Non-asymptotic Heisenberg scaling: experimental metrology for a wide resources range

Adopting quantum resources, like entangled states, for parameter estimation discloses the possibility to realize quantum sensors operating at a sensitivity beyond the standard quantum limit. Such approach promises to reach the fundamental Heisenberg scaling of the precision as a function of the employed resources in the estimation process. An accurate choice of the sizes of the entangled probes and their numbers allows us to reach the Heisenberg scaling with optimal prefactor (F. Belliardo and V. Giovannetti, Phys. Rev. A 102, 042613 (2020)). This optimized procedure has been experimentally applied in the estimation of a rotation angle. We quantitatively verified the Heisenberg scaling for a considerable range of resources by using single-photon states with high-order orbital angular momentum, constructed using q-plates. We achieved an error reduction greater than 10 dB below the standard quantum limit (Cimini et al., arXiv:2110.02908). The q-plates are liquid crystal based waveplates that couples the spin and the spatial degrees of freedom of photons, allowing for efficient manipulations of the orbital angular momentum. With a suitable single-photon input state the output of a q-plate resembles for its metrological properties a multiphoton entangled state, and is therefore used as the quantum resource of our protocol. This estimation strategy can be applied in principle to different platforms, opening the way to the optimization of resources in quantum sensing.