  One of the central goals of quantum optics is to generate large interactions
between single photons. Light interacting with motion in an optomechanical
system can sense minute fluctuations in displacement, and also impart a force
via radiation pressure. Taken together, these two effects mean that two photons
can "sense" each other's presence in an interaction mediated by motion. It is
accepted that for an optomechanical system to mediate strong interactions
between single photons, the mechanical system must respond before the photon is
lost ($\omega_m > \kappa$), and the radiation pressure force must generate a
displacement large enough to change the optical properties of the system ($g_0
> \kappa$). The challenge of achieving this "vacuum strong coupling" has
prevented experiments from demonstrating single-photon interactions. In this
work we show that by adding a coherent feedback channel to a slow mechanical
system ($\omega_m < \kappa$) that is weakly nonlinear ($g_0 < \kappa$), two
spatially separated single photons can be made to effectively interact with
each other deterministically through the mechanical motion of a resonator. To
numerically analyze our system, we must solve Schr\"odinger's equation for the
state of an optomechanical system coupled to a long waveguide feeback channel.
We implement a matrix product state approach to keep track of and evolve the
complete quantum state of the system in an efficient way. We analyze the
process semiclassically and then solve the full quantum dynamics numerically to
find a cross-over between the semiclassical and quantum regimes of
optomechanics. Finally we analyze the experimental prospects for implementing
this protocol.
