  Far out-of-equilibrium many-body quantum dynamics in isolated systems
necessarily generate interferences beyond an Ehrenfest time scale, where
quantum and classical expectation values diverge. Of great recent interest is
the role these interferences play in the spreading of quantum information
across the many degrees of freedom, i.e.~scrambling. Ultracold atomic gases
provide a promising setting to explore these phenomena. Theoretically speaking,
the heavily-relied-upon truncated Wigner approximation leaves out these
interferences. We develop a semiclassical theory which bridges classical and
quantum concepts in many-body bosonic systems and properly incorporates such
missing quantum effects. For mesoscopically populated Bose-Hubbard systems, it
is shown that this theory captures post-Ehrenfest quantum interference
phenomena very accurately, and contains relevant phase information to perform
many-body spectroscopy with high precision.
