Categories:
Electromagnetism ,
Laser theory ,
Physics ,
Quantum mechanics ,
Two-level system .
Maxwell-Bloch equations
For an electron in a two-orbital system { ∣ g ⟩ , ∣ e ⟩ } \{\ket{g}, \ket{e}\} { ∣ g ⟩ , ∣ e ⟩ } ε g \varepsilon_g ε g ε e \varepsilon_e ε e c g c_g c g c g c_g c g t t t
∣ Ψ ⟩ = c g ∣ g ⟩ e − i ε g t / ℏ + c e ∣ e ⟩ e − i ε e t / ℏ \begin{aligned}
\ket{\Psi}
&= c_g \ket{g} e^{-i \varepsilon_g t / \hbar} + c_e \ket{e} e^{-i \varepsilon_e t / \hbar}
\end{aligned} ∣ Ψ ⟩ = c g ∣ g ⟩ e − i ε g t /ℏ + c e ∣ e ⟩ e − i ε e t /ℏ This system is being perturbed by an electromagnetic wave
with electric field E \vb{E} E
E ( t ) ≡ E − ( t ) + E + ( t ) \begin{aligned}
\vb{E}(t)
&\equiv \vb{E}^{-}(t) + \vb{E}^{+}(t)
\end{aligned} E ( t ) ≡ E − ( t ) + E + ( t ) Where the forward-propagating component E + \vb{E}^{+} E + E 0 + e − i ω t \vb{E}_0^{+} e^{-i \omega t} E 0 + e − iω t E 0 + ( t ) \vb{E}_0^{+}(t) E 0 + ( t ) E − ( t ) ≡ E 0 − ( t ) e i ω t \vb{E}^{-}(t) \equiv \vb{E}_0^{-}(t) e^{i \omega t} E − ( t ) ≡ E 0 − ( t ) e iω t E \vb{E} E E 0 + = ( E 0 − ) ∗ \vb{E}_0^{+} \!=\! (\vb{E}_0^{-})^* E 0 + = ( E 0 − ) ∗
For ∣ Ψ ⟩ \ket{\Psi} ∣ Ψ ⟩ density operator
ρ ^ \hat{\rho} ρ ^ ω 0 ≡ ( ε e − ε g ) / ℏ \omega_0 \equiv (\varepsilon_e \!-\! \varepsilon_g) / \hbar ω 0 ≡ ( ε e − ε g ) /ℏ
ρ ^ = ∣ Ψ ⟩ ⟨ Ψ ∣ = [ c e c e ∗ c e c g ∗ e − i ω 0 t c g c e ∗ e i ω 0 t c g c g ∗ ] ≡ [ ρ e e ρ e g ρ g e ρ g g ] \begin{aligned}
\hat{\rho}
= \ket{\Psi} \bra{\Psi}
=
\begin{bmatrix}
c_e c_e^* & c_e c_g^* e^{-i \omega_0 t} \\
c_g c_e^* e^{i \omega_0 t} & c_g c_g^*
\end{bmatrix}
\equiv
\begin{bmatrix}
\rho_{ee} & \rho_{eg} \\
\rho_{ge} & \rho_{gg}
\end{bmatrix}
\end{aligned} ρ ^ = ∣ Ψ ⟩ ⟨ Ψ ∣ = [ c e c e ∗ c g c e ∗ e i ω 0 t c e c g ∗ e − i ω 0 t c g c g ∗ ] ≡ [ ρ ee ρ g e ρ e g ρ gg ] Under the electric dipole approximation
and rotating wave approximation ,
it can be shown that ρ ^ \hat{\rho} ρ ^ optical Bloch equations :
d ρ g g d t = γ e ρ e e − γ g ρ g g + i ℏ ( p 0 + ⋅ E − ρ e g − p 0 − ⋅ E + ρ g e ) d ρ e e d t = γ g ρ g g − γ e ρ e e + i ℏ ( p 0 − ⋅ E + ρ g e − p 0 + ⋅ E − ρ e g ) d ρ g e d t = − ( γ ⊥ − i ω 0 ) ρ g e + i ℏ p 0 + ⋅ E − ( ρ e e − ρ g g ) d ρ e g d t = − ( γ ⊥ + i ω 0 ) ρ e g + i ℏ p 0 − ⋅ E + ( ρ g g − ρ e e ) \begin{aligned}
\dv{\rho_{gg}}{t}
&= \gamma_e \rho_{ee} - \gamma_g \rho_{gg}
