Phonon anharmonicity-induced decoherence slowing down in exciton-phonon systems. 
Based on a generalized Fröhlich model, a time-convolutionless master equation is established for studying the dynamics of an exciton coupled with anharmonic phonons.  Special attention is paid to describing the influence of the phonon anharmonicity on specific elements of the exciton reduced density matrix.  These elements, called coherences, characterize the ability of the exciton to develop quantum states that are superimpositions involving the vacuum and the local one-exciton states.  Whether the phonons are harmonic or not, it is shown that dephasing limited-coherent motion takes place.  The coherences irreversibly decrease with time, the decay rate being the so-called dephasing rate, so that they experience a localization phenomenon and propagate over a finite length scale.  However, it is shown that the phonon anharmonicity softens the influence of the phonon bath and reduces the dephasing rate.  A slowdown in the decoherence process appears so that the coherences are able to explore a larger region along the lattice.  Moreover, the phonon anharmonicity modifies the way the dephasing rate depends on both the adiabaticity and the temperature.  In particular, the dephasing rate increases linearly with the temperature in the weak anharmonicity limit whereas it becomes almost temperature-independent in the strong anharmonicity limit.  Note that the present formalism is applied to describe amide-I excitons (vibrons) in a lattice of H-bonded peptide units.