Abstract

Fully quantum calculations are needed for laser cooling techniques in which the momentum of the atoms is reduced close to, or below, the recoil limit. The first theoretical work on velocity selective coherent population trapping used generalized optical Bloch equations,¹ but this becomes cumbersome for higher dimensions or higher J transitions.² More recently various approaches for modeling dissipation have been developed that consider wavefunctions rather than density matrices.³ In the quantum jump method one important step is to get the delay time between quantum jumps, which may be determined by various ways, and the method used here is an improvement over using the numerical integration of Schrödinger equation to find delay time in which the excited state decay term is incorporated into the Hamiltonian making it a non-Hermitian matrix as described by Arimondo and Cohen- Tannoudji.⁴ If the evolution of the Schrödinger equation for this non-Hermitian Hamiltonian were to be calculated by numerical integration this method would be exactly the same as the usual quantum Monte Carlo method, but it is much better to transform into the basis of eigenstates of the non-Hermitian matrix. This gives an analytic expression for the norm |Ψ(t)|² and enables the delay time between quantum jumps to be calculated quickly. To illustrate the method we first describe its application to the simple A structure that occurs in the J = 1 to 1 transition. The wave function to evolve is loss rates and the light shifts of the coupled state, u_c(p), and quasi-dark state, u_d(p), respectively. Figure 1 shows results of calculation for rubidium using this method in comparison with density matrix calculation, where a satisfactory consistency is reached.

© 1996 Optical Society of America