magnetic optical transitions in quantum dots: part 1

Focused beams of twisted light can be dominated by magnetic fields close the the beam axis. What is the effect of these optical magnetic fields with quantum dots?

Among the new features of TL beams, there is the strong magnetic field that focused antiparallel beams present close to the phase singularity at $r=0$. In particular, the fields with orbital angular momentum index $\ell=2$ has a strong and constant in-plane $B$-field, in contrast to the electric field that vanishes for $r \rightarrow 0$, see here.

Figure 1: Electric field profile for $\ell=2$ in antiparallel and parallel configurations.

The interaction can be brought to a form similar to the well-known dipole approximation
\begin{eqnarray*}
   H_I
&=&
   -  \frac{q}{2m} \mathbf B_\perp(t) \cdot (\mathbf r \times \mathbf p)
=
   i \frac{q B_0}{2m}
   \left(        r_+ p_z - z p_+   \right)
\end{eqnarray*}
with $r_+ = x+iy$ and $p_+ = p_x+i p_y $.

To determine the response of the QD, we study the matrix elements between electronic states. Thanks to the envelope-function approximation we transform coordinates and split the matrix element integral in two portions: a sum over unit-cells (coordinate R) and an integral within the unit-cell (coordinate r'). The splitting of the matrix element integral and the transformation $\mathbf r=\mathbf r'+\mathbf R$ allows different combinations of intra and inter cell coordinates and momenta. For example:

$        r_+ p_z \rightarrow        R_+ P_z $ Producing intraband transitions
$        r_+ p_z \rightarrow        r'_+ P_z $ Producing no common interband transitions
$        r_+ p_z \rightarrow      r'_+ p'_z $  Producing interband transitions to other conduction band with different parity

This shows that the dominant magnetic interaction can help to control in different ways the state of the dot, see also here. In the next post (part 2), we will consider the interband transition of the type
\begin{eqnarray*}
 \langle  {v, i \uparrow} \mid
  H_B
 \mid {c, j \uparrow} \rangle
&=&
   i \frac{q B_0}{2m}
   \sum_R \int_{{\text{cell}}} dr' \,
   {\cal E}_i^* u_{v}^* \left(  R_+ p'_z - z' P_+  \right) {\cal E}_j u_{c}
\end{eqnarray*}
connecting light-hole and conduction bands.