Synthetic dimensions alter one of the most fundamental properties in nature, the dimension of space. They allow, for example, a low-dimensional system to act as effectively higher-dimensional. Experiments on ultracold systems create synthetic dimensions using internal or external degrees of freedom of particles for highly controllable quantum simulation.

We consider two methods to create synthetic dimensions in ultracold gases - momentum states of ultracold atoms, and rotational states of ultracold dipolar molecules. In the atomic system with the momentum-state lattice, which has been realized experimentally in the Gadway group, pairs of Raman lasers drive momentum-state transitions, realizing tunnelings in the synthetic lattice. In the molecular system, microwaves can be used to induce rotational-state transitions, realizing tunnelings in the synthetic lattice which can span hundreds of sites. Both systems can show many-body physics due to strong interactions arising respectively from contact interactions and dipolar interactions. We discuss the many-body physics of these systems, ranging from momentum-dependent self-trapping that has been experimentally observed in the atomic systems, to a novel string phase that is theoretically predicted to occur in the molecular systems.