  Including the effect of lattice anharmonicity on electron-phonon interactions
has recently garnered attention due to its role as a necessary and significant
component in explaining various phenomena, including superconductivity, optical
response, and temperature dependence of mobility. This study focuses on
analytically treating the effects of anharmonic electron-phonon coupling on the
polaron self-energy, combined with numerical Diagrammatic Monte Carlo data.
Specifically, we incorporate a quadratic interaction into the method of
squeezed phonon states, which has proven effective for analytically calculating
the polaron parameters. Additionally, we extend this method to nonparabolic
finite-width conduction bands while maintaining the periodic translation
symmetry of the system. Our results are compared with those obtained from
Diagrammatic Monte Carlo, partially reported in a recent study [Phys. Rev. B
107, L121109(2023)], covering a wide range of coupling strengths for the
nonlinear interaction. Remarkably, our analytic method predicts the same
features as the Diagrammatic Monte Carlo simulation.
