  In 1988, Aharonov, Albert, and Vaidman introduced a new paradigm of quantum
measurement in a paper which had the unwieldy but provocative title "How the
result of a measurement of a component of the spin of a spin-1=2 particle can
turn out to be 100." This paradigm, so-called "weak measurement," has since
been the subject of widespread theoretical and experimental attention, both for
the perspective it offers on quantum reality and for possible applications to
precision measurement. Yet almost all of the weak-measurement experiments
carried out so far could be alternatively understood in terms of the classical
(electro-magnetic wave) theory of optics. Here we present a truly quantum
version, the first in which a measurement apparatus deterministically entangles
two distinct optical beams, enabling us to experimentally ask a question
directly analogous to that of the original proposal: "In a two-arm
interferometer containing one photon in total, can the result of a measurement
of the photon number in one arm turn out to be greater than 1?" Specifically,
we show that a single photon, when properly post-selected, can have an 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_0 on a probe beam, we
measure phase shifts as large as 8\phi_0 for appropriately post-selected single
photons. This is the first deterministic weak-value experiment in optics which
defies classical explanation, and constitutes a realization of our proposal for
weak-value amplification (WVA) of the small optical nonlinearity at the
single-photon level. It opens up a new regime for the study of entanglement of
optical beams, as well as further investigations of the power of WVA for the
measurement of small quantities.
