  Kerr-induced synchronization (KIS) [1] involves the capture of a dissipative
Kerr soliton (DKS) microcomb [2] tooth by a reference laser injected into the
DKS resonator. This phase-locking behavior is described by an Adler equation
whose analogous form describes numerous other physical systems [3], such as
Josephson junctions [4]. We present an AC version of KIS whose behavior is
similar to microwave-driven Josephson junctions, where periodic synchronization
occurs as so-called Shapiro steps. We demonstrate consistent results in the
AC-KIS dynamics predicted by the Adler model, Lugiato-Lefever equation, and
experimental data from a chip-integrated microresonator system. The (integer)
Shapiro steps in KIS can simply be explained as the sideband created through
the reference laser phase modulation triggering the synchronization. Notably,
our optical system allows for easy tuning of the Adler damping parameter,
enabling the further observation of fractional-Shapiro steps, where the
synchronization happens at a fraction of the driving microwave frequency. Here,
we show that the comb tooth is indirectly captured thanks to a four-wave mixing
Bragg-scattering process, leading to sub-comb mode-locking, and we demonstrate
this experimentally through noise considerations. Our work opens the door to
the study of synchronization phenomena in the context of microresonator
frequency combs, synthesis of condensed-matter state analogues with DKSs, and
the use of the fractional Shapiro steps for flexible and tunable access to the
KIS regime.
