  We introduce an ultra-sensitive interferometric protocol that combines weak
value amplification (WVA) with traditional interferometry. This
WVA+interferometry protocol uses weak value amplification of the relative delay
between two paths to enhance the interferometric sensitivity, approaching the
quantum limit for classical light. As an example, we demonstrate a
proof-of-principle experiment that achieves few-attosecond timing resolution
(few-nanometer path length resolution) with a double-slit interferometer using
only common optical components. Since our example uses only the spatial shift
of double-slit interference fringes, its precision is not limited by the timing
resolution of the detectors, but is instead limited solely by the fundamental
shot noise associated with classical light. We experimentally demonstrate that
the signal-to-noise ratio can be improved by one to three orders of magnitude
and approaches the shot-noise limit in the large amplification regime.
Previously, quantum-limited WVA delay measurements were thought to require
imaginary weak values, which necessitate light with a broad spectral bandwidth
and high-resolution spectrometers. In contrast, our protocol highlights the
feasibility of using real weak values and narrowband light. Thus, our protocol
is a compelling and cost-effective approach to enhance interferometry.
