Using the Ermakov-Lewis invariants appearing in KvN mechanics, the time-dependent frequency harmonic oscillator is studied. The analysis builds upon the operational dynamical model, from which it is possible to infer quantum or classical dynamics; thus, the mathematical structure governing the evolution will be the same in both cases. The Liouville operator associated with the time-dependent frequency harmonic oscillator can be transformed using an Ermakov-Lewis invariant, which is also time dependent and commutes with itself at any time. Finally, because the solution of the Ermakov equation is involved in the evolution of the classical state vector, we explore some analytical and numerical solutions.

We show that by using the quantum orthogonal functions invariant, we found a solution to coupled time-dependent harmonic oscillators where all the time-dependent frequencies are arbitrary. This system may be found in many applications such as nonlinear and quantum physics, biophysics, molecular chemistry, and cosmology. We solve the time-dependent coupled harmonic oscillators by transforming the Hamiltonian of the interaction using a set of unitary operators. In passing, we show that N time-dependent and coupled oscillators have a generalized orthogonal functions invariant from which we can write a Ermakov–Lewis invariant.

This contribution has two main purposes. First, using classical optics we show how to model two coupled quantum harmonic oscillators and two interacting quantized fields. Second, we present classical analogs of coupled harmonic oscillators that correspond to anisotropic quadratic graded indexed media in a rotated reference frame, and we use operator techniques, common to quantum mechanics, to solve the propagation of light through a particular type of graded indexed medium. Additionally, we show that the system presents phase transitions.

In this work, we detail a proposal for optical signals to be represented and analyzed in phase-space. Our proposal aims to integrate a series of operations in waveguide realization, as a compact all-together platform that takes an initial wavefield and returns a two-dimensional representation of the information. We show, step by step, that the quantum harmonic oscillator can be considered as a propagator of initial fields, and when a discretized version is implemented, the fractional order Fourier transform emerges. This last is crucial, since the Wigner-Radon theorem is used to establish a path between the propagated wavefield and the phase-space representation. We show by example that this integration offers a direct and efficient method for characterizing optical signals by reconstructing their Wigner phase-space in the scope of integrated optics.