+ \frac{i}{\hbar} \Big( \vb{p}_0^{+} \cdot \vb{E}^{-} \rho_{eg} - \vb{p}_0^{-} \cdot \vb{E}^{+} \rho_{ge} \Big)
\\
\dv{\rho_{ee}}{t}
&= \gamma_g \rho_{gg} - \gamma_e \rho_{ee}
+ \frac{i}{\hbar} \Big( \vb{p}_0^{-} \cdot \vb{E}^{+} \rho_{ge} - \vb{p}_0^{+} \cdot \vb{E}^{-} \rho_{eg} \Big)
\\
\dv{\rho_{ge}}{t}
&= - \Big( \gamma_\perp - i \omega_0 \Big) \rho_{ge}
+ \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} \Big( \rho_{ee} - \rho_{gg} \Big)
\\
\dv{\rho_{eg}}{t}
&= - \Big( \gamma_\perp + i \omega_0 \Big) \rho_{eg}
+ \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} \Big( \rho_{gg} - \rho_{ee} \Big)
\end{aligned} d t d ρ gg d t d ρ ee d t d ρ g e d t d ρ e g = γ e ρ ee − γ g ρ gg + ℏ i ( p 0 + ⋅ E − ρ e g − p 0 − ⋅ E + ρ g e ) = γ g ρ gg − γ e ρ ee + ℏ i ( p 0 − ⋅ E + ρ g e − p 0 + ⋅ E − ρ e g ) = − ( γ ⊥ − i ω 0 ) ρ g e + ℏ i p 0 + ⋅ E − ( ρ ee − ρ gg ) = − ( γ ⊥ + i ω 0 ) ρ e g + ℏ i p 0 − ⋅ E + ( ρ gg − ρ ee ) Where we have defined the transition dipole moment p 0 − \vb{p}_0^{-} p 0 − q < 0 q < 0 q < 0
p 0 − ≡ q ⟨ e ∣ x ^ ∣ g ⟩ p 0 + ≡ ( p 0 − ) ∗ = q ⟨ g ∣ x ^ ∣ e ⟩ \begin{aligned}
\vb{p}_0^{-}
\equiv q \matrixel{e}{\vu{x}}{g}
\qquad \qquad
\vb{p}_0^{+}
\equiv (\vb{p}_0^{-})^*
= q \matrixel{g}{\vu{x}}{e}
\end{aligned} p 0 − ≡ q ⟨ e ∣ x ^ ∣ g ⟩ p 0 + ≡ ( p 0 − ) ∗ = q ⟨ g ∣ x ^ ∣ e ⟩ However, the light wave affects the electron,
so the actual electromagnetic dipole moment p \vb{p} p Laporte’s selection rule
to remove diagonal terms by assuming that
the electron’s orbitals are spatially odd or even:
p = q ⟨ Ψ ∣ x ^ ∣ Ψ ⟩ = q ( c g c g ∗ ⟨ g ∣ x ^ ∣ g ⟩ + c e c e ∗ ⟨ e ∣ x ^ ∣ e ⟩ + c g c e ∗ ⟨ e ∣ x ^ ∣ g ⟩ e i ω 0 t + c e c g ∗ ⟨ g ∣ x ^ ∣ e ⟩ e − i ω 0 t ) = q ( ρ g e ⟨ e ∣ x ^ ∣ g ⟩ + ρ e g ⟨ g ∣ x ^ ∣ e ⟩ ) = p 0 − ρ g e ( t ) + p 0 + ρ e g ( t ) ≡ p − ( t ) + p + ( t ) \begin{aligned}
\vb{p}
&= q \matrixel{\Psi}{\vu{x}}{\Psi}
\\
&= q \Big( c_g c_g^* \matrixel{g}{\vu{x}}{g} + c_e c_e^* \matrixel{e}{\vu{x}}{e}
+ c_g c_e^* \matrixel{e}{\vu{x}}{g} e^{i \omega_0 t} + c_e c_g^* \matrixel{g}{\vu{x}}{e} e^{-i \omega_0 t} \Big)
\\
&= q \Big( \rho_{ge} \matrixel{e}{\vu{x}}{g} + \rho_{eg} \matrixel{g}{\vu{x}}{e} \Big)
\\
&= \vb{p}_0^{-} \rho_{ge}(t) + \vb{p}_0^{+} \rho_{eg}(t)
\\
&\equiv \vb{p}^{-}(t) + \vb{p}^{+}(t)
