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Temperature-resolution anomalies in the reconstruction of time dynamics from energy-loss experiments

Inelastic scattering techniques provide a powerful approach to studying electron and nuclear dynamics, via reconstruction of a propagator that quantifies the time evolution of a system. There is now growing interest in applying such methods to very low energy excitations, such as lattice vibrations, but in this limit the cross section is no longer proportional to a propagator. Significant deviations occur due to the finite temperature Bose statistics of the excitations. Here we consider this issue in the context of high-resolution electron energy loss experiments on the copper-oxide superconductor Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$. We find that simple division of a Bose factor yields an accurate propagator on energy scales greater than the resolution width. However, at low energy scales, the effects of resolution and finite temperature conspire to create anomalies in the dynamics at long times. We compare two practical ways for dealing with such anomalies, and discuss the range of validity of the technique in light of this comparison.