Quantum-enhanced estimation of the optical phase gradient by use of image-inversion interferometry

Abstract

We show that the quantum Cram\'er-Rao bound on the error of measurement of the optical phase gradient with a beam of finite width (or the wavefront tilt within a finite aperture) is consistent with the Fourier-transform uncertainty principle for the single-photon state, and is a factor of $N$ lower for the maximally entangled $N$-photon state. This fundamental bound therefore governs the tradeoff between quantum sensitivity and spatial resolution. Error bounds for a structured configuration using binary projective-field measurements implemented by an image-inversion (II) interferometer are higher, and the factor of $N$ advantage attained by the $N$-photon maximally entangled state is reduced and eventually washed out as the beam width or the phase gradient increases. This reduction is more rapid for larger $N$, so that the quantum advantage is more vulnerable. The precision of the II interferometer is greater than that based on a split detector placed in the focal plane of a lens.