Weak-value amplification of the nonlinear effect of a single photon

In quantum mechanics, the concept of weak measurements allows for the description of a quantum system both in terms of the initial preparation and the final state (post-selection)(1). This paradigm has been extensively studied theoretically and experimentally, but almost all of weak-measurement experiments carried out to date can be understood in terms of the classical (electromagnetic wave) theory of optics. Here, we present a quantum version in which the measurement apparatus deterministically entangles two distinct optical beams. We show that a single photon, when properly post-selected, can have ane effect equal to that of eight photons: that is, in a system where a single photon has been calibrated to write a nonlinear phase shift of phi(o) on a probe beam, we measure phase shifts as large as 8 phi(o) for appropriately post-selected single photons. This opens up a new regime for the study of entanglement of optical beams, as well as further investigations of the power of weak-value amplification for the measurement of small quantities.