Josephson diode effect with Andreev and Majorana bound states

Superconductor-semiconductor hybrids have been shown to be useful for realizing the Josephson diode effect, where nonreciprocity in the supercurrents occurs due to the interplay of the Josephson effect and applied magnetic fields. With the same ingredients, these Josephson junctions can also host Andreev and Majorana bound states, whose interplay with the Josephson diode effect is, however, not fully understood. In this work, we consider short Josephson junctions based on superconductor-semiconductor systems under homogeneous Zeeman fields and investigate the Josephson diode effect in the presence of Andreev and Majorana states. Under generic conditions, the Zeeman field component parallel to the spin-orbit axis promotes an asymmetric low-energy spectrum as a function of the superconducting phase, which persists in the trivial and topological phases hosting Andreev and Majorana bound states, respectively. Interestingly, this spectrum asymmetry originates supercurrents that are not odd functions of the superconducting phase difference as in common Josephson junctions, thereby developing a nonreciprocal behavior that signals the emergence of the Josephson diode effect. We show that the Josephson diode effect is particularly promoted under the presence of both zero-energy Andreev and Majorana bound states, revealing that Josephson diodes can be realized in the trivial and topological phases of superconductor-semiconductor hybrids. We then demonstrate that the Zeeman field evolution of the diode's efficiencies is able to map the topological phase transition and the formation of Majorana bound states via an oscillatory behavior that becomes more visible in long superconductors. While Josephson diodes generally exist in the trivial and topological phases of Josephson junctions, we discover that in the tunneling regime only a Josephson diode effect in the topological phase remains due to the finite contribution of Majorana bound states. Our findings help understand the Josephson diode effect in superconductor-semiconductor hybrids and can also be useful for guiding the realization of Majorana-only Josephson diodes as well as for identifying Majorana states.