We show that weak measurement can be used to "amplify" optical nonlinearities at the singlephoton level, such that the effect of one properly post-selected photon on a classical beam may be as large as that of many un-post-selected photons. We find that "weak-value amplification" offers a marked improvement in the signal-to-noise ratio in the presence of technical noise with long correlation times. Unlike previous weak-measurement experiments, our proposed scheme has no classical equivalent.An interaction between two independent photons could be used to serve as a "quantum logic gate," enabling the development of optical quantum computers [1][2][3], as well as opening up an essentially new field of quantum nonlinear optics [4]. Typical optical nonlinearities are many orders of magnitude too weak to create a π phase shift as required in initial proposals, but more recently it was realized that any phase shift large enough to be measured on a single shot could be leveraged into a quantum logic gate [5]. Much recent work has shown that atomic coherence effects [6][7][8][9] and nonlinearities in microstructured fiber [10,11] can generate greatly enhanced Kerr nonlinearities. While even a very small phase shift can be made larger than the quantum (shot) noise, by using a sufficiently intense probe, present experiments are limited by technical rather than quantum noise and difficult to carry out even with much averaging. For example, in Ref.[11], a phase shift of 10 −7 rad was measured by averaging over 3 × 10 9 classical pulses with singlephoton-level intensities. To date, no one has yet been able to observe the cross-Kerr effect induced by a single propagating photon on a second optical beam [12]. In this Letter, we show that using weak-value amplification (WVA) [13][14][15], a single photon can be made to "act like" many photons, and it is possible to amplify a cross-Kerr phase shift to an observable value, much larger than the intrinsic magnitude of the single-photon-level nonlinearity. In so doing, we also demonstrate quantitatively how WVA may improve the signal-to-noise ratio (SNR) in appropriate regimes, a result of broad general applicability to quantum metrology.Weak measurement is an exciting new approach to understanding quantum systems from a time-symmetric perspective, obtaining information from both their preparation and subsequent post-selection [16]. In the past several years, it has been widely studied to address foundational questions in quantum mechanics [17], as well as for its potential application to ultrasensitive measurements [14,15,18,19]. If a quantum system is coupled only weakly to a probe, then very little information may be obtained from a single measurement, and in compensation, this measurement disturbs the sys-
