0198506961.pdf

7.3 Interaction with monochromatic radiation 129

Usingeqn7.5forΨ(t) gives this dipole moment of the atom as

Dx(t)=

∫

(

c 1 e−iω^1 tψ 1 +c 2 e−iω^2 tψ 2

)∗

x

(

c 1 e−iω^1 tψ 1 +c 2 e−iω^2 tψ 2

)

d^3 r

=c∗ 2 c 1 X 21 eiω^0 t+c∗ 1 c 2 X 12 e−iω^0 t. (7.32)

Hereω 0 =ω 2 −ω 1 .The dipole moment is a real quantity since from
eqn 7.13 we see thatX 21 =(X 12 )∗,andalsoX 11 =X 22 =0.To calculate
this dipole moment induced by the applied field we need to know the
bilinear quantitiesc∗ 1 c 2 andc∗ 2 c 1 .These are some of the elements of the
density matrix^1010 This is the outer product of|Ψ〉and
its Hermitian conjugate〈Ψ|≡|Ψ〉†,the
transposed conjugate of the matrix rep-
resenting|Ψ〉. This way of writing the
information about the two levels is an
extremely useful formalism for treating
quantum systems. However, we have no
need to digress into the theory of den-
sity matrices here and we simply adopt
it as a convenient notation.

|Ψ〉〈Ψ|=

(

c 1

c 2

)

(

c∗ 1 c∗ 2

)

=

(

|c 1 |^2 c 1 c∗ 2

c 2 c∗ 1 |c 2 |^2

)

=

(

ρ 11 ρ 12

ρ 21 ρ 22

)

. (7.33)

Off-diagonal elements of the density matrix are calledcoherencesand
they represent the response of the system at the driving frequency (eqn
7.32). The diagonal elements|c 1 |^2 and|c 2 |^2 are the populations. We
define the new variables

̃c 1 =c 1 e−iδt/^2 , (7.34)

̃c 2 =c 2 eiδt/^2 , (7.35)

whereδ =ω−ω 0 is the detuning of the radiation from the atomic
resonance. This transformation does not affect the populations (ρ ̃ 11 =
ρ 11 and ρ ̃ 22 =ρ 22 ) but the coherences becomeρ ̃ 12 =ρ 12 exp(−iδt)and
̃ρ 21 =ρ 21 exp(iδt)=(ρ ̃ 12 )∗. In terms of these coherences the dipole
moment is^1111 We assume thatX 12 is real. This is
true for transitions between two bound
states of the atom—the radial wave-
functions are real and the discussion
of selection rules shows that the in-
tegration over the angular momentum
eigenfunctions also gives a real contri-
bution to the matrix element—the in-
tegral overφis zero unless the terms
containing powers of exp(−iφ) cancel.

−eDx(t)=−eX 12

{

ρ 12 eiω^0 t+ρ 21 e−iω^0 t

}

=−eX 12

{

ρ ̃ 12 eiωt+ρ ̃ 21 e−iωt

}

=−eX 12 (ucosωt−vsinωt). (7.36)

The coherencesρ ̃ 12 andρ ̃ 21 give the response of the atom atω,the
(angular) frequency of the applied field. The real and imaginary parts
ofρ ̃ 12 (multiplied by 2) are:

u= ̃ρ 12 +ρ ̃ 21 ,

v=−i( ̃ρ 12 − ̃ρ 21 ).

(7.37)

In eqn 7.36 we see thatuandvare the in-phase and quadrature com-
ponents of the dipole in a frame rotating atω. To find expressions for
̃ρ 12 , ̃ρ 21 andρ 22 , and henceuandv, we start by writing eqns 7.25 for
c 1 andc 2 in terms ofδas follows:^1212 All the steps in this lengthy proce-
dure cannot be written down here but
enough information is given for metic-
i ulous readers to fill in the gaps.

.

c 1 =c 2 eiδt

Ω

2

, (7.38)

i

.

c 2 =c 1 e−iδt

Ω

2

. (7.39)

Differentiation of eqn 7.34 yields^13

(^13) Spontaneous decay is ignored here—
this section deals only with coherent
evolution of the states.