Free to Harmonic Unitary Transformations in Quantum and Koopman Dynamics

TL;DR

This paper extends the known quantum coordinate transformation mapping the free particle to the harmonic oscillator to classical Koopman dynamics, including dissipative systems, enabling new classical-quantum procedure translations.

Contribution

It introduces a unitary and time transformation framework that maps quantum and classical dissipative systems, bridging quantum squeezing techniques to classical dynamics.

Findings

Quantum free particle maps to harmonic oscillator via coordinate transformation.

Classical Koopman dynamics can be transformed similarly to quantum systems.

Dissipative quantum and classical systems are related through identical time-dependent scaling.

Abstract

It has long been known that there exists a coordinate transformation which exactly maps the quantum free particle to the quantum harmonic oscillator. Here we extend this result by reformulating it as a unitary operation followed by a time coordinate transformation. We demonstrate that an equivalent transformation can be performed for classical systems in the context of Koopman von-Neumann (KvN) dynamics. We further extend this mapping to dissipative evolutions in both the quantum and classical cases, and show that this mapping imparts an identical time-dependent scaling on the dissipation parameters for both types of dynamics. The derived classical procedure presents a number of opportunities to import squeezing dependent quantum procedures (such as Hamiltonian amplification) into the classical regime.