Interplay of charge and spin coherence in Landau-Zener-St\"uckelberg-Majorana interferometry

We study Landau-Zener dynamics in a double quantum dot filled with two electrons, where the spin states can become correlated with charge states and the level velocity can be tuned in a time-dependent fashion. We show that a correct interpretation of experimental data is only possible when finite-time effects are taken into account. In addition, our formalism allows the study of partial adiabatic dynamics in the presence of phonon-mediated hyperfine relaxation and charge-noise-induced dephasing. Our findings demonstrate that charge noise severely impacts the visibility of Landau-Zener-St\"uckelberg-Majorana interference fringes. This indicates that charge coherence must be treated on an equal footing with spin coherence.