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.
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.