Optimizing Raman Ladder Climbing: Theory and Application in Na 2 Bo Y. Chang, Ignacio R. Sola ´ ,* and Jesu ´ s Santamarı ´a Departamento de Quı ´mica Fı ´sica I, UniVersidad Complutense, 28040 Madrid, Spain ReceiVed: March 22, 2001; In Final Form: June 22, 2001 The theory of vibrational ladder climbing excitation by coherent stimulated nonresonant Raman using chirped laser pulses is developed. We analytically obtain the conditions for inverting the population to a final preselected vibrational state and the restrictions that apply in the linear chirp regime. By controlling both the shape of the laser pulses and the chirp profile, the ladder climbing process can be accelerated without reducing the yield of the selective excitation. Numerical results are presented for selection of moderately excited vibrational levels in Na 2 , where the important contribution of several excited electronic states is also clarified. I. Introduction Exciting molecules in specific high vibrational states has been a long-sought goal, both for a better understanding of the spectroscopic properties of the molecule and for igniting some unimolecular (bond breaking) or bimolecular (reactant preparing) reactions. 1 In molecules with a permanent dipole moment, infrared (IR) laser pulses have been used to pump the vibrational energy. By multiphoton IR processes, it has been possible to excite single modes in regions of high anharmonicity and to observe the subsequent intramolecular vibrational relaxation (IVR). 2 However, due to the very weak transition dipole moments between the ground and high vibrational eigenstates, population inversion is unlikely to succeed. For coherent interactions, the minimum time for population inversion (defin- ing the so-called π pulses) is given by π times the inverse of the Rabi frequency, Ω 0V (t) ) µ 0V E(t)/p, where E(t) is the pulse envelope and µ 0V ) 〈ψ V |µ|ψ 0 〉 is the transition dipole moment between the initial (ψ 0 ) and final (ψ V ) vibrational eigenfunctions. For very weak transitions, µ 0V , 1, and therefore population inversion requires long pulses, with the onset of decoherent and nonradiative processes, or very strong laser sources, usually implying competition between several multiphoton routes, if not directly ionizing the molecule. 3 A possibility proposed by Manz and colleagues 4 is to fraction the overall transition into several sequential steps, each of which is driven by a proper π pulse. The limit of this strategy is to use one π pulse for every single quantum step excitation V f V+ 1. The population then follows a pattern equivalent to climbing a ladder one step after another, which takes advantage of the (usually) larger transition dipole moments involved between adjacent vibrational states, especially in approximately harmonic potentials. Nevertheless, the overall sequence of (V - 1)π pulses is both experimentally difficult to prepare and moreover poorly robust, since the yield of each step is very sensitive to frequency, time, and intensity variations, and the overall yield is the product of the yields of every step. This also explains the inability of optimal control algorithms to obtain this kind of solution, when not properly biased. 5 Chelkowski et al. 6 and Guerin 7 showed the way to circumvent the problem using a single pulse with slowly varying frequency, adapted to the anharmonicity of the potential. The pulse duration must be at least as long as the whole sequence of (V- 1)π pulses and the intensity stronger than the intensity of each one. For sufficiently intense pulses, the population can be adiabatically transferred from the initial state up to dissociation with 100% efficiency at least in principle. The method is both robust and not especially difficult to implement in the laboratory with state- of-the-art technology. 8 For molecules without a permanent dipole moment, the ladder climbing method can be implemented using nonresonant stimulated coherent Raman. 9-12 In this case, either the pump or the Stokes or both pulses must be chirped. The detuning with respect to excited electronic states is required in order to avoid absorption. Although there are a number of schemes which make use of resonant Raman transitions, such as stimulated Raman adiabatic passage (STIRAP) 13 or stimulated emission pumping (SEP), 14 the Raman ladder climbing is an alternative method especially suited for dissociation or high vibrational energy excitation, since it benefits from the larger effective two-photon transition dipole moments between adjacent vibrational levels. In this paper, we develop a general theory of Raman ladder climbing deriving the required conditions for the sequential inversion of population between adjacent levels and obtaining the minimum time for the final excitation of a single, selected vibrational level. We demonstrate the important contribution of highly excited electronic states and the possibility of improving the yield of the process by a suitable election of the laser amplitude and chirp profiles. The validity of the theoretical results is numerically tested showing the efficiency of the Raman ladder climbing method applied to the selective excitation of vibrational states of a nonrotating sodium dimer. In section 2, we present the molecular model and calculate the effective Raman Rabi frequencies. In section 3, we develop the theory of optimal Raman ladder climbing, which is numerically tested in section 4. Section 5 is the conclusion. II. Model for Nonresonant Raman in Na 2 In Raman ladder climbing, the system is exposed to the action of two fully overlapping lasers, the pump E p (t), and the Stokes E s (t) pulses. The pulses have the same amplitude, and the frequency of the pump is negatively chirped; that is, its carrier frequency smoothly decreases. The same results can be obtained by positively chirping the frequency of the Stokes pulse or chirping both. * E-mail: ignacio@tchiko.quim.ucm.es. 8864 J. Phys. Chem. A 2001, 105, 8864-8870 10.1021/jp011077s CCC: $20.00 © 2001 American Chemical Society Published on Web 09/06/2001