  The interaction between photons and a single two-level atom constitutes a
fundamental paradigm in quantum physics. The nonlinearity provided by the atom
means that the light-matter interaction depends strongly on the number of
photons interacting with the two-level system within its emission lifetime.
This nonlinearity results in the unveiling of strongly correlated
quasi-particles known as photon bound states, giving rise to key physical
processes such as stimulated emission and soliton propagation. While signatures
consistent with the existence of photon bound states have been measured in
strongly interacting Rydberg gases, their hallmark excitation-number-dependent
dispersion and propagation velocity have not yet been observed. Here, we report
the direct observation of a photon-number-dependent time delay in the
scattering off a single semiconductor quantum dot coupled to an optical cavity.
By scattering a weak coherent pulse off the cavity-QED system and measuring the
time-dependent output power and correlation functions, we show that single
photons, and two- and three-photon bound states incur different time delays of
144.02\,ps, 66.45\,ps and 45.51\,ps respectively. The reduced time delay of the
two-photon bound state is a fingerprint of the celebrated example of stimulated
emission, where the arrival of two photons within the lifetime of an emitter
causes one photon to stimulate the emission of the other from the atom.
