path: root/source/know/concept/maxwell-bloch-equations/index.md

---
title: "Maxwell-Bloch equations"
sort_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-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 $$t$$:

$$\begin{aligned}
    \ket{\Psi(t)}
    &= c_g(t) \ket{g} e^{-i \varepsilon_g t / \hbar} + c_e(t) \ket{e} e^{-i \varepsilon_e t / \hbar}
\end{aligned}$$

This system is being perturbed by an electromagnetic wave
with [electric field](/know/concept/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](/know/concept/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](/know/concept/electric-dipole-approximation/)
and [rotating wave approximation](/know/concept/rotating-wave-approximation/),
it can be shown that $$\hat{\rho}$$ is governed by
the [optical Bloch equations](/know/concept/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 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:

$$\begin{aligned}
    \vb{p}
    &= q \matrixel{\Psi}{\vu{x}}{\Psi}
    \\
    &= q \Big( c_g c_g^* \matrixel{g}{\vu{x}}{g} + c_e c_e^* \matrixel{e}{\vu{x}}{e}
    + c_g c_e^* \matrixel{e}{\vu{x}}{g} e^{i \omega_0 t} + c_e c_g^* \matrixel{g}{\vu{x}}{e} e^{-i \omega_0 t} \Big)
    \\
    &= q \Big( \rho_{ge} \matrixel{e}{\vu{x}}{g} + \rho_{eg} \matrixel{g}{\vu{x}}{e} \Big)
    \\
    &= \vb{p}_0^{-} \rho_{ge} + \vb{p}_0^{+} \rho_{eg}
    \\
    &\equiv \vb{p}^{-} + \vb{p}^{+}
\end{aligned}$$

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 $$d_0$$:

$$\begin{aligned}
    2 \gamma_g \rho_{gg} - 2 \gamma_e \rho_{ee}
    = \gamma_\parallel (d_0 - d)
    \qquad \qquad
    d_0
    \equiv \frac{\gamma_g - \gamma_e}{\gamma_g + \gamma_e}
\end{aligned}$$


{% include proof/start.html id="proof-inversion-decay" -%}
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:

$$\begin{aligned}
    2 \gamma_g \rho_{gg} - 2 \gamma_e \rho_{ee}
    &= \gamma_g - \gamma_e - (\gamma_g + \gamma_e) d
    \\
    &= (\gamma_g + \gamma_e) \Big( \frac{\gamma_g - \gamma_e}{\gamma_g + \gamma_e} - d \Big)
    \\
    &= \gamma_\parallel ( d_0 - d )
\end{aligned}$$
{% include proof/end.html id="proof-inversion-decay" %}


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](/know/concept/maxwells-equations/);
we will effectively derive a modified form of
the [electromagnetic wave equation](/know/concept/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](https://ocw.mit.edu/courses/6-977-ultrafast-optics-spring-2005/pages/lecture-notes/),
    2005, Massachusetts Institute of Technology.
2.  H. Haken,
    *Light: volume 2: laser light dynamics*,
    1985, North-Holland.