Joint commensuration in moir\'e charge-order superlattices drives shear topological defects

The advent of two-dimensional moir\'e systems has revolutionized the exploration of phenomena arising from strong correlations and nontrivial band topology. Recently, a moir\'e superstructure formed by two coexisting charge density waves (CDWs) with slightly mismatched wavevectors has been realized. These incommensurate CDWs can collectively exhibit commensurability, resulting in the jointly commensurate CDW (JC-CDW) and establishing a new paradigm for controlling moir\'e potential and periodicity. Achieving such functionality, however, hinges on a key open question: how do the amplitude, phase coherence, and periodicity of this order respond to external perturbations? Here, we address this question using a suite of time- and momentum-resolved diffraction and spectroscopic techniques to probe light-induced CDW dynamics in EuTe$_4$. Our time-resolved diffraction measurements distinguish the instantaneous quenching of the JC-CDW amplitude, as verified by time-resolved photoemission spectroscopy, from the much slower evolution of phase fluctuations. Furthermore, while the JC-CDW wavevector remains locked along the CDW direction upon photoexcitation, indicating a preserved moir\'e periodicity, the correlation length of JC-CDW shows an exclusive reduction perpendicular to its wavevector, unveiling the formation of previously unexplored shear-type defects. Together, this multimodal methodology reconstructs the spatiotemporal evolution of the JC-CDW upon excitation. These findings not only highlight the remarkable robustness of JC-CDWs out of equilibrium, but also provide insight into optical manipulation and engineering of moir\'e quantum materials through defect control.