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arXiv:2601.09279

Geometric Hybrid Poincaré Sphere with Variable Poles

Chihiro Tago, Takashi Kakue, Ken Morita

quant-phSubmitted Jan 14, 2026• Announced Jan 15, 2026

Abstract

We propose a geometric hybrid Poincaré sphere (GHPS) as a unified geometrical framework for describing structured photon states with independently controllable spin angular momentum (SAM) and orbital angular momentum (OAM). Unlike the conventional higher-order Poincaré sphere, in which the SAM and OAM are intrinsically coupled through fixed basis states, the GHPS is constructed by defining its poles as direct products of arbitrary orthogonal bases on the Poincaré sphere (PS) and orbital Poincaré sphere (OPS) and by superposing these pole states. Using numerical simulations, we analyze representative GHPS states and show that the GHPS spherical coordinates govern the amplitude ratio and relative phase between the pole bases. This framework enables spatially inhomogeneous polarization distributions and intensity patterns, including nonseparable structures in which polarization and intensity are intrinsically intertwined, and provides a systematic state-space description for the coherent geometrical control of advanced structured light fields.

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Light carries both spin (polarization) and orbital angular momentum. Combining these degrees of freedom produces hybrid spin-orbit states that live in a high-dimensional Hilbert space, offering greater information capacity and robustness for optical communication, quantum technologies, and metrology. However, generating arbitrary states in these spaces and characterizing them efficiently has remained difficult. Here we show a compact metasurface that generates arbitrary spin-orbit states in a four-dimensional Hilbert space, visualized on a Poincaré hypersphere, with straightforward scalability to higher dimensions. Using a tetratomic unit cell, the single-layer device precisely controls complex amplitude, phase, and polarization. We further introduce an efficient interferometric scheme that reconstructs the full density matrix of any N-dimensional spin-orbit state using only three interferograms. This approach uncovers an intrinsic spin-orbit parity order that governs the symmetry of projected intensity patterns, independent of the weighting of the eigenmodes, and enables controlled mode transformations through higher-order geometric phases. These advances establish a versatile platform for high-dimensional photonic technologies.

Using the similarity between spin states on the Bloch sphere (BS) and polarization states on the Poincare sphere (PS), we construct higher-order spin states on the higher-order BS corresponding to higher-order polarization states of photons on the higher-order PS. We investigate the time evolution of higher-order spin states in a magnetic field and establish an extended form of the conventional Larmor precession. The results provide insights on coherent transfer from extended photons to extended spin qubit systems with spin and orbital angular momenta and the operation of extended spin qubits.

High-dimensional photonic states have significantly advanced the fundamentals and applications of light. However, it remains huge challenges to quantify arbitrary states in high-dimensional Hilbert spaces with spin and orbital angular momentum bases. Here we introduce a geometric method to quantify arbitrary states in a 4D Hilbert space by interferometrically mapping them to unified centroid ellipses. Specifically, nine Stokes parameters can be deduced from three ellipses to quantify the 4D spin-orbit states described by SU(4) Poincaré hypersphere. We verify its feasibility by detecting these spin-orbit states gotten by both free-space wave plates and few-mode fibers. For the first time, we completely quantify and reconstruct higher-order modal group evolution of a weakly guiding few-mode fiber under twist perturbation. This geometric quantification, beyond the classical Stokes polarimetry, may pave the way to multi-dimensional optical metrology, sensing, and high-dimensional classical or quantum communications.