COMMUNICATIONSPopulation and coherence control by three-pulse four-wave mixingEmily J. Brown, Igor Pastirk,a)Bruna I. Grimberg, Vadim V. Lozovoy,b)and Marcos Dantusc)Department of Chemistry and Center for Fundamental Materials Research, Michigan State University,East Lansing, Michigan 48824共Received 28 April 1999; accepted 22 June 1999兲Control of coherence and population transfer between the ground and excited states is reported usingthree-pulse four-wave mixing. The inherent vibrational dynamics of the system are utilized intiming the pulse sequence that controls the excitation process. A slight alteration in the pulsesequence timing causes a change in the observed signal from coherent vibration in the ground stateto coherent vibration in the excited state. This control is demonstrated experimentally for moleculariodine. The theoretical basis for these experiments is discussed in terms of the density matrix for amultilevel system. © 1999 American Institute of Physics. 关S0021-9606共99兲03233-X兴The probability of exciting a molecular system from theground,兩g典, to the excited state,兩e典, by applying an electricfield E is written quantum mechanically asPeg⫽兩具e兩␮–E兩g典兩2⫽具e兩␮–E兩g典具g兩␮–E*兩e典, 共1兲where␮is the transition dipole moment that couples bothstates. Population inversion is not usually achieved becauseof the competition between the rates of absorption and stimu-lated emission. Control of the population transfer can beachieved if the two electric fields involved in the transitionprobability in Eq. 共1兲 are different and are correlated in timeor in phase. The three-pulse four-wave mixing 共FWM兲 tech-nique allows one to combine three nonphase-locked electricfields in a phase-matching geometry. The first two fieldscause the population transfer and the third field probes thesystem. The specific timing between the pulses can be usedto control the values of diagonal 共population兲 and off-diagonal 共vibrational coherence兲 matrix elements after theinteraction with the first two electric fields. In this Commu-nication, we briefly describe the three-pulse FWM techniqueand demonstrate on molecular iodine that pulse sequencescan be designed to control the transition probability betweentwo electronic states of a molecule.It has long been recognized that in order to optimize thetransfer of population between two states sophisticated elec-tric fields are required.1–4One can create such electric fieldsby a combination of phase and amplitude masks,5–7or onecan combine phase-locked laser pulses to achieve the desiredfield. Scherer et al.8showed that when two phase-locked la-ser pulses were combined in phase the excited state dynam-ics of molecular iodine could be observed as fluorescenceenhancement; however, when they were combined out ofphase, the signal is observed as fluorescence depletion. Co-herent control of chemical reactions depends on the relativephase of two different laser pulses that interact with thesample. The relative phase of the pulses can be used to con-trol the population transfer from the ground to two differentexcited states.9–11A different approach to controlling popu-lation transfer12and enhancing reaction yields13,14useschirped laser pulses. Recent experiments in our group usingchirped femtosecond three-pulse FWM have demonstratedthat laser chirp can be used to control coherence transferbetween the ground and excited states of I2.15The theoretical foundation for three-pulse FWM is basedon the time evolution of the density matrix in Hilbertspace.16Formulas are derived for a multilevel system withground and excited electronic states with vibrational levels.Ultrafast transform-limited pulses are assumed with a band-width that exceeds the vibrational spacing in the ground andexcited states. Electric field interactions are treated withinthe perturbation limit. A more detailed description of thistheory will be published elsewhere.17,18Initially, the densitymatrix,␳(0), contains the populations of the vibrational lev-els of the ground state with ⌺␳gg(0)⫽ 1, while the populationof each vibrational level in the excited state is taken as␳ee(0)⫽ 0. Here, the index g corresponds to the ground state vibra-tional quantum number,␯⬙, and the index e corresponds tothe excited state vibrational quantum number␯⬘. In our cal-culation, we assume that the ground state vibrational levelsare equally populated. This assumption is justified for a hightemperature Boltzmann distribution. Different initial vibra-tional population distributions yield similar results.15,17Afterthe first interaction with the electric field, the density matrixevolves into a coherence between the ground and excitedstates where all the diagonal terms are zero and no net popu-lation transfer has occurred. The interaction with a secondelectric field completes the population transfer 关Eq. 共1兲兴without electronic coherence between the兩g典and兩e典states.a兲Affiliated with the Institute for Nuclear Sciences ‘‘VINCA,’’ Belgrade, F.R. Yugoslavia.b兲Permanent address: N. N. Semenov Institute of Chemical Physics, RAS,Moscow, Russia.c兲Author to whom correspondence should be addressed; electronic mail:[email protected] OF CHEMICAL PHYSICS VOLUME 111, NUMBER 9 1 SEPTEMBER 199937790021-9606/99/111(9)/3779/4/$15.00 © 1999 American Institute of PhysicsIn general, an odd number of interactions with the electricfields will produce a coherence state, which is also a time-dependent polarization of the molecules. An even number ofinteractions will produce a population state that is character-ized by the population of the vibrational levels in each elec-tronic state 共the diagonal terms兲 and the vibrational coher-ence within each electronic state 共the off-diagonal terms inthe diagonal blocks兲.The changes in the density matrix after each interactionwith an electric field involve different processes that can befollowed using double-sided Feynman diagrams.16,19For fur-ther information about these diagrams and their applicationsto four-wave mixing processes, the reader is referred to Refs.16, 17, and 19–21. A wavy arrow symbolizes each electricfield interaction and time progresses from the bottom to thetop. The arrows pointing towards 共away from兲 the centerrepresent the photon annihilation 共creation兲 operator; absorp-tion or emission of a photon requires two electric field inter-actions. The