  Synthetic dimensions provide a powerful approach for simulating condensed
matter physics in cold atoms and photonics, whereby a set of discrete degrees
of freedom are coupled together and re-interpreted as lattice sites along an
artificial spatial dimension. However, atomic experimental realisations have
been limited so far by the number of artificial lattice sites that can be
feasibly coupled along the synthetic dimension. Here, we experimentally realise
for the first time a very long and controllable synthetic dimension of atomic
harmonic trap states. To create this, we couple trap states by dynamically
modulating the trapping potential of the atomic cloud with patterned light. By
controlling the detuning between the frequency of the driving potential and the
trapping frequency, we implement a controllable force in the synthetic
dimension. This induces Bloch oscillations in which atoms move periodically up
and down tens of atomic trap states. We experimentally observe the key
characteristics of this behaviour in the real space dynamics of the cloud, and
verify our observations with numerical simulations and semiclassical theory.
This experiment provides an intuitive approach for the manipulation and control
of highly-excited trap states, and sets the stage for the future exploration of
topological physics in higher dimensions.
