Simultaneously estimating two phases in three-mode nonlinear interferometer

We propose a multi-mode nonlinear interferometers structure by cascading several multi-mode beam splitters (MBS) and nonlinear phase shifters (NPS), in which multiple phases can be estimated simultaneously. Specially, we mainly discuss two phases estimation in a three-mode nonlinear interferometer with single photons as inputs. The phase sensitivity of the three-mode nonlinear interferometer can be discussed by the quantum Fisher information matrix (QFIM) method after the states undergo the nonlinear phase shifters, and can also be studied by classical Fisher information matrix (CFIM) method in the interferometer's output ports. It is shown that the precision given by CFIM can achieve the bound of QFIM. Finally, we make a comparison with an optimal probe state in terms of the total variance and discuss the relationship of the total variance among three different phase estimation schemes.