Near-Deterministic Collective non-Linear Weak-Value Metrology

Weak-value amplification (WVA) involve post-selection measurements on an evolved state and using the measurements to enhance a parameter of interest. Since post-selection involves rejecting all the undesirable outcomes, there exists a tradeoff between the post-selection probability of the desired measurement and the amplification produced by the apparatus. This tradeoff has always limited the applicability of weak-value metrology. This issue was partially resolved by Pang and Brun who proposed that collective effects can enhance WVA. A full solution where a simultaneous superextensive enhancement in both the success probability and the weak value amplification is still lacking. we resolve this completely by considering non-linear collective Hamiltonians and demonstrate that they enhance existing WVA schemes. We demonstrate a linear enhancement in the weak-value and a simultaneous quadratic enhancement in the success probability