0198506961.pdf

148 The interaction of atoms with radiation

Exercises

(7.1)Averaging over spatial orientations of the atom

(a) Light linearly polarized along the x-axis

gives a dipole matrix element of X 12 =

〈 2 |r| 1 〉cosφsinθ.Show that the average over

all solid angles gives a factor of 1/3, as in

eqn 7.21.

(b) Either show explicitly that the same factor of

1 /3 arises for light linearly polarized along the

z-axis,E=E 0 ̂ezcosωt,or prove this by a

general argument.

(7.2)Rabi oscillations

(a) Prove that eqns 7.25 lead to eqn 7.26 and that

this second-order differential equation has a

solution consistent with eqn 7.27.

(b) Plot|c 2 (t)|^2 for the cases ofω−ω 0 =0,Ω

and 3Ω.

(7.3)π-andπ/ 2 -pulses

(a) For zero detuning,ω=ω 0 , and the initial con-

ditionsc 1 (0) = 1 andc 2 (0) = 0,solve eqns

7.25 to find bothc 1 (t)andc 2 (t).

(b) Prove that aπ-pulse gives the operation in

eqn 7.30.

(c) What is the overall effect of twoπ-pulses act-

ing on| 1 〉?

(d) Show that a π/2-pulse gives | 1 〉→

{| 1 〉−i| 2 〉}/

√

2.

(e) What is the overall effect of twoπ/2-pulses

acting on| 1 〉? When state| 2 〉experiences a

phase shift ofφ, between the two pulses, show

that the probabilities of ending up in states

| 1 〉and| 2 〉are sin^2 (φ/2) and cos^2 (φ/2), re-

spectively.

(f) Calculate the effect of the three-pulse se-

quenceπ/2–π–π/2, with a phase shift ofφ

between the second and third pulses. (The

operators can be written as 2×2 unitary ma-

trices, although this is not really necessary for

this simple case.)

Comment. Without the factors of−i the signals

in the two output ports of the interferometer are

not complementary. The fact that the identity op-

eration is a 4π-pulse rather than 2πstems from

the isomorphism between the two-level atom and

a spin-1/2 system.

(7.4)The steady-state excitation rate with radiative

broadening

An alternative treatment of radiative decay simply

introduces decay terms into eqn 7.25 to give

i

.

c 1 =c 2

Ω

2

ei(ω−ω^0 )t+i

Γ

2

c 2 , (7.94)

i

.

c 2 =c 1

Ω

2

e−i(ω−ω^0 )t−i

Γ

2

c 2. (7.95)

This is a phenomenological model, i.e. a guess that

works. The integrating factor exp (Γt/2) allows

eqn 7.95 to be written as

d

dt

{

c 2 exp

(

Γt

2

)}

=−ic 1

Ω∗

2

exp

{

−i

(

ω−ω 0 +

iΓ

2

)

t

}

.

(a) Show that for Ω = 0 eqn 7.95 predicts that

|c 2 (t)|^2 =|c 2 (t=0)|^2 e−Γt.

(b) For the initial conditions c 1 (0) = 1 and

c 2 (0) = 0,integration of eqn 7.94 givesc 1 

1. For these conditions and weak excitation
   (ΩΓ) show that, after a time which is long
   compared to the radiative lifetime, level 2 has
   a steady-state population given by

|c 2 |^2 = Ω

(^2) / 4
(ω−ω 0 )^2 +Γ^2 / 4
.
(7.5)Saturation of absorption
The 3s–3p resonance line of sodium has a wave-
length ofλ= 589 nm.
(a) Sodium atoms in a magnetic trap form a
spherical cloud of diameter 1 mm. The
Doppler shift and the Zeeman effect of the
field are both small compared to Γ. Calculate
the number of atoms that gives a transmission
of e−^1 =0.37 for a weak resonant laser beam.
(b) Determine the absorption of a beam with in-
tensityI=Isat.