Exponential concentration of fluctuations in mean-field boson dynamics

We study the mean-field dynamics of a system of $N$ interacting bosons starting from an initially condensated state. For a broad class of mean-field Hamiltonians, including models with arbitrary bounded interactions and models with unbounded interaction potentials, we prove that the probability of having $n$ particles outside the condensate decays exponentially in $n$ for any finite evolution time. Our results strengthen previously known bounds that provide only polynomial control on the probability of having $n$ excitations.