0198506961.pdf

9.3 The optical molasses technique 187

The termI/Isat 1 has been neglected in the denominator because
this simple treatment of the optical molasses technique is only valid
for intensities well below saturation where the force from each beam
acts independently.^16 Damping requires a positive value ofαand hence^16 Saturation in two counter-
propagating beams could be taken
into account by the replacement
I/Isat→ 2 I/Isatin the denominator of
the expression forα(and alsoRscatt),
although if I/Isat is not negligible
a simple rate equation treatment is
notaccurate. Wewillseethatfor
real atoms, as opposed to theoretical
two-level atoms, the light field needs
to be considered as a standing wave
even for low intensities.

δ=ω−ω 0 <0, i.e. a red frequency detuning (in accordance with the
physical explanation of the optical molasses technique given above). For
this condition the plots of the force in Fig. 9.6 have a negative gradient
∂F/∂v <0atv=0.
The above discussion of the optical molasses technique applies to one
of pair of a counter-propagating laser beams. For the beams parallel to
thez-axis, Newton’s second law gives

d

dt

(

1

2

Mv^2 z

)

=Mvz

dvz

dt

=vzFmolasses=−αv^2 z. (9.18)

The components of the velocity along thex-andy-directions obey sim-
ilar equations, so that in the region where the three orthogonal pairs of
laser beams intersect the kinetic energyE=^12 M(vx^2 +v^2 y+v^2 z) decreases:

dE

dt

=−

2 α

M

E=−

E

τdamp

. (9.19)

Under optimum conditions the damping timeτdamp=M/(2α)isafew
microseconds (see Exercises 9.7 and 9.8). This gives the time-scale for
the initial cooling of atoms that enter the laser beams with velocities
within the capture range of the optical molasses technique, i.e. velocities
for which the force has a significant value in Fig. 9.6. Equation 9.19 gives
the physically unrealistic prediction that energy tends to zero because we
have not taken into account the heating from fluctuations in the force.

(a)

(b)

Fig. 9.6The force as a function of the

velocity in the optical molasses tech-

nique (solid lines) for (a)δ=−Γ/2,

and (b)δ=−Γ. The damping is pro-

portional to the slope of the force curve

atv= 0. Note that the force is nega-

tive forv>0andpositiveforv<0,

so the force decelerates atoms. The

forces produced by each of the laser

beams separately are shown as dotted

lines—these curves have a Lorentzian

line shape and they are drawn with an

FWHM of Γ appropriate for low inten-

sities. Forδ=0(notshowninthefig-

ure) the forces from the two laser beams

cancel each other for all velocities. The

velocity capture range is approximately

±Γ/k.