We demonstrate the periodic interconversion evolution of the double-charge vortex (DCV) and out-of-phase (OOP) quadruple solitons in two-dimensional (2D) optically induced Lieb photonic lattices by single-site excitation under appropriate self-focusing nonlinearity conditions. The numerical results show that the DCV evolves into a rotating quasi-vortex soliton, utilizing an OOP quadrupole-like beam as a transition state to reverse the topological charge and the direction of rotation periodically; The OOP quadrupole beam whose incident orientation along the crystalline c-axis or the diagonal axes of the lattices self-traps into a stable localized mode nearly invariant during propagation, whose Fourier spectrum reveals that they are localized states close to the high-symmetry points $X$ and $M$ of Brillouin Zone. By imposing an initial orientation slightly twisted relative to the lattices’ crystalline c-axis, the localized quadruple state tends to evolve into a dynamical rotating DCV and undergoes periodic charge-flipping when the direction of rotation is reversed. Additionally, analysis of different twisting angle shows that the oscillation period will become longer when the initial orientation approaches the lattices’ crystalline c-axis. Our findings could give some quantitative insight into the experimental feasibility of observing such phenomena.

我们展示了在适当的自聚焦非线性条件下通过单中心激发在二维（2D）光诱导的利勃光子晶格中双电荷涡（DCV）和异相（OOP）四重孤子的周期互变演化。数值结果表明，DCV利用OOP四极杆状光束作为过渡态周期性地反转拓扑电荷和旋转方向，演变成旋转的准涡旋孤子。OOP四极光束的入射方向沿着晶格c轴或晶格的对角线方向，它们在传播过程中自陷成几乎不变的稳定局域模，其傅立叶光谱表明它们是接近高对称点的局域状态$X$ 和 $\mathrm{中号}$布里渊区。通过施加相对于晶格的晶体c轴略微扭曲的初始方向，局部四重态趋向于演变成动态旋转的DCV，并在旋转方向相反时经历周期性的电荷翻转。另外，对不同扭转角的分析表明，当初始取向接近晶格的晶体c轴时，振荡周期将变长。我们的发现可以为观察这种现象的实验可行性提供一些定量的见解。