# Metriplectic Structure of a Radiation-Matter-Interaction Toy Model
###### Abstract  
A dynamical system defined by a metriplectic structure is a dissipative model characterized by a specific pair of tensors, which defines a Leibniz bracket; and a free energy, formed by a \"Hamiltonian\" and an entropy, playing the role of dynamics generator. Generally, these tensors are a Poisson bracket tensor, describing the Hamiltonian part of the dynamics, and a symmetric metric tensor, that models purely dissipative dynamics. In this paper, the metriplectic system describing a simplified two-photon absorption by a two-level atom is disclosed. The Hamiltonian component is sufficient to describe the free electromagnetic radiation. The metric component encodes the radiation-matter coupling, driving the system to an asymptotically stable state in which the excited level of the atom is populated due to absorption, and the radiation has disappeared. First, a description of the system is used, based on the real-imaginary decomposition of the electromagnetic field phasor; then, the whole metriplectic system is re-written in terms of the phase-amplitude pair, named Madelung variables. This work is intended as a first result to pave the way for applying the metriplectic formalism to many other irreversible processes in nonlinear optics. two-photon absorption; metriplectic systems; dissipative systems; asymptotically stable equilibrium; Madelung variables +
Footnote †: journal:  
0022245066  
energy \(F\) extends the Hamiltonian \(H\) combining it with an entropic quantity \(S\); the two tensors forming the metriplectic bracket are a Jacobi tensor \(J\), and a semimetric tensor \(G\) related to the Onsager coefficients [6]. Metriplectic formalism is perfectly analogous to what is referred to as _General Equation for Non-Equilibrium Reversible-Irreversible Coupling_ (GENERIC), described in [6], in which the role of the tensors \(J\) and \(G\) is strongly stressed. In the literature, there are several examples of irreversible dynamics represented as metriplectic systems, from very simple systems in [PERSON]'s mechanics [7], to hydrodynamics [8] and magneto-hydrodynamics [9]; more delicate, but extremely interesting, are the cases of kinetic equations, the collisional terms of which may be written as a semimetric term, or that of a free rotator driven to a stable rotation axis by a suitably designed servo-engine [3; 10]. In all the cases mentioned, the non-Hamiltonian system is gifted of the transparency of motions generated by Leibniz algebra [11], even if proper symplectic formalism is not applicable; moreover, the energy landscape becomes tractable as the free energy of the system is explicitly written. In the present paper, the metriplectic formalism is used to describe a process in which electromagnetic radiation is absorbed by the atoms of a medium. As the radiation disappears, the medium electrons go to a higher energy level, so that the final state of the system is an excited state of matter with no radiation. This process may be regarded as the irreversible sink of an ordered form of energy, contained in the incident electromagnetic field, into the many atoms of the medium, represented with the collective variable of the excited population. This pictorial view of the radiation absorption as a dissipative process is physically supported by the fact that the terms describing absorption in the equations of electromagnetism are time-asymmetric terms, isomorphic to those ones describing friction in mechanics. The process of _two-photon absorption_ (TPA) by a two-level atom is here described through a _classical_ dynamical system, in which the energy initially located in the radiation variables is _irreversibly_ converted into the energy pertaining to the population of the excited level. The final state, in which no free radiation exists any more while the excited state is populated, is _the asymptotic equilibrium state_ of the system.