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Using Time-Dependent Density Matrix Renormalization Group (TDMRG) we studythe collision of one-dimensional atomic clouds confined in a harmonic trap andevolving with the Lieb-Liniger Hamiltonian. It is observed that the motion isessentially periodic with the clouds bouncing elastically, at least on the timescale of the first few oscillations that can be resolved with high accuracy.This is in agreement with the results of the "quantum Newton cradle" experimentof Kinoshita et al. [Nature 440, 900 (2006)]. We compare the results for thedensity profile against a hydrodynamic description, or generalized nonlinearSchr\"odinger equation, with the pressure term taken from the Bethe Ansatzsolution of the Lieb-Liniger model. We find that hydrodynamics can describe thebreathing mode of a harmonically trapped cloud for arbitrary long times whileit breaks down almost immediately for the collision of two clouds due to theformation of shock waves (gradient catastrophe). In the case of the clouds'collision TDMRG alone allows to extract the oscillation period which is foundto be measurably different from the breathing mode period. Concomitantly withthe shock waves formation we observe a local energy distribution typical ofpopulation inversion, i.e., an effective negative temperature. Our results arean important step towards understanding the hydrodynamics of quantum many-bodysystems out of equilibrium and the role of integrability in their dynamics.

Quantum Shock Waves and Population Inversion in Collisions of Ultracold Atomic Clouds