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

half the population while a π-pulse transfers the whole population from the initial into
the final state.
We have assumed the light shifts ν1 and ν2 to be equal. Experimentally this
can be achieved by tuning the intensities of the incident Raman beams. This becomes
especially easy when driving a resonant transition between the two hyperfine ground
states in Sodium (F=1 and F=2), since both states are shifted equally (see chapter 3).
Thus if both beams have the same power and beam waist, i.e. the incident intensities
are identical, AC-Stark shifts will be equalized.
In addition to the final state probability for the resonant case we must now
consider the non-resonant case. We can solve the amplitude equation for the non-
resonant case using perturbation theory and assuming ai(0) ≈ 1. We are then able to
integrate eq. 5.25, where we again assume the light shifts ν1 and ν2 to be equal. For a
square Raman pulse of length τ the final state amplitude a f becomes
a f (t → ∞) =
τ/2
−τ/2
iβ∗ 2 e−iδt′ dt ′ = − β∗
δ sinδτ.
(5.27)
2
We may center the pulse around t = 0 without loss of generality. The general solution
for the final state probability |a f (t → ∞)| 2 is then
where
P f (t → ∞, δ) = τ2 |β| 2
sinc x ≡
4 sinc2
δτ
, (5.28)
2
sin x
. (5.29)
x
We find the exact solution for δ = 0 to be nearly equal to the approximated solution
given above. Hence we can identify eq. 5.28 as the product of eq. 5.26 with a sinc 2 line
shape for small pulse areas. Nevertheless for pulse areas as large as 4π the line shape
stays approximately constant [29] and we will therefore use
2 |β|τ
P f (t → ∞, δ) = sin
2 sinc2
33
δτ
. (5.30)
2