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We derive a topological classification of the steady states of$d$-dimensional lattice models driven by $D$ incommensurate tones. Mapping to aunifying $(d+D)$-dimensional localized model in frequency space revealsanomalous localized topological phases (ALTPs) with no static analog. While theformal classification is determined by $d+D$, the observable signatures of eachALTP depend on the spatial dimension $d$. For each $d$, with $d+D=3$, weidentify a quantized circulating current, and corresponding topological edgestates. The edge states for a driven wire ($d=1$) function as a quantized,nonadiabatic energy pump between the drives. We design concrete models ofquasiperiodically driven qubits and wires that achieve ALTPs of severaltopological classes. Our results provide a route to experimentally accesshigher dimensional ALTPs in driven low-dimensional systems.

Nonadiabatic Topological Energy Pumps with Quasiperiodic Driving