Scale invariance in integrable quantum oscillators' long-time dynamics

Scale invariance usually occurs in extended systems where correlation functions decay algebraically in space and/or time. Here we introduce a new type of scale invariance, occurring in distribution functions of physical observables. At equilibrium, these functions decay over a typical scale set by the temperature, but they can become scale invariant in a sudden quantum quench. We exemplify this effect through the analysis of an integrable quantum oscillator, and show that the associated critical exponent is universal. Our study opens the possibility to address integrability and its breaking in distribution functions, and has immediate applications to matter-wave interferometers.