Improve knowledge base

1 files changed, 8 insertions, 6 deletions

diff --git a/source/know/concept/laser-rate-equations/index.md b/source/know/concept/laser-rate-equations/index.md
index 1f42f73..c81f02b 100644
--- a/source/know/concept/laser-rate-equations/index.md
+++ b/source/know/concept/laser-rate-equations/index.md
@@ -97,14 +97,14 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Typically, $$\gamma_\perp$$ is much larger than the rate of any other decay process,
-in which case $$\ipdv{}{\vb{P}0^{+}\!}{t}$$ is negligible compared to $$\gamma_\perp \vb{P}_0^{+}$$.
+in which case $$\ipdv{\vb{P}_0^{+}\!}{t}$$ is negligible compared to $$\gamma_\perp \vb{P}_0^{+}$$.
 Effectively, this means that the polarization $$\vb{P}_0^{+}$$
 near-instantly follows the electric field $$\vb{E}^{+}\!$$.
-Setting $$\ipdv{}{\vb{P}0^{+}\!}{t} \approx 0$$, the second MBE becomes:
+Setting $$\ipdv{\vb{P}_0^{+}\!}{t} \approx 0$$, the second MBE becomes:
 
 $$\begin{aligned}
     \vb{P}^{+}
-    = -\frac{i |g|^2}{\hbar (\gamma_\perp + i (\omega_0 \!-\! \omega))} \vb{E}^{+} D
+    = -\frac{i |g|^2}{\hbar (\gamma_\perp + i (\omega_0 - \omega))} \vb{E}^{+} D
     = \frac{|g|^2 \gamma(\omega)}{\hbar \gamma_\perp} \vb{E}^{+} D
 \end{aligned}$$
 
@@ -137,7 +137,7 @@ $$\begin{aligned}
     &= i (\omega - \Omega) \vb{E}_0^{+} + i \frac{|g|^2 \omega \gamma(\omega)}{2 \hbar \varepsilon_0 \gamma_\perp n^2} \vb{E}_0^{+} D
 \end{aligned}$$
 
-Next, we insert our ansatz for $$\vb{E}^{+}\!$$ and $$\vb{P}^{+}\!$$
+Next, we insert our ansatz for $$\vb{E}^{+}$$ and $$\vb{P}^{+}$$
 into the third MBE, and rewrite $$\vb{P}_0^{+}$$ as above.
 Using our identity for $$\gamma(\omega)$$,
 and the fact that $$\vb{E}_0^{+} \cdot \vb{E}_0^{-} = |\vb{E}|^2$$, we find:
@@ -293,11 +293,13 @@ $$\begin{aligned}
     \boxed{
         \begin{aligned}
             \pdv{N_p}{t}
-            &= - (\gamma_\mathrm{out} + \gamma_\mathrm{abs} + \gamma_\mathrm{loss}) N_p + \gamma_\mathrm{spon} N_e + G_\mathrm{stim} N_p N_e
+            &= - (\gamma_\mathrm{out} + \gamma_\mathrm{abs} + \gamma_\mathrm{loss}) N_p
+            + \gamma_\mathrm{spon} N_e + G_\mathrm{stim} N_p N_e
             \\
             \pdv{N_e}{t}
             &= R_\mathrm{pump} + \gamma_\mathrm{abs} N_p
-            - (\gamma_\mathrm{spon} + \gamma_\mathrm{n.r.} + \gamma_\mathrm{leak}) N_e - G_\mathrm{stim} N_p N_e
+            - (\gamma_\mathrm{spon} + \gamma_\mathrm{n.r.} + \gamma_\mathrm{leak}) N_e
+            - G_\mathrm{stim} N_p N_e
         \end{aligned}
     }
 \end{aligned}$$