# Quantum Photonic Simulation of Spin-Magnetic Field Coupling and Atom-Optical Field Interaction
## 2 Mechanical Interacting Systems
### Two-Level Atom Interacting with an Electric (Optical) Field  
On the other hand, let us consider a two-level atom coupled to a harmonic external classical electric field, which describes semiclassical light-matter interaction. As is usually done [9], we label the atomic levels as \(|g\rangle\) (ground) and \(|e\rangle\) (excited). They have energies \(\hbar\omega_{g}\) and \(\hbar\omega_{e}\), respectively. Their energy difference is given by \(h\omega_{0}=\hbar\left(\omega_{\rm e}-\omega_{\rm g}\right)\). The expression for this non-interacting part of the Hamiltonian can be formally written as follows:  
\[\hat{H}_{o}=\hbar\omega_{\rm g}|g\rangle\langle g|+\hbar\omega_{\rm e}|e\rangle \langle e|=\hbar\bar{\omega}{\bf I}+\hbar\omega_{o}\sigma_{z}/2, \tag{10}\]  
where we used the matrix representation of \(|g\rangle\langle g|\) and \(|e\rangle\langle e|\), with \(\bar{\omega}=(\omega_{\rm g}+\omega_{\rm e})/2\), and \({\bf I}\) the two-dimensional identity matrix. As for the interacting part, it is given by the dipole interaction (electric dipole approximation) between an external electric (optical) field and the atom. Indeed, by assuming, for the sake of simplicity, that the field is propagating along \(z\), has a sinusoidal dependence in time with frequency \(\omega\), and is linearly polarized along the \(x\)-direction, then the electric (optical) field can be written as follows \({\bf E}=E_{0}\cos\left(\omega(z/c-t)\,\mathbf{u}_{x}\right)\). Moreover, we disregard the spatial dependence of the electric (optical) field because the wavelength \(\lambda=c/2\pi\omega\) is considered much larger than the atomic dimensions. Therefore, we apply the dipole or long-wavelength approximation, that is by assuming without lost of generality \(\omega z/c=2m\pi\), with \(m\) an integer, then \({\mathbf{E}}=E_{0}\cos\omega t\,\mathbf{u}_{x}\), the electric-dipole interaction is given by \(\hat{H}_{I}=-\mathbf{d}\cdot{\mathbf{E}}=-d_{x}E_{x}=\mathrm{e}xE_{x}\), where \(\mathbf{d}\) is the atomic dipole operator \(\mathbf{d}=q\cdot{\mathbf{r}}\), with \(q=-\mathrm{e}\), and \(\mathbf{r}\) is the electron's position vector (operator). This interaction term allows transitions between the two levels. The form of the dipole operator can be calculated by using twice the closure relationship \(\hat{I}=|g\rangle\langle g|+|e\rangle\langle e|\), that is \(\hat{I}(-\mathrm{e}x)\hat{I}\); therefore:  
\[d_{x}=-\langle g|\mathrm{e}x|g\rangle|g\rangle\langle g|-\langle e|\mathrm{e}x |e\rangle\langle e|-\langle g|\mathrm{e}x|e\rangle(|g\rangle\langle e|+|e \rangle\langle g|)\equiv-c_{+}{\bf I}+c_{-}\sigma_{z}-d_{o}\sigma_{x}, \tag{11}\]  
with \(c_{\pm}=(\langle g|\mathrm{e}x|g\rangle\pm\langle e|\mathrm{e}x|e\rangle)/2\), \(d_{0}=\langle g|\mathrm{e}x|e\rangle\), and where we have used the matrix representations \((|g\rangle\langle e|+|e\rangle\langle g|\equiv\sigma_{x},\,|g\rangle\langle g |\equiv({\bf I}+\sigma_{z})/2\), and \(|e\rangle\langle e|=({\bf I}-\sigma_{z})/2\). We must stress that sometimes, parity arguments [9] narrow the form of the dipole operator, leading up to the expression \(d_{x}=-\langle g|\mathrm{e}x|e\rangle(|g\rangle\langle e|+|e\rangle\langle g |)=-d_{o}\sigma_{x}\). Likewise, it is customary to introduce the Rabi frequency \(\Omega=E_{o}d_{o}/\hbar\). In short, the total Hamiltonian is given by \(\hat{H}_{o}+\hat{H}_{I}\), and therefore, the corresponding Schrodinger equation, by using this new variables, is given by:  
\[i\hbar\frac{\partial}{\partial t}|\Psi(t)\rangle=\left(\hbar(\bar{\omega}+C_{ +})\,{\bf I}-\hbar(\frac{\omega_{0}}{2}-C_{-})\sigma_{z}+\hbar\Omega\sigma_{x }\cos\omega t\right)|\psi(t)\rangle=\hat{H}|\Psi(t)\rangle, \tag{12}\]  
with \(C_{\pm}=c_{\pm}\cos\omega t\). The general solution of this Schrodinger equation can again be obtained by using a time-dependent linear combination of fundamental and excited states, \(|g\rangle\equiv|0\rangle\) and \(|e\rangle\equiv|1\rangle\), of the particle, that is \(|\Psi(t)\rangle=c_{o}(t)|0\rangle+c_{1}(t)|1\rangle\).