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Diffstat (limited to 'content/know/concept/maxwell-bloch-equations')
| -rw-r--r-- | content/know/concept/maxwell-bloch-equations/index.pdc | 66 |
1 files changed, 48 insertions, 18 deletions
diff --git a/content/know/concept/maxwell-bloch-equations/index.pdc b/content/know/concept/maxwell-bloch-equations/index.pdc index e3a3680..2e0cdd9 100644 --- a/content/know/concept/maxwell-bloch-equations/index.pdc +++ b/content/know/concept/maxwell-bloch-equations/index.pdc @@ -7,6 +7,7 @@ categories: - Quantum mechanics - Two-level system - Electromagnetism +- Laser theory date: 2021-09-09T21:17:52+02:00 draft: false @@ -34,10 +35,10 @@ $\hat{H}_1$ is given by: $$\begin{aligned} \hat{H}_1(t) = - \hat{\vb{p}} \cdot \vb{E}(t) - \qquad \quad + \qquad \qquad \vu{p} \equiv q \vu{x} - \qquad \quad + \qquad \qquad \vb{E}(t) = \vb{E}_0 \cos\!(\omega t) \end{aligned}$$ @@ -72,7 +73,7 @@ Similarly, we define the transition dipole moment $\vb{p}_0^{-}$: $$\begin{aligned} \vb{p}_0^{-} \equiv q \matrixel{e}{\vu{x}}{g} - \qquad \quad + \qquad \qquad \vb{p}_0^{+} \equiv (\vb{p}_0^{-})^* = q \matrixel{g}{\vu{x}}{e} @@ -194,7 +195,7 @@ both decay with rate $\gamma_\perp$: $$\begin{aligned} \Big( \dv{\rho_{eg}}{t} \Big)_{\perp} = - \gamma_\perp \rho_{eg} - \qquad \quad + \qquad \qquad \Big( \dv{\rho_{ge}}{t} \Big)_{\perp} = - \gamma_\perp \rho_{ge} \end{aligned}$$ @@ -295,7 +296,7 @@ towards an equilbrium $d_0$: $$\begin{aligned} 2 \gamma_g \rho_{gg} - 2 \gamma_e \rho_{ee} = \gamma_\parallel (d_0 - d) - \qquad \quad + \qquad \qquad d_0 \equiv \frac{\gamma_g - \gamma_e}{\gamma_g + \gamma_e} \end{aligned}$$ @@ -367,37 +368,66 @@ Inserting the definition $\vb{D} = \varepsilon_0 \vb{E} + \vb{P}$ together with Ohm's law $\vb{J}_\mathrm{free} = \sigma \vb{E}$ yields: $$\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{ - \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} + \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}$$ -Where $\sigma$ is the medium's conductivity, if any; -many authors assume $\sigma = 0$. -It is trivial to show that $\vb{E}$ and $\vb{P}$ -can be replaced by $\vb{E}^{+}$ and $\vb{P}^{+}$. - +$\vb{E}$ and $\vb{P}$ can trivially be replaced by $\vb{E}^{+}$ and $\vb{P}^{+}$. It is also simple to convert $\vb{p}^{+}$ and $d$ -into the macroscopic polarization $\vb{P}^{+}$ and total inversion $D$ -by summing over the atoms: +into the macroscopic $\vb{P}^{+}$ and total $D$ +by summing over all two-level systems in the medium: $$\begin{aligned} \vb{P}^{+}(\vb{x}, t) - &= \sum_{n} \vb{p}^{+}_n \: \delta(\vb{x} - \vb{x}_n) + &= \sum_{\nu} \vb{p}^{+}_\nu \: \delta(\vb{x} - \vb{x}_\nu) \\ D(\vb{x}, t) - &= \sum_{n} d_n \: \delta(\vb{x} - \vb{x}_n) + &= \sum_{\nu} d_\nu \: \delta(\vb{x} - \vb{x}_\nu) \end{aligned}$$ We thus arrive at the **Maxwell-Bloch equations**, -which are relevant for laser theory: +which are the foundation of laser theory: $$\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} - \mu_0 \varepsilon_0 \pdv[2]{\vb{E}^{+}}{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}^{+} |