# Metriplectic Structure of a Radiation-Matter-Interaction Toy Model
## 3 Two-Photon Absorption Toy Model  
Nonlinear optical systems such as those mediated by two-photon absorptions are described by a generalized nonlinear Schodinger equation including effects such as linear dispersion, dispersion of nonlinearity, and higher order nonlinear processes [12]. In our toy model, we limit it to consider only nonlinear effects that are the core of our metriplecticanalysis and that can be cast in the simplest formulations. Specifically, we consider an electromagnetic field described by a complex envelope \(A\) (\(I=|A|^{2}\) is the optical intensity) that undergoes a self-phase modulation due to the optical Kerr effect. We also include absorption as a multiphoton process, such that we have a two-level system that absorbs two photons (see Figure 1). Writing the complex refractive index nonlinear optical perturbation as \(\Delta n=n_{2}|A|^{2}+i\alpha_{P}n\) with \(n_{2}\) the Kerr coefficient, and assuming the optical beam propagating in the \(z\) direction, with vacuum wavenumber \(k_{0}=2\pi/\lambda\), with \(\lambda\) the wavelength, the dynamic equation for the field envelope is  
\[\frac{\partial A}{\partial z}=ik_{0}\frac{\Delta n}{n}A=\frac{2\pi}{\lambda}( in_{2}|A|^{2}-\alpha_{P}n)A. \tag{4}\]  
Seemingly, the equation for the level inversion reads  
\[\frac{\partial n}{\partial z}=k_{P}|A|^{4} \tag{5}\]  
and the multiphoton coefficient is \(k_{P}\). Equations (4) and (5) can be cast in dimensionless units by scaling the evolution coordinate \(z\) with \(z_{0}\), and letting \(A=A_{0}\psi\), with \(A_{0}^{2}=\lambda/(2\pi n_{2}z_{0})\), and defining the dimensionless absorption coefficient \(\alpha=2\pi\alpha_{P}z_{0}/\lambda\) and the dimensionless multiphoton coefficient as \(k=2k_{P}z_{0}A_{0}^{4}\). Hence, we consider a very simplified toy model for the two-level atomic system [13; 14]. The TPA, sketched in Figure 1, is expressed by the following differential equations:  
\[\left\{ \begin{array}{l}\dot{\psi}=\imath|\psi|^{2}\psi-\alpha n\psi,\\ \dot{n}=\frac{\kappa}{2}|\psi|^{4},\end{array} \right. \tag{6}\]  
where \(\psi\) is the complex field amplitude, \(n\) the population of the second level, \(k>0\) and \(\alpha>0\). This system is directly derived by the multi-photon absorption model [15; 16; 17], when neglecting several physical phenomena, e.g., the spontaneous emission. Moving to real-valued functions, we define  
\[\psi=\frac{q-\imath p}{\sqrt{2}}, \tag{7}\]  
Figure 1: Pictorial sketch of absorbtion of two photons in a two-level atom. In our system, Equation (6) does not have terms of spontaneous emission, considered negligible. This is here represented by the dashed blue line, not present in our model. so that in terms of the variables \(q\) and \(p\), the system (6) reads:  
\[\left\{ \begin{array}{l}\dot{q}=\frac{1}{2}p(q^{2}+p^{2})-\alpha nq,\\ \dot{p}=-\frac{1}{2}q(q^{2}+p^{2})-\alpha np,\\ \dot{n}=\frac{k}{8}(q^{2}+p^{2})^{2}.\end{array} \right. \tag{8}\]  
The phasor \(\psi\) may be also represented by _Madelung variables_\((\phi,\rho,n)\), as:  
\[\psi=\sqrt{\rho}e^{i\phi}. \tag{9}\]  
These canonically conjugated \(\phi\) and \(\rho\) describe the system as reported in Section 5. It is useful to observe that in the limit  
\[\alpha\to 0,\ k\to 0 \tag{10}\]  
Equation (8) become a Hamiltonian system, so that the conditions (10) will be referred to as _non-dissipative limit_ (NDL); under these conditions, the ODEs in \(q\), \(p\) and \(n\) read:  
\[\left\{ \begin{array}{l}\dot{q}=\frac{1}{2}p(q^{2}+p^{2}),\\ \dot{p}=-\frac{1}{2}q(q^{2}+p^{2}),\\ \dot{n}=0.\end{array} \right. \tag{11}\]  
As one defines the Hamiltonian  
\[H_{0}=\frac{1}{2}\bigg{(}\frac{q^{2}+p^{2}}{2}\bigg{)}^{2}. \tag{12}\]  
and the Poisson bracket  
\[\{q,p\}=1,\ \{q,n\}=0,\ \{p,n\}=0, \tag{13}\]  
any quantity \(f(q,p,n)\) evolves according to:  
\[\dot{f}=\{f,H_{0}\}\]  
along the motion (11). The quantity defined in Equation (12) turns out to be the energy that can be attributed to the free radiation, as it is not interacting with matter.