Chapter 13: Femtosecond Pulse Pair Excitation

The first and second order time dependent perturbation theory formulas have a very appealing interpretation when applied to molecular excitation with ultrashort (femtosecond) pulses. We take H = H0+H1, where H0 is the bare molecular Hamiltonian and H1 is the perturbation, taken to be - μ⋅ε(t) where μ is the transition dipole moment and ε(t) is the instantaneous electric field. Assume that at t = 0 the wavefunction is in v = 0 of the ground electronic state. The first order perturbation theory formula takes the form:
Reading this equation from right to left, the wavefunction evolves from t = 0 until time t' under the ground electronic state Hamiltonian, Hα, accumulating an overall phase factor of
 At t = t' the electric field, of amplitude ε(t') interacts with the transition dipole moment, promoting amplitude to the excited electronic state. This amplitude evolves under the influence of Hb from time t' until time t. The integral dt' indicates that all instants in time t' contribute. Similarly, the second order amplitude is given by
Now there is a second interaction with the electric field and the subsequent evolution is taken to be on a third surface, with Hamiltonian Hc.