# Predictive Modeling of Light-Matter Interaction in One Dimension: A Dynamic Deep Learning Approach
## 2 Methods  
### FDTD-Based LMI Model Based on Two Energy Levels  
We first investigate the case where photons and electrons interact based on two energy levels with the corresponding frequencies \(v_{1}\) and \(v_{2}\). In this case, there are only two transitions, which are \(1\to 2\) and \(2\to 1\). Correspondingly, there is only one transition frequency, which is \(\omega_{R}=2\pi(v_{2}-v_{1})\). Hence, the polarization density goes into resonance only at this frequency. When the interaction material has a single electronic transition frequency, upon excitation of the interaction medium via an optical beam of electric field intensity E, the corresponding wave, polarization, and electron dynamics are modeled as follows [5; 6; 7].  
Numerical model for wave dynamics (the wave equation):  
\[\
abla^{2}E-\mu_{0}\varepsilon_{\infty}\frac{\partial^{2}E}{\partial t^{2}}= \mu_{0}\sigma\frac{\partial E}{\partial t}+\mu_{0}\frac{d^{2}P}{dt^{2}} \tag{1}\]  
Numerical model for polarization dynamics (Lorentz dispersion eq. up to the 3 rd order):  
\[\frac{d^{2}P}{dt^{2}}+\Gamma_{R}\frac{dP}{dt}+\omega_{R}{}^{2}P-\frac{\omega_ {R}{}^{2}P^{2}}{2(N_{1}-N_{2})ed}+\frac{\omega_{R}{}^{2}P^{3}}{6(N_{1}-N_{2})^ {2}e^{2}d^{2}}=\frac{(N_{1}-N_{2})e^{2}E}{m_{e}} \tag{2}\]  
Numerical model for electron dynamics:  
\[\frac{dN_{1}}{dt}=-\Gamma N_{1}+\frac{N_{2}}{\tau_{21}}-\frac{1}{(h/2\pi) \omega_{R}}E\frac{dP}{dt} \tag{3}\]  
\[\frac{dN_{2}}{dt}=\Gamma N_{1}+\frac{1}{(h/2\pi)\omega_{R}}E\frac{dP}{dt}- \frac{N_{2}}{\tau_{21}} \tag{4}\]  
E: Electric field, \(\mu_{0}\): Free space permeability, \(\varepsilon_{\infty}\): Background permittivity, t: Time, h: Planck constant, \(\sigma\): Conductivity, P: Polarization density, \(\gamma_{\rm R}\): Polarization damping rate, \(\omega_{\rm R}\): Angular resonance frequency, N\({}_{1}\): Electron density at the first level, N\({}_{2}\): Electron density at the second level, \(\tau_{21}\): Transition lifetime (level 2 to level 1), d: Atom diameter, m\({}_{\rm e}\): Electron mass, e: Elementary charge, and \(\Gamma\): Electron pump rate.  
Here, Equation (1) represents the electric field wave equation for isotropic media that can be easily derived from equations representing Ampere's Law and Faraday's Law (also known as Maxwell's First and Second Equations) [5]. Equation (2) is the famous Lorentz dispersion equation, which governs the temporal behavior of the polarization density of bound electrons under electric field excitation. The Lorentz dispersion equation is analogous to the spring model, which describes the position of a mass that is attached to a spring on which a certain amount of force is exerted. In this case, the electrons are under damped oscillation due to the force that is exerted by the atomic nucleus and the electric field. The damping occurs mainly due to collisions with other electrons. Equations (3) and (4) represent the electron populations at level 1 (ground level) and level 2 (excited level). Since the polarization density in Equation (2) is created by the electron population difference between level 1 and level 2, and since the electric field intensity in Equation (1) depends on the polarization density (which is the source term), Equations (1)-(4) are coupled to each other and must be solved simultaneously. The situation is more complicated in an interaction medium with more energy levels (see next section).