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

For a perturbed quantum system initially in the ground state, the coefficient c_k(t) of excited state k in the time-dependent wave function separates into adiabatic and nonadiabatic terms. The adiabatic term a_k(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 H₀ without transitions; a_k(t) follows the adiabatic theorem of Born and Fock. The nonadiabatic term b_k(t) describes excitation into another quantum state k; b_k(t) is obtained as an integral containing the time derivative of the perturbation. The true transition probability is given by $$ b_k(t) $$ ², as first stated by Landau and Lifshitz. In this work, we contrast $$ b_k(t) $$ ² and $$ c_k(t) $$ ². 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_k0 is on resonance with the frequency ω of the cosine wave, $$ b_k(t) $$ ² and $$ c_k(t) $$ ² 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_k0 ≠ ω), $$ b_k(t) $$ ² and $$ c_k(t) $$ ² 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_k(t) ≡ 0. If ω_k0 < ω, $$ b_k(t) $$ ² generally exceeds $$ c_k(t) $$ ², while the opposite is true when ω_k0 > ω. While the transition probability is rising, the midpoints between successive maxima and minima fit Gaussian functions of the form a exp[-b(t - d)²]. To our knowledge, this is the first analysis of nonadiabatic transition probabilities during a perturbing pulse.