# Topological Qubits as Carriers of Quantum Information in Optics
## 2 Quantum Momentum-Space Winding Number
If \(C\) is a circle subtending a cone with the origin as apex, and \(\theta\) is the helix pitch angle relative to the local axis of the fiber, then geometric arguments [22] show that the solid angle subtended by \(C\) is given by  
\[\Omega(C)=2\pi\
u(1-\cos\theta). \tag{4}\]  
The geometric phase, Equation (3), is therefore proportional to the winding number. Consider a physical situation in which laser photons enter a non-birefringingent material in such a uniformly helically-wound fiber in the linear polarization superposition state  
\[|\psi\rangle=\frac{1}{\sqrt{2}}\bigg{(}|\uparrow\rangle+|\rightarrow\rangle \bigg{)}. \tag{5}\]  
Ignoring any overall dynamical phase factors, the kinetics of the situation dictate that the exit polarization state of the photons leaving the fiber at the other end will be  
\[|\psi^{\prime}\rangle=\frac{1}{\sqrt{2}}\bigg{(}\exp(i\gamma)|\uparrow\rangle +\exp(-i\gamma)|\rightarrow\rangle\bigg{)}, \tag{6}\]  
Figure 3: (**a**) As the light travels along the helically-wound optical fiber, the tip of its momentum vector \(\mathbf{k}\) is confined to a momentum-space plane, \(\mathcal{P}\), and traces out a curve \(\tilde{C}\) which encircles point \(\mathcal{O}\). The winding number \(\
u(\tilde{C})\) of the light’s path is the number of times \(\tilde{C}\) circles \(\mathcal{O}\). (**b**) As \(\tilde{C}\) repeatedly traces out a circle in momentum space, the quantum state of the light traces out a closed path \(C\) on the Bloch sphere. In the process, the quantum state gains a non-trivial geometrical Berry phase, \(\gamma(C)\), which depends on the winding number \(\
u(\tilde{C})\), the area \(\Omega(C)\) subtended by the curve \(C\) relative to the origin, and the photon helicity \(\sigma\). where \(\exp(\pm i\gamma)\) are the phases picked up during parallel transport of the optical state along the fiber. The physical interpretation of the Berry phase is that it measures the rotation of the linear polarization direction [23]. Using a fixed reference beam for comparison, \(\gamma\) can be measured interferometrically, so that one can find the winding number for fixed pitch angle \(\theta\):  
\[\
u=\frac{\gamma}{2\pi(1-\cos\theta)}. \tag{7}\]  
Accordingly, for example, for \(\gamma=\pi\) and a pitch angle of \(\theta=\frac{\pi}{3}\), one has achieved winding number \(\
u=1\). A coherent superposition can then be constructed via an optical-fiber Mach-Zehnder interferometer (cf., e.g., [21], Section 1.6) with an entry beamsplitter of variable transmittance that provides one path with such a twisted fiber and the other with an untwisted but otherwise identical fiber and an exit beamsplitter on which the two paths are recombined after an appropriate corresponding relative phase shift between the two arms to produce superpositions of the state of Equation (5), which has winding number 0, and that of Equation (6), that is, a variable-amplitude \(\alpha_{i}\) (\(\alpha\in\mathcal{C}\), \(i\in\{0,1\}\), \(|\alpha_{0}|^{2}+|\alpha_{1}|^{2}=1\)) superposition of winding number 0 and winding number 1:  
\[|\psi\rangle=\frac{1}{\sqrt{2}}\bigg{(}\alpha_{0}|0;\
u=0\rangle+\alpha_{1}| \gamma_{+};\
u=1\rangle\bigg{)}\, \tag{8}\]  
that is, a generic winding number qubit state on the surface of the Bloch sphere. The topological aspects of this qubit are in some sense trivial; although it is produced by different windings in the coils, the state that leaves the coil is simply a superposition of two computational basis states with a relative phase shift. There is in no sense any topological protection of the output, despite the topological nature of its generation. In the next section, we consider a second example that partially remedies this deficiency.