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diff --git a/source/know/concept/maxwell-bloch-equations/index.md b/source/know/concept/maxwell-bloch-equations/index.md
index 0252b5c..1214703 100644
--- a/source/know/concept/maxwell-bloch-equations/index.md
+++ b/source/know/concept/maxwell-bloch-equations/index.md
@@ -11,96 +11,43 @@ categories:
layout: "concept"
---
-For an electron in a two-level system with time-independent states
-$$\ket{g}$$ (ground) and $$\ket{e}$$ (excited),
-consider the following general solution
-to the time-dependent Schrödinger equation:
+For an electron in a two-orbital system $$\{\ket{g}, \ket{e}\}$$,
+the Schrödinger equation has the following general solution,
+where $$\varepsilon_g$$ and $$\varepsilon_e$$ are the time-independent eigenenergies,
+and the weights $$c_g$$ and $$c_g$$ are functions of $$t$$:
$$\begin{aligned}
\ket{\Psi}
- &= c_g \ket{g} \exp(-i E_g t / \hbar) + c_e \ket{e} \exp(-i E_e t / \hbar)
+ &= c_g \ket{g} e^{-i \varepsilon_g t / \hbar} + c_e \ket{e} e^{-i \varepsilon_e t / \hbar}
\end{aligned}$$
-Perturbing this system with
-an [electromagnetic wave](/know/concept/electromagnetic-wave-equation/)
-introduces a time-dependent sinusoidal term $$\hat{H}_1$$ to the Hamiltonian.
-In the [electric dipole approximation](/know/concept/electric-dipole-approximation/),
-$$\hat{H}_1$$ is given by:
+This system is being perturbed by an electromagnetic wave
+with [electric field](/know/concept/electric-field/) $$\vb{E}$$ given by:
$$\begin{aligned}
- \hat{H}_1(t)
- = - \hat{\vb{p}} \cdot \vb{E}(t)
- \qquad \qquad
- \vu{p}
- \equiv q \vu{x}
- \qquad \qquad
\vb{E}(t)
- = \vb{E}_0 \cos(\omega t)
-\end{aligned}$$
-
-Where $$\vb{E}$$ is an [electric field](/know/concept/electric-field/),
-and $$\hat{\vb{p}}$$ is the dipole moment operator.
-From [Rabi oscillation](/know/concept/rabi-oscillation/),
-we know that the time-varying coefficients $$c_g$$ and $$c_e$$
-can then be described by:
-
-$$\begin{aligned}
- \dv{c_g}{t}
- &= i \frac{q \matrixel{g}{\vu{x}}{e} \cdot \vb{E}_0}{2 \hbar} \exp\!\big( i \omega t \!-\! i \omega_0 t \big) \: c_e
- \\
- \dv{c_e}{t}
- &= i \frac{q \matrixel{e}{\vu{x}}{g} \cdot \vb{E}_0}{2 \hbar} \exp\!\big(\!-\! i \omega t \!+\! i \omega_0 t \big) \: c_g
-\end{aligned}$$
-
-We want to rearrange these equations a bit.
-Therefore, we split the electric field $$\vb{E}$$ like so,
-where the amplitudes $$\vb{E}_0^{-}$$ and $$\vb{E}_0^{+}$$ may be slowly varying:
-
-$$\begin{aligned}
- \vb{E}(t)
- = \vb{E}^{-}(t) + \vb{E}^{+}(t)
- = \vb{E}_0^{-} \exp(i \omega t) + \vb{E}_0^{+} \exp(-i \omega t)
-\end{aligned}$$
-
-Since $$\vb{E}$$ is real, $$\vb{E}_0^{+} = (\vb{E}_0^{-})^*$$.
