  Though the celebrated spin echoes have been widely used to reverse quantum
dynamics, they are not applicable to systems whose constituents are beyond the
control of the su(2) algebra. Here, we design echoes to reverse quantum
dynamics of breathers in three-dimensional unitary fermions and two-dimensional
bosons and fermions with contact interactions, which are governed by an
underlying su(1,1) algebra. Geometrically, SU(1,1) echoes produce closed
trajectories on a single or multiple Poincare disks and thus could recover any
initial states without changing the sign of the Hamiltonian. In particular, the
initial shape of a breather determines the superposition of trajectories on
multiple Poincare disks and whether the revival time has period multiplication.
Our work provides physicists with a recipe to tailor collective excitations of
interacting many-body systems.
