130 The interaction of atoms with radiation
.
̃c 1 =
.
c 1 e−iδt/^2 −
iδ
2
c 1 e−iδt/^2. (7.40)
Multiplication by i and the use of eqns 7.38, 7.34 and 7.35 yields an
equation for
.
̃c 1 (and similarly we can obtain
.
̃c 2 from eqn 7.39):
i
.
̃c 1 =
1
2
(δ ̃c 1 +Ω ̃c 2 ),
i
.
̃c 2 =
1
2
(Ω ̃c 1 −δ ̃c 2 ).
(7.41)
From these we find that the time derivatives
.
ρ ̃ 12 = ̃c 1
.
̃c
∗
2 +
.
̃c 1 ̃c
∗
2 ,etc.are
dρ ̃ 12
dt
=
(
dρ ̃ 21
dt
)∗
=−iδρ ̃ 12 +
iΩ
2
(ρ 11 −ρ 22 ),
dρ 22
dt
=−
dρ 11
dt
=
iΩ
2
( ̃ρ 21 − ̃ρ 12 ).
(7.42)
The last equation is consistent with normalisation in eqn 7.7, i.e.
ρ 22 +ρ 11 =1. (7.43)
In terms ofuandvin eqns 7.37 these equations become
.
u=δv,
.
v=−δu+Ω(ρ 11 −ρ 22 ),
.
ρ 22 =
Ωv
2
.
(7.44)
(^14) This is appropriate for calculating We can write the population differenceρ 11 −ρ 22 as 14
absorption. Alternatively, ρ 22 −ρ 11
could be chosen as a variable—this pop-
ulation inversion determines the gain in
lasers.
w=ρ 11 −ρ 22 , (7.45)
so that finally we get the following compact set of equations:
.
u=δv,
.
v=−δu+Ωw,
.
w=−Ωv.
(7.46)
These eqns 7.46 can be written in vector notation as:
.
R=R×(Ω̂e 1 +δ̂e 3 )=R×W, (7.47)
by takingu,vandwas the components of theBloch vector
R=ûe 1 +v̂e 2 +ŵe 3 , (7.48)
and defining the vector
W=Ω̂e 1 +δ̂e 3 (7.49)
that has magnitudeW=
√
Ω^2 +δ^2 (cf. eqn 7.28). The cross-product of
the two vectors in eqn 7.47 is orthogonal to bothRandW. This implies