0198506961.pdf

214 Laser cooling and trapping

et al. (1999). Various internet resources and popular descriptions can

be found on the web site of the Nobel foundation.

Exercises

More advanced problems are indicated by a *.

(9.1)Radiation pressure

What force does radiation exert on the head of a

person wearing a black hat of radius 15 cm when

the sun is directly overhead. Estimate the ratio of

this radiation force to the weight of the hat.

(9.2)An argument for photon momentum (due to

Enrico Fermi)

An atom moving at velocityv absorbs a pho-

ton propagating in the opposite direction (as in

Fig. 9.1). In the laboratory frame of reference the

photon has (angular) frequencyωand momentum

qph. In the rest frame of the atom the photon has

(angular) frequencyω 0 ,where
ω 0 =E 2 −E 1 is

the energy of the (narrow line width) transition

between levels 1 and 2. After the absorption the

system has a total energy of^12 M(v−∆v)^2 +
ω 0.

(a) Write down the equations for conservation of

energy and momentum.

(b) Expand the equation for conservation of en-

ergy, neglecting the term of order (∆v)^2 .(The

change in velocity ∆vis small compared tov.)

(c) Use the usual expression for the fractional

Doppler shift (ω−ω 0 )/ω=v/cto find an ex-

pression for the photon momentumqph.

(9.3)Heating from photon recoil

This exercise is based on a treatment of laser cool-

ing by Wineland and Itano (1979). The angular

frequencies of radiation absorbed and emitted by

an atom are given by

ωabs=ω 0 +kabs·v−

1

2

ω 0

(v

c

) 2

+

Er




,

ωem=ω 0 +kem·v′−

1

2

ω 0

(v′

c

) 2

−

Er




,

where|kabs|=ωabs/cand|kem|=ωem/care the

wavevectors of the absorbed and emitted photons,

respectively,vis the velocity of the atom before

the photon is absorbed, and similarlyv′is the ve-

locity of the atom before emission. Prove these re-

sults from conservation of (relativistic) energy and

momentum (keeping terms of order (v/c)^2 in the

atomic velocity andEr/
ω 0 in the recoil energy).

Averaged over many cycles of absorption and emis-

sion, the kinetic energy of the atom changes by

∆Eke=
(ωabs−ωem)=
kabs·v+2Er

for each scattering event. Show that this result

follows from the above equations with certain as-

sumptions, that should be stated. Show that,

when multiplied by the scattering rateRscatt,the

terms
kabs·vand 2Ergive cooling and heating

at comparable rates to those derived in the text

for the optical molasses technique.

(9.4)The angular momentum of light

An atom in a^1 S 0 level is excited to a state with

L=1,ML= 1 by the absorption of a photon (a

σ+transition). What is the change in the atomic

angular momentum?

A laser beam with a power of 1 W and a wave-

length of 600 nm passes through a waveplate that

changes the polarization of the light from linear to

circular. What torque does the radiation exert on

the waveplate?

(9.5)Slowing H and Cs with radiation

Atomic beams of hydrogen and caesium are pro-

duced by sources at 300 K and slowed by counter-

propagating laser radiation. In both cases calcu-

late (a) the stopping distance at half of the max-

imum deceleration, and (b) compare the Doppler

shift at the initial velocity with the natural width

of the transition. (Data are given in Table 9.1.)

(9.6)The Doppler cooling and recoil limits

Calculate the ratio TD/Tr for rubidium (from

eqns 9.28, 9.55 and the data in Table 9.1).

(9.7)Damping in the optical molasses technique

(a) For the particular case of a frequency detuning

ofδ=−Γ/2 the slope of the force versus ve-

locity curve, shown in Fig. 9.6, atv=0equals

the peak force divided by Γ/(2k). Use this to