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

frame is obtained by expanding the sum,
T(1, 0) = T ′ (1, 0) cos θ(r) + 1 √ 2 T ′ (1, −1) sin θ(r) e −iφ(r) + 1 √ 2 T ′ (1, 1) sin θ(r) e iφ(r) . (5.41)
This has to be true for all angles θ and φ. By choosing θ = φ = 0 we therefore find that
T(1, 0) = T ′ (1, 0) = β0,0. (5.42)
We will use the Wigner-Eckart Theorem to determine the remaining effective
Raman-Rabi frequencies in the rotated frame; using eq. A.6 we find them to be
β−1,0 = β1,0 = 1
2 β0,0. (5.43)
After doing the transformation to the lab frame using eq. 5.41 the Raman-Rabi fre-
quency become
β0,0(r) = T ′ (1, 0) cos θ(r) = β0,0 cos θ(r) (5.44)
β−1,0(r) = 1 √ 2 T ′ (1, −1) sin θ(r) = 1
β1,0(r) = 1 √ 2 T ′ (1, 1) sin θ(r) = 1
2 √ 2 β0,0 sin θ(r) (5.45)
2 √ 2 β0,0 sin θ(r). (5.46)
We have neglected phase information in these expressions since only the magnitude
of the Raman-Rabi frequencies matters for the final state probability. The remaining
Raman-Rabi frequencies of the initial states with magnetic quantum number m = ±1
can be treated similarly.
The final state probability can therefore be determined by adding the contribu-
tions from all initially populated states. The transfer efficiency is determined by the
product of the effective Raman-Rabi frequency and the pulse duration, while the prod-
uct of the detuning from resonance due to a magnetic field gradient and the pulse dura-
tion determines the resolution. The magnitude of the external magnetic field influences
the effective detuning (δ − δB(r)), while the direction affects the effective Raman-Rabi
i
frequency β.
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