IOP PUBLISHING JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 195402 (8pp) doi:10.1088/0953-4075/42/19/195402 On the divergence of time-dependent perturbation theory applied to laser-induced molecular transitions Klaus Renziehausen 1 , Philipp Marquetand 2 and Volker Engel 1 1 Universit¨ at W¨ urzburg, Institut f¨ ur Physikalische Chemie, Am Hubland, 97074 W¨ urzburg, Germany 2 ´ Ecole Normale Sup´ erieure, D´ epartement de chimie, 24, rue Lhomond, 75005 Paris, France E-mail: voen@phys-chemie.uni-wuerzburg.de Received 25 June 2009, in final form 30 July 2009 Published 22 September 2009 Online at stacks.iop.org/JPhysB/42/195402 Abstract Population transfer between electronic molecular states can be effectively induced via the interaction with shaped laser pulses. Regarding a numerical example, it is demonstrated that perturbation theory, as is often applied in numerical simulations of field-matter interactions, might lead to divergences. The occurring error accumulating in the norm of the wavefunction can be decomposed into two contributions. The first one is a small numerical error, which is controllable by minimization of the time-propagation step, whereas the second one is related to the order of the perturbative expansion. These two contributions behave differently upon variations of the potential energy surface of the system and also the laser pulse parameters. An improved scheme is proposed in which the first part carrying the numerical error disappears. 1. Introduction Time-dependent perturbation theory is a powerful tool to examine the interaction of atoms [1] and molecules [2] with electromagnetic fields. Although perturbative methods, in general, are not norm-conserving and non-perturbative norm-conserving algorithms are available to solve the time- dependent Schr¨ odinger equation, a perturbative ansatz is interesting because of the following reason: due to the systematic expansion in the field-system interaction, it is possible to clearly decompose a multi-photon process into contributions which stem from different orders. As an example, we mention time-resolved four-wave-mixing spectroscopy, where a theoretical description needs the evaluation of the induced third-order polarization associated with a certain direction of the emitted field [3]. Although there exist methods for extracting these desired contributions to the total polarization [4–6], perturbation theory provides a most straightforward approach [7–9]. Regarding a process where, e.g., a high-intensity laser pulse interacts with a molecule, it is then important to investigate how the results of perturbation theory converge to the exact results. This convergence behaviour depends on numerical parameters and will also change for different molecules and laser interactions. In particular, it is of interest to analyse if occurring numerical errors are due only to the approximate numerical method or the fact that the number of included orders of perturbation is not sufficient. In this work, we address the question in how far perturbation theory implemented via a numerical algorithm is applicable to ultra-short laser pulse-molecule interactions. The physical situation we regard is the temporal evolution of molecular wavefunctions in pump/shaped-dump experiments as have recently been realized [10–12]. The excitation scheme is sketched in figure 1. There, diabatic potential curves along a reaction coordinate are displayed. The inclusion of non- adiabatic couplings will modify these potentials in the region they cross each other and an avoided crossing (or in the more general case a conical intersection [13]) occurs. The laser excitation scheme is as follows: a pump-pulse transfers population from the ground (|0〉) to an excited electronic state (|1〉). There, the prepared wave packet evolves in time and, due to the gradient of the excited state potential surface, moves towards larger distances. A time-delayed interaction with a shaped second pulse (dump), then couples the two electronic states in a region well separated from the curve-crossing, giving rise to multi-photon transitions between them. Extensive recent calculations investigated which pulse- shaping leads to an efficient population transfer to the ground state [11, 14]. Here, we use the same model of a molecular 0953-4075/09/195402+08$30.00 1 © 2009 IOP Publishing Ltd Printed in the UK