\end{aligned} p = q ⟨ Ψ ∣ x ^ ∣ Ψ ⟩ = q ( c g c g ∗ ⟨ g ∣ x ^ ∣ g ⟩ + c e c e ∗ ⟨ e ∣ x ^ ∣ e ⟩ + c g c e ∗ ⟨ e ∣ x ^ ∣ g ⟩ e i ω 0 t + c e c g ∗ ⟨ g ∣ x ^ ∣ e ⟩ e − i ω 0 t ) = q ( ρ g e ⟨ e ∣ x ^ ∣ g ⟩ + ρ e g ⟨ g ∣ x ^ ∣ e ⟩ ) = p 0 − ρ g e ( t ) + p 0 + ρ e g ( t ) ≡ p − ( t ) + p + ( t ) Where we have split p \vb{p} p E \vb{E} E p + ≡ p 0 + ρ e g \vb{p}^{+} \equiv \vb{p}_0^{+} \rho_{eg} p + ≡ p 0 + ρ e g
d p + d t = p 0 + d ρ e g d t = − p 0 + ( γ ⊥ + i ω 0 ) ρ e g + i ℏ p 0 + ( p 0 − ⋅ E + ) ( ρ g g − ρ e e ) \begin{aligned}
\dv{\vb{p}^{+}}{t}
= \vb{p}_0^{+} \dv{\rho_{eg}}{t}
= - \vb{p}_0^{+} \Big( \gamma_\perp + i \omega_0 \Big) \rho_{eg}
+ \frac{i}{\hbar} \vb{p}_0^{+} \Big( \vb{p}_0^{-} \cdot \vb{E}^{+} \Big) \Big( \rho_{gg} - \rho_{ee} \Big)
\end{aligned} d t d p + = p 0 + d t d ρ e g = − p 0 + ( γ ⊥ + i ω 0 ) ρ e g + ℏ i p 0 + ( p 0 − ⋅ E + ) ( ρ gg − ρ ee ) Some authors do not bother multiplying ρ g e \rho_{ge} ρ g e p 0 + \vb{p}_0^{+} p 0 +
d p + d t = − ( γ ⊥ + i ω 0 ) p + − i ℏ ( p 0 − ⋅ E + ) p 0 + d \begin{aligned}
\boxed{
\dv{\vb{p}^{+}}{t}
= - \Big( \gamma_\perp + i \omega_0 \Big) \vb{p}^{+}
- \frac{i}{\hbar} \Big( \vb{p}_0^{-} \cdot \vb{E}^{+} \Big) \vb{p}_0^{+} d
}
\end{aligned} d t d p + = − ( γ ⊥ + i ω 0 ) p + − ℏ i ( p 0 − ⋅ E + ) p 0 + d Where we have defined the population inversion d ∈ [ − 1 , 1 ] d \in [-1, 1] d ∈ [ − 1 , 1 ] ∣ e ⟩ \ket{e} ∣ e ⟩
d ≡ ρ e e − ρ g g \begin{aligned}
d
\equiv \rho_{ee} - \rho_{gg}
\end{aligned} d ≡ ρ ee − ρ gg From the optical Bloch equations,
we find its equation of motion to be:
d d d t = d ρ e e d t − d ρ g g d t = 2 γ g ρ g g − 2 γ e ρ e e + i 2 ℏ ( p − ⋅ E + − p + ⋅ E − ) \begin{aligned}
\dv{d}{t}
&= \dv{\rho_{ee}}{t} - \dv{\rho_{gg}}{t}
= 2 \gamma_g \rho_{gg} - 2 \gamma_e \rho_{ee}
+ \frac{i 2}{\hbar} \Big( \vb{p}^{-} \cdot \vb{E}^{+} - \vb{p}^{+} \cdot \vb{E}^{-} \Big)
\end{aligned} d t d d = d t d ρ ee − d t d ρ gg = 2 γ g ρ gg − 2 γ e ρ ee + ℏ i 2 ( p − ⋅ E + − p + ⋅ E − ) We can rewrite the first two terms in the following intuitive form,
which describes a decay with
rate γ ∥ ≡ γ g + γ e \gamma_\parallel \equiv \gamma_g + \gamma_e γ ∥ ≡ γ g + γ e d 0 d_0 d 0
2 γ g ρ g g − 2 γ e ρ e e = γ ∥ ( d 0 − d ) d 0 ≡ γ g − γ e γ g + γ e \begin{aligned}
2 \gamma_g \rho_{gg} - 2 \gamma_e \rho_{ee}
= \gamma_\parallel (d_0 - d)
\qquad \qquad
d_0
\equiv \frac{\gamma_g - \gamma_e}{\gamma_g + \gamma_e}
\end{aligned} 2 γ g ρ gg − 2 γ e ρ ee = γ ∥ ( d 0 − d ) d 0 ≡ γ g + γ e γ g − γ e
Proof
Proof.
