  Anderson localization is a general phenomenon that applies to a variety of
disordered physical systems. Recently, a novel manifestation of Anderson
localization for wave packets launched with a finite average velocity was
proposed, the Quantum boomerang effect (QBE). This phenomenon predicts that the
disorder-averaged center of mass of a particle initially moves ballistically,
then makes a U-turn, and finally slowly returns to its initial position. The
QBE has been predicted to take place in several Hermitian models with Anderson
localization and has been experimentally observed in the paradigmatic quantum
kicked rotor model. In this work, we investigate the emergence of the QBE in
non-Hermitian systems and clarify the importance of symmetries of the
Hamiltonian and the initial state. We generalize the analytical arguments
available in the literature and show that even in the case of complex spectrum
a boomerang-like behavior can appear in a non-Hermitian system. We confirm our
analytical results through a careful numerical investigation of the dynamics
for several non-Hermitian models. We find that non-Hermiticity leads to the
breakdown of the dynamical relation, though the QBE is preserved. This work
opens up new avenues for future investigations in Anderson localized systems.
The models studied here may be implemented using cold atoms in optical
lattices.
