0198506961.pdf

180 Laser cooling and trapping

Fig. 9.1For an atom moving towards
the laser, each absorbed photon gives
the atom a kick in the direction oppo-
site to its motion and the scattered pho-
tons go in random directions, resulting
in a force that slows the atom.

Oven Laser

forcethatslowstheatomdown. Themagnitudeofthisscattering force

equals the rate at which the absorbed photons impart momentum to the

atom:

Fscatt= (photon momentum)×(scattering rate). (9.2)

The scattering rate isRscatt=Γρ 22 ,andρ 22 , the fraction of the popu-

lation in level 2, is given in eqn 7.69, so that

Rscatt=

Γ

2

Ω^2 / 2

δ^2 +Ω^2 /2+Γ^2 / 4

. (9.3)

The frequencydetuningfrom resonanceδ=ω−ω 0 +kvequals the differ-

ence between the laser frequencyωand the atomic resonance frequency

ω 0 taking into account the Doppler shiftkv. The Rabi frequency and

(^7) Generally speaking, intensity is more saturation intensity are related byI/Isat=2Ω (^2) /Γ (^2) (see eqn 7.86) (^7) and
directly related to experimental param-
eters than the Rabi frequency, but we
shall use bothIand Ω in this chapter.
As noted previously, there are other
definitions of the saturation intensity in
common use that differ by a factor of 2
from the one used here.
photons have momentum
k,sothat^8
(^8) This statement relies on the compre-
hensive description of two-level atoms
interacting with radiation given in
Chapter 7.
Fscatt=
k

Γ

2

I/Isat

1+I/Isat+4δ^2 /Γ^2

. (9.4)

AsI→∞the force tends to a limiting value ofFmax=
kΓ/2. The

rate of spontaneous emission from two-level atoms tends to Γ/2athigh

intensities because the populations in the upper and lower levels both

approach 1/2. This follows from Einstein’s equations for radiation inter-

acting with a two-level atom that has degeneracy factorsg 1 =g 2 =1.

For an atom of massM this radiation force produces a maximum

acceleration that we can write in various forms as

amax=

Fmax

M

=

k

M

Γ

2

=

vr

2 τ

, (9.5)

whereτis the lifetime of the excited state. Therecoil velocityvris the

change in the atom’s velocity for absorption, or emission, of a photon

at wavelengthλ; it equals the photon momentum divided by the atomic

mass:vr=
k/M≡h/(λM). For a sodium atomamax=9× 105 ms−^2 ,

which is 10^5 times the gravitational acceleration. For the situation shown

in Fig. 9.1 the atom decelerates at a rate

dv

dt

=v

dv

dx

=−a, (9.6)