Heisenberg limit in phase measurements: the threshold detection approach

The ultimate precision of phase estimation is limited by the Heisenberg scaling $\mathrm{\Delta }{\phi }_0=K\mathrm{/}N$, where $K\sim 1$ 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 Cramér-Rao bound. We characterize the performance by two complementary metrics: the peak sensitivity $\mathrm{\Delta }{\phi }_0$ 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 $\delta \phi =\pi \mathrm{/}2$.