From 7a2346d3ee81c7c852de85527de056fe0b39aad8 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Thu, 19 Jan 2023 21:28:23 +0100
Subject: More improvements to knowledge base

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+---
+title: "Optical Bloch equations"
+sort_title: "Optical Bloch equations"
+date: 2023-01-19
+categories:
+- Physics
+- Quantum mechanics
+- Two-level system
+layout: "concept"
+---
+
+For an electron in a two-level system with time-independent states
+$$\ket{g}$$ (ground) and $$\ket{e}$$ (excited),
+consider the following general solution
+to the time-dependent Schrödinger equation:
+
+$$\begin{aligned}
+    \ket{\Psi}
+    &= c_g \ket{g} e^{-i \varepsilon_g t / \hbar} + c_e \ket{e} e^{-i \varepsilon_e t / \hbar}
+\end{aligned}$$
+
+Perturbing this system with
+an [electromagnetic wave](/know/concept/electromagnetic-wave-equation/)
+introduces a time-dependent sinusoidal term $$\hat{H}_1$$ to the Hamiltonian.
+In the [electric dipole approximation](/know/concept/electric-dipole-approximation/),
+$$\hat{H}_1$$ is given by:
+
+$$\begin{aligned}
+    \hat{H}_1(t)
+    = - \hat{\vb{p}} \cdot \vb{E}(t)
+    \qquad \qquad
+    \vu{p}
+    \equiv q \vu{x}
+    \qquad \qquad
+    \vb{E}(t)
+    = \vb{E}_0 \cos(\omega t)
+\end{aligned}$$
+
+Where $$\vb{E}$$ is an [electric field](/know/concept/electric-field/),
+and $$\hat{\vb{p}}$$ is the dipole moment operator.
+From [Rabi oscillation](/know/concept/rabi-oscillation/),
+we know that the time-varying coefficients $$c_g$$ and $$c_e$$
+can then be described by:
+
+$$\begin{aligned}
+    \dv{c_g}{t}
+    &= i \frac{q \matrixel{g}{\vu{x}}{e} \cdot \vb{E}_0}{2 \hbar} \: e^{i (\omega - \omega_0) t} \: c_e
+    \\
+    \dv{c_e}{t}
+    &= i \frac{q \matrixel{e}{\vu{x}}{g} \cdot \vb{E}_0}{2 \hbar} \: e^{- i (\omega - \omega_0) t} \: c_g
+\end{aligned}$$
+
+Where $$\omega_0 \equiv (\varepsilon_e \!-\! \varepsilon_g) / \hbar$$ is the resonance frequency.
+We want to rearrange these equations a bit,
+so we split the field $$\vb{E}$$ as follows,
+where the amplitudes $$\vb{E}_0^{-}$$ and $$\vb{E}_0^{+}$$
+may be slowly-varying with respect to the carrier wave $$e^{\pm i \omega t}$$:
+
+$$\begin{aligned}
+    \vb{E}(t)
+    &\equiv \vb{E}^{-}(t) + \vb{E}^{+}(t)
+    \\
+    &\equiv \vb{E}_0^{-} e^{i \omega t} + \vb{E}_0^{+} e^{-i \omega t}
+\end{aligned}$$
+
+Since $$\vb{E}$$ is real, $$\vb{E}_0^{+} = (\vb{E}_0^{-})^*$$.
+Similarly, we define the transition dipole moment $$\vb{p}_0^{-}$$:
+
+$$\begin{aligned}
+    \vb{p}_0^{-}
+    \equiv q \matrixel{e}{\vu{x}}{g}
+    \qquad \qquad
+    \vb{p}_0^{+}
+    \equiv (\vb{p}_0^{-})^*
+    = q \matrixel{g}{\vu{x}}{e}
+\end{aligned}$$
+
+With these, the equations for $$c_g$$ and $$c_e$$ can be rewritten as shown below.
+Note that $$\vb{E}^{-}$$ and $$\vb{E}^{+}$$ include the driving plane wave, and the
+[rotating wave approximation](/know/concept/rotating-wave-approximation/) is still made:
+
+$$\begin{aligned}
+    \dv{c_g}{t}
+    &= \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} e^{- i \omega_0 t} \: c_e
+    \\
+    \dv{c_e}{t}
+    &= \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} e^{i \omega_0 t} \: c_g
+\end{aligned}$$
+
+
+For $$\ket{\Psi}$$ as defined above,
+the corresponding pure [density operator](/know/concept/density-operator/)
+$$\hat{\rho}$$ is as follows:
+
+$$\begin{aligned}
+    \hat{\rho}
+    = \ket{\Psi} \bra{\Psi}
+    =
+    \begin{bmatrix}
+        c_e c_e^* & c_e c_g^* e^{-i \omega_0 t} \\
+        c_g c_e^* e^{i \omega_0 t} & c_g c_g^*
+    \end{bmatrix}
+    \equiv
+    \begin{bmatrix}
+        \rho_{ee} & \rho_{eg} \\
+        \rho_{ge} & \rho_{gg}
+    \end{bmatrix}
+\end{aligned}$$
+
+We take the $$t$$-derivative of the matrix elements,
+and insert the equations for $$c_g$$ and $$c_e$$:
+
+$$\begin{aligned}
+    \dv{\rho_{gg}}{t}
+    &= \dv{c_g}{t} c_g^* + c_g \dv{c_g^*}{t}
+    \\
+    &= \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} c_e c_g^* e^{- i \omega_0 t}
+    - \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} c_g c_e^* e^{i \omega_0 t}
+    \\
+    \dv{\rho_{ee}}{t}
+    &= \dv{c_e}{t} c_e^* + c_e \dv{c_e^*}{t}
+    \\
