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+---
+title: "Maxwell-Bloch equations"
+date: 2021-10-02
+categories:
+- Physics
+- Quantum mechanics
+- Two-level system
+- Electromagnetism
+- Laser theory
+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 full Schrödinger equation:
+
+$$\begin{aligned}
+    \Ket{\Psi}
+    &= c_g \: \Ket{g} \exp(-i E_g t / \hbar) + c_e \: \Ket{e} \exp(-i E_e t / \hbar)
+\end{aligned}$$
+
+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:
+
+$$\begin{aligned}
+    \dv{c_g}{t}
+    &= i \frac{q \matrixel{g}{\vu{x}}{e} \cdot \vb{E}_0}{2 \hbar} \exp\!\big( i \omega t \!-\! i \omega_0 t \big) \: c_e
+    \\
+    \dv{c_e}{t}
+    &= i \frac{q \matrixel{e}{\vu{x}}{g} \cdot \vb{E}_0}{2 \hbar} \exp\!\big(\!-\! i \omega t \!+\! i \omega_0 t \big) \: c_g
+\end{aligned}$$
+
+We want to rearrange these equations a bit.
+Therefore, we split the electric field $\vb{E}$ like so,
+where the amplitudes $\vb{E}_0^{-}$ and $\vb{E}_0^{+}$ may be slowly varying:
+
+$$\begin{aligned}
+    \vb{E}(t)
+    = \vb{E}^{-}(t) + \vb{E}^{+}(t)
+    = \vb{E}_0^{-} \exp(i \omega t) + \vb{E}_0^{+} \exp(-i \omega t)
+\end{aligned}$$
+
+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:
+
+$$\begin{aligned}
+    \dv{c_g}{t}
+    &= \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} \exp(- i \omega_0 t) \: c_e
+    \\
+    \dv{c_e}{t}
+    &= \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} \exp(i \omega_0 t) \: c_g
+\end{aligned}$$
+
+
+## Optical Bloch equations
+
+For $\Ket{\Psi}$ as defined above,
+the corresponding pure [density operator](/know/concept/density-operator/)
+$\hat{\rho}$ is as follows:
+
+$$\begin{aligned}
+    \hat{\rho}
+    = \Ket{\Psi} \Bra{\Psi}
+    =
+    \begin{bmatrix}
+        c_e c_e^* & c_e c_g^* \exp(-i \omega_0 t) \\
+        c_g c_e^* \exp(i \omega_0 t) & c_g c_g^*
+    \end{bmatrix}
+    \equiv
+    \begin{bmatrix}
+        \rho_{ee} & \rho_{eg} \\
+        \rho_{ge} & \rho_{gg}
+    \end{bmatrix}
+\end{aligned}$$
+
+Where $\omega_0 \equiv (E_e \!-\! E_g) / \hbar$ is the resonance frequency.
+We take the $t$-derivative of the matrix elements,
+and insert the equations for $c_g$ and $c_e$:
+
+$$\begin{aligned}
+    \dv{\rho_{gg}}{t}
+    &= \dv{c_g}{t} c_g^* + c_g \dv{c_g^*}{t}
+    \\
+    &= \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} \exp(- i \omega_0 t) \: c_e c_g^*
+    - \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} \exp(i \omega_0 t) \: c_g c_e^*
+    \\
+    \dv{\rho_{ee}}{t}
+    &= \dv{c_e}{t} c_e^* + c_e \dv{c_e^*}{t}
+    \\
+    &= \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} \exp(i \omega_0 t) \: c_g c_e^*
+    - \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} \exp(- i \omega_0 t) \: c_e c_g^*
+    \\
+    \dv{\rho_{ge}}{t}
+    &= \dv{c_g}{t} c_e^* \exp(i \omega_0 t) + c_g \dv{c_e^*}{t} \exp(i \omega_0 t) + i \omega_0 c_g c_e^* \exp(i \omega_0 t)
+    \\
+    &= \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} \: c_e c_e^*
+    - \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} \: c_g c_g^*
+    + i \omega_0 c_g c_e^* \exp(i \omega_0 t)
+    \\
+    \dv{\rho_{eg}}{t}
+    &= \dv{c_e}{t} c_g^* \exp(-i \omega_0 t) + c_e \dv{c_g^*}{t} \exp(-i \omega_0 t) - i \omega_0 c_e c_g^* \exp(- i \omega_0 t)
+    \\
+    &= \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} \: c_g c_g^*
+    - \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} \: c_e c_e^*
+    - i \omega_0 c_e c_g^* \: \exp(- i \omega_0 t)
+\end{aligned}$$
+
+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 (see [Einstein coefficients](/know/concept/einstein-coefficients/)).
