In our previous post we sketched a procedure to derive the kinetics and dynamics of electrons in a solid, taking into account the conservation of momentum. Here, we provide examples and how to deduce complex quantities.
Quantum Ring + TL
1. Write the Hamiltonian in second quantization form:
2. Use the Heisenberg's equations of motion, to set up equations + 3. Convert the operator equations into c-number equations for the so-called coherences and populations:
\begin{eqnarray}
i\hbar \frac{d}{dt}{\rho}_{v, m' \,m}
&=& (\varepsilon_{v m } - \varepsilon_{v m' })
\, {\rho}_{v, m' \,m} +
\xi^*\,
\widetilde{\rho}_{v m', c m+l} -
\xi\,
\widetilde{\rho}_{c m'+l, v m},
\label{eq:rho_v_TL} \\
i\hbar \frac{d}{dt} {\rho}_{c, m' \,m}
&=& (\varepsilon_{c m } - \varepsilon_{c m' })
\, {\rho}_{c, m'\, m} +
\xi\,
\widetilde{\rho}_{c m', v m-l} -
\xi^*\,
\widetilde{\rho}_{v m'-l, c m},
\label{eq:rho_c_TL} \\
i\hbar \frac{d}{dt}\widetilde{\rho}_{v m', c m}
&=& (\varepsilon_{c m } - \varepsilon_{v m' } - \hbar \omega)
\, \widetilde{\rho}_{v m', c m} +
\xi\,
\left(
{\rho}_{v, m' \,m-l} - {\rho}_{c, m'+l\, m}
\right)
\,,
\label{eq:rho_vc_TL}
\end{eqnarray}
4. Derive complex quantities: The orbital angular momentum of the whole system is divided into a coherence and population contributions
\begin{eqnarray*}
{L}_z^{(coh)} &=&
\sum_{m m^\prime }
2\Re\left[
{\cal L}^{(p)}_{vm' cm}
\, \rho_{v m^\prime, c m}
\right] \\
{L}_z^{(pop)}
&=& \sum_{m }
\hbar\,m
\, \rho_{c, m m} +
\sum_{m }
\left(
l_{z} + \hbar \,m
\right)
\, \rho_{v, m m}
\,,
\end{eqnarray*}
where one uses the solutions of the EOM [steps 2 and 3] to obtain explicit expressions for the OAM. [1]
Bulk + TL
1. Write the Hamiltonian in second quantization form: the Hamiltonian looks the same as before, but with matrix elements\begin{eqnarray} \label{eq:Matrix_Element}
\langle{c \mathbf k' m'} |h_I^{(+)} |{v \mathbf k m}\rangle
&=& \xi_{c k_r' m', v k_r m} \, e^{-i\omega t}\, \delta_{l, m' - m } \delta_{q_z, k_z' - k_z } \nonumber \\
\langle{v \mathbf k' m'} |h_I^{(-)} |{c \mathbf k m}\rangle
&=& \xi_{c k_r m, v k_r' m'}^* \, e^{i\omega t}\, \delta_{-l, m' - m } \delta_{-q_z, k_z' - k_z }
\,,
\end{eqnarray}
we note that we used cylindrical electron's states, and this is why there is a $\delta_{\pm l, m' - m }$ on angular quantum numbers $m$.
2. Use the Heisenberg's equations of motion, to set up equations + 3. Convert the operator equations into c-number equations for the so-called coherences and populations using
\begin{eqnarray}
\rho_{c, \alpha' \alpha} =
\langle \hat{\rho}_{c \alpha', c \alpha} \rangle, \nonumber \\
\rho_{v, \alpha' \alpha} =
\langle \hat{\rho}_{v \alpha', v \alpha} \rangle, \nonumber \\
\rho_{v \alpha', c \alpha} =
\langle \hat{\rho}_{v \alpha', c \alpha} \rangle \,. \end{eqnarray}
For example:
\begin{eqnarray}
i\hbar \frac{d}{dt} \rho_{v k_z k_r' m, \, k_z k_r m}
&=&
\Delta_{v k_z k_r m, \, k_z k_r' m}
\, {\rho}_{v k_z k_r' m, \, k_z k_r m} + \nonumber \\
&& \hspace{-25mm}
e^{i\omega t} \sum_{k_r''} \,
\xi_{c k_r'' m+l, \, v k_r m}^* \,
{\rho}_{v k_z k_r' m, \, c k_z+q_z k_r'' m+l} -
e^{-i\omega t} \sum_{k_r''}
\xi_{c k_r'' m+l, \, v k_r m}\,
{\rho}_{c k_z+q_z k_r'' m+l, \, v k_z k_r m} \, .