We introduce some new terms, and reorganize the expression:
2 γ g ρ g g − 2 γ e ρ e e = 2 γ g ρ g g − 2 γ e ρ e e + γ g ρ e e − γ g ρ e e + γ e ρ g g − γ e ρ g g = γ g ( ρ g g + ρ e e ) − γ e ( ρ g g + ρ e e ) + γ g ( ρ g g − ρ e e ) + γ e ( ρ g g − ρ e e ) \begin{aligned}
2 \gamma_g \rho_{gg} - 2 \gamma_e \rho_{ee}
&= 2 \gamma_g \rho_{gg} - 2 \gamma_e \rho_{ee}
+ \gamma_g \rho_{ee} - \gamma_g \rho_{ee}
+ \gamma_e \rho_{gg} - \gamma_e \rho_{gg}
\\
&= \gamma_g (\rho_{gg} + \rho_{ee}) - \gamma_e (\rho_{gg} + \rho_{ee})
+ \gamma_g (\rho_{gg} - \rho_{ee}) + \gamma_e (\rho_{gg} - \rho_{ee})
\end{aligned} 2 γ g ρ gg − 2 γ e ρ ee = 2 γ g ρ gg − 2 γ e ρ ee + γ g ρ ee − γ g ρ ee + γ e ρ gg − γ e ρ gg = γ g ( ρ gg + ρ ee ) − γ e ( ρ gg + ρ ee ) + γ g ( ρ gg − ρ ee ) + γ e ( ρ gg − ρ ee ) Since the total probability ρ g g + ρ e e = 1 \rho_{gg} + \rho_{ee} = 1 ρ gg + ρ ee = 1 d ≡ ρ e e − ρ g g d \equiv \rho_{ee} - \rho_{gg} d ≡ ρ ee − ρ gg
2 γ g ρ g g − 2 γ e ρ e e = γ g − γ e − ( γ g + γ e ) d = ( γ g + γ e ) ( γ g − γ e γ g + γ e − d ) = γ ∥ ( d 0 − d ) \begin{aligned}
2 \gamma_g \rho_{gg} - 2 \gamma_e \rho_{ee}
&= \gamma_g - \gamma_e - (\gamma_g + \gamma_e) d
\\
&= (\gamma_g + \gamma_e) \Big( \frac{\gamma_g - \gamma_e}{\gamma_g + \gamma_e} - d \Big)
\\
&= \gamma_\parallel ( d_0 - d )
\end{aligned} 2 γ g ρ gg − 2 γ e ρ ee = γ g − γ e − ( γ g + γ e ) d = ( γ g + γ e ) ( γ g + γ e γ g − γ e − d ) = γ ∥ ( d 0 − d )
With this, the equation for the population inversion d d d
d d d t = γ ∥ ( d 0 − d ) + i 2 ℏ ( p − ⋅ E + − p + ⋅ E − ) \begin{aligned}
\boxed{
\dv{d}{t}
= \gamma_\parallel (d_0 - d) + \frac{i 2}{\hbar} \Big( \vb{p}^{-} \cdot \vb{E}^{+} - \vb{p}^{+} \cdot \vb{E}^{-} \Big)
}
\end{aligned} d t d d = γ ∥ ( d 0 − d ) + ℏ i 2 ( p − ⋅ E + − p + ⋅ E − ) Finally, we would like a relation between the polarization
and the electric field E \vb{E} E Maxwell’s equations ;
we will effectively derive a modified form of
the electromagnetic wave equation .
