# Quantum Photonic Simulation of Spin-Magnetic Field Coupling and Atom-Optical Field Interaction
## 3 Quantum Photonic Simulations
### Photonic Simulation of Spin-Magnetic Field Interaction
Note that a rotating equivalent vector \((\gamma_{p}\textbf{B})_{eq}\equiv\textbf{B}=\text{B}(\sin\alpha\cos\gamma z,- \sin\alpha\sin\gamma z,\cos\alpha)\) is obtained. In short, we have achieved a photonic simulator of aspin-magnetic field coupling. However, we must stress some limitations for this simulator. The momentum operator \(\hat{M}\) defined by (27) and simulating the coupling spin-magnetic field only is valid under the spatial RWA, that is the values of \(\Delta\beta\) and \(\gamma\) have to be high and not too different. Therefore, we will be able to simulate the interaction with magnetic fields with a large \(z\)-component and oscillating with a frequency \(\omega\) of the same order as the term \(\gamma_{p}B_{o}\cos\theta\). Finally, it is interesting to obtain a constant momentum operator by applying a unitary transformation (rotating reference system) to the quantum state \(|L_{b}(z)\rangle\), that is,  
\[|l(z)\rangle=\hat{U}(z,\gamma)|L_{b}(z)\rangle=\exp\left(-i\gamma\sigma_{z}z/2 \right)|L_{b}(z)\rangle. \tag{28}\]  
Therefore, by using \(|L_{b}(z)\rangle=\exp\left(i\gamma\sigma_{z}z/2\right)|l(z)\rangle\) in Equation (27), we obtain, after a certain, but straightforward calculation, the following momentum operator for the state \(|l(z)\rangle\):  
\[-\hat{M}^{\prime}|l(z)\rangle=\frac{\hbar}{2}\left( \begin{array}{cc}-\Delta \beta+\gamma&-C_{o}\\ -C_{o}&\Delta\beta-\gamma\end{array} \right)|l(z)\rangle=-\frac{\hbar}{2}\text{D }_{o}\left( \begin{array}{cc}\cos\alpha^{\prime}&\sin\alpha^{\prime}\\ \sin\alpha^{\prime}&-\cos\alpha^{\prime}\end{array} \right)|l(z)\rangle, \tag{29}\]  
where \(\cos\alpha^{\prime}=(\Delta\beta-\gamma)/\text{D}_{o}\), \(\sin\alpha^{\prime}=C_{o}/\text{D}_{o}\), and \(\text{D}_{o}=\sqrt{(\Delta\beta-\gamma)^{2}+C_{o}^{2}}\). As shown later, these results are important for simulating both dynamical and topological properties. By comparison between Equations (5) and (29), we obtain the following simulation parameters:  
\[(\omega-\gamma_{p}B_{o}\cos\theta)\leftrightarrow(\Delta\beta-\gamma),\quad \gamma_{p}B_{o}\sin\theta\leftrightarrow C_{o}. \tag{30}\]  
Next, it is interesting to compute the eigenstates \(|l(z)\rangle\) and eigenvalues \(\beta_{l}\) of the momentum operator \(\hat{M}^{\prime}\) given by Equation (6), that is \(|l(z)\rangle=\exp\left(i\beta_{l}z\right)|l(0)\rangle=\exp\left(i\beta_{l}z \right)|l(0)\rangle\). After a standard calculation, we have:  
\[|l_{+}(0)\rangle=\cos\frac{\alpha^{\prime}}{2}|1_{0}\rangle+\sin\frac{\alpha^{ \prime}}{2}|1_{1}\rangle,\quad|l_{-}(0)\rangle=\sin\frac{\alpha^{\prime}}{2}| 1_{0}\rangle-\cos\frac{\alpha^{\prime}}{2}|1_{1}\rangle, \tag{31}\]  
with eigenvalues \(\beta_{\pm}=\pm\text{D}_{o}/2\), that is linear momentums \(p_{\pm}=\pm\frac{\hbar}{2}D_{o}=\pm p_{o}\). Therefore, by taking into account Equation (28), the full evolved states \(|L_{b}(z)\rangle\) can be easily obtained. Note that we are simulating the quantum state \(\Psi(t)\) given by Equation (4), and thus, for example, quantum processing based on NMR could be simulated by this photonic device. For the sake of expositional convenience, we will return to this question in the next subsection.