  We explore Loschmidt echo in two regimes of one-dimensional (1D) interacting
Bose gases: the strongly interacting Tonks-Girardeau (TG) regime, and the
weakly-interacting mean-field regime. We find that the Loschmidt echo of a TG
gas decays as a Gaussian when small perturbations are added to the Hamiltonian
(the exponent is proportional to the number of particles and the magnitude of a
small perturbation squared). In the mean-field regime the Loschmidt echo decays
faster for larger interparticle interactions (nonlinearity), and it shows
richer behavior than the TG Loschmidt echo dynamics, with oscillations
superimposed on the overall decay.
