Heisenberg limit in phase measurements: the threshold detection approach

TL;DR

This work analyzes the fundamental phase‑estimation limits for Gaussian squeezed‑coherent light in single‑ and two‑arm interferometers, comparing homodyne and nonlinear threshold detection. By applying the quantum Cramér–Rao bound and an adapted threshold measurement, it demonstrates that Heisenberg‑limited scaling Δφ ∼ K/N with K ∼ 1 is achievable across configurations, while the high‑precision bandwidth δφ depends on the measurement choice and squeezing. The study quantifies Δφ0 and δφ for both detection schemes and interferometer topologies, showing that two‑arm configurations can offer broader high‑precision ranges, whereas single‑arm setups yield narrower windows. A unifying perspective relates Δφ near φ0 to a quadratic increase with φ, providing guidance for exploiting a priori information in realistic phase‑estimation tasks.

Abstract

We analyze the fundamental limits of phase measurement precision, provided by the standard (single- and two-arm) optical interferometers using the Gaussian (squeezed coherent) quantum states of the probing light. We consider two types of the measurements of the output light -- the homodyne measurement and the non-linear threshold measurement that provides the sensitivity saturating the quantum Cramer-Rao bound. For all considered cases, we calculate the best sensitivity $Δφ_0$ achievable at some given value of phase and the range $δφ$ around this value within which the sensitivity is close to $Δφ_0$. We show that in all cases, the Heisenberg scaling $Δφ= K/N$ can be reached, where $K\sim1$ is a numerical prefactor and $N$ is the mean photon number. We show also that $δφ$ strongly depends on $K$.

Figures (2)

• Figure 1: The single-arm interferometer.
• Figure 2: The antisymmetric two-arm interferometer. BS: 50%/50% beamsplitters.