# Topological Qubits as Carriers of Quantum Information in Optics
## 3 Winding Number and Quantum OAM
Applying the angular momentum operator about the propagation axis, \(\hat{L}_{z}=i\hbar\frac{\partial}{\partial\phi}\), it is clear that the discrete integer value \(l\), often called the topological charge, determines the quantized angular momentum eigenvalue  
\[L_{z}=lh, \tag{11}\]  
where \(l\in\mathbb{Z}\) with the phase front turning counter-clockwise for \(l>0\) and clockwise for \(l<0\). From Equation (10), the phase \(l\phi\) winds around the origin \(l\) times each time the spatial azimuthal angle \(\phi\) circles the axis once; thus, \(l\) is the winding number of the phase function about the phase singularity on the beam axis. Such OAM states are described as solutions of the paraxial Helmholtz equation--i.e., in the approximation of a slowly varying envelope \(A(r)\) of the wave \(U(\vec{r})=A(\vec{r})\exp(ikz)\) in the \(z\) direction, where \(\frac{\partial A}{\partial z}\ll kA\) and \(\frac{\partial^{2}A}{\partial z^{2}}\ll k^{2}A\) and slow means the variation of \(A(\vec{r})\) is slow within the distance \(\lambda=2\pi/k\) of one wavelength along \(z\); cf. e.g., [40], p. 51. As a specific example of such a field, the Laguerre-Gauss (LG) mode functions of the EM field \(\text{LG}_{p}^{l}\) have winding number \(l\) associated with the azimuthal dependence of the phase \(\phi\) as in Equation (10); the linear increase of phase tilts the wavefronts, producing a corkscrew-shaped wavefront that rotates about the axis like a screw as it propagates (Figure 4). The value of \(l\) determines how often the phase completes a full cycle of \(2\pi\) as one winds around the spatial axis once (Figure 5). The LG beam field state with orbital angular momentum \(l\hbar\) with winding number \(l\) and with \(p\) radial nodes can be written explicitly as [41]  
\[E_{lp}(r,z,\phi) = \frac{E_{0}}{w(z)}\left(\frac{\sqrt{2}r}{w(z)}\right)^{||}e^{-r^{2 }/w^{2}(r)}L_{P}^{|l|}\left(\frac{2r^{2}}{w^{2}(r)}\right) \tag{12}\] \[\quad\times\ e^{-ibx^{2}z/\left(2(z^{2}+z_{R}^{2})\right)}e^{-i \phi l+i(2p+||l|+1)\arctan(z/z_{R})},\]  
where \(E_{0}\) is a constant and \(w(z)=w_{0}\sqrt{1+\frac{z}{z_{R}}}\) is the beam radius at distance \(z\); \(z_{R}=\frac{\pi w_{0}^{2}}{\lambda}\) is called Rayleigh range. The \(\arctan(z/z_{R})\) term is the well-known _Gouy phase_ that routinely appears in beam optics. Integer \(p\) indexes the radial properties of the mode, with a central dark spot, \(u_{lp}\) surrounded by \(p\) dark nodal rings; see Figure 6.  
Creation and annihilation operators can be defined that connect quantum states with different \(p\) values, and a differential operator may be defined for which these states serve as eigenstates [42; 43; 44]. Examples of the intensity and phase of these beams are given in Figure 5 at \(p=0\) for different values of \(l\); plots of intensity for different \(p\) at fixed \(l\) are shown in Figure 6. Making use of orbital angular momentum beamsplitters [45], one can form qubit states of ang  
Figure 5: Plots of intensity (top row) and phase (bottom row) in the transverse plane for several \(LG\) modes with different \(l\) values, but with \(p=0\) in each case. For \(l=0\), the phase is constant in each plane; for \(l\
eq 0\), it increases linearly with angle as the axis is circled. The amplitude vanishes on the axis for the \(l\
eq 0\) cases, but is a maximum there for \(l=0\). Figure 4:0 Plots of intensity (**top row**) and phase (**bottom row**) in the transverse plane for several \(l=3\)\(LG\) modes with different \(p\) values. In addition to the intensity zero on the axis, there are \(p\) nodal rings on which the amplitude vanishes (the dark blue rings). The phase is discontinuous in the radial direction at each nodal ring, while increasing linearly in the azimuthal direction in each case. of winding number [46]. In particular, one can make use of devices (analogous to polarization beam splitters) that select the optical path based on OAM, with one path for each of the distinguishable OAM states [45].