Quantum enhanced beam displacement estimation: sensitivity limits, optimal probes, and measurement schemes

We investigate the ultimate sensitivity limits for measuring small transverse beam displacements and identify optimal strategies that attain them, using the framework of mode parameter estimation. We contrast two families of states with different resource constraints, using Hermite-Gauss and Laguerre-Gauss mode pairs. First, for fixed total photon number states, the optimal state is identified and we show that it asymptotically approaches a rotated NOON state. The sensitivity bound is compared to that of an asymptotically optimal pure Gaussian state with fixed average photon number. Finally, we propose concrete measurement schemes using the method of moments and show that the twin-Fock state may provide a viable alternative to the rotated NOON state for this task. Our work provides a broader benchmark for developing practical strategies for spatial sensing at the quantum limit.