Coherent Toroidal Vortices: A Unified Analysis

• Coherence toroidal vortices are toroidal structures characterized by global phase, flow, field, or correlation coherence across a range of physical systems.
• Methodologies span theoretical models, numerical simulations, and experimental setups in Bose–Einstein condensates, optics, and fluid dynamics to elucidate quantized circulation and topological invariants.
• Findings reveal that dissipation-induced stabilization and distinct coherence definitions enable persistent, controllable toroidal states in quantum, optical, and magnetic frameworks.

Coherence toroidal vortices denote toroidal vortex structures whose defining coherence is not confined to a single physical mechanism. In annular Bose–Einstein condensates they appear as quantized, phase-coherent circulation states; in correlation theory they appear as topological defects of two-point coherence functions in higher-dimensional domains; in optics and electromagnetics they appear as toroidal phase singularities, vector toroidal pulses, and hybrid scalar–vector vortex fields; in fluid mechanics they appear as objectively coherent tubular vortices; and in molecular magnetism they appear as toroidal magnetic vortices with vanishing net dipole and finite toroidal moment (Yakimenko et al., 2014, Lundh, 2012, Wang et al., 2024). Taken together, these usages suggest that the common structure is a toroidal organization of phase, flow, field, or correlation, while “coherence” may refer to global phase coherence, statistical correlation topology, propagation-invariant field structure, objective material rotation, or coherent quantum control.

1. Terminological scope and unifying features

The most literal use of the expression occurs in stochastic optics, where “coherence toroidal vortices” are defined as toroidal vortex structures that exist only in the mutual coherence function of a partially coherent optical field, rather than in the instantaneous field itself (Zhou et al., 23 Apr 2026). A closely related matter-wave usage appears in the theory of coherence vortices, where the basic object is not the condensate wavefunction but the two-point correlation function

$$g(\mathbf{r},\mathbf{r}')=\langle \hat{\psi}^\dagger(\mathbf{r})\hat{\psi}(\mathbf{r}')\rangle,$$

so that vortex defects live in a $2D$-dimensional correlation space and can form closed manifolds compatible with toroidal topology (Lundh, 2012).

A different but equally established usage appears in toroidal Bose–Einstein condensates, where coherent toroidal vortices are realized as persistent currents with integer winding number, generated by stirring and stabilized by dissipation in a ring geometry (Yakimenko et al., 2014). In photonics, toroidal vortices may be encoded either in a scalar phase singularity on a torus, in a vector field whose field lines form a toroidal surface, or in a hybrid object that contains both simultaneously (Wan et al., 2021, Wang et al., 2024). In fluid mechanics, by contrast, coherence is defined kinematically: rotationally coherent Lagrangian vortices are material tubes whose elements undergo equal bulk material rotation relative to the mean flow, and these tubes may be cylindrical, cup-shaped, or toroidal (Haller et al., 2015).

This plurality matters because a common misconception is to treat all toroidal vortices as visible intensity donuts with a deterministic phase singularity in ordinary three-dimensional space. The surveyed literature shows that the toroidal object may instead live in correlation space, in an order parameter phase, in a field-line topology, in an objective Lagrangian tube, or in a molecular toroidal moment. The term is therefore best read as a family resemblance rather than a single standardized definition.

2. Correlation-space toroidal vortices

In the coherence-vortex framework for Bose–Einstein condensates, the primary topological object is the two-point function rather than the condensate order parameter itself. A coherence vortex is specified by a zero of $g(\mathbf{r},\mathbf{r}')$ together with a nontrivial phase winding around that zero in the $2D$-dimensional domain of $(\mathbf{r},\mathbf{r}')$. For a 2D condensate, $g$ is defined on a 4D space, so a codimension-2 vortex defect is a 2D surface; in 3D condensates the corresponding defects are 4D manifolds in 6D correlation space. Closed toroidal defect manifolds are therefore topologically natural, even though explicit toroidal examples were not constructed in the original proposal (Lundh, 2012).

