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

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 (φ) (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 φ even when the nonlinearities are applied for times much shorter than required to produce Schrödinger 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 n • J, asymptotically in the number of particles. We consider the following physical settings: all-to-all interactions generated by one-axis twisting J 2 z (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 Cramér-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.