Experiments with Supersonic Beams as a Source of Cold Atoms

where k1 · pi = − k2 · pi, where k1 and k2 are the wave vectors of photons from the
two beams. In this scenario, the Doppler shifts cancel each other out. Since the
photons are counterpropagating, there is no recoil either. This is known as Doppler
free two-photon excitation, and in this regime the laser must satisfy
2ωlaser = E2S pf − E1S pi = ω1S−2S. (5.2)
This means that a laser at λ = 243nm is needed to excite the atoms. Assuming
linearly polarized light of equal intensity I in the counter propogating beams, and
a laser frequency ωlaser = ω1S−2S, the Rabi frequency for Doppler free excitation is
[107]
Ω0(r) =2M 12
3 α
2S,1S
2R∞
1
3π 2 c
I(r) =9.264I(r)cm2 , (5.3)
Ws
where M 12
2S,1S is the sum over the two-photon dipole matrix elements, α is the fine
structure constant, R∞ is the Rydberg constant, and I(r) is the spatial intensity
distribution. For a weak resonant excitation the transition rate is
R =
Ω0(r) 2
Γ
= 85.8
Γ
cm4
I(r)2
W 2 , (5.4)
s
where Γ is the linewidth of the dominant homogeneous broadening mechanism. In
the absence of other broadening mechanisms, the natural linewidth of the transition
determined by the lifetime of the 2S state, sets Γ = 8.2s −1 . Thus, without another
source of broadening, the saturation intensity of the transition is I =0.89 W
cm2 . The
most important aspect of equation 5.4 is that the rate scales as the square of the
intensity of the light, rather than linearly as would be the case for a one-photon
transition. This means that the transition rate is very sensitive to the intensity,
and that two-photon transitions typically require much higher powers to achieve the
desired transition rates.
The limiting homogeneous broadening mechanism for the experiment is likely
to be the laser linewidth. This source of broadening is due to the inherent properties
150