Topological pumping of light governed by Fibonacci numbers

Abstract Topological pumping refers to transfer of a physical quantity governed by the system topology, resulting in quantized amounts of the transferred quantities. It is a ubiquitous wave phenomenon typically considered subject to exactly periodic adiabatic variation of the system parameters. Recently, proposals for generalizing quasi-periodic topological pumping and identifying possible physical settings for its implementation have emerged. In a strict sense, pumping with incommensurate frequencies can only manifest over infinite evolution distances, raising a fundamental question about its observability in real-world finite-dimensional systems. Here we demonstrate that bi-chromatic topological pumping with two frequencies, whose ratio is an irrational number, can be viewed as the convergence limit of pumping with two commensurate frequencies representing the best rational approximations of that irrational number. In our experiment, this phenomenon is observed as the displacement of a light beam center in photorefractive crystals induced by two optical lattices. The longitudinal periods of the lattices, that in the paraxial approximation emulate two pumping frequencies, are related as Fibonacci numbers, successively approaching the golden ratio. We observed that a one-cycle displacement of the beam center at each successive approximation is determined by the relation between successive Fibonacci numbers, while the average direction of propagation (emulating average pumping velocity) of the beam is determined by the golden ratio.