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-rw-r--r--source/know/concept/maxwell-bloch-equations/index.md25
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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 ba8a677..0252b5c 100644
--- a/source/know/concept/maxwell-bloch-equations/index.md
+++ b/source/know/concept/maxwell-bloch-equations/index.md
@@ -12,13 +12,13 @@ layout: "concept"
---
For an electron in a two-level system with time-independent states
-$$\Ket{g}$$ (ground) and $$\Ket{e}$$ (excited),
+$$\ket{g}$$ (ground) and $$\ket{e}$$ (excited),
consider the following general solution
-to the full Schrödinger equation:
+to the time-dependent Schrödinger equation:
$$\begin{aligned}
- \Ket{\Psi}
- &= c_g \: \Ket{g} \exp(-i E_g t / \hbar) + c_e \: \Ket{e} \exp(-i E_e t / \hbar)
+ \ket{\Psi}
+ &= c_g \ket{g} \exp(-i E_g t / \hbar) + c_e \ket{e} \exp(-i E_e t / \hbar)
\end{aligned}$$
Perturbing this system with
@@ -87,15 +87,16 @@ $$\begin{aligned}
\end{aligned}$$
+
## Optical Bloch equations
-For $$\Ket{\Psi}$$ as defined above,
+For $$\ket{\Psi}$$ as defined above,
the corresponding pure [density operator](/know/concept/density-operator/)
$$\hat{\rho}$$ is as follows:
$$\begin{aligned}
\hat{\rho}
- = \Ket{\Psi} \Bra{\Psi}
+ = \ket{\Psi} \bra{\Psi}
=
\begin{bmatrix}
c_e c_e^* & c_e c_g^* \exp(-i \omega_0 t) \\
@@ -159,11 +160,10 @@ $$\begin{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 (see [Einstein coefficients](/know/concept/einstein-coefficients/)).
+e.g. [spontaneous emission](/know/concept/einstein-coefficients/).
-Let $$\rho_{ee}$$ decays with rate $$\gamma_e$$.
-Since the total probability $$\rho_{ee} + \rho_{gg} = 1$$,
-we thus have:
+Suppose $$\rho_{ee}$$ decays with rate $$\gamma_e$$.
+Because the total probability $$\rho_{ee} + \rho_{gg} = 1$$, we have:
$$\begin{aligned}
\Big( \dv{\rho_{ee}}{t} \Big)_{e}
@@ -220,10 +220,11 @@ $$\begin{aligned}
}
\end{aligned}$$
-Many authors simplify these equations a bit by choosing
+Some authors simplify these equations a bit by choosing
$$\gamma_g = 0$$ and $$\gamma_\perp = \gamma_e / 2$$.
+
## Including Maxwell's equations
This two-level system has a dipole moment $$\vb{p}$$ as follows,
@@ -286,7 +287,7 @@ $$\begin{aligned}
We can rewrite the first two terms in the following intuitive form,
which describes a decay with
rate $$\gamma_\parallel \equiv \gamma_g + \gamma_e$$
-towards an equilbrium $$d_0$$:
+towards an equilibrium $$d_0$$:
$$\begin{aligned}
2 \gamma_g \rho_{gg} - 2 \gamma_e \rho_{ee}