Collective photon routing improvement in a dissipative quantum emitter chain strongly coupled to a chiral waveguide QED ladder

We examine the routing scheme of single photons in a one-dimensional periodic chain of two-level quantum emitters (QEs) strongly coupled to two waveguides in a ladder configuration. It is known that for a single-emitter chiral waveguide ladder setting, photons can be redirected from one waveguide to another with a $100\%$ probability (deterministically) provided the resonance condition is met and spontaneous emission is completely ignored. However, when the spontaneous emission is included the routing scheme becomes considerably imperfect. In this paper, we present a solution to this issue by considering a chain of QEs where in addition to the waveguide mediated interaction among emitters, a direct and infinitely long-ranged dipole-dipole interaction (DDI) is taken into account. We show that the collective effects arising from the strong DDI protect the routing scheme from spontaneous emission loss. In particular, we demonstrate that the router operation can be improved from $58\%$ to $95\%$ in a typical dissipative chiral light-matter interface consisting of nanowires modes strongly interacting with a linear chain of 30 quantum dots. With the recent experimental progress in chiral quantum optics, trapped QEs evanescently coupled to tapered nanofibers can serve as a platform for the experimental realization of this work.