Quantum rings: Tilted one-to-one interband transitions

Light carrying orbital angular momentum induces in semiconductor quantum rings a clear electronic transition from a single valence-band state to a single conduction-band state.

Figure 1: Interaction of twisted light with 
a 1D quantum ring. Notice that the field 
is centered on the nanostructure.

In this post we start the description of the twisted light (TL) - quantum ring (QR) interaction. We show how simple the mathematical model is, and we present one of the main results: the one-to-one electronic transitions. This makes clear why we believe the QR is the quintessential semiconductor structure to investigate the twisted light - matter interaction. 

We imagine a TL beam whose symmetry axis coincides with the 1D-QR's center. The light has enough energy to produce an interband transition. Semiconductor QRs with radius on the nanometer and micrometer scale can be fabricated.

Our first task is to determine the light-matter matrix element. In the case of QRs this is easy, and leads to a selection rule: An electron in the valence band moving around the QR with envelope angular momentum $\hbar m$ gains from the light an amount $\hbar \ell$ and therefore ends in a conduction band state with envelope angular momentum $\hbar (m + \ell)$ as we show in Fig. 2. This is a one-to-one transition [1], in contrast to the more complicated cone-like transitions in bulk. Note that it somewhat resembles the transitions in QDs, but there and because the system is 2D the transition is one-to-many --"many" meaning not a quasi-continuum.

Figure 2: Left: Plane waves (Gaussian beam) produce a "vertical" transition, while Right: light with orbital angular momentum produces a "tilted" transition.

We can study the electron's kinetics by inspecting the density matrix. One sets up equations for the interband coherence $\large \rho_{vm,cm}$ and populations in the valence $\large \rho_{v,mm}$ and conduction $\large \rho_{c,mm}$ bands for all quantum number $\large m$, and use the light-matter matrix element.  The remarkable thing is that the set of three equations {$\large \rho_{vm_0,cm_0+\ell}$, $\large \rho_{vm_0,vm_0}$, $\large \rho_{cm_0+\ell,cm_0+\ell}$} for a given $m_0$ are decoupled from the rest of $m$'s, because of the selection rule. Thus, the problem looks like the well-known optical Bloch equations. The dynamics can be solved to all orders in the field amplitude and yields tilted Rabi oscillations for each $m$:

\begin{eqnarray}
\label{eq:Exact_sol}
\large
\rho_{v, m \,m}(t)
&=&
\large
1 +
    \frac{1}{2}\,\left(
    \frac{{\cal R}_0}{{\cal R}} \right)^2
    \left[
        \cos({\cal R}\,t) -1
    \right] \nonumber \\
\large
\rho_{c, m+l \,m+l}(t)
&=&
\large
- \frac{1}{2}\,\left( \frac{{\cal
R}_0}{{\cal R}} \right)^2
    \left[
        \cos({\cal R}\,t) -1
    \right] \nonumber \\
\large
\rho_{v m, c m+l}(t)
&=&
\large
\frac{{\cal R}_0}{2{\cal R}}\,
    e^{-i \omega t}
    \left\{
        \frac{\Delta}{\hbar {\cal R}}
        \left[
            1 - \cos({\cal R}\,t)
        \right] +
        i \sin({\cal R} t)
    \right\}
\,,
\end{eqnarray}
with ${\cal R}$ the Rabi frequency and ${\cal R}_0$ the Rabi frequency at resonance. We can look at extreme cases. For example, if the laser field is strong, we can expand the expressions for short times (typically less than a picosecond)
\begin{eqnarray}
\large
\rho_{v, m \,m}(t)
&=&
\large
1 - \frac{1}{4}   ({\cal R}_0 t)^2
\nonumber \\
\large
\rho_{c, m+l \,m+l}(t)
&=&
\large
\frac{1}{4}
    ({\cal R}_0 t)^2
\nonumber \\
\large
\rho_{v m, c m+l}(t)
&=&
\large
\frac{i}{2} ({\cal R}_0 t) e^{-i \omega t}
\,,
\end{eqnarray}
which shows that the coherence (related to the polarization) is linear in $\large t$, while the populations are quadratic in $\large t$.

We showed that the mathematical description of the solid-TL interaction is the simplest in 1D QRs, and also that it can be solved non-perturbatively.  Being the transitions one-to-one, they are easy to understand and they are the analog to the transitions in bulk induced by plane waves. In future posts we will talk about the photon-drag effect, orbital angular momentum and currents in the QR.

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.