# Quadratic Open Quantum Harmonic Oscillator

## TL;DR

This paper analyzes the dynamics of a quadratic open quantum harmonic oscillator, demonstrating exponential convergence to a unique equilibrium state and providing explicit convergence rates, with implications for two-photon processes.

## Contribution

It introduces a detailed analysis of the quantum open system evolution using a specific Lindblad generator related to the $sl_2$ Lie algebra, including explicit convergence rates.

## Key findings

- Initial density matrices evolve to a fully supported state
- Convergence to equilibrium is exponentially fast
- Explicit convergence rates are computed for various parameters

## Abstract

We study the quantum open system evolution described by a Gorini-Kossakowski-Sudarshan-Lindblad generator with creation and annihilation operators arising in Fock representations of the $sl_2$ Lie algebra. We show that any initial density matrix evolves to a fully supported density matrix and converges towards a unique equilibrium state. We show that the convergence is exponentially fast and we exactly compute the rate for a wide range of parameters. We also discuss the connection with the two-photon absorption and emission process.