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

With the calculation of the detuning from the Raman resonance and the effective
Raman-Rabi frequency we are now able to write down an expression for the final state
probability, depending on position r, Raman detuning δ and pulse duration τ
P f (r, θ, δ, τ) = 1
3 (sin2 √
3
(
4 τ β00 sin θ)·
minitial = −1
minitial = 0
minitial = 1
(sinc 2 ( (δ + 3/2 µB dB/dr · r/) τ
) + sinc
2
2 ( (δ − 3/2 µB dB/dr · r/) τ
))
2
+sin 2 √
3
(
4 τ β00 cos θ)·
(sinc 2 ( (δ + µB dB/dr · r/) τ
) + sinc
2
2 ( (δ − µB dB/dr · r/) τ
))
2
+sin 2 ( 1
√ τ β00 sin θ)·
2 4
(sinc 2 ( (δ + 1/2 µB dB/dr · r/) τ
) + sinc
2
2 ( (δ − 1/2 µB dB/dr · r/) τ
))
2
+sin 2 √
3
( √ τ β00 sin θ)·
2 4
(sinc 2 ( (δ + 1/2 µB dB/dr · r/) τ
) + sinc
2
2 ( (δ − 1/2 µB dB/dr · r/) τ
))
2
+sin 2 ( 1
2 τ β00 cos θ) · sinc 2 δ τ
( )). (6.7)
2
m f inal = −2 m f inal = −1 m f inal = 0 m f inal = 1 m f inal = 2
√ √
3
3
2 sin θ 2 cos θ √
1
2 2 sin θ
√ √ √ √
3
3
2 2 sin θ cos θ 2 2 sin θ
√ √ √
1
3
3
2 2 sin θ 2 cos θ 2 sin θ
Table 6.2: Calculation of the effective Raman-Rabi frequencies in the lab frame; in
multiples of β0,0. θ is defined as in eqs. 5.35 and 5.36
6.4 Numerical results for Sodium atoms
Before we can calculate the spatial resolution we need to find the optimal pulse duration
to ensure a complete transfer of atoms from the |F = 1〉 to the |F = 2〉 state. We therefore
ignore , for the time being, the effects due to the detuning from the Raman resonance
frequency and an external magnetic field and concentrate on the time development.
44