26 March 1999Ž.Chemical Physics Letters 302 1999 405–410Using time-dependent rate equations to describe chirped pulseexcitation in condensed phasesChristopher J. Bardeena,), Jianshu Caob, Frank L.H. Brownc, Kent R. WilsoncaBox 20-6, CLSL, Department of Chemistry, UniÕersity of Illinois, 600 S. Mathews AÕe., Urbana, IL 61801, USAbDepartment of Chemistry, Massachusetts Institute of Technology, 77 Massachusetts AÕe., Cambridge, MA 02139, USAcDepartment of Chemistry and Biochemistry, UniÕersity of California, San Diego, La Jolla, CA 92093-0339, USAReceived 26 August 1998; in final form 2 December 1998AbstractA time-dependent rate equation formalism is developed to describe high-intensity excitation of molecules in condensedphases using broadband laser pulses. When electronic and vibrational dephasing rates are fast relative to the pulse dynamics,the generalized optical Bloch equations can be written as a set of coupled rate equations with coefficients that depend on thetime-dependent overlap of the pulse spectrum with the molecular lineshape. These rate equations are shown to reproducequalitatively the effects observed experimentally in recent quantum control experiments using high-intensity, chirpedfemtosecond pulsed excitation. q 1999 Elsevier Science B.V. All rights reserved.1. IntroductionAdvances in both the theoretical understanding oflight-driven quantum molecular dynamics and theexperimental ability to create and shape light fieldshas resulted in the field known as ‘quantum control’wx1 . Recent experiments have involved the use ofwxchirped pulses 2–7 , in which the phase structure ofan ultrashort pulse is modified in order to control thesample. Chirping the pulse delays some of its fre-quency components with respect to others. A posi-tively chirped pulse has low frequencies leading andhigh frequencies trailing, while a negatively chirpedpulse has the opposite frequency ordering. For a zerochirp or transform-limited pulse, all the frequenciesarrive at the same time. Note that we can modify the)Corresponding author. E-mail: [email protected] structure of the laser pulse, and thus its tempo-ral properties, without changing the pulse energy orthe power spectrum. Recently, the techniques ofquantum control have been applied to large moleculeswx wxin condensed phases 8–10 and even proteins 11 .The analysis of such chirped pulse experiments is thesubject of this Letter.With systems like atoms or small molecules in thegas phase, we can achieve selectivity by controllingthe quantum interferences among amplitudes of awxfew quantum states 1 . This is truly quantum con-trol, since such interferences are purely the result ofquantum mechanics and cannot be understood from aclassical treatment. We consider here the oppositelimit of fast electronic and vibrational dephasing,where such quantum coherences decay very rapidly.Our approach is similar in spirit to a more sophisti-wxcated treatment by Fainberg 12 which describeshigh-power chirped pulse excitation in terms of mov-0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.Ž.PII: S0009-2614 99 00175-X()C.J. Bardeen et al.rChemical Physics Letters 302 1999 405–410406ing wavepackets, but instead describes these experi-ments in terms of eigenstates and rate equations. Therate equations are similar to those used to analyzewxsteady-state experiments 13 , except that they takethe pulsed nature of the excitation into account viatime-dependent rate coefficients. These time-depen-dent rates may be evaluated in terms of the overlapof the Wigner representation of the laser pulse withthe molecular absorption and emission spectra, allexperimentally measurable quantities. Multiple levelsand coherence effects due to nonzero dephasing timescan be incorporated in a straightforward manner.Example calculations on 4-level systems demonstratevarious effects recently observed in high-powerchirped pulse excitation of molecules in solution.2. TheoryWe begin with the 4-level system illustrated in< :Fig. 1, where transitions occur between levels 0< : < : < :and 3 and levels 1 and 2 . For the sake of< : < :clarity, we neglect levels 1 and 2 for the time< : < :being and consider only levels 0 and 3 . This2-level system is described by the following opticalBloch equations:y1i))rt sry1 qmEtrŽ. Ž . Ž.˙00 00 03T "1)ymEtr,1Ž. Ž.03y1i)rt srqmEtrŽ. Ž.˙33 33 03T "1))ymEtr,2Ž. Ž.03y1rt sy iDqrŽ.˙03 03ž/T2iqmEtryr,3Ž.Ž . Ž.33 00"where T is the dephasing time, T is the population21< : < :lifetime in level 3 before it relaxes to level 0 ,Dis the detuning of the laser carrier frequency fromŽ.the resonance frequency,Ds´y´y"vr30 laserŽ.",mis the transition dipole moment, and Etis the electric field in the rotating wave appro-ŽŽ.Ž.Ž.ximation i.e., the full´t s Etexp yivt qFig. 1. The 4-level system used in the calculations in this Letter,with the time-dependent transistion rates derived in the text andthe relaxation rateG, which represents the Stokes shift.)Ž. Ž ..Etexp qivt . Assumingmto be real and con-Ž.stant the Condon approximation , we integrate tosolve the equation for the off-diagonal matrix ele-ment to obtain`yim1XXrt s dt exp y iDq tytŽ. Ž .H03ž/" Ty`2=XX XEtrt yrt ,4Ž. Ž. Ž. Ž.33 00wxand make use of the identity 14`XY1 t qtXY X X)EtEts dvW ,vŽ.Ž .Hž/2p 2y`=XX Yexp ivt yt ,5Ž. Ž.Ž. Ž.where Wt,vis the Wigner transform of Et,`tt)wxWt,vs dtEty Etq exp ivt.Ž.Hž/ž/22y`6Ž.The Wigner transform is a particularly useful repre-sentation of the electric field, since it can be obtainedwxexperimentally 15 and provides a conceptual pic-ture of the laser pulse structure jointly in time andŽ. Ž.frequency space. Substituting Eqs. 4 and 5 intoŽ.XEq. 2 and performing the change of variableststytY, we obtain2``ym2tytXXrt s dtRe dvW ,vŽ.˙HH332ž/½2p"0y`=1Xexp ivyDytrtytŽ. Ž.33½5T2rtŽ.33yrtyty .7Ž. Ž.005T1()C.J. Bardeen et al.rChemical Physics Letters 302 1999 405–410 407Ž.Eq. 7 is exact within the assumptions of the Blochequations. If instead we use the generalized Blochwxequations 16 , where 1rT is replaced by a stochas-2Ž.tically varying frequency perturbationdt , we havetytXXexp ™ exp i dtdt ,8Ž. Ž.H¦;T02wxwhich is the absorption lineshape function 17,18 .Thus this approach is valid beyond the Markovianapproximation inherent in the usual Bloch equations.We now make the approximation that the decay ofthe lineshape function, e.g. T , is much shorter than2any relevant