# Two-level System coupled to Phonons: Full Analytical Solution

## TL;DR

This paper introduces a novel analytical method for solving a two-level system coupled to phonons, including cases with quadratic coupling, which were previously considered analytically intractable.

## Contribution

The authors develop a new analytical approach that handles both linear and quadratic system-phonon couplings by differentiating phonons based on the electronic level.

## Key findings

- Recovered known results for linear coupling
- Derived analytical solutions for quadratic coupling
- Provided a comprehensive framework for two-level phonon interactions

## Abstract

We propose an analytical procedure to fully solve a two-level system coupled to phonons. Instead of using the common formulation in terms of linear and quadratic system-phonon couplings, we introduce different phonons depending on the system electronic level. We use this approach to recover known results for the linear-coupling limit in a simple way. More importantly, we derive results for the quadratic coupling induced by a phonon frequency change, a problem considered up to now as not analytically solvable.