Quantum Optical Skyrmions

• Quantum optical skyrmions are topological mappings in light–matter systems defined by normalized Stokes vectors, complex spinors, or density matrices.
• They are realized through classical electromagnetic, atomic, and heralded quantum regimes, demonstrating quantized invariants and exceptional spatial resolution.
• Their applications span on-chip photonics, quantum information storage, and free-space communication, showcasing robustness against decoherence and turbulence.

Searching arXiv for papers on quantum optical skyrmions and related optical skyrmion platforms. Quantum optical skyrmions are topological mappings realized in optical or light–matter systems whose order parameter is a normalized Stokes vector, a local optical spin angular momentum field, a complex optical spinor, or a coherence–Stokes texture defined directly on a density matrix. In current usage, the term covers several distinct regimes: classical electromagnetic skyrmions that provide the modal template; light-written topological excitations stored in atomic polarization; genuinely quantum states of light, including nonlocal entangled, heralded single-photon, and mixed-state density-matrix skyrmions; and optical protocols that detect quantum skyrmions in matter rather than create skyrmions of light (Shen et al., 2022, Parmee et al., 2021, Ornelas et al., 2022, Koni et al., 30 Jul 2025, Kam et al., 26 Apr 2026, Sharma et al., 20 Jun 2025). Across these regimes, the central invariant is an integer topological charge that can remain stable even when field amplitudes, modal balance, or entanglement degrade, provided the underlying mapping remains well defined (Wang et al., 23 Dec 2025, Guo et al., 6 Sep 2025).

1. Topological definitions and order parameters

The common two-dimensional construction is a map from a physical plane into an order-parameter sphere. In the optical review literature, the skyrmion number is written as

$$s=\frac{1}{4\pi }\iint_\sigma \mathbf{S}\cdot \left( \frac{\partial \mathbf{S}}{\partial x}\times \frac{\partial \mathbf{S}}{\partial y} \right)\,\mathrm{d}x\,\mathrm{d}y,$$

where $\mathbf{S}(x,y)$ is the normalized optical vector field. The same review organizes textures by polarity $p$, vorticity $m$, and helicity $\gamma$, with $s=p\,m$, and identifies Néel-type, Bloch-type, anti-skyrmion, meron, bimeron, skyrmionium, and hopfion families (Shen et al., 2022).

The order parameter is not unique across the field. In free-space and near-field optics it is often the local Stokes vector or the local spin angular momentum density. In atomic media it can be the normalized two-component optical spinor $\hat{\mathsf P}$, whose full state space is $S^3$ rather than the Poincaré sphere $S^2$, so that genuine three-dimensional particle-like textures are classified by $\Pi_3(S^3)=\mathbb Z$. In mixed-state quantum optics, the effective two-dimensional domain can be the pair of density-matrix coordinates $\mathbf{S}(x,y)$0, with a coherence–Stokes field $\mathbf{S}(x,y)$1 defined directly from $\mathbf{S}(x,y)$2 (Parmee et al., 2021, Kam et al., 26 Apr 2026).

Family	Order parameter	Topological invariant
Stokes or SAM skyrmion	$\mathbf{S}(x,y)$3	$\mathbf{S}(x,y)$4
3D atomic optical skyrmion	$\mathbf{S}(x,y)$5	$\mathbf{S}(x,y)$6
Hopfion	Hopf projection of $\mathbf{S}(x,y)$7 to $\mathbf{S}(x,y)$8	$\mathbf{S}(x,y)$9
Mixed-state skyrmion	$p$0	$p$1

A recurrent misconception is that all optical skyrmions are exhausted by transverse polarization textures on the Poincaré sphere. The atomic $p$2 construction shows that this is sufficient for baby-skyrmions but insufficient for genuine $p$3 particle-like topology, where the total electromagnetic phase is essential. The mixed-state construction shows, in a different direction, that a skyrmion need not be encoded in a pure-state wavefunction at all; it can emerge directly in the density matrix (Parmee et al., 2021, Kam et al., 26 Apr 2026).

