Two-Mode Bosonic Quantum Metrology with Number Fluctuations

TL;DR

This paper investigates optimal quantum states for two-mode bosonic interferometry with particle number fluctuations, revealing that fluctuations can enhance measurement precision beyond fixed-particle limits and identifying the best states for such scenarios.

Contribution

It introduces a comprehensive analysis of quantum metrology with fluctuating particle numbers, identifying optimal states and demonstrating improved precision scaling.

Findings

Error diminishes as 1/ΔN with larger fluctuations

Quasi-NOON states are optimal input states

Product states with squeezed vacuum are optimal among Gaussian states

Abstract

We search for the optimal quantum pure states of identical bosonic particles for applications in quantum metrology, in particular in the estimation of a single parameter for the generic two-mode interferometric setup. We consider the general case in which the total number of particles is fluctuating around an average $N$ with variance $\Delta N^2$. By recasting the problem in the framework of classical probability, we clarify the maximal accuracy attainable and show that it is always larger than the one reachable with a fixed number of particles (i.e., $\Delta N=0$). In particular, for larger fluctuations, the error in the estimation diminishes proportionally to $1/\Delta N$, below the Heisenberg-like scaling $1/N$. We also clarify the best input state, which is a "quasi-NOON state" for a generic setup, and for some special cases a two-mode "Schr\"odinger-cat state" with a vacuum component. In addition, we search for the best state within the class of pure Gaussian states with a given average $N$, which is revealed to be a product state (with no entanglement) with a squeezed vacuum in one mode and the vacuum in the other.