Starting from Faraday’s law
and splitting B = μ 0 ( H + M ) \vb{B} = \mu_0 (\vb{H} + \vb{M}) B = μ 0 ( H + M )
∇ × E = − ∂ B ∂ t = − μ 0 ∂ H ∂ t − μ 0 ∂ M ∂ t \begin{aligned}
\nabla \cross \vb{E}
= - \pdv{\vb{B}}{t}
= - \mu_0 \pdv{\vb{H}}{t} - \mu_0 \pdv{\vb{M}}{t}
\end{aligned} ∇ × E = − ∂ t ∂ B = − μ 0 ∂ t ∂ H − μ 0 ∂ t ∂ M We assume that there is no magnetization M = 0 \vb{M} = 0 M = 0 ∇ × H \nabla \cross \vb{H} ∇ × H
∇ × ( ∇ × E ) = − μ 0 ∂ ∂ t ( ∇ × H ) = − μ 0 ∂ ∂ t ( J f r e e + ∂ D ∂ t ) \begin{aligned}
\nabla \cross \big( \nabla \cross \vb{E} \big)
= - \mu_0 \pdv{}{t} \big( \nabla \cross \vb{H} \big)
= - \mu_0 \pdv{}{t} \Big( \vb{J}_\mathrm{free} + \pdv{\vb{D}}{t} \Big)
\end{aligned} ∇ × ( ∇ × E ) = − μ 0 ∂ t ∂ ( ∇ × H ) = − μ 0 ∂ t ∂ ( J free + ∂ t ∂ D ) Inserting the definition D = ε 0 E + P \vb{D} = \varepsilon_0 \vb{E} + \vb{P} D = ε 0 E + P J f r e e = σ E \vb{J}_\mathrm{free} = \sigma \vb{E} J free = σ E
∇ × ( ∇ × E ) = − μ 0 σ ∂ E ∂ t − μ 0 ε 0 ∂ 2 E ∂ t 2 − μ 0 ∂ 2 P ∂ t 2 \begin{aligned}
\nabla \cross \big( \nabla \cross \vb{E} \big)
= - \mu_0 \sigma \pdv{\vb{E}}{t} - \mu_0 \varepsilon_0 \pdvn{2}{\vb{E}}{t} - \mu_0 \pdvn{2}{\vb{P}}{t}
\end{aligned} ∇ × ( ∇ × E ) = − μ 0 σ ∂ t ∂ E − μ 0 ε 0 ∂ t 2 ∂ 2 E − μ 0 ∂ t 2 ∂ 2 P Where σ \sigma σ σ = 0 \sigma = 0 σ = 0
Recall that we are describing the dynamics of a two-level system.
In reality, such a system (e.g. a quantum dot)
is suspended in a passive background medium,
which reacts with a polarization P m e d \vb{P}_\mathrm{med} P med E \vb{E} E P m e d = ε 0 χ E \vb{P}_\mathrm{med} = \varepsilon_0 \chi \vb{E} P med = ε 0 χ E
μ 0 ∂ 2 P ∂ t 2 = − ∇ × ( ∇ × E ) − μ 0 σ ∂ E ∂ t − μ 0 ε 0 ∂ 2 E ∂ t 2 − μ 0 ∂ 2 P m e d ∂ t 2 = − ∇ × ( ∇ × E ) − μ 0 σ ∂ E ∂ t − μ 0 ∂ 2 ∂ t 2 ( ε 0 E + ε 0 χ E ) = − ∇ × ( ∇ × E ) − μ 0 σ ∂ E ∂ t − μ 0 ε 0 ε r ∂ 2 E ∂ t 2 \begin{aligned}
\mu_0 \pdvn{2}{\vb{P}}{t}
&= - \nabla \cross \big( \nabla \cross \vb{E} \big) - \mu_0 \sigma \pdv{\vb{E}}{t}
- \mu_0 \varepsilon_0 \pdvn{2}{\vb{E}}{t} - \mu_0 \pdvn{2}{\vb{P}_\mathrm{med}}{t}
\\
&= - \nabla \cross \big( \nabla \cross \vb{E} \big) - \mu_0 \sigma \pdv{\vb{E}}{t}
- \mu_0 \pdvn{2}{}{t}\Big( \varepsilon_0 \vb{E} + \varepsilon_0 \chi \vb{E} \Big)
\\
&= - \nabla \cross \big( \nabla \cross \vb{E} \big) - \mu_0 \sigma \pdv{\vb{E}}{t}
- \mu_0 \varepsilon_0 \varepsilon_r \pdvn{2}{\vb{E}}{t}
