Asymptotic optimality of twist-untwist protocols for Heisenberg scaling in atom-based sensing

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