2. Classical electromagnetic foundations

The modern field was built on classical structured-light and near-field platforms. In confined electromagnetic fields with orbital angular momentum, the local spin angular momentum density

$p$4

supports photonic skyrmions whose spin texture varies on deep-subwavelength scales down to $p$5 of the wavelength, corresponding to about $p$6 nm, with experimentally resolved features as small as $p$7 and simulated features near $p$8 (Du et al., 2018). That work established the distinction between intensity, which remains diffraction-limited, and local field direction, which can vary much more rapidly.

Chiral multilayers extended the taxonomy beyond the originally observed Néel-type textures. In a metal–chiral-medium–metal trilayer supporting a plasmonic optical vortex, chirality induces a nonzero azimuthal spin component $p$9, while opposite interface-induced radial spins cancel at the center plane, producing a Bloch-type photonic skyrmion. The same framework yields twisted-Néel textures at single chiral interfaces and at the outer surfaces of the trilayer (Zhang et al., 2020). This was the first explicit optical route to the Bloch branch of the magnetic-skyrmion classification.

Another classical mechanism is more minimal. A single uniformly polarized scalar vortex already contains an intrinsic skyrmionic polarization structure once the longitudinal field required by $m$0 is retained. The resulting “Gauss–Stokes” skyrmion is built from the transverse circular component $m$1 and the longitudinal component $m$2, with transverse-axial Stokes parameters

$m$3

For the demonstrated central texture, $m$4, with experimental $m$5 and theoretical $m$6 (Mata-Cervera et al., 28 Jan 2025). This removed the earlier assumption that optical skyrmions must be engineered by superposing different spatial modes and polarizations.

Integrated topological photonics adds directional transport. In a valley photonic crystal waveguide, the evanescent field of topological valley edge states carries optical spin skyrmions defined from the full three-component spin angular momentum density

$m$7

The reported skyrmion numbers are nearly quantized, $m$8 and $m$9, with $\gamma$0 taken as the skyrmion criterion, and the sign is locked to the valley degree of freedom (He et al., 4 May 2026). This is a classical platform, but it directly addresses on-chip transport and defect robustness that later become central in quantum-compatible implementations.

3. Light-written skyrmions in atomic media

A decisive shift occurred when skyrmionic topology was transferred from light alone to optically induced matter excitations. In a driven ensemble of atoms on a narrow $\gamma$1 transition, the relevant field is the induced polarization density

$\gamma$2

which in weak scattering follows the incident field through

$\gamma$3

The topological objects are therefore encoded in the induced atomic optical excitation rather than primarily in the free-space light field itself (Parmee et al., 2021).

This framework contains four families. First are two-dimensional baby-skyrmions, defined through the atomic Stokes vector built from the normalized transverse polarization spinor

$\gamma$4

For a full-Poincaré beam

$\gamma$5

the induced texture is fountain-like and integrates to $\gamma$6.

Second are genuine three-dimensional particle-like skyrmions, the conceptual center of the paper. The normalized complex spinor lives on an optical hypersphere $\gamma$7,

$\gamma$8

and the topological charge is

$\gamma$9

A particularly important rewriting introduces the transverse polarization density current

$s=p\,m$0

for which

$s=p\,m$1

This identifies the skyrmion density with a Chern–Simons-like helicity density of the atomic polarization current.

Third are Hopfions and knotted solitons. The Hopf map

$s=p\,m$2

reduces the full $s=p\,m$3 field to the Stokes vector on $s=p\,m$4, and for the optical constructions considered $s=p\,m$5. The same platform also supports trefoil-knot preimages with $s=p\,m$6.

Fourth are singular defects. In dense arrays with strong dipole–dipole interactions, some such defects coincide almost entirely with a single collective eigenmode, with a sharply defined collective linewidth and line shift. This is one of the clearest cases where a topological optical pattern becomes a natural mode of a strongly coupled light–matter system.

