Technopolar Order in Photonic Systems

• Technopolar order is a framework for classifying photonic states where polarization trajectories in Stokes space are organized into topologically distinct phases characterized by winding and linking numbers.
• It employs the manipulation of Stokes parameters and SU(2) symmetry breaking to generate measurable polarization manifolds such as circles, tori, and Möbius strips.
• This approach enables robust control of photonic states for applications in topologically protected optical communications and sensing through direct experimental observation of state transitions.

Technopolar order denotes a class of topological order realized not in band structures or interacting electron systems, but in the space of spin-expectation values (Stokes parameters) of coherent light. It organizes the polarization trajectories of photons into topologically distinct phases—classified by winding numbers, linking numbers, genus, and edge modes—observable directly in Stokes space and robust under polarization-preserving transformations (Saito, 2023). This paradigm enables the classification and control of photonic states based on their trajectory topology rather than momentum-space Berry curvature or conventional mode indices.

1. Stokes Parameters and Polarization Trajectories

The foundational variables for technopolar order are the Stokes parameters, defined from a normalized Jones vector $$|E\rangle = (E_H, E_V)^\mathrm{T} = (\cos\alpha, e^{i\delta}\sin\alpha)$$ in the horizontal (H)–vertical (V) basis. The four Stokes parameters are specified as:

• $S_0 = |E_H|^2 + |E_V|^2$
• $S_1 = |E_H|^2 - |E_V|^2$
• $S_2 = 2\,\mathrm{Re}(E_H E_V^*)$
• $S_3 = 2\,\mathrm{Im}(E_H E_V^*)$

Under unit intensity ($S_0=1$), the triplet $(S_1, S_2, S_3)$ forms a point on the Poincaré sphere $\mathbb{S}^2$ with coordinates: $$S_1 = \cos\gamma,\quad S_2 = \sin\gamma \cos\delta,\quad S_3 = \sin\gamma \sin\delta$$ where $\gamma=2\alpha$ is the polar angle and $\delta$ the azimuth. Thus, the evolution of polarization states traces trajectories on or within the Poincaré sphere.

2. Symmetry Breaking and Topological Structures in Stokes Space

In a system with unbroken SU(2) symmetry (e.g., only wave-plate rotations), the expectation values trace closed orbits on the sphere. Breaking this symmetry, such as by introducing a phase-shifter in a Mach–Zehnder interferometer, yields qualitatively new topology in polarization trajectories.

2.1 Polarization Circle ($\mathbb{S}^1$)

Introducing a single SU(2) phase-shifter with angle $\varphi$ produces a one-dimensional manifold in the Stokes space: $$S(\varphi) = \left(0,\,R(\varphi)\cos\varphi,\,R(\varphi)\sin\varphi\right)$$ where $R(\varphi) = P\cos(\varphi/2)$ (up to loss factors). As $\varphi$ traverses $[0,2\pi]$, $S(\varphi)$ winds once in the $S_2$–$S_3$ plane: a topological circle characterized by the homotopy group $\pi_1(\mathbb{S}^1) = \mathbb{Z}$ with winding number $n=1$.

2.2 Polarization Torus ($\mathbb{S}^1 \times \mathbb{S}^1$)

Cascading a rotator by an independent angle $\theta$ about $S_3$ sweeps the $S^1$ circle into a toroidal surface: $$S(\varphi,\theta) = \left(-R(\varphi) \cos\varphi\, \sin\theta,\; R(\varphi) \cos\varphi\, \cos\theta,\; R(\varphi) \sin\varphi\right)$$ As both $\varphi$ and $\theta$ are periodic, the trajectory forms a torus $T^2 \equiv \mathbb{S}^1 \times \mathbb{S}^1$ embedded in $\mathbb{R}^3$, with $\pi_1(T^2) = \mathbb{Z} \times \mathbb{Z}$, classified by both poloidal and toroidal winding numbers.

3. Higher-Genus and Linked Topological Manifolds

Further manipulations in the polarization control yield higher-genus and linked structures:

• Möbius Strip: Replacing the full phase interval $\varphi\in[0,2\pi]$ with $\varphi\in[0,\pi]$ and gluing ends with a half-twist (upon completing $\theta \in [0,2\pi]$) produces a non-orientable Möbius strip in Stokes space with $\pi_1 = \mathbb{Z}$ and Euler characteristic $\chi=0$.
• Hopf Link: Constructing two disjoint circles in orthogonal subspaces, e.g., $C_1(\theta)=(0,r_1\cos\theta, r_1\sin\theta)$ and $C_2(\psi)=(r_2\cos\psi,0, r_2\sin\psi)$, yields configurations with linking number $\mathrm{Lk}(C_1, C_2) = 1$.

These topological invariants classify polarization trajectories beyond simple parametric curves.

4. Topological Phase Transitions and Bulk–Edge Correspondence

Introducing a nonunitary amplitude controller in the interferometer allows sculpting of the intensity $S_0(\varphi)$ to vanish at an isolated value $\varphi = \varphi_D$. This creates a Dirac-type linear cone in the $(S_1, S_2, S_3)$ space—the photonic analogue of a Dirac point. The cone mediates a topological transition in the genus of polarization manifolds, serving as the edge state connecting the bulk phases: torus ($g=1$, $\chi=0$) and sphere ($g=0$, $\chi=2$). The Dirac cone thus implements a bulk–edge correspondence analogous to that found in topological condensed matter, but realized in polarization space for coherent bosonic fields.

5. Experimental Realizations and Detection

Technopolar order is observable with fiber-integrated Mach–Zehnder interferometers using passive optical components:

• A polarization-independent splitter divides the beam.
• Phase-shifter (two QWPs and two HWPs) implements SU(2) rotations, producing polarization circles.
• Additional HWP pairs serve as rotators for elevation to tori.
• Amplitude control (nonunitary U(2) element) enables Dirac-cone transitions.

These setups allow direct measurement of Stokes-space trajectories and their topological invariants (winding, linking, Euler characteristic) via polarimetry, with the resulting structures robust to loss and polarization-preserving operations (Saito, 2023).

6. Classification and Topological Invariants

Technopolar order is demarcated by a set of topological invariants:

• Winding number $n\in\mathbb{Z}$: Characterizes trajectories along polarization circles ($\mathbb{S}^1$).
• Fundamental group $\pi_1=\mathbb{Z}\times\mathbb{Z}$: Characterizes polarization tori ($T^2$).
• Linking number $\mathrm{Lk}\in\mathbb{Z}$: Distinguishes Hopf-linked polarization loops.
• Euler characteristic $\chi=2-2g$: Differentiates genus-$g$ manifolds (e.g., $\chi=2$ for sphere, $\chi=0$ for torus).
• Bulk–edge correspondence: Dirac cones act as edge states connecting manifolds of different genus.

7. Significance, Perspective, and Applications

Technopolar order transcends traditional classifications of photonic states. Rather than indexing by modal structure or Berry curvature, this framework classifies polarization dynamics using their embedding topology in real Stokes space. The relevant invariants are directly measurable, robust, and offer a route to topologically protected optical communications and sensing architectures. The methodology generalizes beyond photons; any system whose states traverse a nontrivial manifold in the space of expectation values admits analogous technopolar topological order (Saito, 2023).