0198506961.pdf

Exercises for Chapter 9 217

some of the nitty-gritty details and tests under-

standing of hyperfine structure.^86

(a) Sodium has a nuclear spinI=3/2. Draw an

energy level diagram of the hyperfine structure

of the 3s^2 S 1 / 2 and 3p^2 P 3 / 2 levels and indicate

the allowed electric dipole transitions.

(b) In a laser cooling experiment the transition

3s^2 S 1 / 2 ,F=2to3p^2 P 3 / 2 ,F′=3isex-

cited by light that has a frequency detuning

ofδ=−Γ/ 2 −5 MHz (to the red of this

transition). Selection rules dictate that the

excited state decays back to the initial state,

so there is a nearly closed cycle of absorption

and spontaneous emission, but there is some

off-resonant excitation to theF′= 2 hyperfine

level which can decay toF= 1 and be ‘lost’

from the cycle. TheF′= 2 level lies 60 MHz

below theF′= 3 level. Estimate the aver-

age number of photons scattered by an atom

before it falls into the lower hyperfine level of

the ground configuration. (Assume that the

transitions have similar strengths.)

(c) To counteract the leakage out of the laser

cooling cycle, experiments use an additional

laser beam that excites atoms out of the

3s^2 S 1 / 2 ,F= 1 level (so that they can get

back into the 3s^2 S 1 / 2 ,F= 2 level). Sug-

gest a suitable transition for this ‘repumping’

process and comment on the light intensity re-

quired.^87

(9.14) The gradient force
Figure 9.11 shows a sphere, with a refractive index
greater than the surrounding medium, that feels
a force towards regions of high intensity. Draw
a similar diagram for the casensphere<nmedium
and indicate forces. (This object could be a small
bubble of air in a liquid.)

(9.15) Dipole-force trap
A laser beam^88 propagating along thez-axis has
an intensity profile of

I=^2 P

πw(z)^2

exp

(

−^2 r

2

w(z)^2

)

, (9.60)

wherer^2 =x^2 +y^2 and the waist size isw(z)=

w 0

(

1+z^2 /b^2

) 1 / 2

withb=πw^20 /λ^2 .Thislaser

beam has a power ofP=1Watawavelengthof

λ=1. 06 μm, and a spot size ofw 0 =10μmatthe

focus.

(a) Show that the integral ofI(r, z)overany

plane of constantzequals the total power

of the beamP.

(b) Calculate the depth of the dipole potential

for rubidium atoms, expressing your answer

as an equivalent temperature.

(c) For atoms with a thermal energy much lower

than the trap depth (so thatr^2 w 02 and

z^2 b^2 ), determine the ratio of the size of

the cloud in the radial and longitudinal di-

rections.

∗(d) Show that the dipole force has a maximum

value at a radial distance ofr=w 0 /2. Find

the maximum value of the waist sizew 0 for

which the dipole-force trap supports rubid-

ium atoms against gravity (when the laser

beam propagates horizontally).

(9.16)An optical lattice

In a standing wave of radiation with a wavelength

ofλ=1. 06 μm, a sodium atom experiences a peri-

odic potential as in eqn 9.52 withU 0 = 100Er,

where Eris the recoil energy (for light at the

atom’s resonance wavelength λ 0 =0. 589 μm).

Calculate the oscillation frequency for a cold atom

trapped near the bottom of a potential well in

this one-dimensional optical lattice. What is the

energy spacing between the low-lying vibrational

levels?

(9.17)Thepotentialforthedipoleforce

Show that the force in eqn 9.43 equals the gradient

of the potential

Udipole=−


δ

2

ln

(

1+

I

Isat

+

4 δ^2

Γ^2

)

.

For what conditions does eqn 9.46 give a good ap-

proximation forUdipole?

(^86) The transfer between different hyperfine levels described here is distinct from the transfer between different Zeeman sub-
levels (states of givenMJorMF) in the Sisyphus effect.
(^87) In the Zeeman slowing technique the magnetic field increases the separation of the energy levels (and also uncouples the
nuclear and electronic spins in the excited state) so that ‘repumping’ is not necessary.
(^88) This is a diffraction-limited Gaussian beam.