# Quantum Photonic Simulation of Spin-Magnetic Field Coupling and Atom-Optical Field Interaction
## 3 Quantum Photonic Simulations
### Quantum Study of the Photonic Device  
A canonical quantization procedure [1; 2] proves that the \(a(z)\) coefficients become the photon absorption (or emission) operators \(\hat{a}(z)\) (correspondence principle). Therefore, the coupled mode equations are in fact the Heisenberg equations, which, in general, give the evolution of the operators in time. In this case, they give the spatial evolution of the operators. Moreover, the relevant operator here responsible for the spatial propagation of quantum states of the device is the momentum operator, which is the generator of spatial translations [2; 19], and not the Hamiltonian, which is the generator of temporal translations. In short, we can study the integrated photonic device in a fully quantum mechanical way, by solving the equations for absorption operators (spatial Heisenberg equations). That is, by performing the change \(a(z)\rightarrow\hbar\hat{a}(z)\) in Equation (15), we obtain [2; 20]:  
\[-i\hbar\frac{d\hat{a}_{n}(z)}{dz}=\hbar\tilde{\beta}_{n}\hat{a}_{n}(z)+\hbar \sum_{n\
eq m}C_{nm}(z)\hat{a}_{m}(z). \tag{18}\]  
We must stress that modal coupling preserves energy; therefore, Equation (18) corresponds to a unitary transformation, and accordingly, \(C_{nm}=C_{mn}\). On the other hand, we only consider single-photon states, which is enough for our simulation purposes. We must stress that the linear momentum of a single photon without modal coupling, that is \(\Delta\epsilon(x,y)=0\), is given by \(p_{(0,1)}=\hbar\beta_{(0,1)}\) depending on whether the photon is excited in mode \(\beta_{0}\) or mode \(\beta_{1}\)[19; 20]. Obviously, more general quantum states could be used such as multiphoton states, entangled states of two photons, and so on. These states would give rise to more complex quantum simulations, which fall outside of the scope of this work. In general, the single-photon state is a quantum superposition because the photon can either be excited in the mode \(\beta_{0}\), that is \(|1_{0}\rangle\), or in the mode \(\beta_{1}\), that is \(|1_{1}\rangle\). Hence, the general quantum state is given by:  
\[|L(z)\rangle=a_{0}(z)|1_{0}\rangle+a_{1}(z)|1_{1}\rangle, \tag{19}\]  
where \(a_{0}(z)\) and \(a_{1}(z)\) are the quantum complex amplitudes and fulfil the normalization condition \(|a_{0}(z)|^{2}+|a_{1}(z)|^{2}=1\). States \(|1_{0}\rangle\) and \(|1_{1}\rangle\) must be understood as single-photon states at a distance (plane) \(z\). It can be checked that the solutions of Equation (18) for the spatial propagation of emission operators are the same as for single-photon states \(|L(z)\rangle\)[21]. The main reason is that single-photon states are proportional to emission operators, that is \(|1_{0}\rangle=\hat{a}_{0}^{\dagger}|0\rangle\), \(|1_{1}\rangle=\hat{a}_{1}^{\dagger}|0\rangle\). Indeed, let us consider free propagation, that is non-coupling case \(C_{nm}=0\), then the solutions of Equation (18) are \(\hat{a}_{0,1}(z)=\exp(i\beta_{0,1}z)\hat{a}_{0,1}(0)\). Now, let us consider, for the sake of simplicity, a single-photon state at \(z=0\), for instance \(|1_{0}\rangle\), then the optical propagation can be obtained as follows: \(|1_{0}\rangle=\hat{a}_{0}^{\dagger}(0)|0\rangle\) (one-photon emission), but by taking into account the \(z\)-propagation, we have \(|L(z)\rangle=\exp(i\beta_{0}z)\hat{a}_{0}^{\dagger}(z)|0\rangle=\exp(i\beta_{ 0}z)|1_{0}\rangle\equiv a_{0}(z)|1_{0}\rangle\); therefore, the coefficients \(a_{0,1}(z)\) of a single-photon state have the same optical propagation solution as the operators \(\hat{a}_{0,1}(z)\). This can be proven for a general unitary transformation after a certain algebra. We will take advantage of this property for simulation purposes. Accordingly, for single-photon states, Equation (18) can formally be rewritten as follows:  
\[i\hbar\frac{\mathrm{d}|L(z)\rangle}{\mathrm{d}z}=-\hbar\left( \begin{array}{ cc}\tilde{\beta}_{0}(z)&C_{12}(z)\\ C_{21}(z)&\tilde{\beta}_{1}(z)\end{array}