Echoes and defects in the Calogero model

In this work, we extend the study of the interplay between scaling symmetries and statistics to one-dimensional fluids by studying the Calogero model in a harmonic trap modulated through time. The latter harbors an interpretation in terms of free particles imbued with exclusion statistics and is an example of a scale invariant fluid in one-dimension displaying $\text{SO}(2,1)$ dynamical symmetry preserved by harmonic traps. Taking advantage of the dynamical symmetry, two experimentally relevant drive protocols spanning both quasistatic and nonadiabatic regimes are investigated and universal signatures of the interactions and exclusion statistics are uncovered in the ground-state echo amplitude and closely related ground-state fidelity. In particular, under both periodic modulation and slow drive through the gapless point of the trap frequency, enhanced interactions and exclusion are shown to favor the proliferation of defects and to hinder their annihilation, which leads to a universal decrease of ground-state fidelities and echo amplitudes. We also show that increasing exclusion sparks a sharp suppression of the likelihood of intermediate echoes beyond those imposed by the commensurability of a periodic drive and the natural frequency of the trap.