Unitary Integration: A Numerical Technique Preserving the Structure of the Quantum Liouville Equation

The quantum Liouville equation for an n-level atomic system driven by external fields has a nontrivial kinematic structure; the quantities tr ρ^j, j  =  1,2,…,n remain constant in time, independent of the Hamiltonian. These invariants are physically significant; the qualitative character of the solution depends on their existence. A generic numerical method will not, in general, preserve these invariants. We present a numerical technique which evolves the density matrix via unitary transformations thus exactly preserving these invariants to all orders in the time step.