Two-phonon absorption operator in spectroscopy

While the raising operator â⁺ does not have eigenvectors, coherent states are eigenvectors of the lowering operator â. Therefore, operating with the operator exp[kâ⁺²] on coherent states poses a challenge, where k is an arbitrary constant. The result of this operation on coherent states has not yet been reported in a closed form. A closed-form expression for this problem is derived. The applicability and correctness of the result are tested by calculating the electronic dipole moment time correlation function and the corresponding Franck-Condon factors for systems with distorted Hamiltonian which arises in case of quadratic electron-phonon coupling which is significant in quantifying electronic dephasing.