Spin-Stokes Hybrid Correlations

• Spin–Stokes hybrid correlations are defined as the measurable interdependence between local spin angular momentum and polarization, enabling the formation of hybrid topological structures such as skyrmions.
• They emerge in systems where the electric field’s vector structure, photonic spin, and polarization interlock via spin–orbit coupling to generate complex textures like meron pairs and Bloch-type skyrmions.
• This phenomenon offers applications in high-dimensional optical encoding, topological photonics, and experimental tests of relativistic quantum nonlocality through hybrid photon–fermion entanglement.

Spin–Stokes hybrid correlations describe the interdependence between the local spin angular momentum density and the polarization (as characterized by Stokes parameters) of electromagnetic fields or, more broadly, between spin and polarization observables in hybrid quantum or classical systems. These correlations are notably manifested in systems where multiple degrees of optical or quantum freedom—such as electric field vectorial structure, photonic spin, and polarization states—interact through underlying couplings, enabling the emergence of topologically nontrivial textures such as hybrid skyrmions. They also arise in hybrid quantum systems involving coupled photon and fermion subsystems, demonstrating quantum-entangled spin–polarization observables that exhibit nonclassical correlation patterns and relativistic effects.

1. Mathematical Structure and Definitions

In free-space optical settings, the local electric field vector $\mathbf{E}(\mathbf{r})$ at position $\mathbf{r} = (x, y, z)$ defines the electromagnetic state: $$\mathbf{E}(\mathbf{r}, t) = \mathrm{Re}\{\mathbf{E}(\mathbf{r}) e^{-i\omega t}\},\quad \mathbf{E}(\mathbf{r}) = E_x \hat{\mathbf{x}} + E_y \hat{\mathbf{y}} + E_z \hat{\mathbf{z}}$$ The photon spin angular momentum (SAM) density is given by

$$\mathbf{S}(\mathbf{r}) = \frac{1}{4\omega}\, \mathrm{Im} \bigl\{ \varepsilon\, \mathbf{E}^*(\mathbf{r}) \times \mathbf{E}(\mathbf{r}) + \mu\, \mathbf{H}^*(\mathbf{r}) \times \mathbf{H}(\mathbf{r}) \bigr\}$$

where $\omega$ is the optical frequency, and $\varepsilon$, $\mu$ denote the permittivity and permeability.

The local polarization is compactly described by the Stokes vector $\mathbf{S}_t(\mathbf{r}) = (S_1, S_2, S_3)$, with parameters: $$S_1 = |E_x|^2 - |E_y|^2,\quad S_2 = 2\,\mathrm{Re}(E_x^* E_y),\quad S_3 = 2\,\mathrm{Im}(E_x^* E_y)$$ A normalized Stokes direction is $\mathbf{n}_t = \mathbf{S}_t/|\mathbf{S}_t|$.

In quantum hybrid systems (e.g., photon–fermion EPR pairs), the fermion spin is characterized by a relativistic, frame-invariant operator (such as the Pauli–Lubanski construction), while the photon’s polarization is represented by a Stokes-like operator acting on transverse photon polarization states.

Spin–Stokes hybrid correlations are operationalized as scalar products or cross-correlations of the form: $\mathbf{r} = (x, y, z)$0 or, in the quantum case, as joint expectation values

$\mathbf{r} = (x, y, z)$1

where measurements are performed on spatially separated spin and polarization observables.

2. Topological Skyrmions and Hybrid Textures

Spin–Stokes correlations play a central role in structured light fields supporting multiple topological textures—optical skyrmions—arising from the interplay of field, spin, and polarization degrees of freedom. A skyrmion number (topological charge) is defined for any normalized three-component vector field $\mathbf{r} = (x, y, z)$2 as

$\mathbf{r} = (x, y, z)$3

where, for spin-skyrmions, $\mathbf{r} = (x, y, z)$4, and for Stokes-skyrmions, $\mathbf{r} = (x, y, z)$5.

