  We propose an efficient nonlinear readout scheme for entangled non-Gaussian
spin states (ENGSs) based on the intrinsic quasi-cyclic dynamics of interacting
spin-1/2 systems. We focus on two well-known spin models of twist-and-turn
(TNT) and two-axis-counter-twisting (TACT), where ENGS can be generated by spin
dynamics starting from unstable fixed points. In the TNT model, non-Gaussian
probe state evolves directly back to the vicinity of initial state during the
subsequent time-forward evolution for path recombining, accompanied by quantum
magnification of encoded signal and refocusing of the associated quantum noise.
Based on low-order moment measurement, we find the optimal metrological gain
nearly saturates the quantum Cramer-Rao bound (QCRB) and follows the Heisenberg
scaling. For the TACT case, the QCRB can also be nearly approached when the
state converges to either of the two unstable fixed points, respectively
corresponding to the initial state or its orthogonal coherent state. The latter
case goes beyond previous studies where tracing back to or crossing the initial
states is mostly considered. The present protocol does not require
time-reversal as in typical nonlinear interferometries, and it also avoids
complicated measurement of nonlinear observables or full probability
distributions. The operational approach we discuss presents a practical way for
realizing high-precision and detection-noise-robust quantum metrology with
ENGS.
