  Realizing a sensitive photon-number-dependent phase shift on a light beam is
required both in classical and quantum photonics. It may lead to new
applications for classical and quantum photonics machine learning or pave the
way for realizing photon-photon gate operations. Non-linear phase-shifts
require efficient light-matter interaction, and recently quantum dots coupled
to nanophotonic devices have enabled near-deterministic single-photon coupling.
We experimentally realize an optical phase shift of $0.19 \pi \pm 0.03$ radians
($\approx 34$ degrees) using a weak coherent state interacting with a single
quantum dot in a planar nanophotonic waveguide. The phase shift is probed by
interferometric measurements of the light scattered from the quantum dot in the
waveguide. The nonlinear process is sensitive at the single-photon level and
can be made compatible with scalable photonic integrated circuitry. The work
may open new prospects for realizing high-efficiency optical switching or be
applied for proof-of-concept quantum machine learning or quantum simulation
demonstrations.
