  Conformal symmetry heavily constrains the dynamics of non-relativistic
quantum gases tuned to a nearby quantum critical point. One important
consequence of this symmetry is that entropy production can be absent in far
away from equilibrium dynamics of strongly interacting three-dimensional (3D)
and one-dimensional (1D) quantum gases placed inside a soft harmonic trapping
potential. This can lead to an oscillatory fully revivable many-body dynamic
state, which is reflected in many physical observables. In this article we
further investigate the consequences of conformal symmetry on a) the
zero-temperature auto-correlation function, b) the Wigner distribution
function, and c) the Von Neumann entanglement entropy. A direct calculation of
these quantities for generic strongly interacting systems is usually extremely
difficult. However, we have derived the general structures of these functions
in the non-equilibrium dynamics when their dynamics are constrained by
conformal symmetry. We obtain our results for a) by utilizing an operator-state
correspondence which connects the imaginary time evolution of primary operators
to different initial states of harmonically trapped gases. While the dynamics
of the functions in b) and c) are derived from conformal invariant density
matrices.
