Authors: Felipe Herrera, Kirk W. Madison, Roman V. Krems and Mona BerciuThe study of a single atomic impurity embedded in a BEC or Fermi gas has led to the observation of polaron phenomena in the regime of weak Fröhlich-type particle-boson interaction. The implementation of generalized polaron models in different parameter regimes using cold gases would allow access to physics that is presumably unreachable in the solid state. Using a robust semi-analytic technique, we determine the phase diagram of a polaron model with mixed breathing-mode and Su-Schrieffer-Heeger couplings and show that it has two sharp transitions, in contrast to pure models which exhibit one (for Su-Schrieffer-Heeger coupling) or no (for breathing-mode coupling) transition. Our results indicate that the physics of realistic mixed polaron models is much richer than that of simplified models. We then show that ultracold molecules trapped in optical lattices can be used to study precisely this mixed Hamiltonian, and that the relative contributions of the two couplings can be tuned with external electric fields. The parameters of current experiments with ultra-cold polar molecules place them in the region where one of the transitions occurs. We propose a scheme to measure the polaron dispersion using stimulated Raman spectroscopy.