Poincaré-Bloch Sphere

• Poincaré-Bloch sphere is a unified geometric representation where fully polarized light and qubit states are mapped onto a unit sphere using Jones vectors and Stokes parameters.
• It leverages SU(2)/SO(3) symmetries to model both unitary and non-unitary operations, enabling a consistent method to describe polarization and quantum state evolution.
• Recent generalizations extend the model to hybrid and higher-order modes, facilitating advanced applications in quantum encoding, mode multiplexing, and structured light analysis.

The Poincaré-Bloch sphere refers to the unified geometric representation of polarization states in classical optics and quantum two-level systems ("qubits") as points on a unit sphere in three-dimensional space, capturing orientation and ellipticity for fully polarized light, or the pure-state manifold of a qubit. In polarization optics, the Jones vector and Stokes parameters map one-to-one onto this sphere, isomorphic to the quantum Bloch sphere, and both share the symmetry structure of SU(2)/SO(3) rotations. Recent extensions generalize the concept to hybrid, higher-order, and generalized spheres encoding structured modes with orbital angular momentum (OAM), spatial degrees of freedom, and non-unitary transformations, subsuming classical, quantum, and spin–orbit coupled systems into a single geometric framework (Kuntman et al., 2022, Yi et al., 2014, Ren et al., 2014, Kim et al., 2013, Cardoso-Isidoro et al., 29 Dec 2025).

1. Basic Formalism: Jones, Stokes, and Bloch Sphere Equivalence

In the standard representation, any fully polarized light state is described by a Jones vector $|\psi\rangle = (\alpha, \beta)^\mathrm{T}$ (with $|\alpha|^2 + |\beta|^2 = 1$) or equivalently by the real four-component Stokes vector $S = (S_0, S_1, S_2, S_3)^\mathrm{T}$, where $S_1$, $S_2$, $S_3$ refer to projections onto Cartesian axes and $S_0$ is intensity. Normalized states set $S_0=1$ and define the Bloch vector $r = (S_1,S_2,S_3)^\mathrm{T}$ obeying $|r|=1$, mapping each pure polarization/qubit state to a point on the unit sphere (Kuntman et al., 2022, Castillo et al., 2013).

Standard spherical coordinates (for optics):

$$S_1 = \cos 2\theta \cos 2\varphi,\quad S_2 = \sin 2\theta \cos 2\varphi,\quad S_3 = \sin 2\varphi,$$

with $\theta$ half-longitude and $\varphi$ half-latitude. For qubits, the Bloch state $$|\psi(\theta, \phi)\rangle = \cos(\theta/2)|0\rangle + e^{i\phi}\sin(\theta/2)|1\rangle$$, the Bloch vector components are $$x = \sin\theta \cos\phi, y = \sin\theta \sin\phi, z = \cos\theta$$.

The Poincaré sphere thus provides a direct visual mapping of all pure polarization states, with antipodal points corresponding to orthogonal polarization/qubit states. SU(2) (unitary) operations correspond to rotations of the sphere, realized by optical elements in classical polarization or by Hamiltonian evolution in quantum mechanics (Castillo et al., 2013).

2. Stokes Vector, Lorentz Symmetries, and Mixed-State Geometry

General states (including partial polarization or decoherence) are described by the 2×2 Hermitian coherency (density) matrix $C$ in optics, and by the density matrix $\rho$ for a qubit, both expanded in the Pauli basis. The normalized Stokes vector components $r_i = S_i/S_0$ with $|r| \leq 1$ describe purity; mixed optical or quantum states contract the radius of the sphere’s image.

The Stokes 4-vector $(S_0, S_1, S_2, S_3)$ transforms as a Minkowski four-vector under the Lorentz group O(3,1), with Lorentz-invariant determinant $\det C = S_0^2 - S_1^2 - S_2^2 - S_3^2$ (Kim, 2012, Kim et al., 2013). Depolarization (classical) and decoherence (quantum) uniformly contract the sphere, with Bloch radius $f = \sqrt{1 - 4 \det C / (\mathrm{Tr} C)^2}$, directly related to von Neumann entropy (Kim, 2012). Feynman's "rest of the universe" is represented geometrically as the complementary sphere of lost coherence or entanglement (Kim, 2012).

3. Higher-Order and Hybrid Poincaré-Bloch Spheres

The conventional sphere is extended to account for vector beams and spin–orbit coupled states. In the hybrid-order formalism, states are labeled by both spin angular momentum (SAM) and orbital angular momentum (OAM) quantum numbers: the poles now correspond to orthonormal spin–orbit eigenstates (e.g., $|N_\ell\rangle$ and $|S_m\rangle$), and arbitrary superpositions describe points on a generalized sphere (Yi et al., 2014). Hybrid Stokes parameters are defined analogously, and the sphere’s surface charts all such superpositions, including the familiar polarization sphere (spin only), orbital sphere (OAM only), and high-order spheres for vector beams.

The Berry connection and curvature extend to this context, yielding the Pancharatnam–Berry phase proportional to net change of total angular momentum across a cyclic circuit on the sphere,

$$\gamma(C) = -\frac{\ell-(m+2\sigma)}{4}\,\Omega(C),$$

where $\Omega(C)$ is the enclosed solid angle and $\ell-(m+2\sigma)$ quantifies angular-momentum variation on the evolution path (Yi et al., 2014).

