Dual-Wavelength Quantum Skyrmions

• Dual-wavelength quantum skyrmions are photonic spin–orbit topological structures encoded over two nondegenerate wavelengths, enabling both nonlocal entangled and local heralded implementations.
• The platform employs a dual-wavelength liquid-crystal q-plate to induce wavelength-dependent spin–orbit conversion, validated by state tomography and CHSH Bell tests.
• Voltage tuning drives transitions between skyrmionic and trivial regimes, demonstrating dynamic control over quantum states in advanced photonic systems.

Searching arXiv for papers on dual-wavelength and quantum skyrmions to ground the article in current literature. Dual-wavelength quantum skyrmions are skyrmionic topological structures generated in quantum photonic states in which wavelength is an intrinsic, nonseparable degree of freedom of the topological state rather than an external label. In the formulation introduced with a wavelength-dependent spin–orbit conversion device based on a true liquid-crystal topological defect, the topology can appear either as a nonlocal skyrmion distributed across an entangled photon pair at two different wavelengths or as a local skyrmion in a heralded single photon at a chosen wavelength (Koni et al., 30 Jul 2025). In this sense, the term denotes a specific quantum-optical regime, distinct from the broader literature on magnetic quantum skyrmions, where multi-component, CP$^2$, CP$^3$, and operator-based quantum skyrmion constructions are established but explicit dual-wavelength structure is generally absent (Zhang et al., 2022).

1. Definition and conceptual scope

The central object is a skyrmion encoded in the spin–orbit structure of a quantum state of light. The relevant degrees of freedom are polarization, orbital angular momentum (OAM), spatial mode structure, and wavelength. The topological content is reconstructed from a spatially varying reduced polarization field and quantified by a Skyrme number on the Poincaré sphere. The defining feature of the dual-wavelength regime is that the skyrmionic structure intrinsically involves two nondegenerate wavelengths, rather than a single-wavelength structured beam with an added spectral label (Koni et al., 30 Jul 2025).

This construction differs from earlier quantum-skyrmion work in two ways. First, previous quantum optical skyrmions were described as single-wavelength configurations. Second, the dual-wavelength platform switches between nonlocal and local realizations within one device: the skyrmion may be distributed across an entangled biphoton state at distinct wavelengths, or it may be localized in a heralded single-photon spin–orbit state. The paper introducing the concept characterizes these regimes as “neither previously conceptualized nor demonstrated” (Koni et al., 30 Jul 2025).

In the broader skyrmion literature, “quantum skyrmion” usually denotes a different class of objects: quantized magnetic textures in microscopic spin systems, coherent-state textures in enlarged projective target spaces, or quasiparticle operators dressing a classical skyrmion with a quantum cloud. Those developments are directly relevant to quantum topology, but not to dual-wavelength structure in the photonic sense (Haller et al., 2024).

2. Physical platform and state generation

The experimental platform is a nondegenerate entangled-photon source followed by a dual-wavelength liquid crystal q-plate realized as a unit-charge true liquid-crystal topological defect. The source is type-0 spontaneous parametric down-conversion pumped at $\lambda_0=532$ nm, producing collinear photons at $\lambda_1=1550$ nm and $\lambda_2=810$ nm, with

$$\frac{1}{\lambda_1}+\frac{1}{\lambda_2}=\frac{1}{\lambda_0}.$$

The initial two-photon state is written as

$$\ket{\Psi} = \sum_{l \in \mathbb{Z}} \sum_{k=1,2} c_{l,k} \ket{l}_A \ket{\rm R}_A \ket{\lambda_k}_A \ket{-l}_B \ket{\rm R}_B \ket{\bar\lambda_k}_B,$$

where photon $A$ carries OAM $l\hbar$ and photon $B$ carries $^3$0, while $^3$1 is the complementary wavelength fixed by the pump constraint (Koni et al., 30 Jul 2025).

The liquid-crystal defect performs wavelength- and voltage-dependent spin–orbit conversion. For each photon,

$^3$2

with conversion efficiency $^3$3. The optic axis is azimuthal, $^3$4, with $^3$5, and the birefringent retardation is electrically tunable. Because the same device acts differently at 1550 nm and 810 nm, wavelength enters the skyrmion-generation mechanism dynamically rather than passively (Koni et al., 30 Jul 2025).

