  We investigate the Landau-Zener tunneling (LZT) of a self-interacting
two-level system in which the coupling between the levels is nonreciprocal. In
such a non-Hermitian system, when the energy bias between two levels is
adjusted very slowly, i.e., in the adiabatic limit, we find that a quantum
state can still closely follow the eigenstate solution until it encounters the
exceptional points (EPs) at which two eigenvalues and their corresponding
eigenvectors coalesce. In the absence of the nonlinear self-interaction, we can
obtain explicit expressions for the eigenvectors and eigenvalues and
analytically derive the adiabatic LZT probability from invariants at EPs. In
the presence of the nonlinear interaction, the dynamics of the adiabatic
evolutions are explicitly demonstrated with the help of classical trajectories
in the plane of the two canonical variables of the corresponding classical
Josephson Hamiltonian. We show that the adiabatic tunneling probabilities can
be precisely predicted by the classical action at EPs in the weak nonreciprocal
regime. In a certain region of strong nonreciprocity, we find that
interestingly, the nonlinear interaction effects can be completely suppressed
so that the adiabatic tunneling probabilities are identical to their linear
counterparts. We also obtain a phase diagram for large ranges of nonreciprocity
and nonlinear interaction parameters to explicitly demonstrate where the
adiabaticity can break down, i.e., the emergence of the nonzero tunneling
probabilities even in adiabatic limit.
