Pancharatnam Topological Charge

• Pancharatnam topological charge is a quantized geometric signature that represents the net solid angle swept by a state’s evolution on spherical state spaces.
• It underpins advanced photonics applications by enabling the generation of complex beam profiles, such as optical vortices and polarization textures, through engineered phase gradients.
• Its measurement through interference and polarization-resolved techniques facilitates the creation of robust, topologically protected modes in quantum and condensed matter systems.

The Pancharatnam topological charge is a quantized geometric signature associated with the evolution of a physical system’s state—typically polarization or spin—on its projective (often spherical) state space. Arising fundamentally from the Pancharatnam-Berry phase, it manifests in a wide range of fields, from quantum optics and condensed matter to quantum information and topological photonics. This topological charge encodes the net “winding” or solid angle subtended by the trajectory of the state on its associated sphere and is directly linked to the accumulation of achromatic geometric phases, robustness against local perturbations, and the generation of topologically protected modes or textures.

1. Geometric and Topological Foundations

The Pancharatnam phase, first established in the context of interference between differently polarized light beams, quantifies the geometric phase acquired when a quantum or classical state traces a path on its state space. For a closed evolution, the measurable phase is proportional to half the solid angle Ω subtended by the path on the Poincaré or Bloch sphere: $\varphi_P = \frac{1}{2}\Omega\;.$ This phase is insensitive to the dynamical details of evolution and depends only on the global geometry of the trajectory. The associated Pancharatnam topological charge is the integer (or half-integer) quantization of such a phase, reflecting a discrete, gauge-invariant, and origin-independent property of the completed path (Lages et al., 2013, Vyas et al., 2019, Ferrer-Garcia et al., 2022).

In many practical cases—such as systems possessing inversion or time-reversal symmetry—this charge is strictly quantized (e.g., 0 or π for bands with inversion symmetry), marking topologically distinct states or phases (Vyas et al., 2019).

2. Manifestations in Photonics and Polarization Optics

In photonics, the Pancharatnam topological charge underlies a variety of polarization-dependent phenomena, including spin–orbit interactions, beam steering, and structured light generation. For instance, Pancharatnam–Berry (PB) optical elements and metasurfaces impart position-dependent geometric phase shifts by locally rotating the optical axis or nanostructure orientation, encoding a phase of the form: $\text{PB phase} = \pm 2\theta(x, y)$ where the sign and factor depend on the polarization handedness and local orientation. This locally imparted phase imbues the transmitted or reflected beam with a topological charge that determines its orbital angular momentum or polarization texture (Capaldo et al., 2018, Gao et al., 2021).

Of particular note is the ability to create complex beam profiles, such as optical vortices and skyrmions, by spatially engineering the geometric phase landscape. The topological charge here directly determines the number and type of helical wavefronts or polarization singularities present in the output field (Zou et al., 2 May 2025).

3. Experimental Observation and Engineering

A variety of experimental strategies have been developed to measure and exploit the Pancharatnam topological charge. In intensity interferometry, the charge manifests nonlocally: coincidence detection between spatially separated detectors yields interference patterns modulated by the geometric phase accumulated jointly by pairs of photons, with no signature observable in local (single-photon) measurements. The nonlocal phase is given by the half-solid angle closed by polarization state projections on the Poincaré sphere and is tunable via the relative orientation of polarizers or waveplates (Mehta et al., 2010).

In engineered photonic elements, the spatial structure of the metasurface or waveguide is tuned so the geometric phase gradients impart a spin-dependent transverse momentum—enabling spin–Hall effect beam splitting or robust, index-free waveguiding. For instance, PB phase waveguides with longitudinally periodic optic axis modulation generate effective transverse potentials proportional to the geometric phase gradient, guiding light by “topological” rather than index contrast (Jisha et al., 2022, Jisha et al., 2018).

4. Topological Charge in Quantum and Condensed Matter Systems

In condensed matter, the analog of the Pancharatnam topological charge appears in the context of Berry curvature and Chern numbers for electronic bands. Here, the “Pancharatnam-Berry curvature” is given by the curl of the Berry connection, with quantized topological invariants (Chern numbers) corresponding to net phase windings around the Brillouin zone: $$\mathcal{B}(\mathbf{k}) = \sum_n p_n \pi \delta^{(2)}(\mathbf{k} - \mathbf{K}_n)\;,$$ where $p_n$ is the vorticity or flux associated with Dirac points $\mathbf{K}_n$ (Sriluckshmy et al., 2013). These invariants are robust and characterize topological quantum phases, Lifshitz transitions, and edge state structures.

The Pancharatnam-Zak phase refines the Zak phase for one-dimensional lattices by including the argument of the overlap between initial and final Bloch states, resulting in a gauge-invariant and origin-independent measure of band topology (Vyas et al., 2019). This Pancharatnam-Zak phase is in direct correspondence with a quantized “topological charge” in systems with crystal inversion symmetry.

5. Dynamical Generation, Control, and Measurement

A central feature of the Pancharatnam topological charge is its controllability via system parameters. In quantum optics, the degree of entanglement between photon pairs or classical modes can be tuned by adjusting the geometric phase through waveplate orientation, q-plates, or generalized measurement sequences (Ferrer-Garcia et al., 2022, Perumangatt et al., 2016). In structured light, amplitude–phase decoupled PB elements (checkerboard-encoded designs) provide on-demand longitudinal transformations and the ability to realize evolving polarization textures, including skyrmion lattices with dynamic topological charge (Zou et al., 2 May 2025).

Crucially, the charge can be measured directly via interference experiments (in many cases, via nonlocal correlations) or inferred from the polarization structure using Stokes parameter measurements, polarization-resolved imaging, or through the detection of edge modes and Hall currents in matter-wave systems (Leonard et al., 2017, Zhang et al., 2014).

6. Multibody and Non-Abelian Extensions

Multiparticle generalizations of the Pancharatnam phase—such as those realized in quantum spin Hall systems with spin-polarized electrodes—lead to novel multi-electron (or multi-photon) Aharonov–Bohm effects in spin space. Here, the interference is governed by the geometric phase accumulated around closed loops on the generalized state space (e.g., the Bloch sphere for spins), with topological invariants dictating the modulation of current and noise correlations, robust even in the presence of strong orbital dephasing (Wadhawan et al., 2017, Mehta et al., 2010).

Similarly, when sequences of weak measurements drive a single qubit (polarization state) in cyclic paths on the Bloch sphere, a quantized topological transition in the acquired geometric phase marks a change in the Pancharatnam topological charge (Ferrer-Garcia et al., 2022).

7. Applications and Impact

The concept of Pancharatnam topological charge is now a unifying principle in photonic device design—enabling efficient, deterministic generation of structured light with prescribed modal and polarization properties, quantum logic based on topological invariants, robust quantum state transfer, and meta-optical architectures exploiting polarization-dependent symmetry breaking (Gao et al., 2021, Capaldo et al., 2018). In quantum and classical communication, precise control and detection of topological charge offer routes to dense information encoding, robust transmission, and dynamical reconfiguration.

In summary, the Pancharatnam topological charge encodes the global geometric properties of state-space evolution—often realized as half the solid angle swept on the relevant sphere—imparting quantized, robust phases to physical observables. Its manifestations bridge quantum optics, condensed matter, and photonic engineering, offering a foundation for next-generation topological devices and quantum technologies.