Enhancing a slow and weak optomechanical nonlinearity with delayed   quantum feedback
  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. 
Zhaoyou Wang and Amir H. Safavi-Naeini
quant-ph physics.optics