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arXiv:2608.28862

Regularization of Vortex Core Size in Photon BECs due to Harmonic Trap: An Analytical Approach

Joshua Krauß, Tabitha Bates, Francisco Ednilson Alves dos Santos, Axel Pelster

Published Aug 28, 2026• 0 citations

Abstract

Quantized vortices are a hallmark of superfluidity. However, in a photon Bose--Einstein condensate, the photon-photon interaction is so weak that in a homogeneous system the healing length exceeds the experimentally achievable size of the condensate itself. Here we show that a harmonic confinement regularizes the vortex size to experimentally achievable scales. Moreover, such an external confinement closely resembles the standard experimental setup of a pumped dye-filled microcavity. We model the condensate via a complex Gross--Pitaevskii equation and obtain an approximate dynamical single-vortex solution by applying the recently proposed projection optimization method. The latter generalizes the variational approach of closed systems to open-dissipative systems without assuming a specific form for the condensate phase. The radial photon flow, which is characteristic for driven-dissipative systems, yields a definition of the vortex core size based on the competition of gain and loss. However, this condition reproduces the heuristic closed system definition now based on physical grounds. A subsequent linear stability analysis shows that interaction, as well as pumping and loss can drive the system to an unstable regime. In this way one can fundamentally distinguish between closed and open-dissipative systems.

DOI: 10.48550/arxiv.2608.28862

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Quantized vortices are a hallmark of superfluidity. However, in a photon Bose--Einstein condensate, the photon-photon interaction is so weak that in a homogeneous system the healing length exceeds the experimentally achievable size of the condensate itself. Here we show that a harmonic confinement regularizes the vortex size to experimentally achievable scales. Moreover, such an external confinement closely resembles the standard experimental setup of a pumped dye-filled microcavity. We model the condensate via a complex Gross--Pitaevskii equation and obtain an approximate dynamical single-vortex solution by applying the recently proposed projection optimization method. The latter generalizes the variational approach of closed systems to open-dissipative systems without assuming a specific form for the condensate phase. The radial photon flow, which is characteristic for driven-dissipative systems, yields a definition of the vortex core size based on the competition of gain and loss. However, this condition reproduces the heuristic closed system definition now based on physical grounds. A subsequent linear stability analysis shows that interaction, as well as pumping and loss can drive the system to an unstable regime. In this way one can fundamentally distinguish between closed and open-dissipative systems.

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-Pitaevskii equation, which models 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.

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.

Vortex states of weakly interacting Bose-Einstein condensates confined in three-dimensional rotating harmonic traps are investigated numerically at zero temperature. The ground state in the rotating frame is obtained by propagating the Gross-Pitaevskii equation for the condensate in imaginary time. The total energies between states with and without a vortex are compared, yielding critical rotation frequencies that depend on the anisotropy of the trap and the number of atoms. Vortices displaced from the center of nonrotating traps are found to have long lifetimes for sufficiently large numbers of atoms. The relationship between vortex stability and bound core states is explored.

Trapped Bose-Einstein condensates (BECs) differ considerably from the standard textbook example of a uniform Bose gas. In an isotropic harmonic potential V(r) = ½Mω2 r 2, the single-particle ground state introduces a new intrinsic scale of length [the ground-state size d = √ℏ/(Mω)] and energy [the ground-state energy E 0 =

\frac{3} {2}

ℏω]. When the trap rotates at a low angular velocity, the behavior of a single vortex illustrates the crucial role of discrete quantized vorticity. For more rapid rotation, the condensate contains a vortex array. The resulting centrifugal forces expand the condensate radially and shrink it axially; thus, the condensate becomes effectively two dimensional. If the external rotation speed approaches the frequency of the radial harmonic confining potential, the condensate enters the “lowest-Landau-level” regime, and a simple description again becomes possible. Eventually, the system is predicted to make a quantum phase transition to a highly correlated state analogous to the fractional quantum Hall states of electrons in a strong magnetic field.

Abstract We study the motion of superfluid vortices with filled massive cores. Previous point-vortex models already pointed out the impact of the core mass on the vortex dynamical properties, but relied on an assumption that is questionable in many physical systems where the immiscibility condition is barely satisfied: the fact that the massive core always lays at the very bottom of the effective confining potential constituted by the hosting vortex. Here, we relax this assumption and present a new point-vortex model where quantum vortices are harmonically coupled to their massive cores. We thoroughly explore the new dynamical regimes offered by this improved model; we then show that the functional dependence of the system normal modes on the microscopic parameters can be correctly interpreted only within this new generalized framework. Our predictions are benchmarked against the numerical simulations of coupled Gross–Pitaevskii equations for a realistic mixture of atomic Bose–Einstein condensates.

We consider a superfluid of trapped fermionic atoms and study the single vortex solution in the Ginzburg-Landau regime. We define simple analytical estimates for the main characteristics of the system, such as the vortex core size, temperature regimes for the existence of a vortex, and the effects of rotation and interactions with normal fermions. The parameter dependence of the vortex core size (healing length) is found to be essentially different from that of the healing length in metallic superconductors or in trapped atomic Bose-Einstein condensation in the Thomas-Fermi limit. This is an indication of the importance of the confining geometry for the properties of fermionic superfluids.