The same paper proposed a concrete route to create and detect such defects. Two initially independent condensates are collided and then transferred into a single component, yielding

$$g(\mathbf{r},\mathbf{r}')=\Psi_1^*(\mathbf{r})\Psi_1(\mathbf{r}')+\Psi_2^*(\mathbf{r})\Psi_2(\mathbf{r}'),$$

while an experimentally accessible approximation is obtained from phase reconstruction with a third reference condensate,

$$g_{\mathrm{exp}}(\mathbf{r},\mathbf{r}')=\left\langle e^{i[\theta(\mathbf{r}')-\theta(\mathbf{r})]}\right\rangle.$$

A central point is that coherence vortices can exist even when the condensate wavefunction itself shows no density vortices after ensemble averaging. The topology is stored in coherence, not in density (Lundh, 2012).

The stochastic-optics realization makes this distinction experimentally explicit. There the instantaneous field $E(\mathbf{r})$ is deliberately randomized by multiplying a deterministic Laguerre–Gaussian superposition by a complex random field $R(\mathbf{r})$, so that no stable toroidal vortex remains in any single realization. The toroidal structure reappears only in the mutual coherence function

$2D$0

where zeros with phase winding form a toroidal coherence singularity. Using 2500 speckle realizations, transverse coherence length $2D$1, beam waist $2D$2, and a 3.5% $2D$3 isosurface threshold, the experiment reconstructed both fundamental and hopfionic toroidal correlation textures, with crossing number $2D$4 for longitudinal OAM index $2D$5 (Zhou et al., 23 Apr 2026).

The same work showed that these “statistically veiled correlation topologies” survive a chaotic channel produced by a heated plate. The instantaneous field remains speckled, but the ensemble-averaged $2D$6 retains the toroidal structure and its topological invariant $2D$7. This directly counters the assumption that partial coherence necessarily destroys topological structure: here partial coherence eliminates deterministic first-order signatures while revealing topological order in correlations (Zhou et al., 23 Apr 2026).

3. Matter-wave toroidal superflow and quantized circulation

In a toroidal Bose–Einstein condensate, coherent toroidal vortices are realized as persistent currents pinned in the central hole of the annulus. The relevant order parameter is

$2D$8

and circulation is quantized,

$2D$9

The resulting long-lived state may be identified with an $g(\mathbf{r},\mathbf{r}')$0-charged vortex line pinned at the center, and once mobile annular vortices have left the condensate one finds $g(\mathbf{r},\mathbf{r}')$1 (Yakimenko et al., 2014).

The stirred-annulus study identified two distinct nucleation regimes. For slow stirring, $g(\mathbf{r},\mathbf{r}')$2, vortex–antivortex pairs nucleate in the bulk near the stirrer; the antivortex spirals outward and decays, while the vortex spirals inward and becomes pinned in the central hole, incrementing the winding number by one. For fast stirring above the critical angular velocity

$g(\mathbf{r},\mathbf{r}')$3

an outer edge surface mode is excited, breaks into vortices, and injects many vortices into the annulus, after which dissipation relaxes the vortex gas toward an integer persistent current (Yakimenko et al., 2014).

The rotating weak-link study complements this two-dimensional picture with a three-dimensional dissipative mean-field model. There, phase slips between circulation states are mediated by vortices created at the weak link, and a minimal angular velocity

$g(\mathbf{r},\mathbf{r}')$4

emerges below which the $g(\mathbf{r},\mathbf{r}')$5 phase slip does not occur at any barrier height. At low rotation, the dominant mechanism is a vortex entering from the outer edge, pairing with an inner antivortex, and transferring vorticity into the central hole; at higher rotation, outer and inner surface waves generate turbulent annular vortex dynamics and serial emission of vortices from highly charged currents (Yakimenko et al., 2014).

In both studies, dissipation is not an incidental correction but a structural ingredient. In the narrow-stirrer model with $g(\mathbf{r},\mathbf{r}')$6, the annulus is effectively purged of mobile vortices after about 3 s, leaving a smooth phase winding around the ring and singularities confined to the hole (Yakimenko et al., 2014). This clarifies another common misconception: coherence of the final toroidal current is not opposed to dissipative relaxation. On the contrary, dissipation converts a vortex-rich transient into a metastable coherent circulation state.

