Nonlinear readout of entangled non-Gaussian state without time reversal

Nonlinear readout has been proposed recently to facilitate the characterization of entangled non-Gaussian states (ENGSs). To achieve efficient detection, nonlinear readout protocols usually take advantage of time-reversed dynamics, which disentangle ENGSs by evolving them unitarily back towards the classical-like states before measurements. However, time reversal typically requires a sign-flip of Hamiltonian, which is challenging to realize in a many-body interacting system. Here we show such challenge can be circumvented if the system takes on cyclic interaction dynamics. By adopting two consecutive time-forward evolutions complementary to each other, ENGSs generated by the first one can be driven back to the initial state after the second one in the absence of phase encoding in between. The second evolution therefore behaves as an effective time reversal of the first. We find for rigorous cyclic dynamics, such a looped-evolution readout scheme is as efficient as with the time reversal and can saturate the quantum Cramer-Rao bound (QCRB) through the measurement of Loschmidt echo. Even for the case of approximate quasi-cyclic dynamics where the system does not completely return to the initial state, our scheme can still provide high phase sensitivity of the same order of QCRB. In addition to the spin-1 ⁸⁷Rb atomic BEC, this scheme can also be directly applied to other spin systems with interactions such as twist-and-turn, one-axis twisting, and two-axis counter-twisting squeezing ones. For the latter two cases, we find the total time durations required can be shortened even to half of that required before, at the instant when the system evolves into a coherent spin state orthogonal to the initial one.