+
+Let $\rho_{ee}$ decays with rate $\gamma_e$.
+Since the total probability $\rho_{ee} + \rho_{gg} = 1$,
+we thus 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 $\hat{\rho}$:
+
+$$\begin{aligned}
+    \boxed{
+        \begin{aligned}
+            \dv{\rho_{gg}}{t}
+            &= \gamma_e \rho_{ee} - \gamma_g \rho_{gg}
+            + \frac{i}{\hbar} \Big( \vb{p}_0^{+} \cdot \vb{E}^{-} \rho_{eg} - \vb{p}_0^{-} \cdot \vb{E}^{+} \rho_{ge} \Big)
+            \\
+            \dv{\rho_{ee}}{t}
+            &=  \gamma_g \rho_{gg} - \gamma_e \rho_{ee}
+            + \frac{i}{\hbar} \Big( \vb{p}_0^{-} \cdot \vb{E}^{+} \rho_{ge} - \vb{p}_0^{+} \cdot \vb{E}^{-} \rho_{eg} \Big)
+            \\
+            \dv{\rho_{ge}}{t}
+            &= - \Big( \gamma_\perp - i \omega_0 \Big) \rho_{ge}
+            + \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} \Big( \rho_{ee} - \rho_{gg} \Big)
+            \\
+            \dv{\rho_{eg}}{t}
+            &= - \Big( \gamma_\perp + i \omega_0 \Big) \rho_{eg}
+            + \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} \Big( \rho_{gg} - \rho_{ee} \Big)
+        \end{aligned}
+    }
+\end{aligned}$$
+
+Many 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,
+where we use [Laporte's selection rule](/know/concept/selection-rules/)
+to remove diagonal terms, by assuming that
+the electron's orbitals are odd or even:
+
+$$\begin{aligned}
+    \vb{p}
+    &= \matrixel{\Psi}{\hat{\vb{p}}}{\Psi}
+    \\
+    &= q \Big( c_g c_g^* \matrixel{g}{\vu{x}}{g} + c_e c_e^* \matrixel{e}{\vu{x}}{e}
+    + c_g c_e^* \matrixel{e}{\vu{x}}{g} \exp(i \omega_0 t) + c_e c_g^* \matrixel{g}{\vu{x}}{e} \exp(-i \omega_0 t) \Big)
+    \\
+    &= q \Big( \rho_{ge} \matrixel{e}{\vu{x}}{g} + \rho_{eg} \matrixel{g}{\vu{x}}{e} \Big)
+    = \vb{p}_0^{-} \rho_{ge}(t) + \vb{p}_0^{+} \rho_{eg}(t)
+    \equiv \vb{p}^{-}(t) + \vb{p}^{+}(t)
+\end{aligned}$$
+
+Where we have split $\vb{p}$ analogously to $\vb{E}$
+by defining $\vb{p}^{+} \equiv \vb{p}_0^{+} \rho_{eg}$.
+Its equation of motion can then be found from the optical Bloch equations:
+
+$$\begin{aligned}
+    \dv{\vb{p}^{+}}{t}
+    = \vb{p}_0^{+} \dv{\rho_{eg}}{t}
+    = - \vb{p}_0^{+} \Big( \gamma_\perp + i \omega_0 \Big) \rho_{eg}
+    + \frac{i}{\hbar} \vb{p}_0^{+} \Big( \vb{p}_0^{-} \cdot \vb{E}^{+} \Big) \Big( \rho_{gg} - \rho_{ee} \Big)
+\end{aligned}$$
+
+Some authors do not bother multiplying $\rho_{ge}$ by $\vb{p}_0^{+}$.
