  Realizing a strong interaction between individual optical photons is an
important objective of research in quantum science and technology. Since
photons do not interact directly, this goal requires, e.g., an optical medium
in which the light experiences a phase shift that depends nonlinearly on the
photon number. Once the additional phase shift for two photons reaches pi, such
an ultra-strong nonlinearity could even enable the direct implementation of
high-fidelity quantum logic operations. However, the nonlinear response of
standard optical media is many orders of magnitude too weak for this task.
Here, we demonstrate the realization of an optical fiber-based nonlinearity
that leads to an additional two-photon phase shift close to the ideal value of
pi. Our scheme employs a whispering-gallery-mode resonator, interfaced by an
optical nanofiber, where the presence of a single rubidium atom in the
resonator results in a strongly nonlinear response. We experimentally show that
this results in entanglement of initially independent incident photons. The
demonstration of this ultra-strong nonlinearity in a fiber-integrated system is
a decisive step towards scalable quantum logics with optical photons.
