# Number-unconstrained quantum sensing

## TL;DR

This paper develops a theory for quantum sensing without particle number constraints, revealing optimal particle numbers emerge naturally and demonstrating advantages of squeezed beams over classical strategies.

## Contribution

It introduces a new framework for particle-number-unconstrained quantum sensing and shows how optimal particle numbers arise from interactions, with practical application to optical atomic sensing.

## Key findings

- Optimal particle numbers emerge from interactions without external constraints.
- Squeezed beams provide significant sensitivity advantages over classical strategies.
- Finite optima in sensing performance appear naturally in the unconstrained scenario.

## Abstract

Quantum sensing is commonly described as a constrained optimization problem: maximize the information gained about an unknown quantity using a limited number of particles. Important sensors including gravitational-wave interferometers and some atomic sensors do not appear to fit this description, because there is no external constraint on particle number. Here we develop the theory of particle-number-unconstrained quantum sensing, and describe how optimal particle numbers emerge from the competition of particle-environment and particle-particle interactions. We apply the theory to optical probing of an atomic medium modeled as a resonant, saturable absorber, and observe the emergence of well-defined finite optima without external constraints. The results contradict some expectations from number-constrained quantum sensing, and show that probing with squeezed beams can give a large sensitivity advantage over classical strategies, when each is optimized for particle number.