# Quantum Photonic Simulation of Spin-Magnetic Field Coupling and Atom-Optical Field Interaction
## 3 Quantum Photonic Simulations
### Classical Study of the Photonic Device  
Let us consider a standard integrated photonic device consisting of, for example, an integrated waveguide 1D (one-dimensional) with refractive index \(n(x)\) (slab guide) that supports two optical modes \(e_{0}(x)\) and \(e_{1}(x)\). These modes are collinear and travel in the \(z\)-direction with propagation constants \(\beta_{0}\) and \(\beta_{1}\), that is wave vectors in the \(z\)-direction defined as \(\beta=(\omega/c)N\), where \(\omega\) is the frequency of the mode and \(N\) is the effective index [10]. The mentioned modes are coupled by an integrated grating present in a region of the slab guide, as shown in Figure 1a), where relevant parameters are indicated, that is substrate index \(n_{s}\), film index \(n_{f}\), cover index \(n_{c}=1\), and film thickness \(d\). The grating is represented, for example, by a periodic modulation (perturbation) of the electrical permittivity [10],  
\[\Delta\epsilon(x,z)=\Delta\epsilon(x)\cos(\gamma z+\alpha_{o}), \tag{14}\]  
with \(\Delta\epsilon(x)\) the modulation strength of the optical grating, \(\alpha_{o}\) an initial phase, if required, \(\gamma=2\pi/\Lambda\) the frequency of the perturbation, and \(\Lambda\) its period. This index profile can be obtained by different technologies of integrated optics, for instance ion-exchange in glass [15; 16] could be used, or even by optical fiber technology [17]. Likewise, in crystals such as lithium niobate, these optical gratings can also be reconfigurable due to acousto-optic or electro-optic effects [18].  
As we assume that only two guided modes are excited in our integrated photonic structure, the perturbed electric field amplitude is given by \(e(x,z)=a_{0}(z)e_{0}(x)+a_{1}(z)e_{1}(x)\). It is well known that the general set of equations that describe the coupling between \(n\) copropagating optical modes with propagation constants \(\beta_{n}\) in a perturbed waveguide and that allow calculating the amplitude coefficients \(a_{n}(z)\) is given by (see for instance [10]):  
\[-i\frac{da_{n}(z)}{dz}=\tilde{\beta}_{n}(z)a_{n}(z)+\sum_{n\
eq m}C_{nm}(z)a_{m} (z), \tag{15}\]  
Figure 1: **(a)** Integrated optical grating with period \(\Lambda\) on a two-mode slab waveguide with modes \(e_{0}(x)\) and \(e_{1}(x)\). The inset shows the simulated mechanical device: atom-field interaction. **(b)** Integrated optical grating with period \(\Lambda\) on a two-mode channel waveguide with modes, for example \(e_{00}(x,y)\) and \(e_{10}(x,y)\).  
where \(\tilde{\beta}_{n}(z)=\beta_{n}+C_{nn}(z)\) are the corrected propagation constants due to the self-coupling coefficients \(C_{nn}\) and \(C_{nm}\) are the coupling coefficients between the \(n\) mode and each of the other \(m\) modes. All these coefficients are calculated as follows [10]:  
\[C_{nm}(z)=\frac{\omega}{2}\int\Delta\epsilon(x,z)\,e_{n}(x)\,e_{m}^{*}(x)\, \mathrm{d}x, \tag{16}\]  
with \(\omega\) the temporal frequency of the modes and \(e_{n}(x)\) and \(e_{m}(x)\) the normalized optical \(n\) and \(m\) modes of a planar guide.  
The study with 2D guides (integrated channel guides or optical fibers) can be also made, but no new relevant result would be obtained. Indeed, a channel guide can be defined starting from a planar guide whose width is reduced up to a size \(a\) of the same order as its depth, that is \(a\approx d\), as shown in Figure 1b). In such a case, the optical modes are characterized by two subscripts, one for each spatial direction, that is \(e_{np}(x,y)\) and \(e_{mq}(x,y)\); therefore, the general coupling coefficients are given by:  
\[C_{npmq}(z)=\frac{\omega}{2}\int\Delta\epsilon(x,y,z)\,e_{np}(x,y)\,e_{mq}^{*} (x,y)\,\mathrm{d}x\mathrm{d}y, \tag{17}\]  
where we assumed that the grating modulation can also have a \(y\)-dependence. Finally, the mode coupling equations can be obtained by applying the formal changes \(n\to np\), \(m\to mq\) in Equation (15). Accordingly, the results for planar guides can be easily transferred to channel guides.