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

When these expressions are substituted into the optical Bloch equations and
terms proportional to e −iωt are equated while terms proportional to e −2iωt are
discarded, the equations take on their familiar form.
˙ρ11 = iΩ
2 (σ12 − σ21) + ρ22
T1
˙ρ22 = − iΩ
2 (σ12 − σ21) − ρ22
˙σ12 = −( 1
T2
T1
(2.70a)
(2.70b)
+ i∆)σ12 − iΩ
2 (ρ22 − ρ11) (2.70c)
The off-diagonal terms are complex conjugates of each other ˙σ21 = ˙σ ∗ 12, the
detuning of the applied field from resonance is denoted ∆ ≡ ω − ω0, and
Ω = −µ · E0/ is the Rabi frequency.
If we assume that radiative decay is the only dephasing process then
1/T1 = Γ and 1/T2 = Γ/2, where Γ is the natural decay rate of the excited
state. With this assumption, the steady state solution of the excited popula-
tion is
ρ22 =
(Ω/Γ) 2
1 + 4(∆/Γ) 2 + 2(Ω/Γ) 2.
(2.71)
This can be put in a more useful form for experimentalists by introducing the
transition saturation intensity Isat
so that we may write
I
Isat
= 2 Ω2
Γ 2
(2.72)
ρ22 = 1 (I/Isat)
2 1 + 4(∆/Γ) 2 . (2.73)
+ (I/Isat)
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