130 The interaction of atoms with radiation
.
̃c 1 =.
c 1 e−iδt/^2 −iδ
2c 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 vectorR=ûe 1 +v̂e 2 +ŵe 3 , (7.48)and defining the vectorW=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