  A scheme for the enhanced generation of higher photon-number states is
realized, using an optical time-multiplexing setting that exploits a parametric
down-conversion source for an iterative state generation. We use a quantum
feedback mechanism for already generated photons to induce self-seeding of the
consecutive nonlinear process, enabling us to coherently add photons to the
light that propagates in the feedback loop. The addition can be carried out for
any chosen number of round trips, resulting in a successive buildup of
multiphoton states. Our system is only limited by loop losses. The looped
design is rendered possible by a carefully engineered waveguide source that is
compatible with and preserves the shape of the propagating mode. We compare the
fidelities and success probabilities of our protocol with the common direct
heralding of photon-number states. This comparison reveals that, for same the
fidelity, our feedback-based setup significantly enhances success
probabilities, being vital for an efficient utilization in quantum
technologies. Moreover, quantum characteristics of the produced states are
analyzed, and the flexibility of producing higher photon-number states with our
setup beyond the common direct heralding is demonstrated.
