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authorPrefetch2024-07-17 10:01:43 +0200
committerPrefetch2024-07-17 10:01:43 +0200
commit075683cdf4588fe16f41d9f7b46b9720b42b2553 (patch)
treef200b341b35e9c89aa1030a2f6da94cd0e1e958b /source/know/concept/maxwell-bloch-equations
parentc4d597e8d695eb145755464cffbf88a68fd0c88a (diff)
Improve knowledge base
Diffstat (limited to 'source/know/concept/maxwell-bloch-equations')
-rw-r--r--source/know/concept/maxwell-bloch-equations/index.md22
1 files changed, 12 insertions, 10 deletions
diff --git a/source/know/concept/maxwell-bloch-equations/index.md b/source/know/concept/maxwell-bloch-equations/index.md
index 1214703..28885af 100644
--- a/source/know/concept/maxwell-bloch-equations/index.md
+++ b/source/know/concept/maxwell-bloch-equations/index.md
@@ -17,8 +17,8 @@ where $$\varepsilon_g$$ and $$\varepsilon_e$$ are the time-independent eigenener
and the weights $$c_g$$ and $$c_g$$ are functions of $$t$$:
$$\begin{aligned}
- \ket{\Psi}
- &= c_g \ket{g} e^{-i \varepsilon_g t / \hbar} + c_e \ket{e} e^{-i \varepsilon_e t / \hbar}
+ \ket{\Psi(t)}
+ &= c_g(t) \ket{g} e^{-i \varepsilon_g t / \hbar} + c_e(t) \ket{e} e^{-i \varepsilon_e t / \hbar}
\end{aligned}$$
This system is being perturbed by an electromagnetic wave
@@ -32,8 +32,8 @@ $$\begin{aligned}
Where the forward-propagating component $$\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^{-})^*$$.
+and similarly $$\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}$$ as defined above,
the pure [density operator](/know/concept/density-operator/)
@@ -92,7 +92,7 @@ $$\begin{aligned}
\end{aligned}$$
However, the light wave affects the electron,
-so the actual electromagnetic dipole moment $$\vb{p}$$ is as follows,
+so the true electromagnetic dipole moment $$\vb{p}$$ is as follows,
using [Laporte's selection rule](/know/concept/selection-rules/)
to remove diagonal terms by assuming that
the electron's orbitals are spatially odd or even:
@@ -106,9 +106,9 @@ $$\begin{aligned}
\\
&= 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)
+ &= \vb{p}_0^{-} \rho_{ge} + \vb{p}_0^{+} \rho_{eg}
\\
- &\equiv \vb{p}^{-}(t) + \vb{p}^{+}(t)
+ &\equiv \vb{p}^{-} + \vb{p}^{+}
\end{aligned}$$
Where we have split $$\vb{p}$$ analogously to $$\vb{E}$$
@@ -117,8 +117,9 @@ Its equation of motion can then be found from the optical Bloch equations:
$$\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}
+ &= \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}$$
@@ -147,7 +148,8 @@ we find its equation of motion to be:
$$\begin{aligned}
\dv{d}{t}
&= \dv{\rho_{ee}}{t} - \dv{\rho_{gg}}{t}
- = 2 \gamma_g \rho_{gg} - 2 \gamma_e \rho_{ee}
+ \\
+ &= 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}$$