The atomic proposal is “quantum optical” only in a qualified sense. The atoms are quantum emitters and the transition can be long-lived, but the treatment is classical or mean coherent response: low-intensity light, linear response, coherent dipole amplitudes, and collective modes of coupled oscillating dipoles. It is not a many-body quantum skyrmion state, nor a single-photon topological wavefunction. Its importance lies instead in showing that topological optical excitations can be written into matter and, in principle, stored and manipulated there.

4. Nonlocal, heralded, and mixed-state quantum skyrmions of light

The first explicitly quantum optical skyrmions of light were nonlocal. In hybrid-entangled biphoton states of the form

$s=p\,m$7

photon $s=p\,m$8 contributes the spatial coordinate and photon $s=p\,m$9 contributes the polarization state conditioned on that coordinate. The position-space form induces

$\hat{\mathsf P}$0

which defines a nonlocal map $\hat{\mathsf P}$1. The skyrmion density is

$\hat{\mathsf P}$2

For $\hat{\mathsf P}$3, the reported values are state fidelity $\hat{\mathsf P}$4, Poincaré-sphere coverage $\hat{\mathsf P}$5, and measured skyrmion number $\hat{\mathsf P}$6. The same experiment realizes $\hat{\mathsf P}$7 and shows that nontrivial topology remains intact under controlled entanglement decay until the entanglement itself vanishes (Ornelas et al., 2022).

Dual-wavelength quantum skyrmions add wavelength as a nonseparable resource. A nondegenerate SPDC source at $\hat{\mathsf P}$8 and $\hat{\mathsf P}$9, combined with a voltage-tunable liquid-crystal topological defect of charge $S^3$0, generates both nonlocal skyrmions distributed across two photons of different colors and local heralded single-photon skyrmions at a chosen wavelength. The defect implements

$S^3$1

and the reconstructed spatial Stokes textures yield $S^3$2 with less than $S^3$3 deviation in all topological cases. The preconversion OAM entanglement is verified by a CHSH value $S^3$4 and Bell-state fidelity $S^3$5. Around $S^3$6, the device switches to a trivial regime with $S^3$7 (Koni et al., 30 Jul 2025).

A further generalization abandons pure states altogether. In mixed-state quantum skyrmions, the topology is encoded directly in the density matrix $S^3$8 through the coherence–Stokes quantities

$S^3$9

with normalized texture $S^2$0. A first-order skyrmion can be realized by the rank-2 mixed state

$S^2$1

and, more generally, a sparse rule $S^2$2 suffices to reconstruct a skyrmion of charge $S^2$3. In biphoton systems the same construction yields “nested topology”: skyrmions persist simultaneously in the full state, in local reductions, and in hybrid nonlocal reductions (Kam et al., 26 Apr 2026).

5. Storage, propagation, and topological robustness

The move from generation to interfacing is exemplified by storage in cold atoms. Optical skyrmions built from paraxial vector beams were stored and retrieved in a dual-path EIT memory in a cold $S^2$4Rb vapor using the $S^2$5 line and a three-level $S^2$6-system. Probe pulses had temporal width $S^2$7; the cloud was $S^2$8 cm long and $S^2$9 mm in diameter; the control beam had waist $\Pi_3(S^3)=\mathbb Z$0 mm, power $\Pi_3(S^3)=\mathbb Z$1 mW in the standard setting, and intersected the probes at $\Pi_3(S^3)=\mathbb Z$2. For target topological numbers $\Pi_3(S^3)=\mathbb Z$3, the experimentally calculated skyrmion number remained effectively invariant for storage times $\Pi_3(S^3)=\mathbb Z$4, $\Pi_3(S^3)=\mathbb Z$5, and $\Pi_3(S^3)=\mathbb Z$6, despite unequal storage efficiencies, differential phase, and spin-wave decoherence. The same topology remained robust when the control power was changed by more than $\Pi_3(S^3)=\mathbb Z$7; the maximum power was $\Pi_3(S^3)=\mathbb Z$8 mW, corresponding to approximately $\Pi_3(S^3)=\mathbb Z$9 MHz in the authors’ coupling-scale notation (Wang et al., 23 Dec 2025).

