  We demonstrate a platform for synthetic dimensions based on coupled Rydberg
levels in ultracold atoms, and we implement the single-particle
Su-Schrieffer-Heeger (SSH) Hamiltonian. Rydberg levels are interpreted as
synthetic lattice sites, with tunneling introduced through resonant
millimeter-wave couplings. Tunneling amplitudes are controlled through the
millimeter-wave amplitudes, and on-site potentials are controlled through
detunings of the millimeter waves from resonance. Using alternating weak and
strong tunneling with weak tunneling to edge lattice sites, we attain a
configuration with symmetry-protected topological edge states. The band
structure is probed through optical excitation to the Rydberg levels from the
ground state, which reveals topological edge states at zero energy. We verify
that edge-state energies are robust to perturbation of tunneling-rates, which
preserves chiral symmetry, but can be shifted by the introduction of on-site
potentials.
