  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.