+In any case, we arrive at:
+
+$$\begin{aligned}
+    \boxed{
+        \dv{\vb{p}^{+}}{t}
+        = - \Big( \gamma_\perp + i \omega_0 \Big) \vb{p}^{+}
+        - \frac{i}{\hbar} \Big( \vb{p}_0^{-} \cdot \vb{E}^{+} \Big) \vb{p}_0^{+} d
+    }
+\end{aligned}$$
+
+Where we have defined the **population inversion** $d \in [-1, 1]$ as follows,
+which quantifies the electron's excitedness:
+
+$$\begin{aligned}
+    d
+    \equiv \rho_{ee} - \rho_{gg}
+\end{aligned}$$
+
+From the optical Bloch equations,
+we find its equation of motion to be:
+
+$$\begin{aligned}
+    \dv{d}{t}
+    &= \dv{\rho_{ee}}{t} - \dv{\rho_{gg}}{t}
+    = 2 \gamma_g \rho_{gg} - 2 \gamma_e \rho_{ee}
+    + \frac{i 2}{\hbar} \Big( \vb{p}^{-} \cdot \vb{E}^{+} - \vb{p}^{+} \cdot \vb{E}^{-} \Big)
+\end{aligned}$$
+
+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 $d_0$:
+
+$$\begin{aligned}
+    2 \gamma_g \rho_{gg} - 2 \gamma_e \rho_{ee}
+    = \gamma_\parallel (d_0 - d)
+    \qquad \qquad
+    d_0
+    \equiv \frac{\gamma_g - \gamma_e}{\gamma_g + \gamma_e}
+\end{aligned}$$
+
+<div class="accordion">
+<input type="checkbox" id="proof-inversion-decay"/>
+<label for="proof-inversion-decay">Proof</label>
+<div class="hidden">
+<label for="proof-inversion-decay">Proof.</label>
+We introduce some new terms, and reorganize the expression:
+
+$$\begin{aligned}
+    2 \gamma_g \rho_{gg} - 2 \gamma_e \rho_{ee}
+    &= 2 \gamma_g \rho_{gg} - 2 \gamma_e \rho_{ee}
+    + \gamma_g \rho_{ee} - \gamma_g \rho_{ee}
+    + \gamma_e \rho_{gg} - \gamma_e \rho_{gg}
+    \\
+    &= \gamma_g (\rho_{gg} + \rho_{ee}) - \gamma_e (\rho_{gg} + \rho_{ee})
+    + \gamma_g (\rho_{gg} - \rho_{ee}) + \gamma_e (\rho_{gg} - \rho_{ee})
+\end{aligned}$$
+
+Since the total probability $\rho_{gg} + \rho_{ee} = 1$,
+and $d \equiv \rho_{ee} - \rho_{gg}$, this reduces to:
+
+$$\begin{aligned}
+    2 \gamma_g \rho_{gg} - 2 \gamma_e \rho_{ee}
+    &= \gamma_g - \gamma_e - (\gamma_g + \gamma_e) d
+    \\
+    &= (\gamma_g + \gamma_e) \Big( \frac{\gamma_g - \gamma_e}{\gamma_g + \gamma_e} - d \Big)
+    \\
+    &= \gamma_\parallel ( d_0 - d )
+\end{aligned}$$
+</div>
+</div>
+
+With this, the equation for the population inversion $d$
+takes the following final form:
+
+$$\begin{aligned}
+    \boxed{
+        \dv{d}{t}
+        = \gamma_\parallel (d_0 - d) + \frac{i 2}{\hbar} \Big( \vb{p}^{-} \cdot \vb{E}^{+} - \vb{p}^{+} \cdot \vb{E}^{-} \Big)
+    }
+\end{aligned}$$
+
+Finally, we would like a relation between the polarization
+and the electric field $\vb{E}$,
+for which we turn to [Maxwell's equations](/know/concept/maxwells-equations/).
+We start from Faraday's law,
+and split $\vb{B} = \mu_0 (\vb{H} + \vb{M})$:
+
+$$\begin{aligned}
+    \nabla \cross \vb{E}
+    = - \pdv{\vb{B}}{t}
+    = - \mu_0 \pdv{\vb{H}}{t} - \mu_0 \pdv{\vb{M}}{t}
+\end{aligned}$$
+
+We assume that there is no magnetization $\vb{M} = 0$.
+Then we we take the curl of both sides,
+and replace $\nabla \cross \vb{H}$ with Ampère's circuital law:
+
+$$\begin{aligned}
+    \nabla \cross \big( \nabla \cross \vb{E} \big)
+    = - \mu_0 \pdv{}{t} \big( \nabla \cross \vb{H} \big)
+    = - \mu_0 \pdv{}{t} \Big( \vb{J}_\mathrm{free} + \pdv{\vb{D}}{t} \Big)
+\end{aligned}$$
+
+Inserting the definition $\vb{D} = \varepsilon_0 \vb{E} + \vb{P}$
+together with Ohm's law $\vb{J}_\mathrm{free} = \sigma \vb{E}$ yields:
+
+$$\begin{aligned}
+    \nabla \cross \big( \nabla \cross \vb{E} \big)
+    = - \mu_0 \sigma \pdv{\vb{E}}{t} - \mu_0 \varepsilon_0 \pdvn{2}{\vb{E}}{t} - \mu_0 \pdvn{2}{\vb{P}}{t}
+\end{aligned}$$
+
+Where $\sigma$ is the active material's conductivity, if any;
+almost all authors assume $\sigma = 0$.
