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.

We show that the Markovian dynamics of two coupled harmonic oscillators may be analyzed using a Schrödinger equation and an effective non-Hermitian Hamiltonian. This may be achieved by a non-unitary transformation that involves superoperators; such transformation enables the removal of quantum jump superoperators, which allows us to rewrite the Lindblad master equation in terms of a von Neumann-like equation with an effective non-Hermitian Hamiltonian. This may be generalized to an arbitrary number of interacting fields. Finally, by applying an extra non-unitary transformation, we may diagonalize the effective non-Hermitian Hamiltonian to obtain the evolution of any input state in a fully quantum domain.

We propose a trapped-ion platform to simulate a reconfigurable spin-spin Hamiltonian related to quantum thermodynamic processes. Starting from an experimental model describing two trapped ions under slightly off-resonant first sideband driving with individually controlled driving phases, we follow an operational quantum optics approach to show that it produces an effective model appearing in recent quantum thermodynamics proposals. We show that projection into the vibrational ground-state manifold allows full analytic treatment. As a practical example, we take experimental data from a ${}^{171}\mathrm{Yb}^{+}$ trap and numerically simulate the reversal of heat flow between two thermal spins controlled by their quantum correlations.

Leveraging operator techniques, we address the paraxial wave equation governing a field formed by the multiplication of a Gaussian function and an entire function; notably, the latter adheres to the Laplace equation, ${\nabla }_{\perp }^{2}f(x+iy)=0$, a direct consequence of satisfying the Cauchy-Riemann equations. Our theoretical and experimental exploration brings to light the intrinsic rotation of this field during propagation, elucidated by the incorporation of the quantum (Bohm) potential. This straightforward result holds promise, enabling the analytical deduction of the Fraunhofer or Fresnel diffraction pattern. Essentially, it simplifies the extraction of the Fresnel or Fourier transform from a function satisfying the Cauchy-Riemann equations.