Entanglement in Three Coupled Harmonic Oscillators

TL;DR

This paper presents an exact solution for a three-body harmonic oscillator system with quadratic interactions, analyzing energy spectra and entanglement properties using group theory and unitary transformations.

Contribution

It introduces a novel method combining unitary transformations and $SU(3)$ representation theory to solve and analyze three coupled oscillators with arbitrary parameters.

Findings

Explicit energy spectrum solutions derived.

Purity function characterizing entanglement obtained.

Conditions for minimal and maximal entanglement identified.

Abstract

We develop an approach in solving exactly the problem of three-body oscillators including general quadratic interactions in the coordinates for arbitrary masses and couplings. We introduce a unitary transformation of three independent angles to end up with a diagonalized Hamiltonian. Using the representation theory of the group $SU(3)$, we explicitly determine the solutions of the energy spectrum. Considering the ground state together with reduced density matrix, we derive the corresponding purity function that is giving rise to minimal and maximal entanglement under suitable conditions. The cases of realizing one variable among three is discussed and know results in literature are recovered.