Tunable intertwining via collective excitations

The intertwining of multiple order parameters is a widespread phenomenon in equilibrium condensed matter systems, yet its exploration is often hindered by the complexity of real materials. Here, we present a controlled study of intertwined orders in a minimal driven-dissipative quantum-engineered platform. We consider a Bose-Einstein condensate at the intersection of two optical cavities, realizing two competing copies of a Z₂ symmetry-breaking superradiant phase transition characterized by density wave orders. We show that collective excitations emerging at the order-to-order transition can be harnessed via periodic driving to stabilize a variety of dynamical phases, including intertwined orders. Our drive exploits dynamical symmetry reduction to generate a tunable mass gap in the excitation spectrum, offering a versatile mechanism for engineering intertwined dynamical orders in quantum-optical platforms.