54
3 Microscopic Expressions of Nonlinear Polarization
electric field for nonmagnetic materials (see Appendix A.2). 2 The electric field E(t)
in Eq. (3.18) is expressed by the superposition of oscillating fields like Eq. (1.3),
E q (t) =
k
E q (ω k ) exp(−iω k t)
(q = x ∼ z)
= E q (ω 1 ) exp(−iω 1 t) + E q (ω 2 ) exp(−iω 2 t) + · · · + c.c.
The perturbation by this electric field alters the density matrix from ρ (0) to ρ,
ρ = ρ
(0)
+ ρ
(1)
+ ρ
(2)
+ · · · .
By substituting this perturbation expansion of ρ in the Liouville equation (3.16), we
find that each order of perturbation has to satisfy the following equation.
0-th: i ¯
h
dρ
(0)
mn
dt
=
H 0 , ρ
(0)
mn
= 0
(3.19)
1-st: i ¯
h
dρ
(1)
mn
dt
=
H 0 , ρ
(1)
mn
+
H
, ρ
(0)
mn
− i ¯
hh mn ρ
(1)
mn
(3.20)
2-nd: i ¯
h
dρ
(2)
mn
dt
=
H 0 , ρ
(2)
mn
+
H
, ρ
(1)
mn
− i ¯
hh mn ρ
(2)
mn
(3.21)
. . .
The matrix elements in Eqs. (3.19), (3.20), and (3.21) are represented on the basis set
of energy eigenstates that satisfy H 0 |n = E n |n. The zero-th order equation (3.19)
means that the original state ρ (0) in thermal equilibrium is a steady state. Since
ρ (0) = ρ eq in Eq. (3.15) is diagonal, it is expressed by ρ
(0)
mn = ρ
(0)
n δ mn hereafter.
3.2.2 First-Order Susceptibility
The first-order equation (3.20) is expanded by the energy eigenstates,
i ¯
h
dρ
(1)
mn
dt
=
H 0 , ρ
(1)
mn
+
H
, ρ
(0)
mn
− i ¯
hh mn ρ
(1)
mn
= (E m − E n )ρ
(1)
mn +
m|H
|n
(ρ
(0)
n − ρ
(0)
m ) − i ¯
hh mn ρ
(1)
mn .
2 We extend this treatment to include the interaction with electric quadrupole and magnetic dipole
in Chap. 7.