Simple determination of dark states in a general multi-level system

TL;DR

This paper introduces straightforward criteria and methods to identify dark states in complex multi-level quantum systems by analyzing Hamiltonian submatrices and their singular vectors.

Contribution

It provides a simple, general approach to determine dark states in multi-level systems using Hamiltonian submatrices and singular value decomposition.

Findings

Dark states are right-singular vectors of specific Hamiltonian submatrices.

A determinant-based method simplifies finding dark states.

Criteria apply broadly to multi-level energy systems.

Abstract

In a multi-level energy system with energy transitions, dark states are eigenstates of a Hamiltonian that consist entirely of ground states, with zero amplitude in the excited states. We present several criteria which allows one to deduce the presence of dark states in a general multi-level system based on the submatrices of the Hamiltonian. The dark states can be shown to be the right-singular vectors of the submatrix that connect the ground states to the excited states. Furthermore, we show a simple way of finding the dark state involving the determinant of a matrix constructed from the same submatrix.