Formulation of general dynamical invariants and their unitary relations for time-dependent coupled quantum oscillators

TL;DR

This paper derives an exact invariant operator for time-dependent coupled quantum oscillators and establishes a unitary relation to simpler systems, aiding in understanding complex quantum phenomena like entanglement and decoherence.

Contribution

It introduces a novel method to relate complex coupled oscillators to simpler systems via invariant operators and unitary transformations.

Findings

Derived an exact invariant operator for coupled oscillators.

Established a unitary relation to simple harmonic oscillators.

Enabled analysis of quantum characteristics through inverse transformations.

Abstract

An exact invariant operator of time-dependent coupled oscillators is derived using the Liouville-von Neumann equation. The unitary relation between this invariant and the invariant of two uncoupled simple harmonic oscillators is represented. If we consider the fact that quantum solutions of the simple harmonic oscillator is well-known, this unitary relation is very useful in clarifying quantum characteristics of the original systems, such as entanglement, probability densities, fluctuations of the canonical variables, and decoherence. We can identify such quantum characteristics by inversely transforming the mathematical representations of quantum quantities belonging to the simple harmonic oscillators. As a case in point, the eigenfunctions of the invariant operator in the original systems are found through inverse transformation of the well-known eigenfunctions associated with the simple harmonic oscillators.