+    &= \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} c_g c_e^* e^{i \omega_0 t}
+    - \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} c_e c_g^* e^{- i \omega_0 t}
+    \\
+    \dv{\rho_{ge}}{t}
+    &= \dv{c_g}{t} c_e^* e^{i \omega_0 t} + c_g \dv{c_e^*}{t} e^{i \omega_0 t} + i \omega_0 c_g c_e^* e^{i \omega_0 t}
+    \\
+    &= \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} c_e c_e^*
+    - \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} c_g c_g^*
+    + i \omega_0 c_g c_e^* e^{i \omega_0 t}
+    \\
+    \dv{\rho_{eg}}{t}
+    &= \dv{c_e}{t} c_g^* e^{-i \omega_0 t} + c_e \dv{c_g^*}{t} e^{-i \omega_0 t} - i \omega_0 c_e c_g^* e^{- i \omega_0 t}
+    \\
+    &= \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} c_g c_g^*
+    - \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} c_e c_e^*
+    - i \omega_0 c_e c_g^* e^{- i \omega_0 t}
+\end{aligned}$$
+
+Recognizing the density matrix elements allows us
+to reduce these equations to:
+
+$$\begin{aligned}
+    \dv{\rho_{gg}}{t}
+    &= \frac{i}{\hbar} \Big( \vb{p}_0^{+} \cdot \vb{E}^{-} \rho_{eg} - \vb{p}_0^{-} \cdot \vb{E}^{+} \rho_{ge} \Big)
+    \\
+    \dv{\rho_{ee}}{t}
+    &= \frac{i}{\hbar} \Big( \vb{p}_0^{-} \cdot \vb{E}^{+} \rho_{ge} - \vb{p}_0^{+} \cdot \vb{E}^{-} \rho_{eg} \Big)
+    \\
+    \dv{\rho_{ge}}{t}
+    &= i \omega_0 \rho_{ge} + \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} \big( \rho_{ee} - \rho_{gg} \big)
+    \\
+    \dv{\rho_{eg}}{t}
+    &= - i \omega_0 \rho_{eg} + \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} \big( \rho_{gg} - \rho_{ee} \big)
+\end{aligned}$$
+
+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](/know/concept/einstein-coefficients/).
+
+Suppose $$\rho_{ee}$$ decays with rate $$\gamma_e$$.
+Because the total probability $$\rho_{ee} + \rho_{gg} = 1$$, we have:
+
+$$\begin{aligned}
+    \Big( \dv{\rho_{ee}}{t} \Big)_{e}
+    = - \gamma_e \rho_{ee}
+    \quad \implies \quad
+    \Big( \dv{\rho_{gg}}{t} \Big)_{e}
+    = \gamma_e \rho_{ee}
+\end{aligned}$$
+
+Meanwhile, for whatever reason,
+let $$\rho_{gg}$$ decay into $$\rho_{ee}$$ with rate $$\gamma_g$$:
+
+$$\begin{aligned}
+    \Big( \dv{\rho_{gg}}{t} \Big)_{g}
+    = - \gamma_g \rho_{gg}
+    \quad \implies \quad
+    \Big( \dv{\rho_{gg}}{t} \Big)_{g}
+    = \gamma_g \rho_{gg}
+\end{aligned}$$
+
+And finally, let the diagonal (perpendicular) matrix elements
+both decay with rate $$\gamma_\perp$$:
+
+$$\begin{aligned}
+    \Big( \dv{\rho_{eg}}{t} \Big)_{\perp}
+    = - \gamma_\perp \rho_{eg}
+    \qquad \qquad
+    \Big( \dv{\rho_{ge}}{t} \Big)_{\perp}
+    = - \gamma_\perp \rho_{ge}
+\end{aligned}$$
+
+Putting everything together,
+we arrive at the **optical Bloch equations** governing $$\hat{\rho}$$:
+
+$$\begin{aligned}
+    \boxed{
+        \begin{aligned}
+            \dv{\rho_{gg}}{t}
+            &= \gamma_e \rho_{ee} - \gamma_g \rho_{gg}
+            + \frac{i}{\hbar} \Big( \vb{p}_0^{+} \cdot \vb{E}^{-} \rho_{eg} - \vb{p}_0^{-} \cdot \vb{E}^{+} \rho_{ge} \Big)
+            \\
+            \dv{\rho_{ee}}{t}
+            &=  \gamma_g \rho_{gg} - \gamma_e \rho_{ee}
+            + \frac{i}{\hbar} \Big( \vb{p}_0^{-} \cdot \vb{E}^{+} \rho_{ge} - \vb{p}_0^{+} \cdot \vb{E}^{-} \rho_{eg} \Big)
+            \\
+            \dv{\rho_{ge}}{t}
+            &= - \Big( \gamma_\perp - i \omega_0 \Big) \rho_{ge}
+            + \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} \Big( \rho_{ee} - \rho_{gg} \Big)
+            \\
+            \dv{\rho_{eg}}{t}
+            &= - \Big( \gamma_\perp + i \omega_0 \Big) \rho_{eg}
+            + \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} \Big( \rho_{gg} - \rho_{ee} \Big)
+        \end{aligned}
+    }
+\end{aligned}$$
+
+Some authors simplify these equations a bit by choosing
+$$\gamma_g = 0$$ and $$\gamma_\perp = \gamma_e / 2$$.
+
+
+
+## References
+1.  F. Kärtner,
+    [Ultrafast optics: lecture notes](https://ocw.mit.edu/courses/6-977-ultrafast-optics-spring-2005/pages/lecture-notes/),
+    2005, Massachusetts Institute of Technology.
+2.  H.J. Metcalf, P. van der Straten,
+    *Laser cooling and trapping*,
+    1999, Springer.
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