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9.6 Theory of the dipole force 201

Atoms in

optical molasses

Laser

beam

Dipole

trap

(a) (b)

Fig. 9.14(a) An intense laser beam

alters the energy levels of an atom, as

illustrated for a radial direction across

a laser beam propagating perpendicu-

lar to the plane of the figure. For a

laser frequency less than the atomic res-

onance frequency the a.c. Stark effect

forms a potential well in the ground-

state energy and atoms are attracted

towards regions of high intensity. (b)

Thedipole-forcetrapformedbyafo-

cused laser beam can be loaded with

cold atoms produced by the optical mo-

lasses technique, as described in the

text.

the region of optical molasses because the density of trapped atoms was
greater. When the focused laser beam was first switched on the dipole
trap contained relatively few atoms within its small volume, at a density
comparable with that in the surrounding region, e.g. 10^10 cm−^3 .This
was perceived to be a problem, but atoms that started off outside the
trap executed a random walk^47 that took some of them into the dipole^47 A random walk in space leading to
spatial diffusion, rather than the ran-
dom walk of the velocity leading to
heating that was used to calculate the
Doppler cooling limit—both processes
are caused by scattering.

trap, where they remained. In this way atoms accumulated in the trap
to give a high density.
A dipole-force trap formed by a single laser beam gives tight radial
confinement, but it is weak in the axial direction. Therefore the atoms
in such a trap form an elongated, cigar-shaped cloud. To obtain strong
confinement in all directions, if necessary, one can form a dipole-force
trap at the intersection of two laser beams.^48 Many other configurations^48 The dipole potential is proportional
to the total intensity. Laser beams
with orthogonal polarizations, or sub-
stantially different frequencies, do not
interfere and the total intensity is the
sum of the individual intensities.

are possible and the design of dipole traps is restricted only by the
form of the intensity distributions that can be sculpted from laser light.
Diffraction limits the minimum distance over which the intensity of the
light changes. An ingenious way of creating a high-intensity gradient is
shown in Fig. 9.15. A laser beam that is totally internally reflected at
the surface of glass gives an evanescent wave in which the electric field
falls off exponentially over a distance of the wavelength of the light.^49

(^49) This behaviour of the light closely re-
sembles the quantum reflection at a po-
tential step that is higher than the en-
ergy of the incident particle. The wave-
function falls exponentially to zero in
the classically forbidden region.
Foralaserfrequencytotheblue(δ>0), the repulsive dipole force near
the surface acts like a reflective coating for atoms. This creates a mirror
that reflects low-energy atoms, as shown in Fig. 9.15.

9.6.1 Optical lattice

The dipole force is strong in a standing wave of light because the inten-
sity changes from a maximum (at the anti-nodes) to zero (at the nodes)
over a distance ofλ/2 to give a high gradient of intensity. The physical
explanation for this strong force is stimulated scattering of radiation.
In a standing wave, an atom absorbs light with wavevectorkfrom one
beam and the laser beam in the opposite direction stimulates emission
with wavevectork′=−k; this gives the atom an impulse of 2k.The
rate of this stimulated process does not saturate at high intensities.^50

(^50) More generally, the dipole force
arises from astimulated process of
absorption of a photon of wavevec-
tork 1 and stimulated emission with
wavevectork 2. In this process the
atom receives an impulse
(k 1 −k 2 )
that changes the atom’s momentum.
A tightly-focused laser beam contains
a range of wavevectors and exerts a
dipole force on an atom analogous to
that in the optical tweezers technique.
A dipole force cannot occur in a plane
wave since the stimulated processes
havek 1 =k 2.