  Grusdt et al. [New J. Phys. 19, 103035 (2017)] recently made a
renormalization group study of a one-dimensional Bose polaron in cold atoms.
Their study went beyond the usual Frohlich description, which includes only
single-phonon processes, by including two-phonon processes in which two phonons
are simultaneously absorbed or emitted during impurity scattering [Shchadilova
et al. Phys. Rev. Lett. 117, 113002 (2016)]. We study this same beyond-Frohlich
model, but in the static impurity limit where the ground state is described by
a multimode squeezed state instead of the multimode coherent state in the
static Frohlich model. We solve the system exactly by applying the generalized
Bogoliubov transformation, an approach that can be straightforwardly adapted to
higher dimensions. Using our exact solution, we obtain a polaron energy free of
infrared divergences and construct analytically the polaron phase diagram. We
find that the repulsive polaron is stable on the positive side of the
impurity-boson interaction but is always thermodynamically unstable on the
negative side of the impurity-boson interaction, featuring a bound state, whose
binding energy we obtain analytically. We find that the attractive polaron is
always dynamically unstable, featuring a pair of imaginary energies which we
obtain analytically. We expect the multimode squeezed state to help with
studies that go not only beyond the Frohlich paradigm but also beyond
Bogoliubov theory, just as the multimode coherent state has helped with the
study of Frohlich polarons.
