  In this work, we have studied the non-Hermitian nonlinear LZSM interferometry
in a non-Hermitian N-body interacting boson system in which the non-Hermicity
is from the nonreciprocal tunnelings between the bosons. By using the
mean-field approximation and projective Hilbert space, the effect of
nonreciprocity and nonlinearity on the energy spectrum, the dynamics, and the
formation of the interference fringes have been studied. The different
symmetries and the impact of the two different types of reciprocity, i.e. the
in-phase tunneling and anti-phase tunneling, on the energy spectrum and the
phase transition between the Josephson oscillation and the self-trapping have
been investigated. For the LZSM interferometry, the strength of the
nonreciprocity is found to take an essential role in the population of the
projective state and the strengths of the interference patterns in the
projective space. While the conditions of destructive and constructive
interference under the weak-coupling approximation still only depend on the
strength of nonlinearity. Our result provides an application of the nonlinear
non-Hermitian LZSM interferometry in studying the parameters of a non-Hermitian
nonlinear two-level system which related to the nonlinearity and the
non-Hermicity.
