# Quantum Photonic Simulation of Spin-Magnetic Field Coupling and Atom-Optical Field Interaction
## 4 Quantum Geometric Phases
### Geometric Phases in Spin-Magnetic Field Photonic Simulation  
Let us consider the eigenstates \(|l_{\pm}(0)\rangle\) given by Equation (31). Spatial propagation of these states is given, according to Equations (28) and (31), by \(|L_{b}(z)\rangle\equiv|L_{\pm}(z)\rangle=\exp(i\gamma\sigma_{z}z/2)|l_{\pm}(z )\rangle=\exp(\frac{ip_{\pm}}{\hbar}z)\exp(i\gamma\sigma_{z}z/2)|l_{\pm}(0)\rangle\). For instance, let us take \(|L_{+}(z)\rangle\) after a propagation distance \(z=\
u\Lambda\), that is for a photonic integrated grating with a length \(\
u\Lambda\), where \(\Lambda=2\pi/\gamma\) and \(\
u\) is the number of cycles taken, then we have:  
\[|L_{+}(\
u\Lambda)\rangle=\exp\big{(}\frac{ip_{+}}{\hbar}\
u\Lambda\big{)} \big{(}\cos\frac{\alpha^{\prime}}{2}\exp(i\
u\pi)|1_{0}\rangle+\sin\frac{ \alpha^{\prime}}{2}\exp(-i\
u\pi)|1_{1}\rangle\big{)} \tag{42}\]  
By following the same procedure for the state \(|L_{-}(z=\
u\Lambda)\rangle\) and taking into account that \(\
u\) is an integer, we obtain:  
\[|L_{\pm}(\
u\Lambda)\rangle=\exp\bigg{(}\frac{ip_{\pm}}{\hbar}\
u\Lambda\pm i \
u\pi\bigg{)}|l_{\pm}(0)\rangle=\exp(i\phi^{\pm}))|l_{\pm}(0)\rangle. \tag{43}\]  
The phases \(\phi^{\pm}\) obtained above are the full phases acquired by the states \(|L_{\pm}\rangle\) after \(\
u\) cycles in the optical grating, that is,  
\[\phi^{\pm}=\
u(\pm p_{o}\Lambda/\hbar\pm\pi) \tag{44}\]  
A photonic Bloch sphere is shown on the left in Figure 4 where each point corresponds to a single-photon state given by Equation (19). For comparison purposes, an NMR Bloch sphere is also shown on the right. A single-photon state propagating along \(z\) can be represented by the following general expression \(|L(z)\rangle=c_{0}(z)|1_{0}\rangle+c_{1}(z)|1_{1}\rangle\); therefore, each point \((x,y,z)\) of the photonic Blochsphere is defined as follows: \(x=2\operatorname{Rec}_{0}(z)c_{1}(z)\), \(y=2\operatorname{Im}c_{0}(z)c_{1}(z)\), \(z=|c_{0}(z)|^{2}-|c_{1}(z)|^{2}\), where \(\operatorname{Re}\) and \(\operatorname{Im}\) stand for real and imaginary parts, respectively. It is easy to check that eigenstates \(|L_{\pm}(z)\rangle\) follow the curves \(C_{c}\) and \(C_{c}^{\prime}\) corresponding to spherical caps, as shown in Figure 4. The solid angle subtended by a spherical cap with an angular extension \(\alpha^{\prime}\) is given by \(\mathbf{\Omega}(C)=2\pi(1-\cos\alpha^{\prime})\). As the full phase \(\phi\) can be decomposed into a dynamical part \(\phi_{\text{d}}\) and a geometric one \(\phi_{\text{g}}\), then the geometric phase can be obtained from the relationship:  
\[\phi_{\text{g}}^{\pm}=\phi_{\pm}-\phi_{\text{d}}^{\pm} \tag{45}\]  
We first compute the dynamical phase, that is \(\phi_{\text{d}}^{\pm}=\frac{1}{\hbar}\int_{0}^{\
u\Lambda}\langle L_{\pm}(z)| \hat{M}(z)|L_{\pm}(z)\rangle\text{d}z\); therefore, by taking into account the transformation (28) and expression (29) for \(\hat{M}^{\prime}\), we obtain:  
\[\phi_{\text{d}}^{\pm} =\frac{1}{\hbar}\int_{0}^{n\Lambda}\langle L_{\pm}(z)|\hat{M}(z)| L_{\pm}(z)\rangle\text{d}z=\frac{1}{\hbar}\int_{0}^{\
u\Lambda}\langle l_{\pm}(0)| \hat{M}(0)|l_{\pm}(0)\rangle\text{d}z= \tag{46}\] \[=\frac{1}{\hbar}\int_{0}^{\
u\Lambda}\big{(}\langle l_{\pm}(0)| \hat{M}^{\prime}|l_{\pm}(0)\rangle-\frac{\gamma}{2}\langle l_{\pm}(0)|\sigma_ {z}|l_{\pm}(0)\rangle\big{)}\text{d}z=\] \[=\frac{p_{\pm}}{\hbar}\
u\Lambda\pm\
u\pi\cos\alpha^{\prime},\]  
and by taking into account the total phase given by Equation (43), the geometrical phases acquired by the eigenstates are given by:  
\[\phi_{\text{g}}^{\pm}=\pm\
u\pi(1-\cos\alpha^{\prime})=\pm\
u\pi\Big{(}1-\frac {\Delta\beta-\gamma}{\sqrt{(\Delta\beta-\gamma)^{2}+C_{\text{g}}^{2}}}\Big{)}. \tag{47}\]  
Note that the geometric phase is half the solid angle subtended by the circuit, where the sign \(\pm\) depends on the direction of rotation followed by the state on the Bloch sphere.