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

where m is the atomic mass. By setting ∆xsp = ∆xacc one finds the optimal pulse
duration
Tacc =
2m
F 2
1/3
and the acceleration limited resolution thus becomes
∆xacc =
2
2mF
(5.6)
1/3
. (5.7)
Let us assume that only the final state is shifted by the potential. The transition
between the initial state |i〉 and the final state | f 〉 can occur at any time during the pulse
time T, and so the uncertainty in the final momentum of the atoms is ∆p = FT. Simple
multiplication with the spectral resolution limit (eq. 5.2) leads to the fundamental limit
given by the uncertainty relation between space and momentum
µm
5.3 Raman transitions in a three level atom
∆x∆p = . (5.8)
The total electro-magnetic field driving a stimulated Raman transition can be described
by the sum of the two individual light beams with frequencies ω1 and ω2,
E(r, t) = 1
2 ·
E1 · e i(k·r−ω1t) + E1 · e −i(k·r−ω1t) + E2 · e i(k·r−ω2t) + E2 · e −i(
k·r−ω2t)
. (5.9)
There exist three possible configurations of a three-level system with the initial
state |i〉, an intermediate state |I〉 and a final state | f 〉, that is driven by a two-photon pro-
cess, usually called the ”ladder”, ”V” or ”lambda” configurations, which are schemat-
ically shown in fig. 5.3 [26]. A stimulated Raman transition between two hyperfine
levels of the ground state corresponds to the Λ-configuration.
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