Categories: Physics, Quantum mechanics, Two-level system.

# Optical Bloch equations

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 introduces a time-dependent sinusoidal term $\hat{H}_1$ to the Hamiltonian. In the 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, and $\hat{\vb{p}}$ is the dipole moment operator. From 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 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 $\hat{\rho}$ 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.

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 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, 2005, Massachusetts Institute of Technology.
2. H.J. Metcalf, P. van der Straten, Laser cooling and trapping, 1999, Springer.