Solution of two level system using $3^{rd}$ order Magnus expansion

TL;DR

This paper applies third order Magnus expansion to analytically solve a laser-driven two-level quantum system near resonance, simplifying the Hamiltonian and deriving solutions using eigenvalues and Sylvester's formula.

Contribution

It introduces a third order Magnus expansion approach for solving two-level systems under near-resonance laser driving, providing a new analytical method.

Findings

Derived explicit solutions for the two-level system Hamiltonian.

Showed the Hamiltonian commutes at different times under resonance conditions.

Utilized eigenvalues and Sylvester's formula for solution calculation.

Abstract

The solution of a two level system driven by a Laser in the adiabatic limit is determined using third order Magnus expansion. We made the assumption that the laser is on resonance or close to resonance with the Bohr transition. As a consequence of which we are able to obtain a Hamiltonian which commute with itself at different times. We solve the problem using the Sylvester Formula where we make use of the eigenvalues.