Nonlinear interferometry beyond classical limit enabled by cyclic dynamics

Time-reversed evolution has substantial implications in physics, including applications in refocusing of classical waves or spins and fundamental studies such as quantum information scrambling. In quantum metrology, nonlinear interferometry based on time-reversal protocols supports entanglement-enhanced measurements without requiring low-noise detection. Despite the broad interest in this topic, it remains challenging to reverse the quantum dynamics of an interacting many-body system, which is typically realized by an (effective) sign flip of the system’s Hamiltonian. Here we present an approach that is broadly applicable to cyclic systems for implementing nonlinear interferometry without invoking time reversal. As time-reversed dynamics drives a system back to its starting point, we propose to accomplish the same by forcing the system to travel along a ‘closed loop’ instead of explicitly tracing back its antecedent path. Utilizing the quasiperiodic spin mixing dynamics in a three-mode 87 Rb atomic spinor condensate, we implement such a closed-loop nonlinear interferometer and achieve a metrological gain of 5.01_-0.76^+0.76 decibels over the classical limit for a total of 26,500 atoms. Our approach unlocks the potential of nonlinear interferometry by allowing the dynamics to penetrate into the deep nonlinear regime, which gives rise to highly entangled non-Gaussian states.