How a Bird in the Quantum Hand Can Be Worth 1000(*)
in the Bush

(* - at least in binary)

OR: How the Result of Counting One Photon Can Turn Out to Be a Value of
8

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 φo on a probe beam,
we measure phase shifts as large as 8φo 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.

"Weak-value
amplification of the nonlinear effect of a single photon,"
the technical paper on WVA of postselected photon (here is the published Nature Physics version, but the arXiv version contains the fuller introduction, the shortening of which by the editors distressingly also occasioned the appearance of a misspelling of a researcher's name in the published article.)