# Quantum Photonic Simulation of Spin-Magnetic Field Coupling and Atom-Optical Field Interaction
## 3 Quantum Photonic Simulations
### Implementation of Photonic Simulators
\right)|L_{b}(z)\rangle \tag{40}\]  
where \(\mathbb{I}_{4x4}\) is the identity matrix and the following propagation constants are used: \(\beta_{1}\), \(\beta_{2}\)=\(\beta_{1}\)\(-\Delta_{o}\), \(\beta_{3}\)=\(\beta_{1}\)\(-2\Delta_{o}\), \(\beta_{4}\)=\(\beta_{1}\)\(-3\Delta_{o}\). These values can be achieved by adjusting, for example, the width of the channel waveguides. On the other hand, the coupling coefficients for the gratings fulfil the relationships C\({}_{12}\)=C\({}_{21}\)=C\({}_{34}\)=\(C_{43}\)=\(\sqrt{3}C_{o}\) and C\({}_{23}\)=C\({}_{32}\)=\(2C_{o}\), which can be achieved by adjusting the separation between consecutive guides with optical modes e\({}_{\text{j}}(x,y)\), \(j\)=\(1,2,3,4\). In our case, Guides \(2-3\) are closer than \(1-2\) and \(3-4\) because the coupling between Guides \(2 and3\) is larger, as shown in Figure 2. Finally, by defining \(\cos\theta=\Delta_{o}/\Gamma\) and \(\sin\theta=C_{o}/\Gamma\), with \(\Gamma=(\Delta_{o}^{2}+C_{o}^{2})^{1/2}\), and a fictitious magnetic field \(\mathbf{B}_{f}=(\sin\theta\cos\gamma z,-\sin\theta\sin\gamma z,\cos\theta)\), we can write the above equation system as follows:  
\[-i\hbar\frac{\partial}{\partial z}|L_{b}(z)\rangle=\{-\hbar\Delta_{o}\mathbb{I} _{4x4}+\frac{\hbar}{2}\,\Gamma\,\,\mathbf{J}(3/2)\,\mathbf{B}_{f}\}\,|L_{b}(z)\rangle \tag{41}\]  
where \(\mathbf{J}(3/2)=(J_{x}(3/2),J_{y}(3/2),J_{z}(3/2))\) are the spin matrices for spin \(s=3/2\). In short, we constructed a photonic simulator for interaction between a spin-3/2 particle, or physical system, and a periodic magnetic field. Next, we present a general unitary transformation SU(4) by using optical gratings. With this example, we want to show that OGs allow reducing the number of paths required for constructing a simulator. Indeed, an SU(4) simulator is formed by four paths implemented by SMWs with optical modes \(\mathrm{e}_{00}^{(\mathrm{j})}(x,y)\) (\(j\)=1,...,4), six DCs, and six Phase Shifters (PSs), as shown in Figure 3 (bottom). However, if OGs are used, we only need two paths, that is TMWs with modes \(\mathrm{e}_{00}^{(\mathrm{j})}(x,y)\), \(\mathrm{e}_{10}^{(\mathrm{j})}(x,y)\) (\(j\) = 1, 2) and six OGs, as shown in Figure 3 (up). In this case, we also use Selective Directional Couplers (SDCs) (a SDC always performs an X transformation in one mode, and the other mode undergoes an identity transformation). Note that the number of paths was reduced by half because OGs act on collinear modes, unlike directional couplers, which act on codirectional modes. Obviously, if we had used OGs with four modes, we could reduce the number of paths by 1/4. Overall, we will be able to reduce the number of paths by 1/\(d\) if we use OGs with a number \(d\) of collinear modes. Ultimately, we can take advantage of the OGs to increase integration and thus to implement more flexible and scalable photonic simulators such as for example boson sampling ones [24], where the number of paths would be reduced by half. In short, the number \(N\) of paths of any required SU(\(N\)) transformation [25] could be reduced up to \(N/d\). Finally, it is important to indicate that quantum photonic devices have their own limitations [4]. Thus, the difficulty to implement two-qubit logic gates is well known, which is, at present, an important drawback in general purpose quantum photonic computation. However, quantum photonic simulation, or simply quantum photonic computation, for specific purposes can provide efficient technological  
Figure 3: Standard implementation (bottom) of an arbitrary unitary transformation SU(4) by using Single-Mode channel Waveguides (SMWs) and Directional Couplers (DCs) (**bottom**). Alternative implementation (**top**) by using Two-Mode channel Waveguides (TMWs), Selective Directional Couplers (SDCs) and Optical Gratings (OGs). PS, Phase Shifter. Figure 2: Four non-synchronous Single-Mode channel Waveguides (SMWs) assisted by an optical grating to simulate the interaction between a spin \(s=3/2\) particle and an oscillating magnetic field.