# Quantum Photonic Simulation of Spin-Magnetic Field Coupling and Atom-Optical Field Interaction
## 4 Quantum Geometric Phases
### Geometric Phases in Light-Matter Photonic Simulation  
Finally, we check that light-matter simulation can be also used to obtain geometric phases. For the sake of simplicity, we show geometric phases for wedge circuits, although more general cases can be studied. Let us consider an initial state \(|L(0)\rangle=|1_{0}\rangle\), that is the point \((0,0,1)\) on the photonic Bloch sphere. Next, let us consider an asynchronous optical grating with \(\delta_{s}\ll C_{\circ}\); therefore, according to Equation (37) (phase gate), for \(\delta_{s}z_{o}=3\pi/2\), we obtain \(|L(z_{o})\rangle=(1/\sqrt{2})(|1_{0}\rangle+i|1_{1}\rangle)\), that is the state reaches the Bloch sphere point \((0,1,0)\). Next, we consider that the grating has a greater coupling coefficient \(C_{\circ}\), then the single-photon state is given by the expression:  
\[|L(z)\rangle=\frac{1}{\sqrt{2}}[(\cos\frac{\delta_{r}z}{2}+b\sin\frac{\delta_ {r}z}{2}+ia\sin\frac{\delta_{r}z}{2})|1_{0}\rangle+(a\sin\frac{\delta_{r}z}{2} +i\cos\frac{\delta_{r}z}{2}-ib\sin\frac{\delta_{r}z}{2})|1_{1}\rangle] \tag{59}\]  
where \(a=\delta_{s}/\delta_{r}\) and \(b=C_{\circ}/\delta_{r}\). If we choose a distance \(z=z_{1}\) such as \(\delta_{r}z_{1}/2=\pi/2\), then, after a certain calculation, the following state is obtained:  
\[|L(z_{1})\rangle=\frac{e^{i\phi_{\circ}}}{\sqrt{2}}(|1_{0}\rangle+i|1_{1}\rangle) \tag{60}\]  
where \(\phi_{\circ}=\text{atn}(a/b)=\text{atn}(\delta_{s}/C_{\circ})\), that is \(a=\sin\phi_{\circ}\) and \(b=\cos\phi_{\circ}\). The state has reached the point \((0,-1,0)\) of the photonic Bloch sphere. Now, we show that \(\phi_{\circ}\) is a geometric phase. Indeed, for the sake of symmetry, the state before reaching the above state has crossed the meridian \(y=0\) when \(\delta_{r}z/2=\pi/4\), that is, the state:  
\[|L^{\prime}\rangle=\frac{(1+b+ia)}{2}\,|1_{0}\rangle+\frac{(a+i(1-b))}{2}\,|1 _{1}\rangle\equiv m_{0}\,e^{i\phi_{0}}|1_{0}\rangle+m_{1}\,e^{i\epsilon_{1}}| 1_{1}\rangle. \tag{61}\]  
It is easy to prove that both states have the same phase, that is \(e_{0}=\epsilon_{1}\), and the modulus is given by \(m_{0}=\sin(\phi_{\circ}/2)\) and \(m_{1}=\cos(\phi_{\circ}/2)\); therefore, the state crosses the point \(P=(\sin\phi_{\circ},0,\cos\phi_{\circ})\) of the photonic Bloch sphere as indicated in Figure 4 for \(\varphi=\phi_{\circ}\). Now, we must recall that partialcycles also generate geometric phases, which can be calculated by closing the end points of the open cycle by a geodesic line [28; 30]. In our case, the corresponding geodesic lies along the meridian at the plane \(x=0\), from point \((0,-1,0)\) to initial point \((0,0,1)\). In short, the state has followed a wedge circuit \(\mathcal{C}_{w}\), as shown in Figure 4. Now, we calculate the subtended solid angle by this wedge circuit. It is easy to check that a wedge circuit with angle \(\phi_{o}\) subtends a solid angle \(\mathbf{\Omega}(\mathcal{C}_{w})=2\phi_{o}\); therefore, the geometric phase is \(\Phi_{g}=(1/2)\mathbf{\Omega}(\mathcal{C}_{w})=\phi_{o}=\text{atn}(\delta_{s} /C_{o})\), which is just the global phase obtained in Equation (60). It is worth underlining that in this case, the geometric phase is not hidden, and therefore, the dynamical phase does not have to be eliminated. Moreover, this geometric phase can be also obtained in a atom-optical field system out of resonance by using \(\Theta\)-pulses, with the first optical field with low amplitude \(E_{o}\) (Z gate) and the next with a higher amplitude \(E_{o}\). By using the simulation parameters given by Equation (34), the geometric phase would be \(\Phi_{g}=\text{atn}(\delta/\Omega)\).