diff options
Diffstat (limited to 'content/know/concept/maxwell-bloch-equations')
| -rw-r--r-- | content/know/concept/maxwell-bloch-equations/index.pdc | 60 |
1 files changed, 23 insertions, 37 deletions
diff --git a/content/know/concept/maxwell-bloch-equations/index.pdc b/content/know/concept/maxwell-bloch-equations/index.pdc index 020a120..3f090a2 100644 --- a/content/know/concept/maxwell-bloch-equations/index.pdc +++ b/content/know/concept/maxwell-bloch-equations/index.pdc @@ -34,8 +34,8 @@ $$\begin{aligned} \hat{H}_1(t) = - \hat{\vb{p}} \cdot \vb{E}(t) \qquad \quad - \hat{\vb{p}} - \equiv q \hat{\vb{x}} + \vu{p} + \equiv q \vu{x} \qquad \quad \vb{E}(t) = \vb{E}_0 \cos\!(\omega t) @@ -49,10 +49,10 @@ can then be described by: $$\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 + &= 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}{\hat{\vb{x}}}{g} \cdot \vb{E}_0}{2 \hbar} \exp\!\big(\!-\! i \omega t \!+\! i \omega_0 t \big) \: c_g + &= 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}$$ We want to rearrange these equations a bit. @@ -70,11 +70,11 @@ Similarly, we define the transition dipole moment $\vb{p}_0^{-}$: $$\begin{aligned} \vb{p}_0^{-} - \equiv q \matrixel{e}{\vb{x}}{g} + \equiv q \matrixel{e}{\vu{x}}{g} \qquad \quad \vb{p}_0^{+} \equiv (\vb{p}_0^{-})^* - = q \matrixel{g}{\vb{x}}{e} + = q \matrixel{g}{\vu{x}}{e} \end{aligned}$$ With these, the equations for $c_g$ and $c_e$ can be rewritten as shown below. @@ -238,10 +238,10 @@ $$\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( 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}{\hat{\vb{x}}}{g} + \rho_{eg} \matrixel{g}{\hat{\vb{x}}}{e} \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}$$ @@ -366,41 +366,27 @@ Inserting the definition $\vb{D} = \varepsilon_0 \vb{E} + \vb{P}$ together with Ohm's law $\vb{J}_\mathrm{free} = \sigma \vb{E}$ yields: $$\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} + \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} + } \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}$$ - It is trivial to show that $\vb{E}$ and $\vb{P}$ can be replaced by $\vb{E}^{+}$ and $\vb{P}^{+}$. -It is also simple to convert -the dipole $\vb{p}^{+}$ and inversion $d$ -into their macroscopic versions $\vb{P}^{+}$ and $D$: + +It is also simple to convert $\vb{p}^{+}$ and $d$ +into the macroscopic polarization $\vb{P}^{+}$ and total inversion $D$ +by summing over the atoms: $$\begin{aligned} - \vb{P}^{+}(\vb{r}, t) - = \sum_{n} \vb{p}^{+}_n \: \delta(\vb{r} \!-\! \vb{r}_n) - \qquad \quad - D(\vb{r}, t) - = \sum_{n} d_n \: \delta(\vb{r} \!-\! \vb{r}_n) + \vb{P}^{+}(\vb{x}, t) + &= \sum_{n} \vb{p}^{+}_n \: \delta(\vb{x} - \vb{x}_n) + \\ + D(\vb{x}, t) + &= \sum_{n} d_n \: \delta(\vb{x} - \vb{x}_n) \end{aligned}$$ We thus arrive at the **Maxwell-Bloch equations**, @@ -410,7 +396,7 @@ $$\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} + &= - \nabla \cross \nabla \cross \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}^{+} |