After the defect, spectral separation removes the wavelength ambiguity while preserving the entangled spin–orbit structure. In the representative case $^3$6, projecting photon $^3$7 into the Gaussian mode produces

$^3$8

This state is the parent object from which the nonlocal and heralded local skyrmions are obtained by further projective measurements (Koni et al., 30 Jul 2025).

The experiment verifies the OAM-entangled precursor state before spin–orbit conversion. In the $^3$9 subspace, the reconstructed state has CHSH Bell parameter $\lambda_0=532$0 and fidelity $\lambda_0=532$1 to an ideal Bell state, establishing that the dual-wavelength skyrmion protocol starts from a genuinely entangled nondegenerate biphoton resource (Koni et al., 30 Jul 2025).

3. Topological formulation and tomographic reconstruction

The skyrmion number is not assigned directly to the full biphoton wavefunction. Instead, the construction proceeds through a position-resolved reduced polarization state. From the reconstructed $\lambda_0=532$2 spin–orbit density matrix $\lambda_0=532$3, one defines

$\lambda_0=532$4

with $\lambda_0=532$5. The local Stokes parameters are

$\lambda_0=532$6

The normalized local pseudospin field is then

$\lambda_0=532$7

The Skyrme number is

$\lambda_0=532$8

This is the principal topological invariant used throughout the dual-wavelength construction (Koni et al., 30 Jul 2025).

The same tomographic framework also fixes how the spatial basis enters the reconstruction. Expanding

$\lambda_0=532$9

one obtains

$\lambda_1=1550$0

The spatial basis functions are Laguerre–Gauss modes, so the topology is reconstructed from a hybrid spin–orbit density matrix rather than from direct real-space polarization imaging alone (Koni et al., 30 Jul 2025).

This tomographic definition is conceptually important because it makes the topological observable compatible with both regimes of the platform. In the nonlocal case, the relevant polarization structure is conditional on a measurement on the partner photon; in the heralded case, it becomes a local single-photon polarization texture at a definite wavelength. A plausible implication is that the topological invariant is best understood as a measurement-context-dependent observable of a reduced quantum state, rather than a direct invariant of an unreduced many-photon field.

4. Nonlocal and heralded local realizations

Projective measurements generate two distinct skyrmion regimes. If photon $\lambda_1=1550$1 is projected onto circular polarization, the resulting states are

$\lambda_1=1550$2

and

$\lambda_1=1550$3

These are nonlocal skyrmions: the OAM/spatial structure resides on one photon, the polarization degree of freedom on the other, and the two photons are at distinct wavelengths (Koni et al., 30 Jul 2025).

If instead photon $\lambda_1=1550$4 is projected in circular polarization, the partner photon is heralded into a local spin–orbit skyrmion state:

$\lambda_1=1550$5

$\lambda_1=1550$6

In this regime, both polarization and OAM belong to the same photon at a definite wavelength (Koni et al., 30 Jul 2025).

Regime	Representative state	Topological locality
Nonlocal	$\lambda_1=1550$7	Distributed across the biphoton
Heralded local	$\lambda_1=1550$8	Local to one photon

Experimentally, both regimes produce second-order quantum skyrmions with measured Skyrme number

$\lambda_1=1550$9

with less than 1% deviation in the principal cases. Voltage tuning drives transitions between skyrmionic and trivial sectors. Near $\lambda_2=810$0 V the topology becomes trivial, $\lambda_2=810$1, whereas representative skyrmion data are reported at $\lambda_2=810$2 V for the entangled topology, $\lambda_2=810$3 V for the heralded topology at $\lambda_2=810$4, and $\lambda_2=810$5 V for the heralded topology at $\lambda_2=810$6 (Koni et al., 30 Jul 2025).

The distinction between nonlocal and local topology is operational rather than merely semantic. In the entangled regime, joint tomography and coincidence measurements are essential because the skyrmionic structure is encoded across two photons. In the heralded regime, one photon functions as a trigger and the partner alone carries the reconstructed texture. This duality is the main structural novelty of the platform.

5. Relation to the wider quantum-skyrmion literature

Quantum-skyrmion research outside the dual-wavelength setting has developed several adjacent but different notions of multi-component topology. In spin-1 frustrated magnets, CP$\lambda_2=810$7 skyrmions arise because the local order-parameter manifold is $\lambda_2=810$8 rather than $\lambda_2=810$9, allowing dipolar and quadrupolar sectors to coexist in a single texture. These objects can interpolate between a magnetic core and a nematic periphery, and they may form field-induced skyrmion crystals, but the analysis does not derive two distinct intrinsic wavelengths or two independent modulation magnitudes in one skyrmion (Zhang et al., 2022).

