# Topological Qubits as Carriers of Quantum Information in Optics
## 4 Multiports and Winding Number
Two directionally-unbiased three-ports connected to each other on two edges, with a phase shift between them provides a means to experimentally implement this graph (**right**). Figure 7: The directionally-unbiased three-port (**left**) and four-port (**middle**). The rectangles after the beam splitters represent a vertex mirror unit consisting of a mirror and a phase-shifter. The distance between each beam splitter and the adjacent mirror unit is half the distance between two adjacent beam splitter. **Right** figure: more schematic representations of the three- and four-port units as optical scattering vertices that connect to three or four external legs for use in optical graphs or networks. This system is directly analogous to a one-dimensional lattices of atoms, such as a polyacetylene molecule, in which topological effects are well-known. In such systems, there is an alternation of two distinct types of unit cells, leading to a periodic modulation of hopping amplitudes for electrons moving through the structure. As a result of this periodic modulation in space, the energy levels wrap around the Brillouin zone in a topologically nontrivial manner, leading to a nonzero winding number as the quasi-momentum traverses a full Brillouin zone. Here, similar effects occur in an optical system by having unit cells (pairs of diamond graphs) whose internal phase shifts alternate periodically. The transition amplitudes for photons hopping through the system then become periodically modulated in space. As in the condensed matter case, this leads to energy bands that wind nontrivially around the Brillouin zone. The transition amplitudes depend on the phase shifts in the diamond graphs, which can be arranged to be easily controlled (by varying path lengths or by using electro-optically controlled phase plates); as a result, the topological behavior of the system can also be readily controlled by the experimenter, in contrast to the situation in condensed matter systems. Consider a discrete-time view of such a system. Its behavior can be described by a quantum Hamiltonian \(\hat{H}\) and the corresponding discrete-time evolution matrix \(\hat{U}=e^{-i\hat{H}T}\) that takes the system forward one time-step \(T\). (Here, the units are chosen such that \(\hbar=1\).) There is then one photon collision with a diamond graph per unit time. For initial photon quantum state \(|\psi(0)\rangle\), the state at time \(t=nT\) is  
\[|\psi(nT)\rangle=\hat{U}^{n}|\psi(0)\rangle. \tag{15}\]  
One can then define a quasi-momentum \(k\) on a one-dimensional periodic lattice made from a sequence of repeating unit cells, labeled by an integer, \(m\); the quasi-momentum \(k\) and the position variable \(m\) is dimensionless and discrete. A quantum Hamiltonian expression for a system behaving in this way is  
\[\hat{H} = |t_{a}|\sum_{m=1}^{N}\left(|m,b\rangle\langle m,a|+|m,a\rangle \langle m,b|\right)\] \[+\,|t_{b}|\sum_{m=1}^{N-1}\left(|m+1,a\rangle\langle m,b|+|m,b \rangle\langle m+1,a|\right),\]  
where the diamond graph transmission amplitudes for phase shifts are [51; 52; 53]:  
\[t_{j}(k)=\frac{4(1+e^{-i\phi_{j}})(1-e^{-i(\phi_{j}+4k)})}{e^{-ik(1+e^{-i\phi _{j}})^{2}}-(3e^{-i(\phi_{j}+4k)}-1)^{2}}, \tag{17}\]  
Figure 9: Diamond graphs, made from two directionally unbiased three-ports, as shown in Figure 8; each cell (indicated by the dashed curves) is made from four three-ports. The two diamond graphs at each site may have different internal phase shifts, \(\phi_{a}\) and \(\phi_{b}\). The red rectangles are phase plates that rotate the polarization by 90\({}^{\circ}\), so photons in the right half of the cell have vertical polarization (state b), while those in the left half have horizontal polarization (state a). The size of the diamond graphs is exaggerated for clarity—they should be small compared to the distance separating them. where \(j=a,b\). At each fixed lattice site \(m\) or each fixed \(k\), this Hamiltonian can be written in terms of the identity matrix and the Pauli matrices [21].