Non-Floquet engineering in periodically driven dissipative open quantum systems

Floquet engineering plays a key role in realizing novel dynamical topological states. The conventional Floquet engineering, however, only applies to time-periodic non-dissipative Hermitian systems, and for the open quantum systems, non-Hermitian processes usually occur. So far, it remains unclear how to characterize the topological phases of time-periodic open quantum systems via the frequency space Floquet Hamiltonian. Here, we propose the non-Floquet theory to solve the problem and illustrate it by a continuously time-periodic non-Hermitian bipartite chain. In non-Floquet theory, a temporal non-unitary transformation is exercised on the Floquet states, and the transformed Floquet spectrum restores the form of the Wannier-Stark ladder. Besides, we also show that different choices of the starting points of the driving period can result in different localization behavior, effects of which can reversely be utilized to design quantum detectors of phases in dissipative oscillating fields. Our methods are capable of describing topological features in dynamical open quantum systems with various driving types and can find its applications to construct new types of dynamical topological materials.