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

摘要

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