physics/Optical-transitions.md

@@ -12,12 +12,12 @@ There are two variants for this ansatz common in literature.
 
 ### Variant 1: Atomistic approach
 
-We derive the perturbation Hamiltonian by considering the EM wave as an additional potential for the atom core and the electron, such that $`H = H_0 +H_1`$ with
+We derive the perturbation Hamiltonian by considering the EM wave as an additional potential for the atom core and the electron, such that $`H = H_0 +H_\text{EM}`$ with
 
 ```math
 \begin{equation}
 \begin{split}
-H_1 &= - e \phi(\vec{r}_e) + q \phi(\vec{r}_c) \\
+H_\text{EM} &= - e \phi(\vec{r}_e) + q \phi(\vec{r}_c) \\
     &= e\left[\phi(\vec{r}_c) - \phi(\vec{r}_e)\right] + (q-e)\phi(\vec{r}_c)
 \end{split}
 \end{equation}
@@ -30,19 +30,19 @@ For the first term we use a Taylor series up to first order in the electron-core
 ```math
 \begin{equation}
 \begin{split}
-H_1 &= - e \vec{r} \cdot \vec{\nabla}\phi(\vec{r}_c) \\
+H_\text{EM} &= - e \vec{r} \cdot \vec{\nabla}\phi(\vec{r}_c) \\
     &= - \vec{\mu}_e \cdot \vec{E}(\vec{r}_c)
 \end{split}
 \end{equation}
 ```
-where $`\vec{\mu}_e`$ is an effective electric dipole moment.
+where $`\vec{\mu}_e`$ is an effective electric dipole moment and $`\vec{E}(t) = \vec{E} \cos(\omega t)`$.
 
-Since the commutator $`[H_0,H_1]`$ does not vanish, both Hamiltonians do not have the same eigenstates and $`H_1`$ will introduce transitions between our known eigenstates of $`H_0`$. The strength of such a transition is given by the transition matrix element
+Since the commutator $`[H_0,H_\text{EM}]`$ does not vanish, both Hamiltonians do not have the same eigenstates and $`H_\text{EM}`$ will introduce transitions between our known eigenstates of $`H_0`$. The strength of such a transition is given by the transition matrix element
 
 ```math
 \begin{equation}
 \begin{split}
-v_{fi} &= \langle \psi_f | H_1 | \psi_i \rangle \\
+v_{fi} &= \langle \psi_f | H_\text{EM} | \psi_i \rangle \\
        &= e \vec{E} \langle \psi_f | \vec{r} | \psi_i \rangle \\
        &= \frac{-i \hbar e \vec{E}}{(E_f - E_i)m} \langle \psi_f | \vec{p} | \psi_i \rangle
 \end{split}
@@ -64,7 +64,7 @@ H &= \frac{(\vec{p} +e \vec{A})^2}{2m_0} +V_c\\
 \end{split}
 \end{equation}
 ```
-where we have split the vector potential $`A`$ into the static part $`A_0`$ of the electric and magnetic fields applied to the `kdotpy` state calculation and $`\vec{A}_\text{EM} = \frac{\vec{E}}{\omega} \cos(\omega t - \vec{q}\cdot\vec{r})`$ for the EM wave. We used the identity $`\vec{p} \cdot \vec{A} = \vec{A} \cdot \vec{p}`$ valid for divergence free fields of EM waves and negelected the energy offset terms $`\vec{A}^2_\text{EM} + 2 \vec{A}_\text{0}\cdot\vec{A}_\text{EM}`$. The latter term is by the way the same term missing in the $`\vec{r} \leftrightarrow\vec{p}`$ relation above in [Variant 1](#variant-1-atomistic-approach).
+where we have split the vector potential $`A`$ into the static part $`A_0`$ of the electric and magnetic fields applied to the `kdotpy` state calculation and $`\vec{A}_\text{EM} = \frac{\vec{E}}{\omega} \cos(\omega t - \vec{q}\cdot\vec{r})`$ for the EM wave. We used the identity $`\vec{p} \cdot \vec{A} = \vec{A} \cdot \vec{p}`$ valid for divergence free fields of EM waves and negelected the energy offset terms $`\vec{A}^2_\text{EM} + 2 \vec{A}_\text{0}\cdot\vec{A}_\text{EM}`$. The latter term is by the way the same term missing in the $`\vec{r} \leftrightarrow\vec{p}`$ relation above in [variant 1](#variant-1-atomistic-approach).
 
 Using the dipole approximation ($`\lambda \gg r`$ or $`\vec{q} \to 0`$) and Euler's formula we can rewrite 
 
@@ -73,11 +73,72 @@ Using the dipole approximation ($`\lambda \gg r`$ or $`\vec{q} \to 0`$) and Eule
 \vec{A}_\text{EM} = \frac{\vec{E}}{2\omega} \left(e^{-i\omega t} + \text{c.c.}\right)
 \end{equation}
 ```
-The time varying phase factor and its complex conjugate can be identified as the two terms responsible for absorption and stimulated emission (see Fermi's Golden Rule).
+As in variant 1 we can derive the transition matrix element $`v_{fi}`$.
+The time varying phase factor and its complex conjugate can be identified as the two terms responsible for absorption and stimulated emission (see Fermi's Golden Rule). To fullfill energy conservation, we identify §`\hbar \omega = E_f - E_i`§
 