-Similarly, we define the transition dipole moment $$\vb{p}_0^{-}$$:
-
-$$\begin{aligned}
- \vb{p}_0^{-}
- \equiv q \matrixel{e}{\vu{x}}{g}
- \qquad \qquad
- \vb{p}_0^{+}
- \equiv (\vb{p}_0^{-})^*
- = q \matrixel{g}{\vu{x}}{e}
-\end{aligned}$$
-
-With these, the equations for $$c_g$$ and $$c_e$$ can be rewritten as shown below.
-Note that $$\vb{E}^{-}$$ and $$\vb{E}^{+}$$ include the driving plane wave, and the
-[rotating wave approximation](/know/concept/rotating-wave-approximation/) is still made:
-
-$$\begin{aligned}
- \dv{c_g}{t}
- &= \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} \exp(- i \omega_0 t) \: c_e
- \\
- \dv{c_e}{t}
- &= \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} \exp(i \omega_0 t) \: c_g
+ &\equiv \vb{E}^{-}(t) + \vb{E}^{+}(t)
\end{aligned}$$
-
-
-## Optical Bloch equations
+Where the forward-propagating component $$\vb{E}^{+}$$
+is a modulated plane wave $$\vb{E}_0^{+} e^{-i \omega t}$$
+with slowly-varying amplitude $$\vb{E}_0^{+}(t)$$,
+and similarly $$\vb{E}^{-}(t) \equiv \vb{E}_0^{-}(t) e^{i \omega t}$$;
+since $$\vb{E}$$ is real, $$\vb{E}_0^{+} \!=\! (\vb{E}_0^{-})^*$$.
For $$\ket{\Psi}$$ as defined above,
-the corresponding pure [density operator](/know/concept/density-operator/)
-$$\hat{\rho}$$ is as follows:
+the pure [density operator](/know/concept/density-operator/)
+$$\hat{\rho}$$ is as follows,
+with $$\omega_0 \equiv (\varepsilon_e \!-\! \varepsilon_g) / \hbar$$
+being the transition's resonance frequency:
$$\begin{aligned}
\hat{\rho}
= \ket{\Psi} \bra{\Psi}
=
\begin{bmatrix}
- c_e c_e^* & c_e c_g^* \exp(-i \omega_0 t) \\
- c_g c_e^* \exp(i \omega_0 t) & c_g c_g^*
+ c_e c_e^* & c_e c_g^* e^{-i \omega_0 t} \\
+ c_g c_e^* e^{i \omega_0 t} & c_g c_g^*
\end{bmatrix}
\equiv
\begin{bmatrix}
@@ -109,139 +56,59 @@ $$\begin{aligned}
\end{bmatrix}
\end{aligned}$$
-Where $$\omega_0 \equiv (E_e \!-\! E_g) / \hbar$$ is the resonance frequency.
-We take the $$t$$-derivative of the matrix elements,
-and insert the equations for $$c_g$$ and $$c_e$$:
+Under the [electric dipole approximation](/know/concept/electric-dipole-approximation/)
+and [rotating wave approximation](/know/concept/rotating-wave-approximation/),
+it can be shown that $$\hat{\rho}$$ is governed by
+the [optical Bloch equations](/know/concept/optical-bloch-equations/):
$$\begin{aligned}
\dv{\rho_{gg}}{t}
- &= \dv{c_g}{t} c_g^* + c_g \dv{c_g^*}{t}
- \\
- &= \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} \exp(- i \omega_0 t) \: c_e c_g^*
- - \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} \exp(i \omega_0 t) \: c_g c_e^*
+ &= \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}
- &= \dv{c_e}{t} c_e^* + c_e \dv{c_e^*}{t}
- \\
- &= \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} \exp(i \omega_0 t) \: c_g c_e^*
- - \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} \exp(- i \omega_0 t) \: c_e c_g^*
+ &= \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}
