherein the Magnus force is balanced by the Ginzburg-Landau elastic restorative force, we derive the screening of the effective mass density of the Bose-condensate and the emergent permeability coefficient κ_BEC. Sensitivity analysis conducted for various variational coefficients σ_core = γξ demonstrates that the microscopic permeability of the vortex core lies within the interval 4.20 ≤ κ_individual ≤ 7.01, while the macroscopic permeability lies within 1.25 ≤ κ_BEC ≤ 1.48. Utilizing coarse-grained wave equations and the Lagrangian formalism, we derive the frequency screening formula for Tkachenko waves. The theoretical results are compared with JILA experimental data and Simula's numerical solutions, demonstrating high quantitative agreement.

We develop an approach for calculating stationary states of rotating Bose-Einstein condensates in harmonic traps which is applicable for arbitrary ratios of the rotation frequency to the transverse frequency of the trap ${\ensuremath{\omega}}_{\ensuremath{\perp}}$. Assuming the number of vortices to be large, we write the condensate wave function as the product of a function that describes the structure of individual vortices times an envelope function varying slowly on the scale of the vortex spacing. By minimizing the energy, we derive Gross-Pitaevskii equations that determine the properties of individual vortices and the global structure of the cloud. For low rotation rates, the structure of a vortex is that of an isolated vortex in a uniform medium, while for rotation rates approaching the frequency of the trap (the mean-field lowest-Landau-level regime), the structure is that of the lowest

p

-wave state of a particle in a harmonic trap with frequency ${\ensuremath{\omega}}_{\ensuremath{\perp}}$. The global structure of the cloud is determined by minimizing the energy with respect to variations of the envelope function; for conditions appropriate to most experimental investigations to date, we predict that the transverse density profile of the cloud will be of the Thomas-Fermi form, rather than the Gaussian structure predicted on the assumption that the wave function consists only of components in the lowest Landau level for a regular array of vortices.

When vortices are displaced in Bose-Einstein condensates (BECs), the Magnus force gives the system a momentum transverse in the direction to the displacement. We show that BECs in long channels with vortices exhibit a quantization of the current response with respect to the spatial vortex distribution. The quantization originates from the well-known topological property of the phase around a vortex; it is an integer multiple of $2\ensuremath{\pi}$. In a way similar to that of the integer quantum Hall effect, the current along the channel is related to this topological phase and can be extracted from two experimentally measurable quantities: the total momentum of the BEC and the spatial distribution. The quantization is in units of

m/2h

, where

m

is the mass of the atoms and

h

is Planck's constant. We derive an exact vortex momentum-displacement relation for BECs in long channels under general circumstances. Our results present the possibility that the configuration described here can be used as a novel way of measuring the mass of the atoms in the BEC using a topological invariant of the system. If an accurate determination of the plateaus are experimentally possible, this gives the possibility of a topological quantum mass standard and precise determination of the fine structure constant.

We consider a Gross-Pitaevskii(GP) model of a Bose-Einstein Condensate(BEC). We study a single vortex line in the presence of non-local s-wave interactions. We show that the width of such a vortex is a microscopic length scale of the order of s-wave scattering length and is independent of the healing length. This analysis fixes the problem of arbitrariness in the selection of the width of vortices which is so far approximately considered to be the healing length

\xi_{0}

despite not being selected explicitly. We present a variational method to construct analytical solution for vortices of any quantum of circulation from the core to any finite extent. Using our analysis we predict hydrogen BEC to produce the thinnest vortex line.

We have performed numerical simulations of giant vortex structures in rapidly rotating Bose-Einstein condensates within the Gross-Pitaevskii formalism. We reproduce the qualitative features, such as oscillation of the giant vortex core area, formation of toroidal density hole, and the precession of giant vortices, observed in the recent experiment [Phys. Rev. Lett., ()]]. We provide a mechanism which quantitatively explains the observed core oscillation phenomenon. We demonstrate the clear distinction between the mechanism of atom removal and a repulsive pinning potential in creating giant vortices. In addition, we have been able to simulate the transverse Tkachenko vortex lattice vibrations.

The aim of this paper is to derive the equations of motion of quantum vortices in a weakly non-ideal Bose–Einstein condensate (BEC) at finite temperatures and in the presence of macroscopic superfluid and normal velocities. The motivation for this work is that the dynamics of quantum vortices in superfluid helium has been studied at the phenomenological level, and a number of issues require clarification, which can be addressed by studying the corresponding problem in BEC. In BEC, quantum vortices are located at points where the macroscopic wave function (MWF) ψ(r,T), which describes the BEC dynamics, vanishes ψ(r,T)=0. The intersection of the surfaces Reψ(r,T)=0 and Imψ(r,T)=0 forms a one-dimensional line, which is just the quantum vortex line. During the evolution of the MWF, the position of the zero surfaces changes, that is, the vortex lines move. Conversely, the motion of these vortex filaments determines the evolution of the MWF. Thus, there is a mutual correspondence, and the quantity ψ(r,T) can be considered as a functional of the entire vortex configuration, i.e., {s(ξ,t)}. This viewpoint allows us to derive the equations of motion for vortices from the Gross–Pitaevskii equation (GPE), which describes the dynamics of MWF. The key finding is the confirmation that the equation derived from the fundamental quantum description (the GPE) for a BEC has the same mathematical structure as the equation derived from macroscopic, phenomenological reasoning for superfluid helium.