Optimizing the trade off between precision and probability in post-selection estimation of the coupling of trapped ion system

Abstract

In post-selection quantum measurement, it is a dilemma to obtain both high precision and large post-selection probability. Here,we deeply examine quantum Fisher information and probability of post-selection quantum measurement based on the trapped ion system, where the internal electronic state of the trapped ion acts as an interested system and the external motional state as a pointer. Based on the bichromatic excitation of the trapped ion, in the Lamb-Dicke regime we observe the dynamical behaviors of quantum Fisher information of estimating coupling parameter and post-selection probability under the various phase difference of the bichromatic laser. We find that the oscillations of quantum Fisher information and post-selection probability can be simultaneously enhanced by selecting appropriately the phase difference of the bichromatic laser, the amplitude and the phase of the pointer state. Although the competition between quantum Fisher information and the post-selection probability still exists in the whole coupling range, a trade-off point in the moderate-coupling regime where the precision and the probability surpass their convergence values in the strong coupling regime can be obtained. It means that there is a possibility that the desired precision and post-selection probability can be simultaneously achieved.