  Mobile impurity atoms immersed in Bose-Einstein condensates provide a new
platform for exploring Bose polarons. Recent experimental advances in the field
of ultracold atoms make it possible to realize such systems with highly tunable
microscopic parameters and to explore equilibrium and dynamical properties of
polarons using a rich toolbox of atomic physics. In this paper we present a
detailed theoretical analysis of Bose polarons in one dimensional systems of
ultracold atoms. By combining a non-perturbative renormalization group approach
with numerically exact diffusion Monte Carlo calculations we obtain not only
detailed numerical results over a broad range of parameters but also
qualitative understanding of different regimes of the system. We find that an
accurate description of Bose polarons requires the inclusion of two-phonon
scattering terms which go beyond the commonly used Fr\"ohlich model.
Furthermore we show that when the Bose gas is in the strongly interacting
regime, one needs to include interactions between the phonon modes. We use
several theoretical approaches to calculate the polaron energy and its
effective mass. The former can be measured using radio-frequency spectroscopy
and the latter can be studied experimentally using impurity oscillations in a
harmonic trapping potential. We compare our theoretical results for the
effective mass to the experiments by Catani et al. [PRA 85, 023623 (2012)]. In
the weak-to-intermediate coupling regimes we obtain excellent quantitative
agreement between theory and experiment, without any free fitting parameter. We
supplement our analysis by full dynamical simulations of polaron oscillations
in a shallow trapping potential. We also use our renormalization group approach
to analyze the full phase diagram and identify regions that support repulsive
and attractive polarons, as well as multi-particle bound states.
