# Topological Qubits as Carriers of Quantum Information in Optics
## 2 Quantum Momentum-Space Winding Number  
Consider a complex-valued electromagnetic field, representing an optical beam. Recall that the winding number \(\
u\) indicates the different equivalence classes in the homotopy group. Each equivalence class consists of the set of closed curves in space for which a point with the parametrized phase circles the origin in the complex plane \(\
u\) times, as the angle in physical space circles the beam axis once. From the quantum perspective, the EM field can be viewed as possessing excited states, corresponding to discrete numbers of particles, called photons. The spin angular momentum of the photons gives rise to linear polarization states, which can become correlated with a winding number. Superpositions of polarization states then lead to superposition of winding number, or in other words, to winding number qubits. One context in which such polarization-based winding number qubits can be formed was demonstrated already in the 1980's [22]. For example, consider light traveling along an optical fibre wound helically into a coil characterized by a helix pitch angle \(\theta\) relative to the local axis of the fiber (Figure 2). The photons propagate with momentum \(\vec{k}(\tau)\) parallel to the fiber axis along an optical path parametrized by arclength \(\tau\) along the fiber. These photons have a general spin state \(|\vec{k},\sigma\rangle\) such that  
\[\vec{s}\cdot\vec{k}|\vec{k}(\tau),\sigma\rangle=\sigma|\vec{k}(\tau),\sigma\rangle, \tag{2}\]  
where eigenvalue \(\sigma\) characterizes the photon helicity. Superpositions of photon helicity states lead to linearly polarized light. This polarized light can then be sent, for instance, into a single-mode isotropic fiber that is helically wound; cf., [23]. The momentum \(\vec{k}\) is of fixed magnitude, but its direction rotates as the fiber is traversed; accordingly \(\vec{k}\) takes a path on the intersection of a plane with the momentum 3-sphere, or in other words, on a circle in momentum space (Figure 3a). In this case the number of windings about the axis in momentum space also corresponds to the windings in physical space. Figure 2: A fiber is wound into a helix of pitch angle \(\theta\). The momentum vector \(\vec{k}(\tau)\) of the light points along the axis of the fiber at each point, and rotates as one moves along the fiber, causing the accumulation of a geometric phase. \(\tau\) parameterized the position along the fiber. One can consider a non-trivial geometrical phase known as the _[PERSON] phase_[24; 25; 26; 27], which arises as a system is taken around a closed loop in parameter space. In his original paper on the subject [24], [PERSON] showed that this phase takes the form  
\[\gamma(C)=-\sigma\Omega(C)\, \tag{3}\]  
where \(C\) is the closed path traced out on the Bloch sphere by the quantum state of the photon as it winds along the curve, and where \(\Omega(C)\) is the solid angle subtended by the curve \(C\) relative to the origin of a relevant Hilbert space. This proportionality between geometric phase and areas or solid angles is generic behavior for geometric phases. In the case of interest here, the parameter being varied adiabatically is the direction of the momentum vector as the light winds around the helix, and the polarization state will trace out a curve \(C\) on the Bloch sphere as in Figure 3b. The Bloch sphere is often used to describe qubit states in a two-state system. In the current case, the basis qubits \(|0\rangle\) and \(|1\rangle\) are represented by vertically and horizontally polarized optical states, respectively. These lie at the poles of the Bloch sphere (which is often called the Poincare sphere in the optical context), with diagonal linear polarizations, circular polarizations, and other equal superpositions of the horizontal and vertical states all lying on the equator.