From 8a9fb5fef2a97af3274290e512816e1a4cac0c02 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Mon, 24 Jan 2022 19:29:00 +0100
Subject: Rewrite "Lindhard function", split off "dielectric function"

---
 .../know/concept/maxwell-bloch-equations/index.pdc | 60 +++++++++-------------
 1 file changed, 23 insertions(+), 37 deletions(-)

(limited to 'content/know/concept/maxwell-bloch-equations/index.pdc')

diff --git a/content/know/concept/maxwell-bloch-equations/index.pdc b/content/know/concept/maxwell-bloch-equations/index.pdc
index 020a120..3f090a2 100644
--- a/content/know/concept/maxwell-bloch-equations/index.pdc
+++ b/content/know/concept/maxwell-bloch-equations/index.pdc
@@ -34,8 +34,8 @@ $$\begin{aligned}
     \hat{H}_1(t)
     = - \hat{\vb{p}} \cdot \vb{E}(t)
     \qquad \quad
-    \hat{\vb{p}}
-    \equiv q \hat{\vb{x}}
+    \vu{p}
+    \equiv q \vu{x}
     \qquad \quad
     \vb{E}(t)
     = \vb{E}_0 \cos\!(\omega t)
@@ -49,10 +49,10 @@ can then be described by:
 
 $$\begin{aligned}
     \dv{c_g}{t}
-    &= i \frac{q \matrixel{g}{\hat{\vb{x}}}{e} \cdot \vb{E}_0}{2 \hbar} \exp\!\big( i \omega t \!-\! i \omega_0 t \big) \: c_e
+    &= i \frac{q \matrixel{g}{\vu{x}}{e} \cdot \vb{E}_0}{2 \hbar} \exp\!\big( i \omega t \!-\! i \omega_0 t \big) \: c_e
     \\
     \dv{c_e}{t}
-    &= i \frac{q \matrixel{e}{\hat{\vb{x}}}{g} \cdot \vb{E}_0}{2 \hbar} \exp\!\big(\!-\! i \omega t \!+\! i \omega_0 t \big) \: c_g
+    &= i \frac{q \matrixel{e}{\vu{x}}{g} \cdot \vb{E}_0}{2 \hbar} \exp\!\big(\!-\! i \omega t \!+\! i \omega_0 t \big) \: c_g
 \end{aligned}$$
 
 We want to rearrange these equations a bit.
@@ -70,11 +70,11 @@ Similarly, we define the transition dipole moment $\vb{p}_0^{-}$:
 
 $$\begin{aligned}
     \vb{p}_0^{-}
-    \equiv q \matrixel{e}{\vb{x}}{g}
+    \equiv q \matrixel{e}{\vu{x}}{g}
     \qquad \quad
     \vb{p}_0^{+}
     \equiv (\vb{p}_0^{-})^*
-    = q \matrixel{g}{\vb{x}}{e}
+    = q \matrixel{g}{\vu{x}}{e}
 \end{aligned}$$
 
 With these, the equations for $c_g$ and $c_e$ can be rewritten as shown below.
@@ -238,10 +238,10 @@ $$\begin{aligned}
     \vb{p}
     &= \matrixel{\Psi}{\hat{\vb{p}}}{\Psi}
     \\
-    &= q \Big( c_g c_g^* \matrixel{g}{\hat{\vb{x}}}{g} + c_e c_e^* \matrixel{e}{\hat{\vb{x}}}{e}
-    + c_g c_e^* \matrixel{e}{\hat{\vb{x}}}{g} \exp\!(i \omega_0 t) + c_e c_g^* \matrixel{g}{\hat{\vb{x}}}{e} \exp\!(-i \omega_0 t) \Big)
+    &= q \Big( c_g c_g^* \matrixel{g}{\vu{x}}{g} + c_e c_e^* \matrixel{e}{\vu{x}}{e}
+    + c_g c_e^* \matrixel{e}{\vu{x}}{g} \exp\!(i \omega_0 t) + c_e c_g^* \matrixel{g}{\vu{x}}{e} \exp\!(-i \omega_0 t) \Big)
     \\
-    &= q \Big( \rho_{ge} \matrixel{e}{\hat{\vb{x}}}{g} + \rho_{eg} \matrixel{g}{\hat{\vb{x}}}{e} \Big)
+    &= q \Big( \rho_{ge} \matrixel{e}{\vu{x}}{g} + \rho_{eg} \matrixel{g}{\vu{x}}{e} \Big)
     = \vb{p}_0^{-} \rho_{ge}(t) + \vb{p}_0^{+} \rho_{eg}(t)
     \equiv \vb{p}^{-}(t) + \vb{p}^{+}(t)
 \end{aligned}$$
@@ -366,41 +366,27 @@ Inserting the definition $\vb{D} = \varepsilon_0 \vb{E} + \vb{P}$
 together with Ohm's law $\vb{J}_\mathrm{free} = \sigma \vb{E}$ yields:
 
