Momentum-entangled two-photon interference for quantum-limited transverse-displacement estimation

We propose a scheme achieving the ultimate quantum precision for the estimation of the transverse displacement between two interfering photons. Such a transverse displacement could be caused, for example, by the refracting properties of the propagation medium or by the orientation of a system of mirrors. By performing transverse-momentum sampling interference between polarization-entangled pairs of photons that propagate with different momenta, we show that it is possible to perform transverse-displacement estimation with a precision that increases with the difference of the transverse momenta of the photons. We show that the precision achieved with our scheme is independent of the value of the displacement, useful when tracking a variable displacement. Moreover, only for small displacements, we show that the estimation can be performed without the need for transverse-momentum-resolving detectors. More fundamentally, we demonstrate that it is the quantum interference arising from two-photon entanglement in the transverse momenta at the very heart of the foreseen quantum-limited sensitivity in the spatial domain.