7.1 Setting up the equations 125Ω=
eX 12 |E 0 |
, (7.12)
where
X 12 =〈 1 |x| 2 〉. (7.13)To solve the coupled differential equations forc 1 (t)andc 2 (t)weneed
to make further approximations.
7.1.2 The rotating-wave approximation
When all the population starts in the lower level,c 1 (0) = 1 andc 2 (0) =
0 ,integration of eqns 7.9 and 7.10 leads to
c 1 (t)=1,c 2 (t)=Ω
2
∗{ 1 −exp [i(ω
0 +ω)t]
ω 0 +ω+
1 −exp [i(ω 0 −ω)t]
ω 0 −ω}
.
(7.14)
This gives a reasonable first-order approximation whilec 2 (t)remains
small. For most cases of interest, the radiation has a frequency close to
the atomic resonance atω 0 so the magnitude of the detuning is small,
|ω 0 −ω|ω 0 , and henceω 0 +ω∼ 2 ω 0 .Therefore we can neglect the
term with denominatorω 0 +ωinside the curly brackets. This is the
rotating-wave approximation.^5 The modulus-squared of the co-rotating^5 This is not true for the interaction of
atoms with radiation at 10.6μmfroma
CO 2 laser. This laser radiation has a
frequency closer to d.c. than to the res-
onance frequency of the atoms, e.g. for
rubidium with a resonance transition in
the near infra-red (780 nm) we find that
ω 0
15 ω, henceω 0 +ω ω 0 −ω.Thus
the counter-rotating term must be kept.
The quasi-electrostatic traps (QUEST)
formed by such long wavelength laser
beams are a form of the dipole-force
traps described in Chapter 10.
term gives the probability of finding the atom in the upper state at time
tas
|c 2 (t)|^2 =∣
∣
∣
∣Ω
sin{(ω 0 −ω)t/ 2 }
ω 0 −ω∣
∣
∣
∣
2
, (7.15)or, in terms of the variablex=(ω−ω 0 )t/2,
|c 2 (t)|^2 =1
4
|Ω|^2 t^2sin^2 x
x^2. (7.16)
The sinc function (sinx)/xhas a maximum atx=0,andthefirst
minimum occurs atx=πorω 0 −ω=± 2 π/t, as illustrated in Fig. 7.1;
the frequency spread decreases as the interaction timetincreases.
Fig. 7.1The excitation probability
function of the radiation frequency has
a maximum at the atomic resonance.
The line width is inversely proportional
to the interaction time. The function
sinc^2 also describes the Fraunhofer dif-
fraction of light passing through a sin-
gle slit—the diffraction angle decreases
as the width of the aperture increases.
The mathematical correspondence be-
tween these two situations has a natural
explanation in terms of Fourier trans-
forms.