 $$\begin{aligned}
-    \nabla \cross \big( \nabla \cross \vb{E} \big)
-    = - \mu_0 \sigma \pdv{\vb{E}}{t} - \mu_0 \varepsilon_0 \pdv[2]{\vb{E}}{t} - \mu_0 \pdv[2]{\vb{P}}{t}
+    \boxed{
+        \nabla \cross \big( \nabla \cross \vb{E} \big)
+        = - \mu_0 \sigma \pdv{\vb{E}}{t} - \mu_0 \varepsilon_0 \pdv[2]{\vb{E}}{t} - \mu_0 \pdv[2]{\vb{P}}{t}
+    }
 \end{aligned}$$
 
 Where $\sigma$ is the medium's conductivity, if any;
 many authors assume $\sigma = 0$.
-Next, we rewrite the left side using a vector identity,
-and assume no net charge $\nabla \cdot \vb{E} = 0$:
-
-$$\begin{aligned}
-    \nabla^2 \vb{E} - \nabla \big( \nabla \cdot \vb{E} \big)
-    = \nabla^2 \vb{E}
-    = \mu_0 \sigma \pdv{\vb{E}}{t} + \mu_0 \varepsilon_0 \pdv[2]{\vb{E}}{t} + \mu_0 \pdv[2]{\vb{P}}{t}
-\end{aligned}$$
-
-After some rearranging,
-we arrive at a variant of the electromagnetic wave equation:
-
-$$\begin{aligned}
-    \nabla^2 \vb{E} - \mu_0 \sigma \pdv{\vb{E}}{t} - \mu_0 \varepsilon_0 \pdv[2]{\vb{E}}{t}
-    &= \mu_0 \pdv[2]{\vb{P}}{t}
-\end{aligned}$$
-
 It is trivial to show that $\vb{E}$ and $\vb{P}$
 can be replaced by $\vb{E}^{+}$ and $\vb{P}^{+}$.
-It is also simple to convert
-the dipole $\vb{p}^{+}$ and inversion $d$
-into their macroscopic versions $\vb{P}^{+}$ and $D$:
+
+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:
 
 $$\begin{aligned}
-    \vb{P}^{+}(\vb{r}, t)
-    = \sum_{n} \vb{p}^{+}_n \: \delta(\vb{r} \!-\! \vb{r}_n)
-    \qquad \quad
-    D(\vb{r}, t)
-    = \sum_{n} d_n \: \delta(\vb{r} \!-\! \vb{r}_n)
+    \vb{P}^{+}(\vb{x}, t)
+    &= \sum_{n} \vb{p}^{+}_n \: \delta(\vb{x} - \vb{x}_n)
+    \\
+    D(\vb{x}, t)
+    &= \sum_{n} d_n \: \delta(\vb{x} - \vb{x}_n)
 \end{aligned}$$
 
 We thus arrive at the **Maxwell-Bloch equations**,
@@ -410,7 +396,7 @@ $$\begin{aligned}
     \boxed{
         \begin{aligned}
             \mu_0 \pdv[2]{\vb{P}^{+}}{t}
-            &= \nabla^2 \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} - \mu_0 \varepsilon_0 \pdv[2]{\vb{E}^{+}}{t}
             \\
             \pdv{\vb{P}^{+}}{t}
             &= - \Big( \gamma_\perp + i \omega_0 \Big) \vb{P}^{+}
-- 
cgit v1.2.3