0198506961.pdf

188 Laser cooling and trapping

9.3.1 The Doppler cooling limit

We can write the force from a single laser beam as

F=Fabs+δFabs+Fspont+δFspont. (9.20)

The average of the force from the absorption of photons is the scattering

force that we have already derived,Fabs=Fscatt, and the random kicks

from spontaneously-emitted photons average to zero, i.e.Fspont=0.

What we have not considered previously is the effect of the fluctuations

in these two processes, i.e.δFspontandδFabs.

The spontaneous emission that always accompaniesFscattcauses the

atom to recoil in random directions. These recoil kicks lead to a random

walk of the velocity, as shown in Fig. 9.7 (analogous to the Brownian

motion of microscopic particles in air). A random walk ofNsteps gives a

mean displacement proportional to

√

N, or equivalently the mean square

displacement equalsN times the square of the step length. During a

timetan atom scatters a mean number of photons

N=Rscattt. (9.21)

Spontaneous emission causes the mean square velocity to increase as

v^2 =Rscattt×vr^2 ,oralongthez-axis

(

v^2 z

)

spont=ηv

2

rRscattt. (9.22)

Each spontaneous photon gives a recoil kick in thez-direction of
kcosθ

and the factorη =〈cos^2 θ〉is the angular average, e.g. for isotropic

(^17) The spontaneously-emitted photons spontaneous emissionη=1/3. 17
that go in the x-andy-directions
lead to heating in those directions,
that must be taken into account
in a full three-dimensional treatment.
For isotropic spontaneous emission the
heating would be the same in all direc-
tions, i.e.v^2 xandv^2 ywould increase at
thesamerateasv^2 z. Radiation from an
electric dipole oscillator is not isotropic,
but this turns out not to be important
for reasons discussed below.
The fluctuationsδFabsarise because the atom does not always absorb
the same number of photons in a time periodt. Each absorption is
followed by spontaneous emission and the mean number of such events
in timetis given by eqn 9.21. Assuming that the scattering obeys
Poissonian statistics, the fluctuations about the mean have a standard
deviation of

√

N and cause a random walk of the velocity along the

laser beam, on top of the change in velocity (acceleration or deceleration)

caused by the mean force. This one-dimensional random walk caused by

the fluctuationsδFabsleads to an increase in the velocity spread similar

Fig. 9.7The recoil of an atom from
each spontaneous emission causes the
atomic velocity to change by the re-
coil velocity in a random direction.
Thus the atom undergoes a random
walk in velocity space with steps of
lengthvr. The equilibrium tempera-
ture is determined by the balance be-
tween this diffusive heating and the
cooling. (For simplicity, only two di-
rections are shown here.)