Single-Photon Atomic Cooling - Raizen Lab - The University of ...

excited state occupation probability and the natural decay rate
Rscatt = Γ
2
(I/Isat)
1 + 4(∆/Γ) 2 . (2.76)
+ (I/Isat)
Since each photon scattered carries momentum k the force on the atom due
to this process is
Fscatt = k Γ
2
(I/Isat)
1 + 4(∆/Γ) 2 . (2.77)
+ (I/Isat)
We can use this result and the first order Doppler shift ω ′ = ω − kv,
where ω ′ is the frequency seen by an atom moving at velocity v with respect to
a laser beam at frequency ω, to find the force on the atom in this arrangement.
If we consider the 1-D case, with a single pair of counter propagating beams,
then the total force is just the sum of forces due to each beam
Fmolasses = Fscatt(ω − ω0 − kv) − Fscatt(ω − ω0 + kv). (2.78)
We can approximate this result for small velocities through Taylor expansion
of Fscatt
Fmolasses ≈ Fscatt(ω − ω0) − kv ∂F
∂ω − [Fscatt(ω − ω0 + kv)] ≈ −2 ∂F
kv. (2.79)
∂ω
We note that this may be written
Fmolasses = −αv, (2.80)
showing that this force mimics classical viscous damping, the reason it is called
optical molasses. The value of α is found through differentiation of Eq. 2.77
to be
2 I −2∆/Γ
α = 4k
Isat [1 + (2∆/Γ) 2 ] 2,
58
(2.81)