\label{eq:TL-SBE_v}
\end{eqnarray}
We note that, because in these examples there are no 4 operators' terms, it is trivial to go from step 2 to 3, and so we have put together both steps. For a situation with 4 operators see Ref. [2].
4. Derive complex quantities: The paramagnetic electric current is \begin{eqnarray} \hat{\mathbf j}^{(p)}(\mathbf r,t) &=& -i\,\frac{q\,\hbar}{2\,m_e} \sum_{\scriptsize \begin{array}{c} b' \mathbf k' m' \\ b \mathbf k m \end{array} } \left[ \psi_{b' \mathbf k' m'}^*(\mathbf r) \, \nabla \psi_{b \mathbf k m}(\mathbf r) - \psi_{b \mathbf k m}(\mathbf r) \, \nabla \psi_{b' \mathbf k' m'}^*(\mathbf r) \right] \, {\rho}_{b' \mathbf k' m',b \mathbf k m}^\dagger (t)\,, \label{eq:jparamag} \end{eqnarray}
See Ref. [3]
REFERENCES
[1] Quinteiro, G. F., and J. Berakdar. Electric currents induced by twisted light in Quantum Rings, Optics express 17, no. 22 (2009): 20465-20475.
[2] G. F. Quinteiro, Below-bandgap excitation of bulk semiconductors by twisted light, EPL 91 27002 (2010).
[3] G. F. Quinteiro and P. I. Tamborenea, Twisted-light-induced optical transitions in semiconductors: Free-carrier quantum kinetics, Phys. Rev. B 82, 125207 (2010).
For example:
\begin{eqnarray}
i\hbar \frac{d}{dt} \rho_{v k_z k_r' m, \, k_z k_r m}
&=&
\Delta_{v k_z k_r m, \, k_z k_r' m}
\, {\rho}_{v k_z k_r' m, \, k_z k_r m} + \nonumber \\
&& \hspace{-25mm}
e^{i\omega t} \sum_{k_r''} \,
\xi_{c k_r'' m+l, \, v k_r m}^* \,
{\rho}_{v k_z k_r' m, \, c k_z+q_z k_r'' m+l} -
e^{-i\omega t} \sum_{k_r''}
\xi_{c k_r'' m+l, \, v k_r m}\,
{\rho}_{c k_z+q_z k_r'' m+l, \, v k_z k_r m} \, .
\label{eq:TL-SBE_v}
\end{eqnarray}
We note that, because in these examples there are no 4 operators' terms, it is trivial to go from step 2 to 3, and so we have put together both steps. For a situation with 4 operators see Ref. [2].
4. Derive complex quantities: The paramagnetic electric current is \begin{eqnarray} \hat{\mathbf j}^{(p)}(\mathbf r,t) &=& -i\,\frac{q\,\hbar}{2\,m_e} \sum_{\scriptsize \begin{array}{c} b' \mathbf k' m' \\ b \mathbf k m \end{array} } \left[ \psi_{b' \mathbf k' m'}^*(\mathbf r) \, \nabla \psi_{b \mathbf k m}(\mathbf r) - \psi_{b \mathbf k m}(\mathbf r) \, \nabla \psi_{b' \mathbf k' m'}^*(\mathbf r) \right] \, {\rho}_{b' \mathbf k' m',b \mathbf k m}^\dagger (t)\,, \label{eq:jparamag} \end{eqnarray}
See Ref. [3]
REFERENCES
[1] Quinteiro, G. F., and J. Berakdar. Electric currents induced by twisted light in Quantum Rings, Optics express 17, no. 22 (2009): 20465-20475.
[2] G. F. Quinteiro, Below-bandgap excitation of bulk semiconductors by twisted light, EPL 91 27002 (2010).
[3] G. F. Quinteiro and P. I. Tamborenea, Twisted-light-induced optical transitions in semiconductors: Free-carrier quantum kinetics, Phys. Rev. B 82, 125207 (2010).
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