We calculate the beating of $h/2e$ and $h/e$ periodic oscillations of the flux-dependent critical supercurrent $I_c(Phi)$ through a quantum spin-Hall insulator between two superconducting electrodes. A conducting pathway along the superconductor conn
ects the helical edge channels via a non-helical channel, allowing an electron incident on the superconductor along one edge to be Andreev reflected along the opposite edge. In the limit of small Andreev reflection probability the resulting even-odd effect is described by $I_cpropto|cos(ePhi/hbar)+f|$, with $|f|ll 1$ proportional to the probability for phase-coherent inter-edge transmission. Because the sign of $f$ depends on microscopic details, a sample-dependent inversion of the alternation of large and small peaks is a distinctive feature of the beating mechanism for the even-odd effect.
We introduce an all-electrical measurement technique, which makes it possible to prepare and detect the ground and excited many-particle states in self-assembled InAs QDs at 4K. This way, the pure-electron spectra of QD-hydrogen, -helium and -lithium
are resolved. Comparison with detailed many-body calculations enables us to identify the different charge configurations and in particular detect the singlet and triplet spin states of QD helium. Furthermore, the time-resolved evolution of the density of states from non-equilibrium to equilibrium charge occupation is shown.