4. Generalization: G Sphere and Structured Light

The "G sphere" is a more flexible geometric object that unifies all vector-field states: it employs a radial coordinate $R$ labeling shells of given ellipticity/SAM and OAM quantum number $m$, such that every vector field is indexed by $(R, 2\theta, 2\varphi)$. Each shell is a conventional Poincaré sphere, with poles representing orthogonal vector basis states of different ellipticity and opposite OAM; continuous $R$ allows seamless description of hybrid and exotic polarization patterns (Ren et al., 2014).

Generalized Stokes ("G") parameters are defined: $$G_{1R}^m = R \sin2\beta \cos2\phi_0,\quad G_{2R}^m = R \sin2\beta \sin2\phi_0,\quad G_{3R}^m = -R \cos2\beta,$$ with the vector’s position $(G_1, G_2, G_3)$ lying on a sphere of radius $R$ (Ren et al., 2014). Pancharatnam–Berry phase is generalized to arbitrary paths within (or across) shells, with the phase given by half the solid angle in shell $R$.

This G sphere construction supports high-dimensional quantum encoding, robust geometric gates for mode multiplexing, and classification of all structured light fields (Ren et al., 2014).

5. Non-Unitary Operations: Z Matrix Formalism

Optical elements enact both unitary (birefringent) and non-unitary (diattenuating/polarizer) transformations on the Poincaré sphere. The 4×4 $\mathbf{Z}$ matrix generalizes the Jones matrix action to the Stokes space: for a nondepolarizing medium,

$\mathbf{M} = \mathbf{Z}\mathbf{Z}^*,$

with $\mathbf{Z}$ acting linearly on the four-component Stokes vector, storing both amplitude and phase (Kuntman et al., 2022). Compact forms for pure polarizer and retarder $Z$ matrices are given in terms of sphere coordinates, with rank-one projectors for diattenuators and SO(3) rotations for birefringent elements.

Unitary $Z$ operations rotate the Stokes vector; non-unitary operations project it onto the diattenuation axis. The geometric phase acquired by closed sequences is computed as $-\frac{1}{2}\,\Omega$, unifying the treatment of unitary and non-unitary processes (Kuntman et al., 2022).

6. Structured Spatial Modes and Orbital Poincaré Spheres

Optical beams with spatial structure (e.g., Laguerre–Gaussian modes, doughnut beams) can be organized via an OAM Poincaré sphere, with basis modes $u_{\pm\ell, 0}$ and general state superpositions mapped onto sphere coordinates $(\theta, \phi)$. Nonlinear wave-mixing processes (e.g., four-wave mixing) induce specific symmetries: reflection, doubling, and complex mapping of input-output mode coordinates, illustrated in experimental observations (Motta et al., 2023). This sphere is mathematically equivalent to the polarization sphere, but built on spatial-mode degrees of freedom.

Geodesics, concatenations of SU(2) rotations (e.g., $\pi$-converters, Dove prisms), and nontrivial topological phase effects extend the utility of the sphere to robust interferometric and quantum-gate schemes (Voitiv et al., 2022).

7. Experimental and Quantum Information Applications

Full sphere access has been demonstrated with time-dependent polarization sweeps employing Rabi oscillations, mapping engineered pulse delays and polarizations to arbitrary Poincaré trajectories (Colas et al., 2014). CHSH Bell tests spanning the full sphere (not just linear bases) verify quantum nonlocality and entanglement witness protocols for arbitrary polarization qubits, with Tsirelson bound violations observed for both Bell and certain non-Bell maximally entangled states (Cardoso-Isidoro et al., 29 Dec 2025).

High-dimensional spheres underlie quantum media conversion (photon-spin transfer), novel qudit encodings, spatial-mode multiplexing in optical communications, and metrological schemes exploiting SAM-OAM hybridization (Yi et al., 2014, Sato et al., 2023).

8. Group Structure and Extensions: Lorentz and de Sitter Symmetries

The two-by-two matrix formalism underpinning both the Poincaré and Bloch spheres maps directly to the Lorentz group (O(3,1)) acting on the Stokes four-vector. Quantum decoherence in the Poincaré sphere is linked to mass mixing in a de Sitter (O(3,2)) symmetry extension, with sphere contractions corresponding to trajectories in higher-dimensional coset spaces (Kim, 2015, Kim, 2012). This framework intertwines polarization optics, Lorentz-group symmetries, internal space-time degrees, and entropy in a unified geometric language.

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In summary, the Poincaré-Bloch sphere is the central geometric object unifying polarization optics (Jones, Stokes, Mueller matrix), quantum two-level systems, and angular-momentum-carrying modes via SU(2)/SO(3) spheres and their generalizations. Its extensions—hybrid, higher-order, generalized spheres—increase the descriptive and operational capacity to treat spin–orbit interactions, spatial vector modes, geometric phase, decoherence, and advanced quantum information applications (Kuntman et al., 2022, Yi et al., 2014, Ren et al., 2014, Kim et al., 2013, Cardoso-Isidoro et al., 29 Dec 2025, Sato et al., 2023).