Hybrid optical skyrmions are realized by diffracting vector vortex beams (with controlled spin $\mathbf{r} = (x, y, z)$6 and orbital charge $\mathbf{r} = (x, y, z)$7) through an annular aperture. Through spin–orbit coupling in free space, it is possible to simultaneously generate electric field skyrmions, spin skyrmions (e.g., Bloch-type, $\mathbf{r} = (x, y, z)$8), and higher-order Stokes skyrmions ($\mathbf{r} = (x, y, z)$9) in the same light field, with substantial spatial overlap between spin and Stokes topologies (Yao et al., 2024).

The local spin–Stokes cross-correlation $$\mathbf{E}(\mathbf{r}, t) = \mathrm{Re}\{\mathbf{E}(\mathbf{r}) e^{-i\omega t}\},\quad \mathbf{E}(\mathbf{r}) = E_x \hat{\mathbf{x}} + E_y \hat{\mathbf{y}} + E_z \hat{\mathbf{z}}$$0 is maximized when strong spin–orbit coupling interlocks the handedness of the polarization ellipse with the orientation of the SAM. For example, incident beams with $$\mathbf{E}(\mathbf{r}, t) = \mathrm{Re}\{\mathbf{E}(\mathbf{r}) e^{-i\omega t}\},\quad \mathbf{E}(\mathbf{r}) = E_x \hat{\mathbf{x}} + E_y \hat{\mathbf{y}} + E_z \hat{\mathbf{z}}$$1 yield nearly perfect core-region alignment, while $$\mathbf{E}(\mathbf{r}, t) = \mathrm{Re}\{\mathbf{E}(\mathbf{r}) e^{-i\omega t}\},\quad \mathbf{E}(\mathbf{r}) = E_x \hat{\mathbf{x}} + E_y \hat{\mathbf{y}} + E_z \hat{\mathbf{z}}$$2 produces negligible coupling.

3. Experimental and Numerical Realizations

Structured beams are generated by illuminating annular apertures (e.g., inner radius $$\mathbf{E}(\mathbf{r}, t) = \mathrm{Re}\{\mathbf{E}(\mathbf{r}) e^{-i\omega t}\},\quad \mathbf{E}(\mathbf{r}) = E_x \hat{\mathbf{x}} + E_y \hat{\mathbf{y}} + E_z \hat{\mathbf{z}}$$3, outer radius $$\mathbf{E}(\mathbf{r}, t) = \mathrm{Re}\{\mathbf{E}(\mathbf{r}) e^{-i\omega t}\},\quad \mathbf{E}(\mathbf{r}) = E_x \hat{\mathbf{x}} + E_y \hat{\mathbf{y}} + E_z \hat{\mathbf{z}}$$4) with laser sources (e.g., $$\mathbf{E}(\mathbf{r}, t) = \mathrm{Re}\{\mathbf{E}(\mathbf{r}) e^{-i\omega t}\},\quad \mathbf{E}(\mathbf{r}) = E_x \hat{\mathbf{x}} + E_y \hat{\mathbf{y}} + E_z \hat{\mathbf{z}}$$5). The incident vector vortex beam is parameterized by spin angular momentum $$\mathbf{E}(\mathbf{r}, t) = \mathrm{Re}\{\mathbf{E}(\mathbf{r}) e^{-i\omega t}\},\quad \mathbf{E}(\mathbf{r}) = E_x \hat{\mathbf{x}} + E_y \hat{\mathbf{y}} + E_z \hat{\mathbf{z}}$$6 (circular or linear polarization), and orbital angular momentum $$\mathbf{E}(\mathbf{r}, t) = \mathrm{Re}\{\mathbf{E}(\mathbf{r}) e^{-i\omega t}\},\quad \mathbf{E}(\mathbf{r}) = E_x \hat{\mathbf{x}} + E_y \hat{\mathbf{y}} + E_z \hat{\mathbf{z}}$$7. The resulting field distribution beyond the aperture is computed using finite-difference time-domain (FDTD) simulations and measured on planes at specific distances (e.g., $$\mathbf{E}(\mathbf{r}, t) = \mathrm{Re}\{\mathbf{E}(\mathbf{r}) e^{-i\omega t}\},\quad \mathbf{E}(\mathbf{r}) = E_x \hat{\mathbf{x}} + E_y \hat{\mathbf{y}} + E_z \hat{\mathbf{z}}$$8).

