When a mobile impurity interacts with a surrounding bath of bosons, it forms a polaron. Numerous methods have been developed to calculate how the energy and the effective mass of the polaron are renormalized by the medium for equilibrium situations. Here we address the much less studied non-equilibrium regime and investigate how polarons form dynamically in time. To this end, we develop a time-dependent renormalization group approach which allows calculations of all dynamical properties of the system and takes into account the effects of quantum fluctuations in the polaron cloud. We apply this method to calculate trajectories of polarons following a sudden quench of the impurity-boson interaction strength, revealing how the polaronic cloud around the impurity forms in time. Such trajectories provide additional information about the polarons properties which are challenging to extract directly from the spectral function measured experimentally using ultracold atoms. At strong couplings, our calculations predict the appearance of trajectories where the impurity wavers back at intermediate times as a result of quantum fluctuations. Our method is applicable to a broader class of non-equilibrium problems. We also apply it to calculate the spectral function and find good agreement with experimental results. At very strong couplings, we predict that quantum fluctuations lead to the appearance of a dark continuum with strongly suppressed spectral weight at low energies. While our calculations start from an effective Frohlich Hamiltonian describing impurities in a three-dimensional Bose-Einstein condensate, we also calculate the effects of additional terms in the Hamiltonian beyond the Frohlich paradigm. We demonstrate that the main effect of these additional terms on the attractive side of a Feshbach resonance is to renormalize the coupling strength of the effective Frohlich model.
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