4. Electromagnetic and photonic toroidal vortices

Photonic toroidal vortices were introduced as spatiotemporal optical fields whose vortex core is a closed ring and whose phase twists helically around that ring. In the scalar-envelope description they satisfy

$g(\mathbf{r},\mathbf{r}')$7

and are generated experimentally by conformal mapping of a spatiotemporal Laguerre–Gaussian tube into a torus. The resulting field has a ring-shaped singularity in $g(\mathbf{r},\mathbf{r}')$8 space, a helical phase around the ring, and an azimuthal local orbital angular momentum density (Wan et al., 2021).

Recent electromagnetic work broadened this picture in two directions. First, strongly longitudinally polarized toroidal pulses were constructed as exact Maxwell solutions with a propagation-invariant space–time spectrum. At $g(\mathbf{r},\mathbf{r}')$9, the ratio $2D$0 is approximately 11, the spectral support lies near a $2D$1 line $2D$2, and normalized concurrence and normalized entanglement of formation exceed 0.98 and 0.94, respectively. These pulses preserve toroidal and skyrmionic topology while reorganizing polarization so that the longitudinal field dominates (Wang et al., 2024).

Second, hybrid electromagnetic toroidal vortices showed that scalar and vector toroidal structures are not mutually exclusive. In that construction, a compact coaxial horn emitter and radial asymmetric metasurface generate a scalar toroidal vortex in the radial electric field component and, through Gauss’s law and radial polarization, a family of vector toroidal vortices with saddle points, electromagnetic vortex streets, and skyrmion textures. The metasurface transfer function is

$2D$3

and measurements gave OAM per photon close to $2D$4 on four toroidal slices and skyrmion number $2D$5 (Wang et al., 2024).

A further dynamical development concerns photonic toroidal vortices with both transverse and longitudinal OAM. In dispersive media the toroidal structure may split or vanish, but in vacuum a renascent toroidal vortex was observed: the original structure first destabilizes, then the vortex line reforms with reversed polarity and propagates robustly while maintaining its toroidal structure (Liu et al., 10 May 2025). This suggests that coherence of a toroidal electromagnetic vortex need not mean strict profile invariance; it may instead mean persistence or recovery of the toroidal singular topology under propagation.

5. Fluid and geometric notions of toroidal coherence

In fluid mechanics, coherence is defined kinematically and objectively rather than through phase or correlation functions. Rotationally coherent Lagrangian vortices are material tubes whose elements complete equal bulk material rotation relative to the mean rotation of the surrounding flow. Their initial positions coincide with tubular level surfaces of the Lagrangian-Averaged Vorticity Deviation,

$2D$6

and the intrinsic dynamic rotation angle satisfies $2D$7. The corresponding tubular structures may be cylindrical, cup-shaped, or toroidal, and their objectivity is the central point: the detected toroidal vortex tube is invariant under time-dependent translations and rotations of the frame (Haller et al., 2015).

A superfluid realization of toroidal coherence appears in toroidal bundles of vortex filaments. Numerical simulations of bundles of quantized rings arranged in a torus showed generalized leapfrogging of constituent rings, followed by Kelvin waves and reconnections, yet the bundle retained its coherence over a relatively large distance compared to its size. Across cases with $2D$8, the structures traveled on the order of 10 times their initial diameter before major degradation, consistent with experiments in superfluid helium (Wacks et al., 2014).

The hydrodynamic picture on a torus further clarifies which clusters remain coherent and which do not. On a two-dimensional toroidal fluid film, the point-vortex Hamiltonian contains not only pairwise interactions through the toroidal Green function but also a geometric self-force from the Robin function. In this setting, a randomly initialized chiral cluster of same-sign vortices remains geometrically confined and undergoes collective drift along the toroidal direction; a fast–slow chiral cluster develops a core–halo structure in which fast vortices occupy the core and slow vortices are expelled to the periphery; by contrast, an achiral zero-circulation cluster disperses over the entire torus (R et al., 16 Jun 2025).

