General solution of the time evolution of two interacting harmonic oscillators

TL;DR

This paper provides an exact analytical solution for the dynamics of two coupled harmonic oscillators with arbitrary interaction strength, including complex regimes like ultrastrong coupling and higher order interactions, revealing phase transitions and practical quantum optical implementations.

Contribution

It introduces a comprehensive analytical framework for the time evolution of two interacting oscillators with arbitrary coupling, including phase transition analysis and extension to multiple oscillators.

Findings

Predicts a second order phase transition with critical exponents

Provides an exact decoupling of evolution into simple quantum optical operations

Extends techniques to systems with more oscillators and higher order interactions

Abstract

We study the time evolution of an ideal system composed of two harmonic oscillators coupled through a quadratic Hamiltonian with arbitrary interaction strength. We solve its dynamics analytically by employing tools from symplectic geometry. In particular, we use this result to completely characterize the dynamics of the two oscillators interacting in the ultrastrong coupling regime with additional single-mode squeezing on both oscillators, as well as higher order terms. Furthermore, we compute quantities of interest, such as the average number of excitations and the correlations that are established between the two subsystems due to the evolution. We find that this model predicts a second order phase transition and we compute the critical exponents and the critical value. We also provide an exact decoupling of the time evolution in terms of simple quantum optical operations, which can be used for practical implementations and studies. Finally, we show how our techniques can be extended to include more oscillators and higher order interactions.