# Creating lattice Hamiltoninans in multidimensional Floquet spaces

Driving a system with multiple incommensurate frequencies leads to Floquet lattices in multiple dimensions. Various lattice Hamiltonians can be implemented, including topologically non-trivial ones. The number of bands can be controlled by choosing appropriate building blocks -- two level systems will lead to two-band Hamiltonians, 3-level -- to 3 band, etc.

The distinguishing feature of such Floquet lattice Hamiltonians is that they are **tilted**, with the tilt potential $U(n_1, n_2, ...) = n_1\omega_1 + n_2\omega_2 + ...$. That has several consequences:

- the wave-functions are localized in the direction perpendicular to the hyperplane $U(n_1, n_2, ...) = const$.
- for weak drive the wavefunction will be localized on the lattice points closest to the hyperplanes. The lattice potential at these points is quasiperiodic.
- for strong drive, w.f. will be weakly confined, as if they are near a smooth edge of a sample

- if band structure is topologically non-trivial, every hyperplane is also a locus of topologically protected "edge-modes." For instance, in the case of two-dim Floquet space, there will be helical modes propagating along the lines $n_1\omega_1 + n_2\omega_2 = const$. Chiral propagation corresponds to preferential absorption of one frequency photon and emission of the other.
- Even when the drive frequencies are commensurate, the pumping between the modes persists, as can be directly verified by the transfered power calculation.