E.1 Raman transitions 311The perturbation produces the admixture of two terms into the initial
state| 1 〉that both have the same small amplitude Ωi 1 /|2∆|1. We
will find that the term with angular frequencyωirepresents real exci-
tation to|i〉. The term with angular frequencyω 1 +ωL1corresponds
to avirtual level, i.e. in the mathematics this term acts as if there is a
level at an energyωL1above the ground state that has the symmetry
properties of|i〉, but in reality there is no such level.
To determine the effect of the field oscillating atωL2on the perturbed
atom, we take eqn 7.10, which states that for a single-photon transition
i
.
c 2 =Ωcos(ωt)eiω^0 tc 1 , and make the replacementsω →ωL2,ω 0 →
ω 2 −ωiand Ω→Ω 2 ito obtain^33 The possibility that radiation at an-
gular frequencyωL1could drive this
transition betweeniand 2 is considered
later for two-photon transitions.
i.
c 2 (t)=Ω 2 icos (ωL2t)ei(ω^2 −ωi)tci(t). (E.5)Insertion of the expression forci(t) from eqn E.2 yields
i.
c 2 (t)=−Ω 2 icos (ωL2t)e−i(ωi−ω^2 )t×Ωi 1
2∆[
1 −ei(ωi−ω^1 −ωL1)t]
=−
Ω 2 iΩi 1
4∆[
eiωL2t+e−iωL2t]
·
[
e−i(ωi−ω^2 )t−ei{(ω^2 −ω^1 )−ωL1}t]
.
(E.6)
Integration and the rotating-wave approximation lead to
c 2 (t)=Ω 2 iΩi 1
4∆[
1 −e−i(ωi−ω^2 −ωL2)t
ωi−ω 2 −ωL2+
1 −ei{(ω^2 −ω^1 )−(ωL1−ωL2)}t
(ω 2 −ω 1 )−(ωL1−ωL2)]
=
Ω 2 iΩi 1
4∆ (∆ +δ)[
1 −e−i(∆+δ)t]
−
Ω 2 iΩi 1
4∆δ[
1 −e−iδt]
, (E.7)
where
δ=(ωL1−ωL2)−(ω 2 −ω 1 )(E.8)
is the frequency detuning from the Raman resonance condition that the
difference in the laser frequencies matches the energy difference between
levels 1 and 2, over(see Fig. 9.20).^4 Equation E.7 looks complicated^4 Notice that this condition does not de-
pend onωi. The Raman transition can
be viewed as a coupling between| 1 〉and
| 2 〉via a virtual level, whose origin can
be traced back to the term (with a small
amplitude) at frequencyω 1 +ωL1 in
eqn E.4. However, although a virtual
level gives a useful physical picture it
is entirely fictitious—during a Raman
transition there is negligible population
in the excited state and hence negligible
spontaneous emission.
but its two parts have a straightforward physical interpretation—we can
find the conditions for which each part is important by examining their
denominators. Raman transitions are important whenδ0and∆is
large (|δ||∆|) so that the second part of eqn E.7 dominates (and the
individual single-photon transitions are far from resonance). Defining
an effective Rabi frequency as
Ωeff=Ω 2 iΩi 1
2∆=
〈 2 |er·EL2|i〉〈i|er·EL1| 1 〉
^2 (ωi−ω 1 −ωL1), (E.9)
we can write eqn E.7 as
c 2 (t)=Ωeff
21 −e−i(∆+δ)t
∆+δ−
Ωeff
21 −e−iδt
δ. (E.10)
The first term can be neglected when|δ||∆|to yield
|c 2 (t)|^2 =1
4
Ω^2 efft^2 sinc^2(
δt
2