Post contributed by:
Björn Sbierski,
Freie Universität Berlin,
Germany
Blog post based on the journal article "Twistedlightinduced intersubband transitions in quantum wells at normal incidence", J. Phys. Cond. Mat. 25 (2013), 385301 by B. Sbierski, G. F. Quintero and P. I. Tamborenea.
Figure 1: Twisted light beam normally
incident on a semiconductor quantum
well causes intersubband transitions.

The transitions generated in the electronic configurations of the QW are due to a finite outofplane electric field of a TL beam. It is the purpose of this post to explain the structure of these transitions, and shed light on the question how exactly angular momentum carried by the TL is transferred to the QW electrons. Detailed calculations can be found in in our journal article [2].
We consider a beam of collimated TL at normal incidence to a semiconductor QW. The TL is characterized by its vector potential $\vec{A}$ that encodes its circular polarization $\sigma=\pm 1$ and integer orbital angular momentum $l$. The QW is modeled as a disk of radius $r_{0}$ and height $z_{0}$, the envelope wavefunctions for QW electrons in this cylindrical geometry are
\begin{equation}
\psi_{nm\nu}\left(r,\phi,z\right)\propto J_{m}\left(x_{m\nu}\frac{r}{r_{0}}\right)e^{im\phi}\sin\left(\frac{n\pi z}{z_{0}}\right)\label{eq:QW electrons} ~~~~~ (1)
\end{equation}
with Formula $J_{m}$ a Bessel function and $x_{m\nu}$ its $\nu$th zero. The integer quantum number $n$ is also called the subband index, the subband energy spacing is typically on the order of $100meV$ and dominates over corrections due to the orbital angular momentum $m\in\mathbb{Z}$ or radial quantum number $\nu\in\mathbb{N}$.
For the rest of the calculation we focus on TL induced transitions between subbands changing $n$ between $n=1$ and $n=2$. This is justified by an appropriate choice of TL frequency and the observation that intrasubband transitions have strongly reduced matrix elements. For the relevant matrix element, we obtain
\begin{equation}
\left\langle n^{\prime}=2,m^{\prime},\nu^{\prime}A_{z}p_{z}n=1,m,\nu\right\rangle \propto\eta_{m^{\prime}\nu^{\prime},m\nu}\left(\delta_{m^{\prime}m,l+\sigma}e^{i\omega t}\delta_{mm^{\prime},l+\sigma}e^{i\omega t}\right)\label{eq:matel} ~~~~~ (2)
\end{equation}
where $\eta_{m^{\prime}\nu^{\prime},m\nu}$ is a complicated radial integral over Bessel functions that can be somewhat simplified and calculated numerically if the beam waist far exceeds the radius $r_{0}$ of the QW which we assume. The $\delta$functions in Eq. (2) reflect the conservation of angular momentum, i.e. electrons do not only absorb energy to jump from subband 1 to 2 but also get boosted (or slowed down) in their inplane 'rotation'.
It is interesting to study the magnitude $\eta_{m^{\prime}\nu^{\prime},m\nu}$ for fixed $m^{\prime}=m+l+\sigma$ (as required by angular momentum conservation) as a function of $\nu^{\prime}$, $m$ and $l$ for some fixed $\nu$ and $\sigma$ (we take $\nu=30$ and $\sigma=1$, see Fig. 2). The resulting plot of the squared amplitude $\eta^{2}$ in Fig. 2 shows that for small $l$ and $m$ the TL induced transitions are nearly 'vertical' (i.e. preserving the radial quantum number $\nu$), a situation realized for plane waves, for more discussions see Ref. [3]. For increasing $l$ more states with a broader range of $\nu^{\prime}$ couple in, but with an overall decreased coupling strength. Additionally, there is an $l$induced shift of the maximum of $\eta^{2}\left(\nu^{\prime}\right)$, however in Fig. 2 these shift has been counterbalanced by a corresponding change of $m$.
Finally, we estimate, based on reasonable experimental parameters of the TL beam that the population of the second subband under bright TL irradiation will be well detectable in relative absorbance measurements. We believe that such an experimental study is well within reach and that the above calculations will prove useful in aiding interpretations of obtained absorption data.
References
[1] Allen L, Beijersbergen M W, Spreeuw R J C and Woerdman J P, Phys. Rev. A 45 (1992), 8185.
[2] Sbierski B, Quintero G F and Tamborenea P F, J. Phys. Cond. Mat. 25 (2013), 385301.
[3] Quinteiro G F and Tamborenea P I, Europhys. Lett. (2009), 47001
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