Effective-action formulation of $\mathbf{k}$-odd bianisotropy and unidirectional polarization rotation in nonlocal temporal metamaterials

We derive the effective action of temporal metamaterials up to first order in spatial derivatives through a multiscale analysis of the electromagnetic Lagrangian density. We show that the first-order spatial correction gives rise to an effective bianisotropic coupling tensor that is odd in the wavevector $\mathbf{k}$. Remarkably, chiral, pseudochiral, and omega-type bianisotropic responses emerge as distinct directional manifestations of a single nonlocal coupling tensor, even when the underlying electric and magnetic response tensors remain symmetric. Furthermore, we establish a framework for analyzing Lorentz reciprocity based on the Hessian of the effective action. We demonstrate that Lorentz reciprocity is preserved in the presence of omega-type coupling but is broken when chiral or pseudochiral coupling emerges. This nonreciprocity manifests as the unidirectional rotations of normalized Stokes vectors incurred by Floquet-engineered chirality and pseudochirality.