0198506961.pdf

312 Appendix E: Raman and two-photon transitions

This is the same as eqn 7.15 for one-photon transitions, but Ωeffhas re-

placed Ω. This presentation of Raman transitions has assumed a weak

perturbation and we have found results analogous to those for the weak

excitation of a single-photon transition (Section 7.1); a more comprehen-

sive treatment of the Raman coupling between| 1 〉and| 2 〉with effective

Rabi frequency Ωeff, analogous to that in Section 7.3, shows that Ra-

man transitions give rise to Rabi oscillations, e.g. aπ-pulse transfers

all the population from 1 to 2, or the reverse. Raman transitions are

coherent in the same way as radio-frequency, or microwave, transitions

directly between the two-levels, e.g. a Raman pulse can put the atomic

(^5) Raman transitions can impart mo- wavefunction into a coherent superposition stateA| 1 〉+B| 2 〉. 5
mentum to the atoms and this makes
them extremely useful for manipulating
atoms and ions (see Section 9.8).
It is vital to realise that the Raman transition has a quite distinct na-
ture from a transition in two steps, i.e. a single-photon transition from
level 1 toiand then a second step fromito 2. The two-step process
would be described by rate equations and have spontaneous emission
from the real intermediate state. This process is more important than
the coherent Raman process when the frequency detuning ∆ is small
so thatωL1matches the frequency of the transition between| 1 〉and
(^6) Similarly, the quantity ∆ +δ=ωi− |i〉. (^6) The distinction between a coherent Raman process (involving si-
ω 2 −ωL2(in eqn E.7) is small when
ωL2matches the frequency of the tran-
sition between|i〉and| 2 〉.Forthiscon-
dition the single-photon transition be-
tween levelsiand 2 is the dominant
process. The single-photon processes
can be traced back to the small ampli-
tude with time dependence exp(iωit)in
eqn E.4.
multaneous absorption and stimulated emission) and two single-photon
transitions can be seen in the following example.
Example E.1 The duration of aπ-pulse (that contains both frequen-
ciesωL1andωL2)isgivenby
Ωefftπ=π. (E.12)
For simplicity, we shall assume that both Raman beams have similar in-
tensities so that Ωi 1 Ω 2 iΩ and hence ΩeffΩ^2 /∆ (neglecting small
(^7) The generalisation to the case where factors). (^7) From eqn 9.3 we find that the rate of scattering of photons
Ωi 1
=Ω 2 iis straightforward. on the transition| 1 〉to|i〉is approximately ΓΩ^2 /∆^2 ,sinceforaRaman
transition ∆ is large (∆Γ). Thus the number of spontaneously-
emitted photons during the Raman pulse is
Rscatttπ

ΓΩ^2

∆^2

π∆

Ω^2



πΓ

∆

. (E.13)

This shows that spontaneous emission is negligible when ∆Γ. As

a specific example, consider the Raman transition between the two hy-

(^8) The hyperfine splitting isω 2 −ω 1 = perfine levels in the 3s ground configuration of sodium (^8) driven by two
2 π× 1 .7 GHz but we do not need to
know this for this calculation.
laser beams with frequencies such that ∆ = 2π×3GHz. The 3p^2 P 1 / 2
level acts as the intermediate level and has Γ = 2π× 107 s−^1 .Thus
Rscatttπ 0 .01—this means that the atoms can be subjected to many
π-pulses before there is a spontaneous emission event that destroys the
coherence (e.g. in an interferometer as described in Section 11.7). Ad-
mittedly, this calculation is crude but it does indicate the relative im-
portance of the coherent Raman transitions and excitation of the in-
termediate level by single-photon processes. (It exemplifies the order-
of-magnitude estimate that should precede calculations.) The same ap-