Heisenberg limit in phase measurements: the threshold detection approach

Abstract

The ultimate precision of phase estimation is limited by the Heisenberg scaling $\Delta\phi_0 = K/N$, where $K\sim1$ is a numerical prefactor and $N$ is the mean number of photons interacting with the phase shifting object(s). However, achieving this fundamental limit often comes at the cost of an extremely narrow high-sensitive range, rendering schemes impractical. We analyze the precision limits of phase measurements in single- and two-arm optical interferometers with input Gaussian states. We consider two detection methods: conventional homodyne measurement and non-Gaussian threshold detection that saturates the quantum Cram\'er-Rao bound. We characterize the performance by two complementary metrics: the peak sensitivity $\Delta\phi_0$ and the width $\delta\phi$ of the high-sensitivity range. We demonstrate that Heisenberg scaling is attainable in all configurations considered. However, we reveal that $\delta\phi$ strongly depends on $K$. We derive an approximate analytic expression that describes this trade-off. We show also that the two-arm interferometer with antisymmetrically squeezed inputs exhibits exceptional performance, simultaneously achieving Heisenberg-limited sensitivity and a broad high-sensitivity range $\delta\phi=\pi/2$.