Two-phonon absorption operator in spectroscopy

While the raising operator a ˆ + does not have eigenvectors, coherent states are eigenvectors of the lowering operator a ˆ . Therefore, operating with the operator exp [ k a ˆ + 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.