Atmospheric turbulence tests show a parallel robustness in free-space channels. For both classical vector-beam skyrmions and quantum hybrid-entangled skyrmions, turbulence is modeled as a one-sided channel acting on the spatial subsystem,

$\mathbf{S}(x,y)$00

while the topological charge remains invariant under the induced smooth coordinate deformation of the Stokes texture. In the quantum experiment, the state $\mathbf{S}(x,y)$01 retained its skyrmion number while concurrence decreased with turbulence strength. In the classical experiment, near-field robustness persisted up to $\mathbf{S}(x,y)$02, far-field robustness remained strong up to about $\mathbf{S}(x,y)$03, and multiple-phase-screen simulations showed only small deviations beginning around $\mathbf{S}(x,y)$04 (Guo et al., 6 Sep 2025).

On-chip transport has an analogous, though classical, counterpart in valley photonic crystal waveguides. There the skyrmionic texture is not stored but carried as an eigenstate property of a topological edge channel, with sign reversal upon reversing propagation direction and persistence through a Z-bend and two classes of defects (He et al., 4 May 2026). Taken together, these works indicate that skyrmion number can be more robust than mode fidelity, waveform fidelity, or entanglement monotones.

6. Adjacent meanings, detection protocols, and conceptual boundaries

The phrase “quantum optical skyrmions” also touches two adjacent literatures that should not be conflated with quantum states of light carrying skyrmionic topology. One concerns optical detection of quantum skyrmions in matter. In frustrated magnets, a quantum skyrmion is a nanoscale magnetic skyrmion whose helicity $\mathbf{S}(x,y)$05 and spin-flip operator $\mathbf{S}(x,y)$06 satisfy $\mathbf{S}(x,y)$07. Brillouin light scattering then probes the operator $\mathbf{S}(x,y)$08, and sideband asymmetry in the cross-polarized BLS spectrum,

$\mathbf{S}(x,y)$09

is proposed as a witness of quantized skyrmion helicity levels once classical asymmetries are removed by geometry (Sharma et al., 20 Jun 2025). This is quantum optics applied to magnetic quantum skyrmions, not an optical skyrmion of light.

A second adjacent literature concerns quantum fluids of light. In driven-dissipative exciton-polariton condensates, a scalar complex field $\mathbf{S}(x,y)$10 can generate a skyrmion through the derived displacement field

$\mathbf{S}(x,y)$11

There, nonresonant pumping can spontaneously generate isolated scalar skyrmions with $\mathbf{S}(x,y)$12 and self-organized skyrmion lattices in a polariton condensate, while nonlinearities allow switching and moiré-lattice reconfiguration (Wingenbach et al., 1 Jun 2026). These are optical skyrmions in a quantum-fluid platform, but they are neither polarization Stokes skyrmions nor few-photon quantum states.

Several distinctions therefore define the present field. A true three-dimensional particle-like optical skyrmion requires the full complex spinor on $\mathbf{S}(x,y)$13, not the reduced Stokes vector on $\mathbf{S}(x,y)$14 (Parmee et al., 2021). A genuinely quantum optical skyrmion may live in entangled biphoton correlations, in heralded single-photon spin–orbit structure, or directly in a mixed-state density matrix (Ornelas et al., 2022, Koni et al., 30 Jul 2025, Kam et al., 26 Apr 2026). By contrast, many influential platforms remain classical or mean-field, even when they are highly relevant to quantum technologies (Du et al., 2018, He et al., 4 May 2026, Wang et al., 23 Dec 2025). This suggests that the subject is best understood not as a single object class but as a hierarchy of topological optical constructions, connected by common invariants and increasingly quantum realizations.