Copyright by Kirsten Viering 2006 - Raizen Lab - The University of ...

to adiabatically reduce the three-level amplitude equations to an effective two-level
system,
˙ci = −iν1ci − β
2 c f e iδt
(5.18)
˙c f = −iν2c f − β∗
2 cie −iδt , (5.19)
where ∆1 ≈ ∆2 ≡ ∆ and δ = ∆1 − ∆2. The AC-Stark shifts νm and the effective Raman-
Rabi frequency β are then defined by
νm ≡
|Ωm| 2
4∆
1 Ω2
β ≡ − Ω∗
2∆
Eqs. 5.18 and 5.19 can be solved by substituting
ci = aie −iν1t
Doing this we can rewrite the set of differential equations as
and
˙ai =
(5.20)
. (5.21)
(5.22)
c f = a f e −iν2t . (5.23)
i β
2 a f e i(δ−(ν2−ν1))t
(5.24)
i β∗
˙a f =
2 ai e −i(δ−(ν2−ν1))t
. (5.25)
If both light shifts are equal, ν2 = ν1, and the detuning δ from the resonant Raman
transition vanishes, the solution of these equations is trivial. The final state probability
then is
P f (t → ∞) = |c f (t → ∞)| 2 = |a f (t → ∞)| 2 = sin 2
|β|
2 τ
, (5.26)
where we have assumed the initial population of state | f 〉 to be zero and a Raman pulse
duration of length τ. |β|τ is commonly known as pulse area. A π
2 -pulse will transfer
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