  We propose a systematic approach to the non-equilibrium dynamics of strongly
interacting many-body quantum systems, building upon the standard perturbative
expansion in the Coulomb interaction. High order series are derived from the
Keldysh version of determinantal diagrammatic Quantum Monte Carlo, and the
reconstruction beyond the weak coupling regime of physical quantities is
obtained by considering them as analytic functions of a complex-valued
interaction $U$. Our advances rely on two crucial ingredients: i) a conformal
change of variable, based on the approximate location of the singularities of
these functions in the complex $U$-plane; ii) a Bayesian inference technique,
that takes into account additional known non-perturbative relations, in order
to control the amplification of noise occurring at large $U$. This general
methodology is applied to the strongly correlated Anderson quantum impurity
model, and is thoroughly tested both in- and out-of-equilibrium. In the
situation of a finite voltage bias, our method is able to extend previous
studies, by bridging with the regime of unitary conductance, and by dealing
with energy offsets from particle-hole symmetry. We also confirm the existence
of a voltage splitting of the impurity density of states, and find that it is
tied to a non-trivial behavior of the non-equilibrium distribution function.
Beyond impurity problems, our approach could be directly applied to
Hubbard-like models, as well as other types of expansions.
