# Scale-invariant spin dynamics and the quantum limits of field sensing

## TL;DR

This paper establishes fundamental quantum limits for magnetic field sensing using spin ensembles, demonstrating scale-invariant spin dynamics that set a universal bound on energy resolution per bandwidth.

## Contribution

It introduces a new quantum limit for spin-based magnetometers based on scale-invariant spin dynamics, independent of particle number, extending the Tesche and Clarke bound.

## Key findings

- Quantum limits relate noise, geometry, measurement duration, and fundamental constants.
- Spin dynamics are scale invariant, depolarizing without external reservoirs.
- Numerical results show the energy resolution per bandwidth approaches the quantum limit.

## Abstract

We describe quantum limits to field sensing that relate noise, geometry and measurement duration to fundamental constants, with no reference to particle number. We cast the Tesche and Clarke (TC) bound on dc-SQUID sensitivity as such a limit, and find analogous limits for volumetric spin-precession magnetometers. We describe how randomly-arrayed spins, coupled to an external magnetic field of interest and to each other by the magnetic dipole-dipole interaction, execute a spin dynamics that depolarizes the spin ensemble even in the absence of coupling to an external reservoir. We show the resulting spin dynamics are scale invariant, with a depolarization rate proportional to spin number density and thus a number-independent quantum limit on the energy resolution per bandwidth $E_R$. Numerically, we find $E_R \ge \alpha \hbar$, $\alpha \sim 1$, in agreement with the TC limit, for paradigmatic spin-based measurements of static and oscillating magnetic fields.