E.2 Two-photon transitions 313proximations give the duration of aπ-pulse as
tπ=
π
Ωeffπ∆
Ω^22 π∆
Γ^2Isat
I, (E.14)
where eqn 7.86 has been used to relate Ω^2 to intensity. Thus in a Raman
experiment carried out with two laser beams, each of intensity 3Isat,and
the above values of ∆ and Γ for sodium, the pulse has a duration of
tπ=10μs.
E.2 Two-photon transitions
The intuitive description of Raman transitions as two successive appli-
cations of the first-order time-dependent perturbation theory result for
a single-photon transition can also be applied to two-photon transitions
between levels 1 and 2 viai,whereω 2 >ωi>ω 1. The two-photon rate
between levels 1 and 2 is
R 12 =
∣
∣
∣
∣
∑
i{
〈 2 |er·EL2|i〉〈i|er·EL1| 1 〉
^2 (ωi−ω 1 −ωL1)+
〈 2 |er·EL1|i〉〈i|er·EL2| 1 〉
^2 (ωi−ω 1 −ωL2)}∣∣
∣
∣
2
×g(ωL1+ωL2).
(E.15)
This has the form of the modulus squared of a sum of amplitudes mul-
tiplied by the line shape functiong(ωL1+ωL2). There are two con-
tributing amplitudes from (a) the process where the atom interacts with
the beam whose electric field isEL1and then with the beam whose
field isEL2, and (b) the process where the atom absorbs photons from
the two laser beams in the opposite order.^9 The energy increases by^9 Only one of these paths is near res-
onance for Raman transitions because
ωL1−ωL2 =ωL2−ωL1.
(ωL1+ωL2) independent of the order in which the atom absorbs the
photons, and the amplitude in the excited state is the sum of the am-
plitudes for these two possibilities. In Doppler-free two-photon spectro-
scopy (Section 8.4) the two counter-propagating laser beams have the
same frequencyωL1=ωL2=ω, and we shall also assume that they
have the same magnitude of electric field (as would be the case for the
apparatus shown in Fig. 8.8). This leads to an excitation rate given by
R 12
∣
∣
∣
∣
∣
2
∑
iΩ 2 iΩi 1
ωi−ω 1 −ω∣
∣
∣
∣
∣
2
·
Γ ̃/(2π)
(ω 12 − 2 ω)^2 +Γ ̃^2 / 4, (E.16)
with Ωi 1 and Ω 2 ias defined in the previous section. The transition has
a homogeneous width ̃Γ greater than, or equal to, the natural width
of the upper levelΓ ̃Γ; this Lorentzian line shape function has a
similar form to that in eqn 7.77, with a maximum at the two-photon
resonance frequencyω 12 =ω 2 −ω 1 (as in Section 8.4). The constraint
that the two photons have the same frequency means that the frequency
detuning from the intermediate level ∆ =ωi−(ω 1 +ω) is generally
much larger than in Raman transitions, e.g. for the 1s–2s transition in