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 equations¹ but this becomes cumbersome for higher dimensions or higher J transitions² Various quantum Monte-Carlo approaches have been developed which consider wavefunctions rather than density matrices.³ In 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 a more efficient improvement than that 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 Λ-structure which occurs in the J = 1 to 1 transition. The wavefunction to evolve is where Γ'(p), Γ"(p), δ_c(p), and δ_d(p) are the 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.

© 1995 IEEE