Topological charge pumping with subwavelength Raman lattices

Abstract

Recent experiments demonstrated deeply subwavelength lattices using atoms with $N$ internal states Raman coupled with lasers of wavelength $\ensuremath{\lambda}$. The resulting unit cell was $\ensuremath{\lambda}/2N$ in extent, an $N$-fold reduction compared to the usual $\ensuremath{\lambda}/2$ periodicity of an optical lattice. For resonant Raman coupling, this lattice consists of $N$ independent sinusoidal potentials (with period $\ensuremath{\lambda}/2$) displaced by $\ensuremath{\lambda}/2N$ 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.