A related extension appears in frustrated quantum dimer systems, where the local dimer Hilbert space leads to a $$\frac{1}{\lambda_1}+\frac{1}{\lambda_2}=\frac{1}{\lambda_0}.$$0 target space and to two field-induced $$\frac{1}{\lambda_1}+\frac{1}{\lambda_2}=\frac{1}{\lambda_0}.$$1 skyrmion crystal phases. There again the texture is genuinely multicomponent and can interpolate between singlet and triplet sectors, but the stabilized crystals are triple-$$\frac{1}{\lambda_1}+\frac{1}{\lambda_2}=\frac{1}{\lambda_0}.$$2 rather than explicitly dual-wavelength (Williams et al., 27 Jun 2025). In operator-based approaches to magnetic quantum skyrmions, the emphasis shifts from internal projective geometry to a bosonic quasiparticle representation in which a classical skyrmion texture is dressed by a “quantum cloud” of spin-flip excitations. That framework clearly separates a core radius $$\frac{1}{\lambda_1}+\frac{1}{\lambda_2}=\frac{1}{\lambda_0}.$$3 and a domain-wall width $$\frac{1}{\lambda_1}+\frac{1}{\lambda_2}=\frac{1}{\lambda_0}.$$4, and it provides a controlled rotated-Fock expansion, but it still does not introduce dual-wavelength skyrmions as such (Haller et al., 2024).

These comparisons delimit the meaning of the optical term. In magnetic systems, “multi-component,” “multi-$$\frac{1}{\lambda_1}+\frac{1}{\lambda_2}=\frac{1}{\lambda_0}.$$5,” “CP$$\frac{1}{\lambda_1}+\frac{1}{\lambda_2}=\frac{1}{\lambda_0}.$$6,” or “CP$$\frac{1}{\lambda_1}+\frac{1}{\lambda_2}=\frac{1}{\lambda_0}.$$7” usually refer to enlarged target manifolds, symmetry-related ordering vectors, or coupled dipolar and non-dipolar sectors. In the photonic dual-wavelength case, by contrast, wavelength itself is promoted to an intrinsic nonseparable resource of the topological state. The common thread is not a shared microscopic mechanism, but the broader idea that skyrmionic topology can survive beyond a single classical field channel.

6. Boundaries, misconceptions, and future directions

A recurring misconception is to equate dual-wavelength with any two-channel skyrmion protocol. The optical-memory work on storing and retrieving optical skyrmions in a cold $$\frac{1}{\lambda_1}+\frac{1}{\lambda_2}=\frac{1}{\lambda_0}.$$8Rb EIT memory used a dual-path architecture, but the two channels were spatial rails rather than two different wavelengths. That experiment established robustness of the skyrmion number under dual-path imbalance, not dual-wavelength topology (Wang et al., 23 Dec 2025). Likewise, proposals for optical detection of quantum skyrmions via Brillouin light scattering analyze asymmetric Stokes and anti-Stokes sidebands around a single input carrier and thereby suggest dual-frequency readout possibilities, but they do not implement a two-color skyrmion state (Sharma et al., 20 Jun 2025).

The dual-wavelength photonic realization also has clear internal limits. The simple state formulas are written for the representative case $$\frac{1}{\lambda_1}+\frac{1}{\lambda_2}=\frac{1}{\lambda_0}.$$9; the paper does not provide a closed-form expression for $$\ket{\Psi} = \sum_{l \in \mathbb{Z}} \sum_{k=1,2} c_{l,k} \ket{l}_A \ket{\rm R}_A \ket{\lambda_k}_A \ket{-l}_B \ket{\rm R}_B \ket{\bar\lambda_k}_B,$$0, nor a separate topological invariant defined directly on the full biphoton Hilbert space. Its principal topological diagnostic remains the reduced Stokes-field Skyrme number. A plausible implication is that a fuller many-body topological classification of nonlocal photonic skyrmions remains open.

At the same time, the demonstrated platform points toward several extensions already suggested in the source material. Because the same liquid-crystal defect is wavelength dependent and voltage tunable, it supports switching among nonlocal skyrmions, heralded local skyrmions, and trivial topology within one architecture. The same framework is further linked to a GHZ-like proposal involving polarization, OAM, and wavelength, indicating that dual-wavelength quantum skyrmions may be one instance of a broader class of multipartite topological quantum states (Koni et al., 30 Jul 2025).