  When two indistinguishable photons are each incident on separate input ports
of a beamsplitter they `bunch' deterministically, exiting via the same port as
a direct consequence of their bosonic nature. This two-photon interference
effect has long-held the potential for application in precision measurement of
time delays, such as those induced by transparent specimens with unknown
thickness profiles. However, the technique has never achieved resolutions
significantly better than the few femtosecond (micron)-scale other than in a
common-path geometry that severely limits applications. Here we develop the
precision of HOM interferometry towards the ultimate limits dictated by
statistical estimation theory, achieving few-attosecond (or nanometre path
length) scale resolutions in a dual-arm geometry, thus providing access to
length scales pertinent to cell biology and mono-atomic layer 2D materials.
