Analytical study of static beyond-Fröhlich Bose polarons in one dimension

F. Grusdt et al. [New J. Phys. 19, 103035 (2017)NJOPFM1367-263010.1088/1367-2630/aa8a2e] recently made a renormalization-group study of a one-dimensional Bose polaron in cold atoms. Their study went beyond the usual Fröhlich description, which includes only single-phonon processes, by including two-phonon processes in which two phonons are simultaneously absorbed or emitted during impurity scattering [Y. E. Shchadilova, Phys. Rev. Lett. 117, 113002 (2016)PRLTAO0031-900710.1103/PhysRevLett.117.113002]. We study this same beyond-Fröhlich 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 Fröhlich 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 Fröhlich paradigm but also beyond Bogoliubov theory, just as the multimode coherent state has helped with the study of Fröhlich polarons.