  Understanding quantum many-body systems with long-range or infinite-range
interactions is of relevance across a broad set of physical disciplines,
including quantum optics, nuclear magnetic resonance and nuclear physics. From
a theoretical viewpoint, these systems are appealing since they can be
efficiently studied with numerics, and in the thermodynamic limit are expected
to be governed by mean-field equations of motion. Over the past years the
capabilities to experimentally create long-range interacting systems have
dramatically improved permitting their control in space and time. This allows
to induce and explore a plethora of nonequilibrium dynamical phases, including
time-crystals and even chaotic regimes. However, establishing the emergence of
these phases from numerical simulations turns out to be surprisingly
challenging. This difficulty led to the assertion that mean-field theory may
not be applicable to time-dependent infinite-range interacting systems. Here,
we rigorously prove that mean-field theory in fact exactly captures their
dynamics, in the thermodynamic limit. We further provide bounds for finite-size
effects and their dependence on the evolution time.
