  We study inter-chain pair tunnelling dynamics based on an exact two-particle
solution for a two-leg ladder. We show that the Hermitian Hamiltonian shares a
common two-particle eigenstate with a corresponding non-Hermitian Hubbard
Hamiltonian in which the non-Hermiticity arises from an on-site interaction of
imaginary strength. Our results provides that the dynamic processes of
two-particle collision and across-legs tunnelling are well described by the
effective non-Hermitian Hubbard Hamiltonian based on the eigenstate
equivalence. We also find that any common eigenstate is always associated with
the emergence of spectral singularity in the non-Hermitian Hubbard model. This
result is valid for both Bose and Fermi systems and provides a clear physical
implication of the non-Hermitian Hubbard model.
