is the magnetic field dispersion of the extracted single-particle energies for two different QDs. Fig. \[fig:mechanismFull\](d,e) shows the single-particle dispersion for the two QDs shown in the main text. We measure an $s$-to-$p$-splitting of $17.7\ \text{meV}$ respectively $21.0\ \text{meV}$ on the InGaAs QDs; and $13.8\ \text{meV}$ respectively $17.6\ \text{meV}$ on the GaAs QDs. We can determine many further parameters of the single-particle spectrum by fitting the data to a model which assumes states of an asymmetric harmonic confinement potential. The red lines in Fig. \[fig:mechanismFull\](b-e) represent the model that is developed in this section. It reproduces the data very well.

For a symmetric two-dimensional confinement potential, the magnetic field dependence of the single-particle states forms the Fock-Darwin spectrum [@Fock1928; @Darwin1930]. The eigenergies $E_{n,L}$ depend on two quantum numbers, the radial quantum number, $n$, and the angular momentum quantum number, $L$ [@Kouwenhoven2001]. In this model, the two $p$-shells are degenerate at zero magnetic field. This is clearly not the case in our experiments. To describe the single-particle dispersions, we therefore assume an asymmetric harmonic confinement potential of the form $V(x,y)=\frac{1}{2}m_e^*\left(\omega_x^2x^2+\omega_y^2y^2\right)$. When the radial symmetry is broken, angular momentum is no longer a good quantum number, and the eigenenergies are $E_{n_x, n_y}=\hbar\omega_x\left(n_x+\frac{1}{2}\right)+\hbar\omega_y\left(n_y+\frac{1}{2}\right)$, 