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9.8 Raman transitions 209

2

1

Fig. 9.20A Raman transition between

levels 1 and 2 driven by two laser beams

of (angular) frequenciesωL1andωL2.

For a resonant Raman process the fre-

quency detuningδ 0, and the de-

tuning ∆ from the intermediate state

remains large, so that excitation by

single-photon absorption is negligible

in comparison to the coherent transfer

from| 1 〉to| 2 〉.Inthisexamplethe

atom has velocityvalong the direction

of the laser beam with frequencyωL2,

and the laser beam with frequencyωL1

propagates in the opposite direction.

is illustrated in Fig. 9.20. For two beams of frequenciesωL1andωL2the
condition for resonant excitation is

ωL1+k 1 v−(ωL2−k 2 v)=ωL1−ωL2+

v

c

(ωL1+ωL2)=ω 12. (9.56)

For counter-propagating beams the Doppler shifts add to make the Ra-
man transition sensitive to the velocity—about twice as sensitive as a
single-photon transition.^66 Direct excitation of the transition by radio-^66 In contrast, two counter-propagating
laser beams of the same frequency give
Doppler-free two-photon spectra:
ωL+kv+(ωL−kv)=2ωL,
as in eqn 8.20. If a two-photon transi-
tion is excited by two laser beams with
different frequencies then the Doppler
shifts do not cancel exactly.

frequency radiation, or microwaves, at angular frequencyω 12 is insen-
sitive to the motion. The great advantage of the Raman technique for
velocity selection (and cooling) arises from its extremely narrow line
width (comparable with that of radio-frequency methods) of Raman
transitions between levels that have long lifetimes, e.g. hyperfine levels
in the ground configuration of atoms for which spontaneous decay is
negligible. To fully exploit the advantage of this narrow line width, the
difference in frequency between the two laser beams ∆ω=ωL1−ωL2
must be controlled very precisely. This can been achieved by taking
two independent lasers and implementing sophisticated electronic servo-
control of the frequency difference between them, but it is technically
easier to pass a single laser beam through a phase modulator—the resul-
tant frequency spectrum contains ‘sidebands’ whose difference from the
original laser frequency equals the applied modulation frequency from a
microwave source.^67 The selected velocityvis determined by

(^67) For a laser beam with (angular) fre-
quencyω, phase modulation at fre-
quency Ω leads to a spectrum contain-
ing the frequenciesω±nΩ, withnin-
teger. This can be used to carry out
Raman excitation, e.g. withωL1=ω
andωL2=ω−Ω.
2 kv=ω 12 −(ωL1−ωL2), (9.57)
wherek=(ωL1+ωL2)/cis the mean wavevector.
Raman transitions between levels with negligible broadening from
spontaneous decay or collisions have a line width determined by the
interaction time: for a pulse of durationτpulsethe Fourier transform
limit gives^68
(^68) Similar to that for the single-photon
transition in Section 7.1.2.
∆v
λ



1

τpulse

. (9.58)

For a visible transition with a wavelength of 600 nm^69 a pulse of duration

(^69) Asinthecaseofsodiumthatwas
used in the first Raman experiments
with cold atoms. In sodium, levels
1 and 2 are the hyperfine levels with
F= 1 and 2 of the 3s configuration,
and the intermediate leveliis 3p^2 P 3 / 2.