Projection Optimization Method for Open-Dissipative Quantum Fluids and its Application to a Single Vortex in a Photon Bose-Einstein Condensate

TL;DR

This work develops a projection optimization method that extends variational techniques to open-dissipative quantum fluids and applies it to a 2D complex Gross–Pitaevskii equation describing a photon Bose–Einstein condensate. By projecting the equation of motion onto a trial manifold, the authors derive an analytic vortex solution characterized by a density profile $n(r)=\frac{r^2}{r^2+\alpha^2}$ and a spiral velocity field, with the vortex width given by $\alpha=2\xi\sqrt{\left(\frac{g}{\Gamma}\right)^2\left[1-\sqrt{1-\left(\frac{\Gamma}{g}\right)^2}\right]}$, where $\xi=\hbar/\sqrt{2mgn_s}$ and $n_s=\gamma/\Gamma$. These results, combined with a numerical solution of the cGPE, reveal finite-size effects and a three-region flow (circular near core, spiral at intermediate distances, radial far from the core), demonstrating good agreement between analytics and numerics for modest losses. The method thus provides a valuable analytical tool for open-dissipative quantum fluids, offering insights into vortex structure and dynamics beyond purely numerical studies and suggesting directions for improved density ansätze and connections to vortex dynamics and BKT-like phenomena.

Abstract

Open dissipative systems of quantum fluids have been well studied numerically. In view of a complementary analytical description we extend here the variational optimization method for Bose-Einstein condensates of closed systems to open-dissipative condensates. The resulting projection optimization method is applied to a complex Gross-Pitaevski equation, which models phenomenologically a photon Bose-Einstein condensate. Together with known methods from hydrodynamics we obtain an approximate vortex solution, which depends on the respective open system parameters and has the same properties as obtained numerically in the literature.

Figures (5)

• Figure 1: Illustration of projection optimization method for $N=3$ and $M=2$. Approximative ansatz ${\bf X}\approx {\bf x}({\hbox{\boldmath $\alpha$}})$ implies that velocity $\dot{\bf x}$ and vector field ${\bf F}({\bf x})$ roughly lie in tangent plane of manifold ${\bf x}({\hbox{\boldmath $\alpha$}})$ at point ${\bf x}$ spanned by tangent vectors $\partial {\bf x}/\partial \alpha^1$ and $\partial {\bf x}/\partial \alpha^2$.
• Figure 2: Finite-size effects and radial phase divergence for $\sigma = 0.3$. Each curve shows the radial phase obtained from numerical simulations with varying box sizes $L$. The dotted line represents a second-order polynomial fit in $\ln(L)$, as described by Eq. \ref{['eq:fitting_func']}, illustrating the phase growth with increasing box size. Inset: Leading order coefficient as function of loss parameter $\sigma$ with black dots and red line representing the numerical result and the expected behaviour extracted from asymptotic analysis of the analytically obtained phase field, see Eq. (\ref{['eq:phase_inf_asymptotic']}).
• Figure 3: Contour map of the phase around the vortex with dimensionless loss parameter $\sigma = 0.40$. The contour lines, shown in white, demonstrate the spiral nature of the vortex. Here a) follows from the projection optimization method, while b) is obtained from the numerical solution.
• Figure 4: Current around vortex with dimensionless loss parameter $\sigma=0.40$. Left column (a-c-e) determined analytically using the projection optimization method and right column (b-d-f) obtained by numerically solving cGPE (\ref{['eq:cGPEndim']}). From top to bottom different characteristics of the flow are visible for varying system length scales. a), b) illustrate that, near the vortex core, the flow is mostly circular similar to the behaviour of vortices in closed system BECs. c), d) depict spiral behaviour at some intermediate distance from the vortex, whereas e), f) show mostly radial behaviour far away from the vortex core.
• Figure 5: Profiles of a) dimensionsless density, defined in (\ref{['madelung']}), shifted horizontally for the purpose of illustration, and b) radial velocity for different loss parameters $\sigma$ obtained from projection optimization method (solid lines) and from solving cGPE (\ref{['eq:cGPEndim']}) numerically (dashed lines). Insets in a) show vortex width $\alpha$ defined in (\ref{['ansatz']}) and magnified density profile for $\sigma=0.5$. Inset in b) depicts that maximal radial velocity changes linearly with the losses.