76
3 Microscopic Expressions of Nonlinear Polarization
and c 2 . If the phases of the two coefficients have no correlation, the off-diagonal
element disappears via the ensemble average.
In summary, a finite off-diagonal element ρ 12 indicates that there involves a
definite quantum superposition between the states φ 1 and φ 2 with some phase
relation. In such case the coherence is present between the two states φ 1 and φ 2 .
A.2 Interaction Energy of Nonmagnetic Materials
In Chap. 3, the perturbation Hamiltonian by the irradiated light is given by H =
−μ · E(t) in Eq. (7.2), which represents the interaction to the electric field E.
This is based on the assumption that interaction energy with the magnetic field of
light is negligible compared to that with the electric field for ordinary nonmagnetic
materials. Here we estimate their relative orders of magnitude to justify this
assumption.
The interaction energy of material with the magnetic field H is
U m = −mH = −χ m H
2 ,
where m is the magnetization, and χ m is the magnetic susceptibility, which is
dimensionless in the cgs Gauss units. (A possible factor 1/2 is omitted for simplicity
to estimate the order of magnitude.) Typical values of χ m for nonmagnetic materials
are in the range of |χ m | = 10 −4 ∼ 10 −6 for paramagnetic materials, and |χ m | =
10 −6 ∼ 10 −7 for diamagnetic materials [5]. On the other hand, the interaction
energy with the electric field E is analogously presented by
U e = −P E = −χ e E
2 ,
where χ e is the electric susceptibility, also dimensionless in the cgs Gauss units.
Typical range of χ e is in the order |χ e | = 10 −1 ∼ 10 −2 . For example, χ m and χ e of
liquid benzene are roughly estimated to
χ m −5.48 × 10
−5 cm
3 /mol ·
0.8765 g/cm 3
78.11 g/mol
= −6.1 × 10
−7 ,
χ e
1
4π
(2.2825 − 1) = 1.0 × 10
−1 ,
using the experimental values of molar magnetic susceptibility (−5.48 ×
10 −5 cm 3 /mol), density (0.8765 g/cm 3 ), molecular weight (78.11 g/mol), and
dielectric constant (2.2825) [5].
The light field consists of electric and magnetic fields, whose amplitudes are
related to |E| | |H |, as seen in Eq. (2.14). Therefore, the ratio of electric and
magnetic interactions U m /U e is evaluated to
3 Microscopic Expressions of Nonlinear Polarization
and c 2 . If the phases of the two coefficients have no correlation, the off-diagonal
element disappears via the ensemble average.
In summary, a finite off-diagonal element ρ 12 indicates that there involves a
definite quantum superposition between the states φ 1 and φ 2 with some phase
relation. In such case the coherence is present between the two states φ 1 and φ 2 .
A.2 Interaction Energy of Nonmagnetic Materials
In Chap. 3, the perturbation Hamiltonian by the irradiated light is given by H =
−μ · E(t) in Eq. (7.2), which represents the interaction to the electric field E.
This is based on the assumption that interaction energy with the magnetic field of
light is negligible compared to that with the electric field for ordinary nonmagnetic
materials. Here we estimate their relative orders of magnitude to justify this
assumption.
The interaction energy of material with the magnetic field H is
U m = −mH = −χ m H
2 ,
where m is the magnetization, and χ m is the magnetic susceptibility, which is
dimensionless in the cgs Gauss units. (A possible factor 1/2 is omitted for simplicity
to estimate the order of magnitude.) Typical values of χ m for nonmagnetic materials
are in the range of |χ m | = 10 −4 ∼ 10 −6 for paramagnetic materials, and |χ m | =
10 −6 ∼ 10 −7 for diamagnetic materials [5]. On the other hand, the interaction
energy with the electric field E is analogously presented by
U e = −P E = −χ e E
2 ,
where χ e is the electric susceptibility, also dimensionless in the cgs Gauss units.
Typical range of χ e is in the order |χ e | = 10 −1 ∼ 10 −2 . For example, χ m and χ e of
liquid benzene are roughly estimated to
χ m −5.48 × 10
−5 cm
3 /mol ·
0.8765 g/cm 3
78.11 g/mol
= −6.1 × 10
−7 ,
χ e
1
4π
(2.2825 − 1) = 1.0 × 10
−1 ,
using the experimental values of molar magnetic susceptibility (−5.48 ×
10 −5 cm 3 /mol), density (0.8765 g/cm 3 ), molecular weight (78.11 g/mol), and
dielectric constant (2.2825) [5].
The light field consists of electric and magnetic fields, whose amplitudes are
related to |E| | |H |, as seen in Eq. (2.14). Therefore, the ratio of electric and
magnetic interactions U m /U e is evaluated to