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We propose a `Floquet engineering' formalism to systematically design aperiodic driving protocol in order to stroboscopically realize the desiredsystem starting from a given static Hamiltonian. The formalism is applicable toquantum systems which have an underlying closed Lie-algebraic structure, forexample, solid-state systems with noninteracting particles moving on a latticeor its variant described by the ultra-cold atoms moving on an optical lattice.Unlike previous attempts at Floquet engineering, our method produces thedesired Floquet Hamiltonian at any driving frequency and is not restricted tothe fast or slow driving regimes. The approach is based on Wei-Norman ansatz,which was originally proposed to construct a time-evolution operator for anyarbitrary driving. Here, we apply this ansatz to the micro-motion dynamics,defined within one period of the driving, and obtain the driving protocol byfixing the gauge of the micro-motion. To illustrate our idea, we use a two-bandsystem or the systems consisting of two sub-lattices as a testbed.Particularly, we focus on engineering the cross-stitched lattice model that hasbeen a paradigmatic flat-band model.

Floquet Engineering of Lie Algebraic Quantum Systems