Breathing Modes as a Probe of Energy Fluctuations in a Unitary Fermi Gas

Abstract

Directly accessing energy fluctuations in interacting quantum many-body systems remains a long-standing challenge, especially far from equilibrium. Here we show that in scale-invariant quantum gases with SO$(2,1)$ dynamical symmetry, the amplitude of the breathing mode provides a direct and quantitative probe of energy fluctuations. We establish an exact and universal relation between the oscillation amplitude and the energy fluctuation, with a dimensionless ratio fixed solely by the Bargmann index $k$, which labels the irreducible representation of the underlying SU$(1,1)$ algebra and thereby determines the structure of the many-body spectrum and dynamics. As a consequence, this relation is fully dictated by symmetry and remains independent of microscopic details and excitation protocols. Furthermore, we show that the excitation of breathing-mode states follows a universal statistical distribution governed by a single parameter, independent of the specific driving protocol. Our findings demonstrate that energy fluctuations, typically encoded in the many-body spectrum, can be directly accessed through collective dynamics, offering a symmetry-based route to probe nonequilibrium energy statistics in strongly interacting quantum systems.