 ## Matrix elements derived from $`k \cdot p`$ Hamiltonian
 
-TBF: Use p := [r,H] := dH/dk
+Both variants above yield the transition matrix element
+
+```math
+\begin{equation}
+\begin{split}
+v_{fi} &= \langle \psi_f | H_\text{EM} | \psi_i \rangle \\
+       &= \frac{ e \vec{E}}{2m\omega} \langle \psi_f | \vec{p} | \psi_i \rangle
+\end{split}
+\end{equation}
+```
+up to a global phase factor ($`-i`$), which can be neglected, as we will only deal with absolute squares $`|v_{fi}|^2`$ later on.
+
+Let's split the components 
+
+```math
+\begin{equation}
+\begin{split}
+\vec{E}\cdot\vec{p} &= E (\textbf{e}_x p_x +\textbf{e}_y p_y +\textbf{e}_z p_z) \\
+&= \frac{E}{\sqrt{2}}(\textbf{e}_x p_x +\textbf{e}_y p_y) + E\textbf{e}_z p_z
+\end{split}
+\end{equation}
+```
+with linear and circular polarization unit vectors $`\textbf{e}_i`$, $`\textbf{e}_\pm = \frac{1}{\sqrt{2}} (\textbf{e}_x \pm i\textbf{e}_y)`$ and $`p_\pm = p_x \pm ip_y`$.
+
+A typical experimental setup of magneto-optical investigations of layered sample structures is the Faraday configuration, where the EM wave is incident onto the sample parallel to a static magnetic field vector. These two vectors are usually oriented perpendicular to the sample surface, i.e. along the layer growth direction ($`z`$). The $`\textbf{e}_z`$ component of an EM wave traveling in $`z`$ direction is zero and we only need to consider the transition matrices
+
+```math
+\begin{equation}
+\begin{split}
+v_x \pm i v_y = v_\pm = \frac{ e E}{2\sqrt{2}m\omega} p_\pm
+\end{split}
+\end{equation}
+```
+describing the interaction of the system with left and right circular polarized EM waves. We will see later on, that the circular basis is the most convenient polarization basis choice for our system.
+
+We can derive the matrix elements $`v_{fi}`$ of the $`v`$ matrices by using the commutator relation $`[x,H_0] = \frac{i\hbar}{m} p_x `$ (and analogeously for other spatial coordinates) again. For any Hamiltonian the commutator acts like a derivation with respect to the momentum in the same spatial direction. This follows from the $`[x_i,p_j] \propto \delta_{ij}`$ commutator relations and the product rule for the commutator of operator products.
+
+```math
+\begin{equation}
+\begin{split}
+[x,H_0] = \frac{\text{d}H_0}{\text{d}p_x}
+\end{split}
+\end{equation}
+```
+
+This means that we can get $`v_{x,fi}`$ directly from the matrix reprentation of the $`k \cdot p`$ matrix $`H_{0,fi}`$ by taking the derivation to the respective $`k_x`$ momentum. Note that in the 2D periodic case only $`p_z`$ is an operator (central finite difference) in the Hamiltonian matrix, while $`p_{x,y}`$ just act on the plane wave parts $`e^{i(k_x x + k_y y)}`$ of the wave function and are replaced by $`k_{x,y}`$ momentum values.
+
+Therefore, we get
+
+```math
+\begin{equation}
+\begin{split}
+v_{x,fi} &= \frac{\text{d}H_{0,fi}}{\text{d}k_x}\\
+v_{\pm,fi} &= 2 \frac{\text{d}H_{0,fi}}{\text{d}k_\mp} = \frac{ e E}{\sqrt{2}m\omega_{fi}} \frac{\text{d}H_{0,fi}}{\text{d}k_\mp}
+\end{split}
+\end{equation}
+```
+where we have used $`\frac{\text{d}}{\text{d}k_\pm} = \frac{\text{d} k_x}{\text{d}k_\pm} \frac{\text{d}}{\text{d}k_x} + \frac{\text{d} k_y}{\text{d}k_\pm} \frac{\text{d}}{\text{d}k_y} = \frac{1}{2}\left(\frac{\text{d}}{\text{d}k_x} \mp i\frac{\text{d}}{\text{d}k_y} \right)`$.
+
+As we already have a symbolic Hamiltonian defintion in `kdotpy`, we can use the method `deriv(to)` to construct suitable symbolic transition matrices which just need to be evaluated for certain Landau Levels (when using `kdotpy-ll`).
+
 
 ## Spectra