Spin-orbit interaction in quantum rings

So far we have seen that twisted light affects only the orbital motion of electrons. It would be interesting and useful to convert the light's orbital angular momentum to electron's spin. Here we explore the optical excitation by TL for quantum rings with Rashba spin-orbit interaction.

Unlike our previous posts, we will consider the irradiation by a THz field of twisted light (TL). Because of its energy, the lateral size of the THz beam -even under strong focusing- is large; thus, we have to consider the interaction with large-radius semiconductor QRs, in order to maximize the intensity of the beam on the system. The THz field causes intraband transitions in electrons living in the conduction band, electrons that could be placed there by for example modulation doping or interband optical excitation. 

The Hamiltonian of the QR with Rashba spin-orbit interaction ($\large \propto \sigma \times \mathbf p$ with $\large \sigma$ the vector of Pauli matrices and $\large \mathbf p$ the momentum) can be diagonalized analytically, and leads to eigenfunctions of the total angular momentum $n$. By the way, these eigenfunctions are quite interesting: For example 

\begin{equation}
\large
\psi_\uparrow (\varphi)
=
\frac{1}{2 \pi} e^{i(n+1/2)\varphi}
\nu_\uparrow (\varphi)
\end{equation}
because the spinor
\begin{equation}
\large
\nu_\uparrow (\varphi) =  \begin{pmatrix}
  \cos(\gamma/2) e^{-i \varphi/2} \\
  \sin(\gamma/2) e^{i \varphi/2}
 \end{pmatrix}
\end{equation}
depends on the position (polar coordinate $\varphi$) on the QR -compare with no SO-interaction wavefunctions.

The light perturbs the states, and we can see the change in wavefunction using time-dependent perturbation theory. Let's study what happens to a single electron in a conduction band state $\large |{n_0, \uparrow}\rangle$ when we shine TL with $\large \ell = 1$ and right circular polarization ($\large \sigma=1$). See Fig. 1.

Figure 1: An initial state $\large |{n_0, \uparrow}\rangle$ evolves into a superposition state of up and down electron states.

The final state is a superposition of states differing in their total angular momentum. There is spin-flip (transitions 1 and 2), however it is not related to the fact that the light carries orbital angular momentum; it is the polarization of light that couples to the spin (this is well explained in Ref 1, see specially Eq. 18).

We have found another system (see the post by Nathan Clayburn) where there is no transfer of the light's orbital AM to the spin state of electrons.

References

[1] G. F. Quinteiro, P. I. Tamborenea, and J. Berakdar, Orbital and spin dynamics of intraband electrons in quantum rings driven by twisted light, Vol. 19, No. 27 / OPTICS EXPRESS 26733 (2011).