  We propose a realistic cold-atom quantum setting where nontrivial energy-band
topology induces non-reciprocal pumping. This is an intriguing non-Hermitian
phenomenon that illustrates how topology, when assisted with atom loss, can act
as a "switch" for non-Hermitian skin effect (NHSE), rather than as a passive
property that is modified by the NHSE. In particular, we present a lattice
shaking scenario to realize a two-dimensional cold-atom platform, where
non-reciprocity is switched on only in the presence of both atom loss and
topological localization due to time-reversal symmetry breaking. Such
spontaneous non-reciprocal pumping is manifested by asymmetric dynamical
evolution, detectable by atomic populations along the system edges. Our results
may trigger possible applications in non-reciprocal atomtronics, where loss and
topological mechanisms conspire to control atomic transport.
