Electrodynamics of Vortices in Quasi-2D Scalar Bose-Einstein Condensates

TL;DR

The paper develops a duality that maps vortices in a quasi-$2$D scalar BEC to Maxwell-like electrodynamics in $2+1$ dimensions, accommodating inhomogeneous, time-dependent densities, dissipation, and rotation. By defining vortex charge density $ ho_v$ and an effective electric field $oldsymbol{E}_{ m sf}$ along with displacement $oldsymbol{D}_{ m sf}$ and magnetic-like fields, it derives a complete set of effective Maxwell equations that govern vortex dynamics beyond the point-vortex model. A generalized Lorentz-force description fuses the vortex motion with dissipation and external rotation, and the theory yields a framework to compute time-dependent circulation via a generalized Reynolds theorem. The work also connects static vortices to the 2D Coulomb gas, suggesting BKT-like behavior, and discusses implications for vortex patterns, turbulence, and potential extensions to other quantum fluids and higher-dimensional settings. Overall, the approach provides a rigorous, broadly applicable tool for understanding nonequilibrium vortex dynamics in BECs and beyond, highlighting both foundational insights and practical pathways for future research.

Abstract

In two spatial dimensions, vortex-vortex interactions approximately vary with the logarithm of the inter-vortex distance, making it possible to describe an ensemble of vortices as a Coulomb gas. We introduce a duality between vortices in a quasi-two-dimensional (quasi-2D) scalar Bose-Einstein condensates (BEC) and effective Maxwell's electrodynamics. Specifically, we address the general scenario of inhomogeneous, time-dependent BEC number density with dissipation or rotation. Starting from the Gross-Pitaevskii equation (GPE), which describes the mean-field dynamics of a quasi-2D scalar BEC without dissipation, we show how to map vortices in a quasi-2D scalar BEC to 2D electrodynamics beyond the point-vortex approximation, even when dissipation is present or in a rotating system. The physical meaning of this duality is discussed.

Figures (1)

• Figure 1: Duality between the vortices in the quasi-2D scalar BEC and the electrodynamics in the matter. The derivations are in Sec. \ref{['VortexBEC_electrodynamics']}, and the full effective Maxwell's equations are shown in Table \ref{['table:vortex-Maxwell']}. On the right-hand side, $\boldsymbol{D} \left( \boldsymbol{r}, t \right)$ is the electric displacement field, $\boldsymbol{H} \left( \boldsymbol{r}, t \right)$ is the magnetic field strength, $Q_j \left( t \right)$ is the free electric charge with index $j$ at time $t$, and $\boldsymbol{J}_f \left( \boldsymbol{r}, t \right)$ is the free electric current density. For other definitions of symbols, refer to Tables \ref{['SuppSymbolDef']} and \ref{['SuppEDSymbol']}.