Bloch oscillations, Landau-Zener transition and the evolution of topological phase in the bulk of classical pendula arrays

We analyze theoretically and experimentally the dynamics of a one-dimensional array of pendula that have a mild gradient in their self-frequency and where neighboring pendula are linearly coupled with a weak and alternating coupling strength. We show that the dynamics of this classical system map onto the quantum Su-Schrieffer-Heeger (SSH) model of electrons on a lattice in the presence of an electric field. As a result, an initial localized excitation evolves according to a Schr"{o}dinger-like equation and shows phenomena such as Bloch oscillations, Landau-Zener (LZ) transition, and coupling between the isospin and the spatial degrees of freedom. The mapping of the pendula system in the weak coupling limit to the SSH model quantitatively predicts these phenomena over a broad range of the parameter values. Consequently, we use Bloch oscillations in the adiabatic regime to directly measure the non-trivial topological phase winding of the band in two ways: by comparing the phase evolution in two systems with interchanged couplings, and by comparing the phase evolution of the two isospin components in a single system's wave evolution.