Diode effect in Shapiro steps in an asymmetric SQUID with a superconducting nanobridge

We investigate the Josephson diode effect in an asymmetric SQUID consisting of a sinusoidal Josephson junction formed by a ${\mathrm{Bi}}_{2}{\mathrm{Te}}_{2}\mathrm{Se}$ flake and a superconducting Nb nanobridge with a linear and multivalued current-phase relation (CPR). Current-voltage characteristics were measured both in the absence (dc regime) and presence (ac regime) of external microwave irradiation. Our dc measurements reveal only weak critical current asymmetry (i.e. weak Josephson diode effect), while confirming the multivalued behavior of the SQUID. At the same time, the key finding of this work is the observation of strong Shapiro step asymmetry (concerning the dc current direction) in the ac regime at finite magnetic flux. This peculiarity oscillates as a function of magnetic field with the SQUID's periodicity and varies nonmonotonically with the increase in microwave power. Our theoretical model shows that the pronounced Shapiro step asymmetry, despite the small diode effect in critical current, arises from the interplay between the sinusoidal and multivalued CPRs of the junctions.