Two-parameter estimation in a matter-wave Sagnac interferometer

We explore the general characteristics of a matter-wave Sagnac interferometer in a two-parameter estimation scheme. We find that the measurement precisions of both parameters cannot reach the Heisenberg limit (HL) simultaneously when the input state is maximally entangled. Only one of the parameters' uncertainties can approach the HL while the other is scaled by the standard quantum limit. We provide the conditions with which the measurement precision of the specific parameter can reach the HL. We also discuss and figure out the concrete expressions of the constraint conditions for saturating the quantum Cramer-Rao bound. To satisfy these constraint conditions, the evolution time has to be a series of discrete values. Additionally, we calculate the variances of the parameters through some examples under these constraint conditions. The results provided in our work show some intrinsic features of the matter-wave Sagnac interferometer for the two-parameter estimation, which can be valuable in actual experiments.