From dedb366c3a78f61c64f6be627ea091e71e009f7d Mon Sep 17 00:00:00 2001 From: Prefetch Date: Sat, 2 Oct 2021 15:40:20 +0200 Subject: Expand knowledge base --- .../know/concept/maxwell-bloch-equations/index.pdc | 427 +++++++++++++++++++++ 1 file changed, 427 insertions(+) create mode 100644 content/know/concept/maxwell-bloch-equations/index.pdc (limited to 'content/know/concept/maxwell-bloch-equations') diff --git a/content/know/concept/maxwell-bloch-equations/index.pdc b/content/know/concept/maxwell-bloch-equations/index.pdc new file mode 100644 index 0000000..3a0df1b --- /dev/null +++ b/content/know/concept/maxwell-bloch-equations/index.pdc @@ -0,0 +1,427 @@ +--- +title: "Maxwell-Bloch equations" +firstLetter: "M" +publishDate: 2021-10-02 +categories: +- Physics +- Quantum mechanics +- Electromagnetism + +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: + +$$\begin{aligned} + \hat{H}_1(t) + = - \hat{\vb{p}} \cdot \vb{E}(t) + \qquad \quad + \hat{\vb{p}} + \equiv q \hat{\vb{x}} + \qquad \quad + \vb{E}(t) + = \vb{E}_0 \cos\!(\omega t) +\end{aligned}$$ + +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: + +$$\begin{aligned} + \dv{c_g}{t} + &= i \frac{q \matrixel{g}{\hat{\vb{x}}}{e} \cdot \vb{E}_0}{2 \hbar} \exp\!\big( i \omega t \!-\! i \omega_0 t \big) \: c_e + \\ + \dv{c_e}{t} + &= i \frac{q \matrixel{e}{\hat{\vb{x}}}{g} \cdot \vb{E}_0}{2 \hbar} \exp\!\big(\!-\! i \omega t \!+\! i \omega_0 t \big) \: c_g +\end{aligned}$$ + +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 $\vb{p}_0^{-}$: + +$$\begin{aligned} + \vb{p}_0^{-} + \equiv q \matrixel{e}{\vb{x}}{g} + \qquad \quad + \vb{p}_0^{+} + \equiv (\vb{p}_0^{-})^* + = q \matrixel{g}{\vb{x}}{e} +\end{aligned}$$ + +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: + +$$\begin{aligned} + \dv{c_g}{t} + &= \frac{i}{\hbar} \vb{p}_0^{+} \cdot \vb{E}^{-} \exp\!(- i \omega_0 t) \: c_e + \\ + \dv{c_e}{t} + &= \frac{i}{\hbar} \vb{p}_0^{-} \cdot \vb{E}^{+} \exp\!(i \omega_0 t) \: c_g +\end{aligned}$$ + + +## 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 $\gamma_\perp$: + +$$\begin{aligned} + \Big( \dv{\rho_{eg}}{t} \Big)_{\perp} + = - \gamma_\perp \rho_{eg} + \qquad \quad + \Big( \dv{\rho_{ge}}{t} \Big)_{\perp} + = - \gamma_\perp \rho_{ge} +\end{aligned}$$ + +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: + +$$\begin{aligned} + \vb{p} + &= \matrixel{\Psi}{\hat{\vb{p}}}{\Psi} + \\ + &= q \Big( c_g c_g^* \matrixel{g}{\hat{\vb{x}}}{g} + c_e c_e^* \matrixel{e}{\hat{\vb{x}}}{e} + + c_g c_e^* \matrixel{e}{\hat{\vb{x}}}{g} \exp\!(i \omega_0 t) + c_e c_g^* \matrixel{g}{\hat{\vb{x}}}{e} \exp\!(-i \omega_0 t) \Big) + \\ + &= q \Big( \rho_{ge} \matrixel{e}{\hat{\vb{x}}}{g} + \rho_{eg} \matrixel{g}{\hat{\vb{x}}}{e} \Big) + = \vb{p}_0^{-} \rho_{ge}(t) + \vb{p}_0^{+} \rho_{eg}(t) + \equiv \vb{p}^{-}(t) + \vb{p}^{+}(t) +\end{aligned}$$ + +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 $d_0$: + +$$\begin{aligned} + 2 \gamma_g \rho_{gg} - 2 \gamma_e \rho_{ee} + = \gamma_\parallel (d_0 - d) + \qquad \quad + d_0 + \equiv \frac{\gamma_g - \gamma_e}{\gamma_g + \gamma_e} +\end{aligned}$$ + +
+ + + +
+ +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 medium's conductivity, if any; +many authors assume $\sigma = 0$. +Next, we rewrite the left side using a vector identity, +and assume no net charge $\nabla \cdot \vb{E} = 0$: + +$$\begin{aligned} + \nabla^2 \vb{E} - \nabla \big( \nabla \cdot \vb{E} \big) + = \nabla^2 \vb{E} + = \mu_0 \sigma \pdv{\vb{E}}{t} + \mu_0 \varepsilon_0 \pdv[2]{\vb{E}}{t} + \mu_0 \pdv[2]{\vb{P}}{t} +\end{aligned}$$ + +After some rearranging, +we arrive at a variant of the electromagnetic wave equation: + +$$\begin{aligned} + \nabla^2 \vb{E} - \mu_0 \sigma \pdv{\vb{E}}{t} - \mu_0 \varepsilon_0 \pdv[2]{\vb{E}}{t} + &= \mu_0 \pdv[2]{\vb{P}}{t} +\end{aligned}$$ + +It is trivial to show that $\vb{E}$ and $\vb{P}$ +can be replaced by $\vb{E}^{+}$ and $\vb{P}^{+}$. +It is equally trivial to convert +the dipole $\vb{p}^{+}$ and inversion $d$ +into their macroscopic versions $\vb{P}^{+}$ and $D$, +simply by summing over all atoms in the medium. +We thus arrive at the **Maxwell-Bloch equations**: + +$$\begin{aligned} + \boxed{ + \begin{aligned} + \mu_0 \pdv[2]{\vb{P}^{+}}{t} + &= \nabla^2 \vb{E}^{+} - \mu_0 \sigma \pdv{\vb{E}^{+}}{t} - \mu_0 \varepsilon_0 \pdv[2]{\vb{E}^{+}}{t} + \\ + \pdv{\vb{P}^{+}}{t} + &= - \Big( \gamma_\perp + i \omega_0 \Big) \vb{P}^{+} + - \frac{i}{\hbar} \Big( \vb{p}_0^{-} \cdot \vb{E}^{+} \Big) \vb{p}_0^{+} D + \\ + \pdv{D}{t} + &= \gamma_\parallel (D_0 - D) + \frac{i 2}{\hbar} \Big( \vb{P}^{-} \cdot \vb{E}^{+} - \vb{P}^{+} \cdot \vb{E}^{-} \Big) + \end{aligned} + } +\end{aligned}$$ + + + +## 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. -- cgit v1.2.3