- &= \dv{c_g}{t} c_e^* \exp(i \omega_0 t) + c_g \dv{c_e^*}{t} \exp(i \omega_0 t) + i \omega_0 c_g c_e^* \exp(i \omega_0 t)
- \\
- &= \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} \: c_e c_e^*
- - \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} \: c_g c_g^*
- + i \omega_0 c_g c_e^* \exp(i \omega_0 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}
- &= \dv{c_e}{t} c_g^* \exp(-i \omega_0 t) + c_e \dv{c_g^*}{t} \exp(-i \omega_0 t) - i \omega_0 c_e c_g^* \exp(- i \omega_0 t)
- \\
- &= \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} \: c_g c_g^*
- - \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} \: c_e c_e^*
- - i \omega_0 c_e c_g^* \: \exp(- i \omega_0 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}$$
-Recognizing the density matrix elements allows us
-to reduce these equations to:
+Where we have defined the transition dipole moment $$\vb{p}_0^{-}$$,
+with $$q < 0$$ the electron charge:
$$\begin{aligned}
- \dv{\rho_{gg}}{t}
- &= \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}
- &= \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}
- &= i \omega_0 \rho_{ge} + \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} \big( \rho_{ee} - \rho_{gg} \big)
- \\
- \dv{\rho_{eg}}{t}
- &= - i \omega_0 \rho_{eg} + \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} \big( \rho_{gg} - \rho_{ee} \big)
-\end{aligned}$$
-
-These equations are correct if nothing else is affecting $$\hat{\rho}$$.
-But in practice, these quantities decay due to various processes,
-e.g. [spontaneous emission](/know/concept/einstein-coefficients/).
-
-Suppose $$\rho_{ee}$$ decays with rate $$\gamma_e$$.
-Because the total probability $$\rho_{ee} + \rho_{gg} = 1$$, we have:
-
-$$\begin{aligned}
- \Big( \dv{\rho_{ee}}{t} \Big)_{e}
- = - \gamma_e \rho_{ee}
- \quad \implies \quad
- \Big( \dv{\rho_{gg}}{t} \Big)_{e}
- = \gamma_e \rho_{ee}
-\end{aligned}$$
-
-Meanwhile, for whatever reason,
-let $$\rho_{gg}$$ decay into $$\rho_{ee}$$ with rate $$\gamma_g$$:
-
-$$\begin{aligned}
- \Big( \dv{\rho_{gg}}{t} \Big)_{g}
- = - \gamma_g \rho_{gg}
- \quad \implies \quad
- \Big( \dv{\rho_{gg}}{t} \Big)_{g}
- = \gamma_g \rho_{gg}
-\end{aligned}$$
-
-And finally, let the diagonal (perpendicular) matrix elements
-both decay with rate $$\gamma_\perp$$:
-
-$$\begin{aligned}
- \Big( \dv{\rho_{eg}}{t} \Big)_{\perp}
- = - \gamma_\perp \rho_{eg}
+ \vb{p}_0^{-}
+ \equiv q \matrixel{e}{\vu{x}}{g}
\qquad \qquad
- \Big( \dv{\rho_{ge}}{t} \Big)_{\perp}
- = - \gamma_\perp \rho_{ge}
-\end{aligned}$$
-
-Putting everything together,
-we arrive at the **optical Bloch equations** governing $$\hat{\rho}$$:
-
-$$\begin{aligned}
- \boxed{
- \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}
- }
+ \vb{p}_0^{+}
+ \equiv (\vb{p}_0^{-})^*
+ = q \matrixel{g}{\vu{x}}{e}
\end{aligned}$$
-Some authors simplify these equations a bit by choosing
-$$\gamma_g = 0$$ and $$\gamma_\perp = \gamma_e / 2$$.
