206 Laser cooling and trapping
Fig. 9.19The intensities of the com-
ponents of theJ=1/2toJ=3/ 2
transition are represented bya,band
c(as in Example 7.3) and their rel-
ative values can be determined from
sum rules. The sum of the intensities
from each of the upper states is the
same:a=b+c; because normally the
lifetime of an atom does not depend
on its orientation. (A similar rule ap-
plies to the states in the lower level but
this does not yield any further informa-
tion in this case.) When the states of
the upper level are equally populated
the atom emits unpolarized radiation;
hencea+c=2b.Wenowhavetwo
simultaneous equations whose solution
isb=^23 aandc=^13 a,sotherelative
intensities area:b:c=3:2:1.
Excited state:Ground state:aabbcc(^55) This is the same convention for de- a position where the light hasσ+polarization; (^55) here the interaction
scribing polarization that we used for
the magneto-optical trap;σ+andσ−
refer to transitions that the radiation
excites in the atom, ∆MJ =±1, re-
spectively. In laser cooling we are
mainly interested in determining what
transitions occur, and this depends on
the sense of rotation of the electric
field around the quantization axis of the
atom (as described in Section 2.2). As
stated previously, the electric field of
the radiation drives the bound atomic
electron(s) around in the same sense
as the electric field; the circularly-
polarized radiation travelling parallel to
the quantization axis that is labelled
σ+imparts positive angular momen-
tum to the atom. The handedness of
the polarization can be deduced from
this statement for a given direction of
propagation, if necessary.
for theMJ =1/2toMJ′ =3/2 transition is stronger than that for
theMJ=− 1 /2toMJ′=1/2. (The squares of the Clebsch–Gordan
coefficients are 2/3 and 1/3, respectively, for these two transitions, as
determined from the sum rules as shown in Fig. 9.19). For light with
a frequency detuning to the red (δ<0), both of theMJstates in the
lower level (J =1/2) are shifted downwards; theMJ =+1/2 state
is shifted to a lower energy than theMJ =− 1 /2 state. Conversely,
at a position ofσ−polarization theMJ =− 1 /2 state is lower than
theMJ=1/2 state. The polarization changes fromσ−toσ+over a
distance of ∆z=λ/4, so that the light shift varies along the standing
wave, as shown in Fig. 9.18(d). An atom moving over these hills and
valleys in the potential energy speeds up and slows down as kinetic and
potential energy interchange, but its total energy does not change if it
stays in the same state.
To cool the atom there must be a mechanism for dissipating energy
and this occurs through absorption and spontaneous emission—the pro-
cess in which an atom absorbs light at the top of a hill and then decays
spontaneously back down to the bottom of a valley has a higher proba-
bility than the reverse process. Thus the kinetic energy that the atom
converts into potential energy in climbing the hill is lost (taken away
by the spontaneously-emitted photon); the atom ends up moving more
slowly, at the bottom of a valley. This was dubbed the ‘Sisyphus’ effect
after a character in Greek mythology who was condemned by the gods