  Pumps are transport mechanisms in which direct currents result from a cyclic
evolution of the potential. As Thouless has shown, the pumping process can have
topological origins, when considering the motion of quantum particles in
spatially and temporally periodic potentials. However, the periodic evolution
that drives these pumps has always been assumed to be imparted from outside, as
was the case in the experimental systems studied so far. Here we report on an
emergent mechanism for pumping in a quantum gas coupled to an optical
resonator, where we observe a particle current without applying a periodic
drive. The pumping potential experienced by the atoms is formed by the
self-consistent cavity field interfering with the static laser field driving
the atoms. Due to dissipation, the cavity field evolves between its two
quadratures, each corresponding to a different centrosymmetric crystal
configuration. This self-oscillation results in a time-periodic potential
analogous to that describing the transport of electrons in topological
tight-binding models, like the paradigmatic Rice-Mele pump. In the experiment,
we directly follow the evolution by measuring the phase winding of the cavity
field with respect to the driving field and observing the atomic motion
in-situ. The discovered mechanism combines the dynamics of topological and open
systems, and features characteristics of continuous dissipative time crystals.
