Asymptotic optimality of twist-untwist protocols for Heisenberg scaling in atomic interferometry

TL;DR

This paper proves that twist-untwist protocols in atomic interferometry are asymptotically optimal for Heisenberg scaling, achieving near-quantum limit precision with minimal nonlinear evolutions and measurement constraints.

Contribution

It establishes the asymptotic optimality of twist-untwist protocols under realistic physical interactions and measurement limitations, providing bounds close to the quantum Cramér-Rao limit.

Findings

Protocols asymptotically achieve 85% and 92% of quantum Cramér-Rao bounds.

Error reduction by factor of L with noiseless iteration of protocols.

Optimal protocols utilize two calls to weak nonlinear evolution and simple spin measurements.

Abstract

Twist-untwist protocols for quantum metrology consist of a serial application of: 1. unitary nonlinear dynamics (e.g., spin squeezing or Kerr nonlinearity), 2. parameterized dynamics $U(\phi)$ (e.g., a collective rotation or phase space displacement), 3. time reversed application of step 1. Such protocols are known to produce states that allow Heisenberg scaling for experimentally accessible estimators of $\phi$ even when the nonlinearities are applied for times much shorter than required to produce Schr\"{o}dinger cat states. In this work, we prove that twist-untwist protocols provide the lowest estimation error among quantum metrology protocols that utilize two calls to a weakly nonlinear evolution and a readout involving only measurement of a spin operator $\vec{n}\cdot \vec{J}$, asymptotically in the number of particles. We consider the following physical settings: all-to-all interactions generated by one-axis twisting $J_{z}^{2}$ (e.g., interacting Bose gases), constant finite range spin-spin interactions of distinguishable or bosonic atoms (e.g., trapped ions or Rydberg atoms, or lattice bosons). In these settings, we further show that the optimal twist-untwist protocols asymptotically achieve $85\%$ and $92\%$ of the respective quantum Cram\'{e}r-Rao bounds. We show that the error of a twist-untwist protocol can be decreased by a factor of $L$ without an increase in the noise of the spin measurement if the twist-untwist protocol can be noiselessly iterated as an $L$ layer quantum alternating operator ansatz.