Optical precursors in waveguide quantum electrodynamics*

When a broadband signal propagates through a dispersive medium some frequency components move faster than the main pulse. This leads to the appearance of precursors, rapidly oscillating waves that emerge from the medium earlier than the main signal and seem to propagate superluminally. Here, we investigate the microscopic origin of precursors in a minimal setup: an array of qubits coupled to a waveguide. For large qubit numbers, our result for the linear transmission function converges to that of a continuous medium. The continuous description breaks down for small samples. Nevertheless, oscillations in the transmitted field persist down to only two qubits, which is the minimal number of elements required for the emergence of precursors. Precursors are best observed under conditions of electromagnetically-induced transparency, as the main signal is significantly delayed. Under these conditions, just a single qutrit is enough to generate a precursor. Our results pave the way towards dispersion engineering of light with just a few qubits, and can be realized with superconducting qubits coupled to transmission lines or atoms coupled to fibers.