  We investigate the mean-field energy spectrum and dynamics in a Bose-Einstein
condensate in a double-well potential with non-Hermiticity from the
nonreciprocal hopping, and show that the interplay of nonreciprocity and
nonlinearity leads to exotic properties. Under the two-mode and mean-field
approximations, the nonreciprocal generalization of the nonlinear
Schr\"{o}dinger equation and Bloch equations of motion for this system are
obtained. We analyze the PT phase diagram and the dynamical stability of fixed
points. The reentrance of PT-symmetric phase and the reformation of stable
fixed points with increasing the nonreciprocity parameter are found. Besides,
we uncover a linear self-trapping effect induced by the nonreciprocity. In the
nonlinear case, the self-trapping oscillation is enhanced by the nonreciprocity
and then collapses in the PT-broken phase, and can finally be recovered in the
reentrant PT-symmetric phase.
