Single-Photon Atomic Cooling - Raizen Lab - The University of ...

In this equation we let H = H0 + Hcoh and neglect Hdamp because its exact
form is unknown and its effect can be added in later by hand. With this
approximation we find the time evolution of the off-diagonal elements to be
˙ρ21 = −iω0ρ21 + i µE(t)
(ρ11 − ρ22) (2.67)
with ρ12 ˙ = ρ21 ˙ ∗ . The diagonal elements are found to be
˙ρ11 = − µE(t)
i (ρ21 − ρ12) = − ˙ρ22. (2.68)
Damping is a property of the interaction between individual dipoles
and is therefore a property of the ensemble and not individual atoms. Damp-
ing can be included phenomenologically in the off-diagonal elements from the
knowledge that when E(t) → 0 the ensemble average dipole moment must
decay to 0. There are many causes of dephasing which lead to the decay of
the off-diagonal element such as population relaxation, collisions, and dipole-
dipole interactions, the net effect of which can be encompassed in a transverse
decay rate 1/T2. Likewise, processes such as spontaneous decay and collisional
de-excitation lead to the decay (growth) of the excited (ground) state popu-
lation. The rate of these processes can be encompassed by the longitudinal
decay rate 1/T1.
Before including the effects of damping in the optical Bloch equations,
I would like to briefly discuss the commonly used rotating wave approxima-
tion, which is a result of writing the off-diagonal elements as a slowly varying
envelope function multiplied by a response at the applied optical frequency.
ρ21(t) = σ21(t)e −iωt = ρ ∗ 12(t) (2.69)
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