# Quantum Photonic Simulation of Spin-Magnetic Field Coupling and Atom-Optical Field Interaction
## 3 Quantum Photonic Simulations
### Implementation of Photonic Simulators  
In this subsection, we present more complex simulators by using several integrated optical gratings along with other integrated components as Directional Couplers (DCs) made with Single-Mode (SMWs) or Two-Mode channel Waveguides (TMWs). We present a simulator of the interaction between an atom with four levels and \(\Theta\)-pulses, next the interaction of a particle (or physical system) with spin \(3/2\) and a magnetic field, and finally, an arbitrary unitary transformation SU(4) implemented with optical gratings what allows reducing the number of paths by half or even to a quarter, which can be generalized to SU(N) transformations, which can be of interest for photonic simulators. Let us consider, as the first example, the Optical Grating (OG) studied above, but with a number \(d=4\) of collinear guided modes whose propagation constants are \(\beta_{j}\) with \(j=1,2,3,4\). This simple device can simulate a four-level atom, which can be used to implement a CNOT logic gate operation under atom-laser interaction [22]. The optical grating fulfills \(\gamma=\beta_{2}-\beta_{3}\); therefore, it will produce an efficient transition between Modes 2 and 3 if \(z=\pi/C_{o}\) (\(\pi\)-pulse in atom-laser interaction) according to Equation (35). We can identify each single-photon state excited in each optical mode as a computational state (two-qubit single-photon [23]); thus, we have the state \(|L\rangle=c_{00}|00\rangle+c_{01}|01\rangle+c_{10}|10\rangle+c_{11}|11\rangle\). When this state goes through the optical grating, the output state is \(|L\rangle=c_{00}|00\rangle+c_{01}|01\rangle+c_{11}|10\rangle+c_{10}|11\rangle\), that is a CNOT operation is obtained. Obviously, this is not the best way to implement logic gates for two-qubits, but it makes clear how OGs can be used to simulate interaction between an atom with several levels and the so-called \(\Theta\)-pulses. The second example consists of using single-mode channel guides coupled by OGs in order to simulate the interaction between a particle (or physical system) with spin \(s\) and a magnetic field. This can be implemented by using a number \(N\)=\(2s\)+1 of optical modes excited in \(N\) channel waveguides and coupled by OGs, as shown in Figure 2 for the particular case of four guides (spin \(s\)=3/2). We must stress that coupling between parallel SMWs can be also described by the Heisenberg quantum equations given by Equation (18). In order to simulate spin-magnetic field interaction, either the strength of the OGs or the separation between consecutive SMWs has to be adjusted, and therefore, proper coupling strength values \(C_{ij,i+1}\) with \(j\)=\(1,...N-1\), between consecutive guides, are obtained. Likewise, the SMWs have to be designed with different propagation constants \(\beta_{j}\) (asynchronous guides) by changing, for example, the depth of each channel waveguide. Finally, it is well known in integrated optics that for asynchronous DCs assisted by optical gratings, the coupling due to the overlapping fields is negligible. As a particular example, let us consider spin \(s=3/2\). The general equation system is given by:  
\[-i\hbar\frac{d\dot{a}_{j}(z)}{dz}=\hbar\dot{\beta}_{j}(z)\dot{a}_{j}(z)+\hbar \sum_{j^{\prime}\
eq j}C_{jj^{\prime}}(z)\dot{a}_{j^{\prime}}(z). \tag{39}\]  
with \((|j-j^{\prime}|<2)\), that is coupling only exists between consecutive channel guides. Next, by using the same procedure followed for the case of spin-1/2, we obtain, after a long, but straightforward calculation, the following equation system:  
\[-i\hbar\frac{\partial}{\partial z}|L_{b}(z)\rangle=-\Delta_{o}\mathbb{I}_{4x4}+\]  
\[+\left( \begin{array}{cccc}3\Delta_{o}&C_{12}\exp{(i\gamma z)}&0&0\\ C_{21}\exp{(-i\gamma z)}&\Delta_{o}&C_{23}\exp{(i\gamma z)}&0\\ 0&C_{32}\exp{(-i\gamma z)}&-\Delta_{o}&C_{34}\exp{(i\gamma z)}\\ 0&0&C_{43}\exp{(-i\gamma z)}&3\Delta_{o}\end{array}