# Topological Qubits as Carriers of Quantum Information in Optics
## 3 Winding Number and Quantum OAM  
In the study of quantum electromagnetic states of nonzero orbital angular momentum, it is most common to consider those arising in paraxial laser light beams, that is, beams propagating nearly parallel to a given axis. It is convenient to consider these states in the basis formed by the Laguerre-Gauss (LG) mode functions [28] of the field; these can be seen to depend on the winding number associated with the angle about the propagation axis \(\phi\); see Figure 4. In this context, the wave has a phase proportional to the azimuthal angle, \(\psi\sim e^{il\phi}\). The wavefunction is required to be single-valued, which forces parameter \(l\) to be integer-valued. It is in fact a conserved quantum number for the system: it characterizes the orbital angular momentum (OAM) eigenvalue of the corresponding field eigenstate and plays the role of winding number, \(l\equiv\
u\) (see below). Figure 4: An optical wavefront with nonzero OAM. The phase factor of the wave, \(e^{il\phi}\), depends on the azimuthal angle \(\phi\), giving the surfaces of constant phase a characteristic corkscrew shape that clearly exhibits winding about the point of the plane penetrated by axis of propagation, \(z\). The Poynting vector, \(\vec{S}\) is perpendicular to the wavefront at all points, rotating as the wave propagates along the \(z\)-axis. Recall that, in general, the EM field possesses energy, momentum, and angular momentum, the last being associated with either spin or spatial degrees of freedom [29]. To see directly the connection to winding number, first note that, in free space, the Poynting vector \(\vec{S}=c^{2}\vec{P}\) of the EM field, indicating the direction and magnitude of the field momentum \(\vec{P}\) is  
\[\vec{P}=\vec{E}\times\vec{B}\, \tag{9}\]  
the vector cross-product of the electric (\(\vec{E}\)) and magnetic (\(\vec{B}\)) field intensities. It has been theoretically demonstrated [30] and experimentally observed [31] that LG modes of a laser beam carry an OAM for quantum states of linearly polarized light that are (in the paraxial case) _independent_ of the corresponding spin-angular momentum (polarization) of the beam photons. Whereas the polarization is a result of the intrinsic spin-1 nature of the light, the OAM is a result of spatial structure imposed on the shape of its wavefronts. It has also been shown experimentally that the OAM carried by LG modes is conserved in spontaneous parametric down conversion (SPDC), with the OAM of the incoming pump beam determining the combined OAM of the two outgoing daughter modes [32]. In general, fields of laser beams can be described by the product of a Hermite polynomial and a spatial Gaussian, that is, they are described via the Hermite-Gaussian (HG) basis. However, it is convenient and most common to consider the states under consideration in this case in the basis formed by the Laguerre-Gauss (LG) mode functions \(L_{p}^{a}(x)\) of the field. If the phase front of the propagating field is helical, the Poynting vector of the beam possesses a non-trivial azimuthal component corresponding to the OAM circulating about the beam axis (along the \(z\) direction), tracing out a continuous closed curve in the \(x\)-\(y\) plane and forming an optical vortex on the axis with annular intensity profiles with radii increasing with the magnitude \(|l|\) of the winding number, the modes being describable in that plane. For the EM field in the paraxial case, only the region close to the propagation axis need be considered [33; 34; 35; 36; 37; 38; 39]; the LG laser mode functions of a paraxial cylindrical beam have the helical phase factor  
\[\exp\left(-il\phi\right). \tag{10}\]  
This phase becomes undefined on the beam axis, so the axis corresponds to a singular vortex line.