-
-
-
-## Including Maxwell's equations
-
-This two-level system has a dipole moment $$\vb{p}$$ as follows,
-where we use [Laporte's selection rule](/know/concept/selection-rules/)
-to remove diagonal terms, by assuming that
-the electron's orbitals are odd or even:
+However, the light wave affects the electron,
+so the actual electromagnetic dipole moment $$\vb{p}$$ is as follows,
+using [Laporte's selection rule](/know/concept/selection-rules/)
+to remove diagonal terms by assuming that
+the electron's orbitals are spatially odd or even:
$$\begin{aligned}
\vb{p}
- &= \matrixel{\Psi}{\hat{\vb{p}}}{\Psi}
+ &= 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} \exp(i \omega_0 t) + c_e c_g^* \matrixel{g}{\vu{x}}{e} \exp(-i \omega_0 t) \Big)
+ + 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)
+ \\
+ &= \vb{p}_0^{-} \rho_{ge}(t) + \vb{p}_0^{+} \rho_{eg}(t)
+ \\
+ &\equiv \vb{p}^{-}(t) + \vb{p}^{+}(t)
\end{aligned}$$
Where we have split $$\vb{p}$$ analogously to $$\vb{E}$$
@@ -256,7 +123,7 @@ $$\begin{aligned}
\end{aligned}$$
Some authors do not bother multiplying $$\rho_{ge}$$ by $$\vb{p}_0^{+}$$.
-In any case, we arrive at:
+In our case, we arrive at a prototype of the first of three Maxwell-Bloch equations:
$$\begin{aligned}
\boxed{
@@ -266,8 +133,8 @@ $$\begin{aligned}
}
\end{aligned}$$
-Where we have defined the **population inversion** $$d \in [-1, 1]$$ as follows,
-which quantifies the electron's excitedness:
+Where we have defined the **population inversion** $$d \in [-1, 1]$$ like so,
+to quantify the electron's "excitedness" i.e. its localization to $$\ket{e}$$:
$$\begin{aligned}
d
@@ -325,8 +192,8 @@ $$\begin{aligned}
{% include proof/end.html id="proof-inversion-decay" %}
-With this, the equation for the population inversion $$d$$
-takes the following final form:
+With this, the equation for the population inversion $$d$$ takes the form below,
+namely the second Maxwell-Bloch equation's prototype:
$$\begin{aligned}
\boxed{
@@ -337,9 +204,11 @@ $$\begin{aligned}
Finally, we would like a relation between the polarization
and the electric field $$\vb{E}$$,
-for which we turn to [Maxwell's equations](/know/concept/maxwells-equations/).
-We start from Faraday's law,
-and split $$\vb{B} = \mu_0 (\vb{H} + \vb{M})$$:
+for which we turn to [Maxwell's equations](/know/concept/maxwells-equations/);
+we will effectively derive a modified form of
+the [electromagnetic wave equation](/know/concept/electromagnetic-wave-equation/).
+Starting from Faraday's law
+and splitting $$\vb{B} = \mu_0 (\vb{H} + \vb{M})$$:
$$\begin{aligned}
\nabla \cross \vb{E}
@@ -391,7 +260,8 @@ $$\begin{aligned}
Where $$\varepsilon_r \equiv 1 + \chi_e$$ is the medium's relative permittivity.
The speed of light $$c^2 = 1 / (\mu_0 \varepsilon_0)$$,
and the refractive index $$n^2 = \mu_r \varepsilon_r$$,
-where $$\mu_r = 1$$ due to our assumption that $$\vb{M} = 0$$, so:
+where $$\mu_r = 1$$ due to our assumption that $$\vb{M} = 0$$,
+so the third Maxwell-Bloch equation's prototype is:
$$\begin{aligned}
\boxed{
@@ -436,11 +306,8 @@ $$\begin{aligned}
## References
1. F. Kärtner,
- [Ultrafast optics: lecture notes](https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-977-ultrafast-optics-spring-2005/lecture-notes/),
- 2005, MIT.
+ [Ultrafast optics: lecture notes](https://ocw.mit.edu/courses/6-977-ultrafast-optics-spring-2005/pages/lecture-notes/),
+ 2005, Massachusetts Institute of Technology.
2. H. Haken,
*Light: volume 2: laser light dynamics*,
1985, North-Holland.
-3. H.J. Metcalf, P. van der Straten,
- *Laser cooling and trapping*,
- 1999, Springer.