+
+Recall that we are describing the dynamics of a two-level system.
+In reality, such a system (e.g. a quantum dot)
+is suspended in a passive background medium,
+which reacts with a polarization $\vb{P}_\mathrm{med}$
+to the electric field $\vb{E}$.
+If the medium is linear, i.e. $\vb{P}_\mathrm{med} = \varepsilon_0 \chi \vb{E}$,
+then:
+
+$$\begin{aligned}
+    \mu_0 \pdvn{2}{\vb{P}}{t}
+    &= - \nabla \cross \big( \nabla \cross \vb{E} \big) - \mu_0 \sigma \pdv{\vb{E}}{t}
+    - \mu_0 \varepsilon_0 \pdvn{2}{\vb{E}}{t} - \mu_0 \pdvn{2}{\vb{P}_\mathrm{med}}{t}
+    \\
+    &= - \nabla \cross \big( \nabla \cross \vb{E} \big) - \mu_0 \sigma \pdv{\vb{E}}{t}
+    - \mu_0 \pdvn{2}{}{t}\Big( \varepsilon_0 \vb{E} + \varepsilon_0 \chi \vb{E} \Big)
+    \\
+    &= - \nabla \cross \big( \nabla \cross \vb{E} \big) - \mu_0 \sigma \pdv{\vb{E}}{t}
+    - \mu_0 \varepsilon_0 \varepsilon_r \pdvn{2}{\vb{E}}{t}
+\end{aligned}$$
+
+Where $\varepsilon_r \equiv 1 + \chi_e$ is the medium's relative permittivity.
+The speed of light $c^2 = 1 / (\mu_0 \varepsilon_0)$,
+and the refractive index $n^2 = \mu_r \varepsilon_r$,
+where $\mu_r = 1$ due to our assumption that $\vb{M} = 0$, so:
+
+$$\begin{aligned}
+    \boxed{
+        \mu_0 \pdvn{2}{\vb{P}}{t}
+        = - \nabla \cross \big( \nabla \cross \vb{E} \big) - \mu_0 \sigma \pdv{\vb{E}}{t} - \frac{n^2}{c^2} \pdvn{2}{\vb{E}}{t}
+    }
+\end{aligned}$$
+
+$\vb{E}$ and $\vb{P}$ can trivially be replaced by $\vb{E}^{+}$ and $\vb{P}^{+}$.
+It is also simple to convert $\vb{p}^{+}$ and $d$
+into the macroscopic $\vb{P}^{+}$ and total $D$
+by summing over all two-level systems in the medium:
+
+$$\begin{aligned}
+    \vb{P}^{+}(\vb{x}, t)
+    &= \sum_{\nu} \vb{p}^{+}_\nu \: \delta(\vb{x} - \vb{x}_\nu)
+    \\
+    D(\vb{x}, t)
+    &= \sum_{\nu} d_\nu \: \delta(\vb{x} - \vb{x}_\nu)
+\end{aligned}$$
+
+We thus arrive at the **Maxwell-Bloch equations**,
+which are the foundation of laser theory:
+
+$$\begin{aligned}
+    \boxed{
+        \begin{aligned}
+            \mu_0 \pdvn{2}{\vb{P}^{+}}{t}
+            &= - \nabla \cross \nabla \cross \vb{E}^{+} - \mu_0 \sigma \pdv{\vb{E}^{+}}{t} - \frac{n^2}{c^2} \pdvn{2}{\vb{E}^{+}}{t}
+            \\
+            \pdv{\vb{P}^{+}}{t}
+            &= - \Big( \gamma_\perp + i \omega_0 \Big) \vb{P}^{+}
+            - \frac{i}{\hbar} \Big( \vb{p}_0^{-} \cdot \vb{E}^{+} \Big) \vb{p}_0^{+} D
+            \\
+            \pdv{D}{t}
+            &= \gamma_\parallel (D_0 - D) + \frac{i 2}{\hbar} \Big( \vb{P}^{-} \cdot \vb{E}^{+} - \vb{P}^{+} \cdot \vb{E}^{-} \Big)
+        \end{aligned}
+    }
+\end{aligned}$$
+
+
+
+## References
+1.  F. Kärtner,
+    [Ultrafast optics: lecture notes](https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-977-ultrafast-optics-spring-2005/lecture-notes/),
+    2005, MIT.
+2.  H. Haken,
+    *Light: volume 2: laser light dynamics*,
+    1985, North-Holland.
+3.  H.J. Metcalf, P. van der Straten,
+    *Laser cooling and trapping*,
+    1999, Springer.