  Tunnelling of material particles through a classically impenetrable barrier
constitutes one of the hallmark effects of quantum physics. When interactions
between the particles compete with their mobility through a tunnel junction,
intriguing novel dynamical behaviour can arise where particles do not tunnel
independently. In single-electron or Bloch transistors, for example, the
tunnelling of an electron or Cooper pair can be enabled or suppressed by the
presence of a second charge carrier due to Coulomb blockade. Here we report on
the first direct and time-resolved observation of correlated tunnelling of two
interacting atoms through a barrier in a double well potential. We show that
for weak interactions between the atoms and dominating tunnel coupling,
individual atoms can tunnel independently, similar to the case in a normal
Josephson junction. With strong repulsive interactions present, two atoms
located on one side of the barrier cannot separate, but are observed to tunnel
together as a pair in a second order co-tunnelling process. By recording both
the atom position and phase coherence over time, we fully characterize the
tunnelling process for a single atom as well as the correlated dynamics of a
pair of atoms for weak and strong interactions. In addition, we identify a
conditional tunnelling regime, where a single atom can only tunnel in the
presence of a second particle, acting as a single atom switch. Our work
constitutes the first direct observation of second order tunnelling events with
ultracold atoms, which are the dominating dynamical effect in the strongly
interacting regime. Similar second-order processes form the basis of
superexchange interactions between atoms on neighbouring lattice sites of a
periodic potential, a central component of quantum magnetism.
