# Thermal Entanglement Phase Transition in Coupled Harmonic Oscillators with Arbitrary Time-Dependent Frequencies

## TL;DR

This paper derives the thermal state of two coupled harmonic oscillators with time-dependent parameters, analyzes their entanglement properties, and identifies how frequency changes influence thermal entanglement phase transitions.

## Contribution

It provides explicit analytical expressions for the thermal state and entanglement measures in a time-dependent coupled oscillator system, highlighting how frequency differences affect critical temperature.

## Key findings

- Critical temperature increases with frequency difference.
- Large frequency differences can protect entanglement against temperature.
- Analytical formulas for purity, entropy, and mutual information.

## Abstract

We derive explicitly the thermal state of the two coupled harmonic oscillator system when the spring and coupling constants are arbitrarily time-dependent. In particular, we focus on the case of sudden change of frequencies. In this case we compute purity function, R\'{e}nyi and von Neumann entropies, and mutual information analytically and examine their temperature-dependence. We also discuss on the thermal entanglement phase transition by making use of the negativity-like quantity. Our calculation shows that the critical temperature $T_c$ increases with increasing the difference between the initial and final frequencies. In this way we can protect the entanglement against the external temperature by introducing large difference of initial and final frequencies.