# Quantum Photonic Simulation of Spin-Magnetic Field Coupling and Atom-Optical Field Interaction
## 3 Quantum Photonic Simulations
### Photonic Simulation of Light-Matter Interaction
\right)\left( \begin{array}{c}l_{0} (0)\\ l_{1}(0)\end{array} \right)\equiv\text{Z}(\Phi/2), \tag{37}\]  
where \(\Phi=\delta_{s}z\). Therefore, a Z quantum logic gate is obtained from Equation (37) if \(\delta_{s}z/2=\pi/2\), an S-gate if \(\delta_{s}z/2=\pi/4\), a T-gate if \(\delta_{s}z/2=\pi/8\), and so on. Moreover, by using an optical gratingimplementing a transformation \(X(\Theta/2)\) and two phase gates \(Z(\Phi/2)\), a ZXZ-factorization of SU(2) is obtained; therefore, any unitary transformation can be implemented with the photonic device, and consequently, any one-qubit can be generated. Next, we present solutions for the asynchronous case, that is \(\Delta\beta\
eq\gamma\), which provides a most general solution and will be very useful to obtain geometric phases. By using standard methods to solve a linear equation system, the general solution of Equation (33) is given by:  
\[\left( \begin{array}{c}l_{0}(z)\\ l_{1}(z)\end{array} \right)=\left( \begin{array}{cc}\cos\frac{\delta_{x}}{2}z+i \frac{\delta_{x}}{\delta_{r}}\sin\frac{\delta_{y}}{2}z&-i\frac{\zeta_{x}}{ \delta_{y}}\sin\frac{\delta_{y}}{2}z\\ -i\frac{\zeta_{x}}{\delta_{r}}\sin\frac{\delta_{y}}{2}z&\cos\frac{\delta_{y}}{ 2}z-i\frac{\delta_{x}}{\delta_{y}}\sin\frac{\delta_{y}}{2}z\end{array} \right) \left( \begin{array}{c}l_{0}(0)\\ l_{1}(0)\end{array} \right), \tag{38}\]  
with \(\delta_{r}=\sqrt{\delta_{s}^{2}+C_{g}^{2}}\) and where, once more, the irrelevant global phase \(e^{i\beta z}\) is omitted. Note that for \(\delta_{s}=0\), the matrix given by Equation (35) is recovered. Likewise, under the condition \(|\delta_{s}|\ll C_{o}\), the matrix solution given by Equation (37) is obtained. We must recall that all these transformations are obtained in a spatial rotating reference system defined by Equation (28) and induced by the RWA approximation, as occurs in atom-light temporal interactions. Likewise, we must stress that by using the simulation parameters given by Equation (34), mechanical solutions for light-matter (atom) interaction are directly obtained. Finally, it is also easy to check that the same solutions are obtained for the spin-magnetic field interaction simulation given by Equation (29); therefore, such an interaction has the same quantum processing properties as the atom-optical field interaction. Finally, it is worth paying attention to the problem of properly initializing a quantum state. For that, let us consider an SPDC (Spontaneous Parametric Down Conversion) source of biphotons \(|1_{\mathbf{k}_{a}}1_{\mathbf{k}_{b}}\rangle\), that is twin photons excited in two spatial modes propagating along directions \(\mathbf{k}_{a}\) and \(\mathbf{k}_{b}\). Photon \(|1_{\mathbf{k}_{b}}\rangle\) is directed towards an APD device, and the other one \(|1_{\mathbf{k}_{a}}\rangle\) is directed to the prism-waveguide coupler, as shown in Figure 1a). The direction \(\mathbf{k}_{a}\) is chosen in such a way that, for example, if the fundamental mode of the planar guide is excited, that is \(\mathbf{k}_{a}=\mathbf{k}_{0}\), then we obtain the single-photon state (or register) \(|1_{0}\rangle\). Alternatively, direction \(\mathbf{k}_{a}=\mathbf{k}_{1}\) can be chosen to excite the mode \(\mathrm{e}_{1}\) of the planar guide, and thus, the single-photon state (or register) \(|1_{1}\rangle\) is obtained. If we had channel waveguides, a similar procedure can be used, for example before the channel waveguide, there would be a planar waveguide with a prism-waveguide coupler and, next, an integrated lens or a similar integrated optical element [15] in such a way that the excited mode is focused onto the channel waveguide to excite the desired single-photon state. Next, by using optical gratings, different quantum transformations can be performed, and consequently, a general output state \(|L\rangle=a_{0}(z)|1_{\mathbf{k}_{0}}\rangle+a_{1}(z)|1_{\mathbf{k}_{1}}\rangle\) is obtained. Finally, at a distance \(z\), another prism-waveguide coupler can be placed at the end of the device to detect the quantum state.