  The system of a charged impurity in an interacting Bose gas has gained
significant attention due to the long-range ion-atom interactions and the study
of transport properties. Here, the ground state energy of a charged Bose
polaron is calculated within the Bogoliubov approximation for both the
Fr\"ohlich and beyond-Fr\"ohlich Hamiltonians using a generalized Feynman
variational path-integral approach, which obtained accurate results for other
polaron problems. The generalized approach, which was used to improve the
energy result for the neutral polaron, has resulted in a minor improvement,
indicating that Feynman's approach is sufficient when the impurity-boson
interaction is long-range. Beyond-Fr\"ohlich corrections results in the
emergence of a divergence in the polaronic energy indicating a transition
between the repulsive and attractive polaron regime. The path-integral approach
with the beyond-Fr\"ohlich Hamiltonian is also compared to a field-theory
calculation from Christensen et al, 2021. The validity of the Bogoliubov
approximation is investigated. The optical absorption has also been calculated
within the Bogoliubov approximation for weak ion-atom interactions, and the
effect of finite temperature has been studied. We show that the coupling of the
ion to an oscillating external electric field offers a straigtforward
experimental probe for the charged polaron in a Bose gas, different from but
complementary to existing spectroscopic techniques.
