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

---
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 $$\hat{\rho}$$,
which are the basis of the
[Maxwell-Bloch equations](/know/concept/maxwell-bloch-equations/)
and by extension all laser theory:

$$\begin{aligned}
    \boxed{
        \begin{aligned}
            \dv{\rho_{gg}}{t}
            &= \gamma_e \rho_{ee} - \gamma_g \rho_{gg}
            + \frac{i}{\hbar} \Big( \vb{p}_0^{+} \cdot \vb{E}^{-} \rho_{eg} - \vb{p}_0^{-} \cdot \vb{E}^{+} \rho_{ge} \Big)
            \\
            \dv{\rho_{ee}}{t}
            &=  \gamma_g \rho_{gg} - \gamma_e \rho_{ee}
            + \frac{i}{\hbar} \Big( \vb{p}_0^{-} \cdot \vb{E}^{+} \rho_{ge} - \vb{p}_0^{+} \cdot \vb{E}^{-} \rho_{eg} \Big)
            \\
            \dv{\rho_{ge}}{t}
            &= - \Big( \gamma_\perp - i \omega_0 \Big) \rho_{ge}
            + \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} \Big( \rho_{ee} - \rho_{gg} \Big)
            \\
            \dv{\rho_{eg}}{t}
            &= - \Big( \gamma_\perp + i \omega_0 \Big) \rho_{eg}
            + \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} \Big( \rho_{gg} - \rho_{ee} \Big)
        \end{aligned}
    }
\end{aligned}$$

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.