\end{aligned} μ 0 ∂ t 2 ∂ 2 P = − ∇ × ( ∇ × E ) − μ 0 σ ∂ t ∂ E − μ 0 ε 0 ∂ t 2 ∂ 2 E − μ 0 ∂ t 2 ∂ 2 P med = − ∇ × ( ∇ × E ) − μ 0 σ ∂ t ∂ E − μ 0 ∂ t 2 ∂ 2 ( ε 0 E + ε 0 χ E ) = − ∇ × ( ∇ × E ) − μ 0 σ ∂ t ∂ E − μ 0 ε 0 ε r ∂ t 2 ∂ 2 E Where ε r ≡ 1 + χ e \varepsilon_r \equiv 1 + \chi_e ε r ≡ 1 + χ e c 2 = 1 / ( μ 0 ε 0 ) c^2 = 1 / (\mu_0 \varepsilon_0) c 2 = 1/ ( μ 0 ε 0 ) n 2 = μ r ε r n^2 = \mu_r \varepsilon_r n 2 = μ r ε r μ r = 1 \mu_r = 1 μ r = 1 M = 0 \vb{M} = 0 M = 0
μ 0 ∂ 2 P ∂ t 2 = − ∇ × ( ∇ × E ) − μ 0 σ ∂ E ∂ t − n 2 c 2 ∂ 2 E ∂ t 2 \begin{aligned}
\boxed{
\mu_0 \pdvn{2}{\vb{P}}{t}
= - \nabla \cross \big( \nabla \cross \vb{E} \big) - \mu_0 \sigma \pdv{\vb{E}}{t} - \frac{n^2}{c^2} \pdvn{2}{\vb{E}}{t}
}
\end{aligned} μ 0 ∂ t 2 ∂ 2 P = − ∇ × ( ∇ × E ) − μ 0 σ ∂ t ∂ E − c 2 n 2 ∂ t 2 ∂ 2 E E \vb{E} E P \vb{P} P E + \vb{E}^{+} E + P + \vb{P}^{+} P + p + \vb{p}^{+} p + d d d P + \vb{P}^{+} P + D D D
P + ( x , t ) = ∑ ν p ν + δ ( x − x ν ) D ( x , t ) = ∑ ν d ν δ ( x − x ν ) \begin{aligned}
\vb{P}^{+}(\vb{x}, t)
&= \sum_{\nu} \vb{p}^{+}_\nu \: \delta(\vb{x} - \vb{x}_\nu)
\\
D(\vb{x}, t)
&= \sum_{\nu} d_\nu \: \delta(\vb{x} - \vb{x}_\nu)
\end{aligned} P + ( x , t ) D ( x , t ) = ν ∑ p ν + δ ( x − x ν ) = ν ∑ d ν δ ( x − x ν ) We thus arrive at the Maxwell-Bloch equations ,
which are the foundation of laser theory:
μ 0 ∂ 2 P + ∂ t 2 = − ∇ × ∇ × E + − μ 0 σ ∂ E + ∂ t − n 2 c 2 ∂ 2 E + ∂ t 2 ∂ P + ∂ t = − ( γ ⊥ + i ω 0 ) P + − i ℏ ( p 0 − ⋅ E + ) p 0 + D ∂ D ∂ t = γ ∥ ( D 0 − D ) + i 2 ℏ ( P − ⋅ E + − P + ⋅ E − ) \begin{aligned}
\boxed{
\begin{aligned}
\mu_0 \pdvn{2}{\vb{P}^{+}}{t}
&= - \nabla \cross \nabla \cross \vb{E}^{+} - \mu_0 \sigma \pdv{\vb{E}^{+}}{t} - \frac{n^2}{c^2} \pdvn{2}{\vb{E}^{+}}{t}
\\
\pdv{\vb{P}^{+}}{t}
&= - \Big( \gamma_\perp + i \omega_0 \Big) \vb{P}^{+}
- \frac{i}{\hbar} \Big( \vb{p}_0^{-} \cdot \vb{E}^{+} \Big) \vb{p}_0^{+} D
\\
\pdv{D}{t}
&= \gamma_\parallel (D_0 - D) + \frac{i 2}{\hbar} \Big( \vb{P}^{-} \cdot \vb{E}^{+} - \vb{P}^{+} \cdot \vb{E}^{-} \Big)
\end{aligned}
}
\end{aligned} μ 0 ∂ t 2 ∂ 2 P + ∂ t ∂ P + ∂ t ∂ D = − ∇ × ∇ × E + − μ 0 σ ∂ t ∂ E + − c 2 n 2 ∂ t 2 ∂ 2 E + = − ( γ ⊥ + i ω 0 ) P + − ℏ i ( p 0 − ⋅ E + ) p 0 + D = γ ∥ ( D 0 − D ) + ℏ i 2 ( P − ⋅ E + − P + ⋅ E − ) References
F. Kärtner,
Ultrafast optics: lecture notes ,
2005, Massachusetts Institute of Technology.
H. Haken,
Light: volume 2: laser light dynamics ,
1985, North-Holland.