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1 files changed, 48 insertions, 18 deletions

diff --git a/content/know/concept/maxwell-bloch-equations/index.pdc b/content/know/concept/maxwell-bloch-equations/index.pdc
index e3a3680..2e0cdd9 100644
--- a/content/know/concept/maxwell-bloch-equations/index.pdc
+++ b/content/know/concept/maxwell-bloch-equations/index.pdc
@@ -7,6 +7,7 @@ categories:
 - Quantum mechanics
 - Two-level system
 - Electromagnetism
+- Laser theory
 
 date: 2021-09-09T21:17:52+02:00
 draft: false
@@ -34,10 +35,10 @@ $\hat{H}_1$ is given by:
 $$\begin{aligned}
     \hat{H}_1(t)
     = - \hat{\vb{p}} \cdot \vb{E}(t)
-    \qquad \quad
+    \qquad \qquad
     \vu{p}
     \equiv q \vu{x}
-    \qquad \quad
+    \qquad \qquad
     \vb{E}(t)
     = \vb{E}_0 \cos\!(\omega t)
 \end{aligned}$$
@@ -72,7 +73,7 @@ Similarly, we define the transition dipole moment $\vb{p}_0^{-}$:
 $$\begin{aligned}
     \vb{p}_0^{-}
     \equiv q \matrixel{e}{\vu{x}}{g}
-    \qquad \quad
+    \qquad \qquad
     \vb{p}_0^{+}
     \equiv (\vb{p}_0^{-})^*
     = q \matrixel{g}{\vu{x}}{e}
@@ -194,7 +195,7 @@ both decay with rate $\gamma_\perp$:
 $$\begin{aligned}
     \Big( \dv{\rho_{eg}}{t} \Big)_{\perp}
     = - \gamma_\perp \rho_{eg}
-    \qquad \quad
+    \qquad \qquad
     \Big( \dv{\rho_{ge}}{t} \Big)_{\perp}
     = - \gamma_\perp \rho_{ge}
 \end{aligned}$$
@@ -295,7 +296,7 @@ towards an equilbrium $d_0$:
 $$\begin{aligned}
     2 \gamma_g \rho_{gg} - 2 \gamma_e \rho_{ee}
     = \gamma_\parallel (d_0 - d)
-    \qquad \quad
+    \qquad \qquad
     d_0
     \equiv \frac{\gamma_g - \gamma_e}{\gamma_g + \gamma_e}
 \end{aligned}$$
@@ -367,37 +368,66 @@ Inserting the definition $\vb{D} = \varepsilon_0 \vb{E} + \vb{P}$
 together with Ohm's law $\vb{J}_\mathrm{free} = \sigma \vb{E}$ yields:
 
 $$\begin{aligned}
+    \nabla \cross \big( \nabla \cross \vb{E} \big)
+    = - \mu_0 \sigma \pdv{\vb{E}}{t} - \mu_0 \varepsilon_0 \pdv[2]{\vb{E}}{t} - \mu_0 \pdv[2]{\vb{P}}{t}
+\end{aligned}$$
+
+Where $\sigma$ is the active material's conductivity, if any;
+almost all authors assume $\sigma = 0$.
+
+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 $\vb{P}_\mathrm{med}$
+to the electric field $\vb{E}$.
+If the medium is linear, i.e. $\vb{P}_\mathrm{med} = \varepsilon_0 \chi \vb{E}$,
+then:
+
+$$\begin{aligned}
+    \mu_0 \pdv[2]{\vb{P}}{t}
+    &= - \nabla \cross \big( \nabla \cross \vb{E} \big) - \mu_0 \sigma \pdv{\vb{E}}{t}
+    - \mu_0 \varepsilon_0 \pdv[2]{\vb{E}}{t} - \mu_0 \pdv[2]{\vb{P}_\mathrm{med}}{t}
+    \\
+    &= - \nabla \cross \big( \nabla \cross \vb{E} \big) - \mu_0 \sigma \pdv{\vb{E}}{t}
+    - \mu_0 \pdv[2]{t} \Big( \varepsilon_0 \vb{E} + \varepsilon_0 \chi \vb{E} \Big)
+    \\
+    &= - \nabla \cross \big( \nabla \cross \vb{E} \big) - \mu_0 \sigma \pdv{\vb{E}}{t}
+    - \mu_0 \varepsilon_0 \varepsilon_r \pdv[2]{\vb{E}}{t}
+\end{aligned}$$
+
+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:
+
+$$\begin{aligned}
     \boxed{
-        \nabla \cross \big( \nabla \cross \vb{E} \big)
-        = - \mu_0 \sigma \pdv{\vb{E}}{t} - \mu_0 \varepsilon_0 \pdv[2]{\vb{E}}{t} - \mu_0 \pdv[2]{\vb{P}}{t}
+        \mu_0 \pdv[2]{\vb{P}}{t}
+        = - \nabla \cross \big( \nabla \cross \vb{E} \big) - \mu_0 \sigma \pdv{\vb{E}}{t} - \frac{n^2}{c^2} \pdv[2]{\vb{E}}{t}
     }
 \end{aligned}$$
 
-Where $\sigma$ is the medium's conductivity, if any;
-many authors assume $\sigma = 0$.
-It is trivial to show that $\vb{E}$ and $\vb{P}$
-can be replaced by $\vb{E}^{+}$ and $\vb{P}^{+}$.
-
+$\vb{E}$ and $\vb{P}$ can trivially be replaced by $\vb{E}^{+}$ and $\vb{P}^{+}$.
 It is also simple to convert $\vb{p}^{+}$ and $d$
-into the macroscopic polarization $\vb{P}^{+}$ and total inversion $D$
-by summing over the atoms:
+into the macroscopic $\vb{P}^{+}$ and total $D$
+by summing over all two-level systems in the medium:
 
 $$\begin{aligned}
     \vb{P}^{+}(\vb{x}, t)
-    &= \sum_{n} \vb{p}^{+}_n \: \delta(\vb{x} - \vb{x}_n)
+    &= \sum_{\nu} \vb{p}^{+}_\nu \: \delta(\vb{x} - \vb{x}_\nu)
     \\
     D(\vb{x}, t)
-    &= \sum_{n} d_n \: \delta(\vb{x} - \vb{x}_n)
+    &= \sum_{\nu} d_\nu \: \delta(\vb{x} - \vb{x}_\nu)
 \end{aligned}$$
 
 We thus arrive at the **Maxwell-Bloch equations**,
-which are relevant for laser theory:
+which are the foundation of laser theory:
 
 $$\begin{aligned}
     \boxed{
         \begin{aligned}
             \mu_0 \pdv[2]{\vb{P}^{+}}{t}
-            &= - \nabla \cross \nabla \cross \vb{E}^{+} - \mu_0 \sigma \pdv{\vb{E}^{+}}{t} - \mu_0 \varepsilon_0 \pdv[2]{\vb{E}^{+}}{t}
+            &= - \nabla \cross \nabla \cross \vb{E}^{+} - \mu_0 \sigma \pdv{\vb{E}^{+}}{t} - \frac{n^2}{c^2} \pdv[2]{\vb{E}^{+}}{t}
             \\
             \pdv{\vb{P}^{+}}{t}
             &= - \Big( \gamma_\perp + i \omega_0 \Big) \vb{P}^{+}