--- title: "Maxwell-Bloch equations" firstLetter: "M" publishDate: 2021-10-02 categories: - Physics - Quantum mechanics - Two-level system - Electromagnetism - Laser theory date: 2021-09-09T21:17:52+02:00 draft: false markup: pandoc --- # Maxwell-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 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 $d_0$: $$\begin{aligned} 2 \gamma_g \rho_{gg} - 2 \gamma_e \rho_{ee} = \gamma_\parallel (d_0 - d) \qquad \qquad d_0 \equiv \frac{\gamma_g - \gamma_e}{\gamma_g + \gamma_e} \end{aligned}$$
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: $$\begin{aligned} \nabla \cross \big( \nabla \cross \vb{E} \big) = - \mu_0 \sigma \pdv{\vb{E}}{t} - \mu_0 \varepsilon_0 \pdv[2]{\vb{E}}{t} - \mu_0 \pdv[2]{\vb{P}}{t} \end{aligned}$$ 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: $$\begin{aligned} \mu_0 \pdv[2]{\vb{P}}{t} &= - \nabla \cross \big( \nabla \cross \vb{E} \big) - \mu_0 \sigma \pdv{\vb{E}}{t} - \mu_0 \varepsilon_0 \pdv[2]{\vb{E}}{t} - \mu_0 \pdv[2]{\vb{P}_\mathrm{med}}{t} \\ &= - \nabla \cross \big( \nabla \cross \vb{E} \big) - \mu_0 \sigma \pdv{\vb{E}}{t} - \mu_0 \pdv[2]{t} \Big( \varepsilon_0 \vb{E} + \varepsilon_0 \chi \vb{E} \Big) \\ &= - \nabla \cross \big( \nabla \cross \vb{E} \big) - \mu_0 \sigma \pdv{\vb{E}}{t} - \mu_0 \varepsilon_0 \varepsilon_r \pdv[2]{\vb{E}}{t} \end{aligned}$$ 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: $$\begin{aligned} \boxed{ \mu_0 \pdv[2]{\vb{P}}{t} = - \nabla \cross \big( \nabla \cross \vb{E} \big) - \mu_0 \sigma \pdv{\vb{E}}{t} - \frac{n^2}{c^2} \pdv[2]{\vb{E}}{t} } \end{aligned}$$ $\vb{E}$ 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: $$\begin{aligned} \boxed{ \begin{aligned} \mu_0 \pdv[2]{\vb{P}^{+}}{t} &= - \nabla \cross \nabla \cross \vb{E}^{+} - \mu_0 \sigma \pdv{\vb{E}^{+}}{t} - \frac{n^2}{c^2} \pdv[2]{\vb{E}^{+}}{t} \\ \pdv{\vb{P}^{+}}{t} &= - \Big( \gamma_\perp + i \omega_0 \Big) \vb{P}^{+} - \frac{i}{\hbar} \Big( \vb{p}_0^{-} \cdot \vb{E}^{+} \Big) \vb{p}_0^{+} D \\ \pdv{D}{t} &= \gamma_\parallel (D_0 - D) + \frac{i 2}{\hbar} \Big( \vb{P}^{-} \cdot \vb{E}^{+} - \vb{P}^{+} \cdot \vb{E}^{-} \Big) \end{aligned} } \end{aligned}$$ ## 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.