A final hydrodynamic refinement concerns helicity. For thin toroidal vortices with swirl, helicity can be written as $2D$9, where $(\mathbf{r},\mathbf{r}')$0 and $(\mathbf{r},\mathbf{r}')$1 are circulations along the small and large linked circles of the torus and the coefficient $(\mathbf{r},\mathbf{r}')$2 depends on the swirl distribution. Homogeneous swirl gives $(\mathbf{r},\mathbf{r}')$3, reproducing the usual Moffatt-type relation, whereas non-homogeneous swirl alters the effective linkage structure and changes $(\mathbf{r},\mathbf{r}')$4 (Bannikova et al., 2016). This does not define coherence by itself, but it shows that the internal distribution of toroidal vorticity constrains the topological robustness of the structure.

6. Molecular toroidal moments and coherent control

In molecular systems, the toroidal vortex is a magnetic rather than hydrodynamic or optical object. The basic quantity is the toroidal moment

$(\mathbf{r},\mathbf{r}')$5

which describes a head-to-tail loop of local magnetic moments with vanishing net magnetic dipole. In Dy$(\mathbf{r},\mathbf{r}')$6 and MDy$(\mathbf{r},\mathbf{r}')$7 single-molecule toroics, the relevant states are clockwise and anticlockwise magnetic vortices, $(\mathbf{r},\mathbf{r}')$8 and $(\mathbf{r},\mathbf{r}')$9, related by time reversal. Because they are magnetically dark and sensitive to magnetoelectric probes, they are natural candidates for coherent manipulation (Hymas et al., 11 Apr 2025).

A realistic protocol for coherent preparation was proposed using pulsed microwave radiation. In Dy$g$0, a single resonant microwave pulse at $g$1 GHz transfers population from a magnetic state to a toroidal state by a one-Dy-flip transition. In AlDy$g$2, three resonant microwave pulses at $g$3 GHz, $g$4 GHz, and $g$5 GHz, delivered either sequentially or simultaneously, drive the antiferrotoroidic state into a ferrotoroidic state with maximal $g$6. The simultaneous protocol reaches complete population transfer in 35 ns and remains robust for spin–phonon relaxation constants $g$7 (Hymas et al., 11 Apr 2025).

Fe$g$8Dy$g$9 extends this program to a much larger molecular ring. The system hosts a toroidal excitation manifold spanning a “$$g(\mathbf{r},\mathbf{r}')=\Psi_1^*(\mathbf{r})\Psi_1(\mathbf{r}')+\Psi_2^*(\mathbf{r})\Psi_2(\mathbf{r}'),$$0 billion dimensional toroidal space,” and its thermodynamics can be reduced to a 24×24 transfer-matrix problem. To quantify bulk toroidal polarization, the paper introduced the molar toroidal susceptibility

$$g(\mathbf{r},\mathbf{r}')=\Psi_1^*(\mathbf{r})\Psi_1(\mathbf{r}')+\Psi_2^*(\mathbf{r})\Psi_2(\mathbf{r}'),$$1

which measures the finite-temperature response of the toroidal moment to a magnetic field with small non-vanishing curl. Direct calculation revealed a significant finite-temperature ground-state toroidal polarization, and the large magnitude of $$g(\mathbf{r},\mathbf{r}')=\Psi_1^*(\mathbf{r})\Psi_1(\mathbf{r}')+\Psi_2^*(\mathbf{r})\Psi_2(\mathbf{r}'),$$2 relative to smaller single-molecule toroics was argued to make direct observation amenable to experiments using spatially focused magnetic field curls, including focused femtosecond laser pulses (Soncini et al., 5 Sep 2025).

Taken together, these molecular results suggest that coherent toroidal vortices need not be macroscopic flow structures or optical singularities. They can also be many-body quantum states in which a magnetic vortex is the operative degree of freedom, coherence is encoded in a protected low-energy manifold, and control is exerted by structured microwave or curl-field perturbations. This broadens the concept from toroidal flow and phase topology to toroidal quantum order itself.