Tomonaga–Luttinger liquids (TLLs) can be used to effectively describe one-dimensional quantum many-body systems such as ultracold atoms, charges in nanowires, superconducting circuits, and gapless spin chains. Their properties are given by two parameters, the propagation velocity and the Luttinger parameter. Here we study inhomogeneous TLLs where these are promoted to functions of position and demonstrate that they profoundly affect the dynamics: in general, besides curving the light cone, we show that propagation is no longer ballistically localized to the light-cone trajectories, different from standard homogeneous TLLs. Specifically, if the Luttinger parameter depends on position, the dynamics features pronounced spreading into the light cone, which cannot be understood via a simple superposition of waves as in the Huygens–Fresnel principle. This is the case for ultracold atoms in a parabolic trap, which serves as our main motivation, and we discuss possible experimental observations in such systems.
