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We introduce a general method to engineer arbitrary Hamiltonians in theFloquet phase space of a periodically driven oscillator, based on thenon-commutative Fourier transformation (NcFT) technique. We establish therelationship between an arbitrary target Floquet Hamiltonian in phase space andthe periodic driving potential in real space. We obtain analytical expressionsfor the driving potentials in real space that can generate novel Hamiltoniansin phase space, e.g., rotational lattices and sharp-boundary well. Our protocolcan be realised in a range of experimental platforms for nonclassical statesgeneration and bosonic quantum computation.

Engineering Arbitrary Hamiltonians in Phase Space