Evolution of an N-level system via automated vectorization of the Liouville equations and application to optically controlled polarization rotation

TL;DR

This paper introduces an automated algorithm to efficiently generate the evolution matrix for N-level quantum systems, demonstrated on a 15-level system for optical polarization control, enabling scalable analysis of complex atomic systems.

Contribution

The paper presents a novel automated method to construct the Liouville evolution matrix for any N-level system, simplifying analysis of large quantum systems.

Findings

Automated algorithm successfully generates evolution matrices for large N-level systems.

Applied method to a 15-level atomic system for polarization rotation.

Method can be extended to systems with arbitrary number of energy levels.

Abstract

The Liouville equation governing the evolution of the density matrix for an atomic/molecular system is expressed in terms of a commutator between the density matrix and the Hamiltonian, along with terms that account for decay and redistribution. For finding solutions of this equation, it is convenient first to reformulate the Liouville equation by defining a vector corresponding to the elements of the density operator, and determining the corresponding time-evolution matrix. For a system of N energy levels, the size of the evolution matrix is N2xN2. When N is very large, evaluating the elements of these matrices becomes very cumbersome. We describe a novel algorithm that can produce the evolution matrix in an automated fashion for an arbitrary value of N. As a non-trivial example, we apply this algorithm to a fifteen-level atomic system used for producing optically controlled polarization rotation. We also point out how such a code can be extended for use in an atomic system with arbitrary number of energy levels.