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Floquet eigenspectra of a nonlinear two-mode system under periodic driving: the emergence of "ring" structures

We study Floquet eigenspectra of a nonlinear two-mode system under a periodic driving of the off-diagonal coupling. By solving the Gross-Pitaevskii equation numerically, we obtain triangular and loop structures near the crossings of different Floquet branches. At lower driving frequencies, we find "ring" and "double-ring" structures which are distinct from the well-known loop structure. The mechanism of the emergence of these structures is discussed and the parameter windows of their existence are obtained analytically. In addition, we study the evolution of the system under the driving with an adiabatic sweep and find there are some dynamically unstable states in the Floquet eigenspectra which break the quantum adiabaticity.