Quantum transition probabilities during a perturbing pulse: Differences between the nonadiabatic results and Fermi's golden rule forms.

For a perturbed quantum system initially in the ground state, the coefficient c(t) of excited state k in the time-dependent wave function separates into adiabatic and nonadiabatic terms. The adiabatic term a(t) accounts for the adjustment of the original ground state to form the new ground state of the instantaneous Hamiltonian H(t), by incorporating excited states of the unperturbed Hamiltonian H without transitions; a(t) follows the adiabatic theorem of Born and Fock. The nonadiabatic term b(t) describes excitation into another quantum state k; b(t) is obtained as an integral containing the time derivative of the perturbation. The true transition probability is given by b(t) , as first stated by Landau and Lifshitz. In this work, we contrast b(t) and c(t) . The latter is the norm-square of the entire excited-state coefficient which is used for the transition probability within Fermi's golden rule. Calculations are performed for a perturbing pulse consisting of a cosine or sine wave in a Gaussian envelope. When the transition frequency ω is on resonance with the frequency ω of the cosine wave, b(t) and c(t) rise almost monotonically to the same final value; the two are intertwined, but they are out of phase with each other. Off resonance (when ω ≠ ω), b(t) and c(t) differ significantly during the pulse. They oscillate out of phase and reach different maxima but then fall off to equal final values after the pulse has ended, when a(t) ≡ 0. If ω < ω, b(t) generally exceeds c(t) , while the opposite is true when ω > ω. While the transition probability is rising, the midpoints between successive maxima and minima fit Gaussian functions of the form exp[-(t - )]. To our knowledge, this is the first analysis of nonadiabatic transition probabilities during a perturbing pulse.