Topological charge pumping with subwavelength Raman lattices

Recent experiments demonstrated deeply subwavelength lattices using atoms with N internal states Raman coupled with lasers of wavelength lambda. The resulting unit cell was lambda/2N in extent, an N-fold reduction compared to the usual lambda/2 periodicity of an optical lattice. For resonant Raman coupling, this lattice consists of N independent sinusoidal potentials (with period lambda/2) displaced by lambda/2N from each other. We show that detuning from Raman resonance induces tunneling between these potentials. Temporally modulating the detuning couples the s and p bands of the potentials, creating a pair of coupled subwavelength Rice-Mele chains. This operates as a topological charge pump that counterintuitively can give half the displacement per pump cycle of each individual Rice-Mele chain separately. We analytically describe this behavior in terms of infinite-system Chern numbers and numerically identify the associated finite-system edge states.