Key empirical findings include

• $$\mathbf{E}(\mathbf{r}, t) = \mathrm{Re}\{\mathbf{E}(\mathbf{r}) e^{-i\omega t}\},\quad \mathbf{E}(\mathbf{r}) = E_x \hat{\mathbf{x}} + E_y \hat{\mathbf{y}} + E_z \hat{\mathbf{z}}$$9: emergence of Néel-type electric field skyrmions.
• $$\mathbf{S}(\mathbf{r}) = \frac{1}{4\omega}\, \mathrm{Im} \bigl\{ \varepsilon\, \mathbf{E}^*(\mathbf{r}) \times \mathbf{E}(\mathbf{r}) + \mu\, \mathbf{H}^*(\mathbf{r}) \times \mathbf{H}(\mathbf{r}) \bigr\}$$0: formation of electric field meron pairs (topological charge $$\mathbf{S}(\mathbf{r}) = \frac{1}{4\omega}\, \mathrm{Im} \bigl\{ \varepsilon\, \mathbf{E}^*(\mathbf{r}) \times \mathbf{E}(\mathbf{r}) + \mu\, \mathbf{H}^*(\mathbf{r}) \times \mathbf{H}(\mathbf{r}) \bigr\}$$1), with coexisting Bloch-type spin skyrmion ($$\mathbf{S}(\mathbf{r}) = \frac{1}{4\omega}\, \mathrm{Im} \bigl\{ \varepsilon\, \mathbf{E}^*(\mathbf{r}) \times \mathbf{E}(\mathbf{r}) + \mu\, \mathbf{H}^*(\mathbf{r}) \times \mathbf{H}(\mathbf{r}) \bigr\}$$2) and second-order Stokes skyrmion ($$\mathbf{S}(\mathbf{r}) = \frac{1}{4\omega}\, \mathrm{Im} \bigl\{ \varepsilon\, \mathbf{E}^*(\mathbf{r}) \times \mathbf{E}(\mathbf{r}) + \mu\, \mathbf{H}^*(\mathbf{r}) \times \mathbf{H}(\mathbf{r}) \bigr\}$$3).

Spin and Stokes textures rotate in opposite senses as the observation plane $$\mathbf{S}(\mathbf{r}) = \frac{1}{4\omega}\, \mathrm{Im} \bigl\{ \varepsilon\, \mathbf{E}^*(\mathbf{r}) \times \mathbf{E}(\mathbf{r}) + \mu\, \mathbf{H}^*(\mathbf{r}) \times \mathbf{H}(\mathbf{r}) \bigr\}$$4 is scanned, directly demonstrating underlying spin–orbit coupling (Yao et al., 2024).

4. Relativistic Hybrid Correlations in Quantum Systems

In relativistic quantum settings, spin–Stokes hybrid correlations arise naturally in the decay of a spin-½ parent particle into a photon (with well-defined polarization) and a massive fermion (with spin). The joint state after decay is constructed as a Lorentz-covariant bispinor projecting onto a joint photon–fermion Hilbert space.

The two-particle relativistic correlation function,

$$\mathbf{S}(\mathbf{r}) = \frac{1}{4\omega}\, \mathrm{Im} \bigl\{ \varepsilon\, \mathbf{E}^*(\mathbf{r}) \times \mathbf{E}(\mathbf{r}) + \mu\, \mathbf{H}^*(\mathbf{r}) \times \mathbf{H}(\mathbf{r}) \bigr\}$$5

