Exercises for Chapter 9 215estimate∂F/∂vand hence to determine the
damping coefficientαforanatominapairof
counter-propagating laser beams, under these
conditions.
(b) Estimate the damping time for a sodium atom
in the optical molasses technique when each
laser beam has intensityIsatandδ=−Γ/2.
(9.8)Laser cooling of a trapped ion
AtrappedCa+ion undergoes simple harmonic
motion with an oscillation frequency of Ω = 2π×
100 kHz. The ion experiences a radiation force
from laser light of wavelength 393 nm and inten-
sityIthat excites a transition with Γ = 2π× 23 ×
106 s−^1. The frequency detuningδdoes not de-
pend on the ion’s position within the trap.
(a) Show that the force on the ion has the form
F=−κ(z−z 0 )−αv. Describe the ion’s mo-
tion.
(b) Find the static displacementz 0 of the ion from
the centre of the harmonic potential, along the
direction of the laser beam, forδ=−Γ/2and
I=2Isat.
(c) Show that, to a good approximation, the
damping coefficient can be written in the form
α∝
xy
(1 +y+x^2 )^2, (9.59)where the variablesxandyare proportional
toδandI, respectively. Maximise this func-
tion of two variables and hence determine the
intensity and frequency detuning that give the
maximum value ofα.
(d) The kinetic energy of small oscillations about
z 0 decays with a damping time ofτdamp=
M/α. Show that this damping time is in-
versely proportional to the recoil energy.^83
Evaluate this minimum value ofτdampfor a
calcium ion of massM40 a.m.u.
Comment.This treatment of Doppler cooling for
a single laser beam is accurate for any intensity
(even aboveIsat), whereas the approximation that
two laser beams (as in the optical molasses tech-
nique) give twice as much damping as a single
beam is not accurate at high intensities.
(9.9)The properties of a magneto-optical trap
(a) Obtain an expression for the damping co-
efficient α foranatomintwocounter-
propagating laser beams (each of intensityI),taking into account saturation. (Use the re-
sults of the previous exercise with the modifi-
cationI→ 2 Iin both the numerator and de-
nominator, or otherwise.) Determine the min-
imum damping time (defined in eqn 9.19) of
a rubidium atom in the optical molasses tech-
nique (with two laser beams).
(b) TheforceonanatominanMOTisgivenby
eqn 9.30. Assume the worst-case scenario in
the calculation of the damping and the restor-
ing force, along a particular direction, i.e. that
the radiation force arises from two counter-
propagating laser beams (each of intensityI)
but the saturation of the scattering rate de-
pends on the total intensity 6Iof all six laser
beams. Show that the damping coefficient can
bewrittenintheformα∝
xy
(1 +y+x^2 )^2,wherex=2δ/Γandy=6I/Isat.Usingthe
results of the previous exercise, or otherwise,
determine the nature of the motion for a ru-
bidium atom in an MOT with the values of
Iandδthat give maximum damping, and a
field gradient 0.1Tm−^1 (in the direction con-
sidered).(9.10)Zeeman slowing in a magneto-optical trap(a) Instead of the optimum magnetic field profile
given in eqn 9.11, a particular apparatus to
slow sodium atoms uses a linear rampB(z)=B 0(
1 −
z
L)for 0zL,andB(z) = 0 outside this
range. Explain why a suitable value forB 0
is the same as in eqn 9.12. Show that the
minimum value ofLis 2L 0 ,whereL 0 is the
stopping distance for the optimum profile.
(b) The capture of atoms by a magneto-optical
trap can be considered as Zeeman slowing in
a uniform magnetic field gradient, as in part
(a). In this situation the maximum veloc-
itycapturedbyanMOTwithlaserbeams
of radius 0.5 cm is equivalent to the velocity
of atoms that come to rest in a distance of(^83) Surprisingly, the damping time does not depend on the line width of the transition Γ, but narrow transitions lead to a
small velocity capture range.