  The manipulation and metrology of light beams are pivotal for optical science
and applications. In particular, achieving ultra-high precision in the
measurement of light beam rotations has been a long-standing challenge. Instead
of utilizing quantum probes like entangled photons, we address this challenge
by incorporating a quantum strategy called "indefinite time direction" into the
parameterizing process of quantum parameter estimation. Leveraging this quantum
property of the parameterizing dynamics allows us to maximize the utilization
of OAM resources for measuring ultra-small angular rotations of beam profile.
Notably, a nanoradian-scale precision of light rotation measurement is finally
achieved in the experiment, which is the highest precision by far to our best
knowledge. Furthermore, this scheme holds promise in various optical
applications due to the diverse range of manipulable resources offered by
photons.