quantifies the expectation value for measuring the Dirac spin along direction $$\mathbf{S}(\mathbf{r}) = \frac{1}{4\omega}\, \mathrm{Im} \bigl\{ \varepsilon\, \mathbf{E}^*(\mathbf{r}) \times \mathbf{E}(\mathbf{r}) + \mu\, \mathbf{H}^*(\mathbf{r}) \times \mathbf{H}(\mathbf{r}) \bigr\}$$6 and the photon Stokes observable at angle $$\mathbf{S}(\mathbf{r}) = \frac{1}{4\omega}\, \mathrm{Im} \bigl\{ \varepsilon\, \mathbf{E}^*(\mathbf{r}) \times \mathbf{E}(\mathbf{r}) + \mu\, \mathbf{H}^*(\mathbf{r}) \times \mathbf{H}(\mathbf{r}) \bigr\}$$7. The vectors $$\mathbf{S}(\mathbf{r}) = \frac{1}{4\omega}\, \mathrm{Im} \bigl\{ \varepsilon\, \mathbf{E}^*(\mathbf{r}) \times \mathbf{E}(\mathbf{r}) + \mu\, \mathbf{H}^*(\mathbf{r}) \times \mathbf{H}(\mathbf{r}) \bigr\}$$8 and $$\mathbf{S}(\mathbf{r}) = \frac{1}{4\omega}\, \mathrm{Im} \bigl\{ \varepsilon\, \mathbf{E}^*(\mathbf{r}) \times \mathbf{E}(\mathbf{r}) + \mu\, \mathbf{H}^*(\mathbf{r}) \times \mathbf{H}(\mathbf{r}) \bigr\}$$9 encode both the spin-polarization of the parent and the momenta of the emitted particles (Caban et al., 2010).

Importantly, relativistic effects induce nontrivial momentum dependence: local extrema in $\omega$0 appear at intermediate fermion velocities (e.g., $\omega$1), leading to observable modulation and even enhancement of quantum nonlocality indicators such as the CHSH inequality.

5. Parameter Dependence and Optimization

The magnitude and spatial character of spin–Stokes hybrid correlations are strongly contingent on experimental parameters:

• Aperture parameters and beam quantum numbers: Adjusting the annular aperture dimensions or tuning the spin–orbital quantum numbers $\omega$2 controls the emergence and sign of skyrmionic textures, as well as the degree of spin–Stokes coupling. Reversal of either spin or orbital quantum number flips the topological indices ($\omega$3) for all associated skyrmions (Yao et al., 2024).
• Observation plane and phase delay: Skyrmion numbers and texture radii (e.g., $\omega$4 for spin, $\omega$5 for Stokes) are functions of phase delay and propagation distance, with reported $\omega$6 and $\omega$7 values maintaining stability above 0.99 across practical regions.
• Relativistic velocity: In the hybrid photon–fermion scenario, the degree of Bell violation (CHSH S-parameter) varies nonmonotonically with fermion velocity, reaching maxima (e.g., $\omega$8 at $\omega$9) before converging to classical bounds at ultrarelativistic or nonrelativistic limits (Caban et al., 2010).

These dependencies enable tuning of hybrid correlations for specific operational criteria or physical regimes.

6. Applications and Implications

Spin–Stokes hybrid correlations underpin several prospective advances:

• High-dimensional optical information encoding: The simultaneous support of field, spin, and Stokes skyrmions in a single structured field creates multi-degree-of-freedom states robust to perturbations. These can be leveraged for protected information channels, subwavelength imaging, or quantum information schemes combining polarization and orbital degrees of freedom (Yao et al., 2024).
• Topological photonics and Hall-effect-like routing: Hybrid skyrmions suggest new routes for nonreciprocal or protected light routing in photonic chips, with spin–Stokes coupling serving as a handle for topological control.
• Quantum foundations and relativistic entanglement: Observation of nonclassical Bell violations in photon–fermion hybrid states at moderate relativistic energies provides a platform for testing quantum nonlocality with distinctly relativistic, hybrid-spin–polarization coupling (Caban et al., 2010).

A plausible implication is that optical systems engineered to maximize $\varepsilon$0 through spin–orbit interaction can serve as experimental testbeds for both classical and quantum topological phenomena.

7. Comparative Table of Key Results

System	Topological Metrics	Maximal Correlation Parameter
Free-space optics	$\varepsilon$1, $\varepsilon$2	$\varepsilon$3 near spin–orbit "hot spot" (Yao et al., 2024)
Quantum hybrid (photon–fermion)	$\varepsilon$4 (CHSH)	At $\varepsilon$5, correlation extremum (Caban et al., 2010)

Spin–Stokes hybrid correlations thus constitute a unifying principle in structured optics and hybrid quantum systems, with substantial ramifications for topological photonics, quantum information, and fundamental experiments on relativistic entanglement.