Appendix E: Raman and

two-photon transitions

E

E.1 Raman transitions 310
E.2 Two-photon transitions 313

E.1 Raman transitions

This appendix gives an explanation of Raman (and two-photon transi-

tions) by adapting the treatment of single-photon transitions given in

Chapter 7—this approach gives much more physical insight than simply

quoting the theoretical formulae from second-order time-dependent per-

(^1) More rigorous treatments can be turbation theory. (^1) A Raman transition involves two laser beams with
found in quantum mechanics texts. frequenciesωL1andωL2, and the atom interacts with an electric field
that has two frequency components:
E=EL1cos (ωL1t)+EL2cos (ωL2t). (E.1)
A Raman transition between two atomic levels, labelled 1 and 2, in-
volves a third atomic level, as shown in Fig. 9.20. This third level is
labelledifor intermediate, but it is very important to appreciate that
atoms are not really excited to leveli. The treatment presented here
emphasises that Raman transitions are fundamentally different from a
process comprised of two single-photon transitions (1→ifollowed by
i→2). As in Section 9.8, we take the frequencies of the levels to be
related byωiω 2 >ω 1.
Firstly, we consider the perturbation produced by the light atωL1for
the transition between levels 1 andi. This is the same situation as for
a two-level atom interacting with an oscillating electric field that was
described in Chapter 7, but here the upper level is labellediinstead of
having the label 2 (and we writeωL1rather thanω). For a weak pertur-
bation, the lower state| 1 〉has amplitudec 1 (0) = 1 and the amplitude
of|i〉, from eqn 7.14, is
ci(t)=
Ωi 1
2

[

1 −exp{i(ωi−ω 1 −ωL1)t}

ωi−ω 1 −ωL1

]

. (E.2)

Here the Rabi frequency for the transition Ωi 1 is defined in terms ofEL1

(^2) We assume that the Rabi frequency is as in eqn 7.12. (^2) We define the difference between the laser frequencyωL1
real, i.e. Ω∗i 1 =Ωi 1. and the frequency of the transition between levels 1 andias
∆=ω 1 +ωL1−ωi. (E.3)
From eqn 7.76 the wavefunction of the atom is
Ψn(r,t)=e−iω^1 t| 1 〉−
Ωi 1
2∆
e−iωit|i〉+
Ωi 1
2∆
e−i(ω^1 +ωL1)t|i〉. (E.4)