# Maxwell-Bloch equations

For an electron in a two-orbital system $\{\ket{g}, \ket{e}\}$, the Schrödinger equation has the following general solution, where $\varepsilon_g$ and $\varepsilon_e$ are the time-independent eigenenergies, 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} \end{aligned}

This system is being perturbed by an electromagnetic wave with electric field $\vb{E}$ given by:

\begin{aligned} \vb{E}(t) &\equiv \vb{E}^{-}(t) + \vb{E}^{+}(t) \end{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, $\vb{E}_0^{+} \!=\! (\vb{E}_0^{-})^*$.

For $\ket{\Psi}$ as defined above, the pure density operator $\hat{\rho}$ is as follows, with $\omega_0 \equiv (\varepsilon_e \!-\! \varepsilon_g) / \hbar$ being the transition’s resonance frequency:

\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}

Under the electric dipole approximation and rotating wave approximation, it can be shown that $\hat{\rho}$ is governed by the optical Bloch equations:

\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}

Where we have defined the transition dipole moment $\vb{p}_0^{-}$, with $q < 0$ the electron charge:

\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}

However, the light wave affects the electron, so the actual electromagnetic dipole moment $\vb{p}$ is as follows, using Laporte’s selection rule to remove diagonal terms by assuming that the electron’s orbitals are spatially odd or even:

\begin{aligned} \vb{p} &= q \matrixel{\Psi}{\vu{x}}{\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} e^{i \omega_0 t} + c_e c_g^* \matrixel{g}{\vu{x}}{e} e^{-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 our case, we arrive at a prototype of the first of three Maxwell-Bloch equations:

\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 $d \in [-1, 1]$ like so, to quantify the electron’s “excitedness” i.e. its localization to $\ket{e}$:

\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 equilibrium $d_0$:

\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}

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}

With this, the equation for the population inversion $d$ takes the form below, namely the second Maxwell-Bloch equation’s prototype:

\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; we will effectively derive a modified form of the electromagnetic wave equation. Starting from Faraday’s law and splitting $\vb{B} = \mu_0 (\vb{H} + \vb{M})$:

\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 the third Maxwell-Bloch equation’s prototype is:

\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, 2005, Massachusetts Institute of Technology.
2. H. Haken, Light: volume 2: laser light dynamics, 1985, North-Holland.