  We show that two-body losses induce a collective excitation in a harmonically
trapped one-dimensional Bose gas, even in the absence of a quench in the trap
or any other external perturbation. Focusing on the dissipatively fermionized
regime, we perform an exact mode expansion of the rapidity distribution
function and characterize the emergence of the collective motion. We find clear
coherent oscillations in both the potential and kinetic energies as well as in
the phase space quadrupole mode of the gas. We also discuss how this loss
induced collective mode differs from the well known breathing mode studied in
the absence of dissipation.
