Analytical Approximations to the Dynamics of Nonlinear Level Crossing Models

TL;DR

This paper develops analytical approximations for the dynamics of a nonlinear two-level crossing model with cubic modifications, expressing solutions via special functions and validating them against numerical simulations.

Contribution

It introduces new analytical approximations for nonlinear Landau-Zener models using bi-confluent Heun functions and provides closed-form expressions for transition probabilities.

Findings

Analytical solutions expressed in terms of special functions.

Closed-form approximations valid over large parameter regions.

Good agreement between analytical and numerical results.

Abstract

We study the dynamics of a nonlinear two-level crossing model with a cubic modification of the linear Landau-Zener diabatic energies. The solutions are expressed in terms of the bi-confluent Heun functions --- the generalization of the confluent hypergeometric functions. We express the finial transition probability as a convergent series of the parameters of the nonlinear laser detuning, and derive analytical approximations for the state populations in terms of parabolic cylinder and Whittaker functions. Tractable closed-form expressions are derived for a large part of the parameter space. We also provide simple method to determine the transition point which connects local solutions in different physical limits. The validity of the analytical approximations is shown by comparison with numerical results of simulations.