  Exploring non-Hermitian phenomenology is an exciting frontier of modern
physics. However, the demonstration of a non-Hermitian phenomenon that is
quantum in nature has remained elusive. Here, we predict quantum non-Hermitian
phenomena: the fractional quantum Zeno (FQZ) effect and FQZ-induced photon
antibunching. We consider a quantum optics platform with reservoir engineering,
where nonlinear emitters are coupled to a bath of decaying bosonic modes whose
own decay rates form band structures. By engineering the dissipation band, the
spontaneous emission of emitters can be suppressed by strong dissipation
through an algebraic scaling with fractional exponents - the FQZ effect. This
fractional scaling originates uniquely from the divergent dissipative density
of states near the dissipation band edge, different from the traditional
closed-bath context. We find FQZ-induced strong photon antibunching in the
steady state of a driven emitter even for weak nonlinearities. Remarkably, we
identify that the sub-Poissonian quantum statistics of photons, which has no
classical analogs, stems here from the key role of non-Hermiticity. Our setup
is experimentally feasible with the techniques used to design lattice models
with dissipative couplings.
