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3 Microscopic Expressions of Nonlinear Polarization
3.1.1 Definition
First we begin with a simple case that the quantum state of material is expressed
by a wavefunction ψ(r, t), which is a proper superposition of energy eigenstates,
{φ n (r) exp(−iE n t/ ¯
h)};
ψ(r, t) =
n
c
n (t) exp
−iE n t
¯
h
φ n (r) =
n
c n (t)φ n (r).
(3.1)
We suppose that the set of eigenstates {φ n } are orthonormal and the wavefunctions
ψ is normalized. The normalization condition for ψ is
n
|c n (t)|
2
= 1.
(3.2)
The expectation value of an arbitrary physical quantity A at this state ψ is
A(t) = ψ(t)|A|ψ(t) =
n
m
c n (t)
∗ c m (t) n|A|m .
(3.3)
Now we introduce the density matrix ρ(t). Its matrix element ρ mn is defined to be
ρ mn (t) = c n (t)
∗ c m (t)
(3.4)
on the basis of the energy eigenfunctions {φ n }. Note the order of suffixes m, n in
ρ mn and its Hermitian character, ρ mn = ρ ∗
nm . According to Eqs. (3.3) and (3.4), the
expectation value A(t) could be written using the density matrix ρ by
A(t) =
n
m
ρ mn A nm = Tr [ρA].
(3.5)
The diagonal element ρ nn = |c n | 2 means the probability to find the state n. The
normalization condition (3.2) is written by
n
ρ nn = Tr [ρ] = 1.
(3.6)
On the other hand, the off-diagonal element ρ mn (m = n) represents the coherence
between the states m and n, as we discuss later (see Appendix A.1).
The time development of the density matrix ρ is defined from the Schrödinger
equation for ψ, i ¯
h
∂ψ
∂t
= ˆ
H ψ. Substituting the expression of ψ in Eq. (3.1) into this
Schrödinger equation, we get the equations for the coefficients {c m (t)},
i ¯
h
dc m (t)
dt
=
n
H mn c n (t).
(3.7)