  Activating transitions between a set of atomic internal states has emerged as
an elegant scheme by which lattice models can be designed in ultracold atomic
gases. In this approach, the internal states can be viewed as fictitious
lattice sites defined along a synthetic dimension, hence offering a powerful
method by which the spatial dimensionality of the system can be extended.
Inter-particle collisions generically lead to infinite-range interactions along
the synthetic dimensions, which a priori precludes the design of
Bose-Hubbard-type models featuring on-site interactions. In this article, we
solve this obstacle by introducing a protocol that realizes strong and tunable
"on-site" interactions along an atomic synthetic dimension. Our scheme is based
on pulsing strong intra-spin interactions in a fast and periodic manner, hence
realizing the desired "on-site" interactions in a digital (Trotterized) manner.
We explore the viability of this protocol by means of numerical calculations,
which we perform on various examples that are relevant to ultracold-atom
experiments. This general method, which could be applied to various atomic
species by means of fast-response protocols based on Fano-Feshbach resonances,
opens the route for the exploration of strongly-correlated matter in synthetic
dimensions.
