0198506961.pdf

198 Laser cooling and trapping

simple harmonic motion gives very useful insight into the frequency de-

pendence of the force, but the quantum treatment is required to find the

(^39) As shown in Section 7.5.1, the classi- correct intensity dependence. (^39) This section presents both approaches,
cal model does not account for satura-
tion.
starting with the classical one—because of the very close parallels be-
tween them this requires little extra effort.
Classically, the displacement of the electronxby an electric field is
calculated by modelling the atom as a harmonic oscillator with a driving
term. Expressingxin terms of its components in phase (U)andin
quadrature (V) to the applied field (cf. eqn 7.56), we find
Fz=−e{Ucos (ωt−kz)−Vsin (ωt−kz)}
×

{

∂E 0

∂z

cos (ωt−kz)+E 0 ksin (ωt−kz)

}

.

(9.37)

(^40) Usingsin (^2) =cos (^2) = 21. The time average over many oscillation periods gives 40
Fz=
−e
2

{

U

∂E 0

∂z

−VkE 0

}

=

e^2

4 mω

{

−(ω−ω 0 )E 0

(ω−ω 0 )^2 +(β/2)^2

∂E 0

∂z

+

(β/2)kE^20

(ω−ω 0 )^2 +(β/2)^2

}

,

(9.38)

using eqns 7.58 and 7.59 forUandV. The intensity of the light is

I=^12  0 cE^20 and, by a simple extension of the derivation given above

to thex-andy-directions, the radiation force can be written in vector

notation as

F=

e^2

2  0 mc

{

−(ω−ω 0 )

(ω−ω 0 )^2 +(β/2)^2

∇I

ω

+

β/ 2

(ω−ω 0 )^2 +(β/2)^2

I

c

k

|k|

}

.

(9.39)

The in-phase component of the dipole (U) leads to a force proportional

to the gradient of the intensity. The frequency dependence of this com-

ponent follows a dispersive line shape that is closely related to the re-

(^41) In optics, it is generally the effect fractive index, (^41) as shown in Fig. 9.12. (The dependence on 1/ωhas
of the medium on the light that is of
interest, e.g. the angle through which
the medium refracts, or bends, a light
beam, but this implies that the medium
feels a force equal to the rate of change
of the momentum of the light. The
refractive index, and absorption coeffi-
cient, describe bulk properties, whereas
it is the effect of light on individual
atoms that is of interest here.
a negligible effect on narrow transitionsβω 0 .) At the atomic reso-
nance frequencyω=ω 0 the componentU= 0. The quadrature term,
fromV, has a Lorentzian line shape and this force, arising from ab-
sorption, is proportional toIand points along the wavevector of the
radiationk. This classical model gives a simple way of understanding
various important features of the forces on atoms and shows how they
relate to radiation forces on larger objects (such as those discussed in
the introductory Sections 9.1 and 9.5); however, we shall not use it for
quantitative calculations.
To find the force quantum mechanically, we use eqn 7.36 for the dipole
moment in terms of the components of the Bloch vectoruandv.Sub-
stitution into eqn 9.36, and taking the time average as above, gives (cf.
eqn 9.38)
Fz=
−eX 12
2

{

u

∂E 0

∂z

−vE 0 k

}

(9.40)

=Fdipole+Fscatt. (9.41)