# Analytical Results for the Dynamics of Parabolic Level-Crossing Model

## TL;DR

This paper derives analytical solutions for a two-level parabolic crossing model using special functions, providing approximations valid over extensive parameter ranges and validated by numerical comparisons.

## Contribution

It introduces new analytical approximations for the model's dynamics using Heun, Airy, and Bessel functions, expanding understanding of parabolic level-crossing phenomena.

## Key findings

- Analytical solutions expressed via special functions.
- Approximate formulas valid across large parameter spaces.
- Validation through numerical simulations.

## Abstract

We study the dynamics of a two-level crossing model with a parabolic separation of the diabatic energies. The solutions are expressed in terms of the tri-confluent Heun equations --- the generalization of the confluent hypergeometric equations. We obtain analytical approximations for the state populations in terms of Airy and Bessel functions. Applicable expressions are derived for a large part of the parameter space. We also provide simple formulas which connect local solution in different time regimes. The validity of the analytical approximations is shown by comparing them to numerical simulations.