Topological frequency conversion

We study the problem of arbitrarily strong multi-tonal drive applied to non-linear systems. The dynamics has a natural representation in terms of ``transport" in a multi-dimensional Floquet space, with an applied ``electric" field (whose components are proportional to the drive frequencies). The number of the Floquet space dimensions equals the number of {\em irrationally} related drive frequencies. In particular, for two-tone drive, when the band structure in the 2D Floquet space is topologically non-trivial (has non-zero Chern number, $C$), we find that there is a topological pumping of energy between the frequencies $\omega_1$ and $\omega_2$, with the rate $P_{12} = -P_{21}= (C/2\pi)\hbar \omega_1\omega_2$. This pumping is the analog of the transverse response in a conventional topological insulator.