THE JOURNAL OF CHEMICAL PHYSICS 126, 174104 2007
Wave packet theory of dynamic stimulated Raman spectra in femtosecond pump-probe spectroscopy
Zhigang Sun and Zhongqi Jin
Division of Chemistry and Biological Chemistry, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637616, Singapore
Department of Chemistry, Fudan University, Shanghai 200433, Peoples Republic of China

Dong H. Zhang

State Key Laboratory of Molecular Reaction Dynamics and Center for Theoretical and Computational Chemistry, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Dalian 116023, Peoples Republic of China

Soo-Y. Leea

Received 8 January 2007; accepted 16 February 2007; published online 3 May 2007 The quantum theory for stimulated Raman spectroscopy from a moving wave packet using the third-order density matrix and polarization is derived. The theory applies, in particular, to the new technique of femtosecond broadband stimulated Raman spectroscopy FSRS. In the general case, a femtosecond actinic pump pulse rst prepares a moving wave packet on an excited state surface which is then interrogated with a coupled pair of picosecond Raman pump pulse and a femtosecond Raman probe pulse and the Raman gain in the direction of the probe pulse is measured. It is shown that the third-order polarization in the time domain, whose Fourier transform governs the Raman gain, is given simply by the overlap of a rst-order wave packet created by the Raman pump on the upper electronic state with a second-order wave packet on the initial electronic state that is created by the coupling of the Raman pump and probe elds acting on the molecule. Calculations are performed on model potentials to illustrate and interpret the FSRS spectra. 2007 American Institute of Physics. DOI: 10.1063/1.2715593

I. INTRODUCTION

The availability of femtosecond pulses and pump-probe techniques has made it possible to directly prepare and monitor coherent vibrational wave packets in the ground and excited electronic states of polyatomic molecules.1,2 A femtosecond pump pulse that is signicantly shorter than the period of a vibrationally active mode of a molecule can produce a localized wave packet which evolves on the excited state surface. The dynamic wave packet can be monitored, for example, with a femtosecond probe pulse in absorption or emission,35 or with a simultaneous pair of picosecond pump and femtosecond probe pulses in stimulated Raman scattering.6,7 On the theoretical front, time-dependent wave packet methods provide a physically intuitive understanding as well as being simple and effective for calculating molecular spectra particularly in situations where short time molecular dynamics of a few vibrational periods prevail.4,814 The development of accurate and efcient numerical methods for time-dependent wave packet propagation has led to its widespread use.1520
Author to whom correspondence should be addressed. Electronic mail: sooying@ntu.edu.sg
The wave packet theory of dynamic absorption spectra in femtosecond pump-probe experiments in the regime where the pump and probe pulses are temporally separated has earlier been presented.4,5 A dynamic wave packet is prepared on an excited state surface by a femtosecond pump pulse and a delayed femtosecond probe pulse is applied whose dispersed spectrum provides information about dynamic absorption of the wave packet to a second excited state and/or dynamic stimulated emission of the wave packet down to the ground state surface. The dynamic probe spectrum is determined by the third-order polarization which is derived using a time-dependent perturbation formalism in Liouville space for the evolution of the multilevel vibronic density matrix of the system.4 The polarization terms can be understood diagrammatically in terms of time-dependent overlap of bra and ket wave packets. With some simplication, the dynamic absorption of the probe pr pulse can be viewed as a rst-order spectroscopy of the wave packet created by the pump pulse. The absorption cross section is governed by a correlation function that is given by the timedependent overlap of wave packets evolving on both the nal and initial potential surfaces leading to a rst- order polarization, P1t, whose Fourier transform, P1, is used to pr pr calculate the absorption cross section,4 ,

* ei+iti21eih2t/12i.

The key element is the Raman amplitude within the modulus which is the half-Fourier transform of a correlation function given by the overlap of a wave packet M i propagated on the excited state surface with Hamiltonian h2, given by eih2/2t/M i, with the nal vibrational state mediated by the transition dipole, M f. The delta function is simply an energy conservation term, where the incident and scattered light have angular frequencies l and s, respectively, while i and f are the energies of the stationary incident and nal vibrational states on the lower state surface with Hamiltonian h1. Equation 4 is equivalent and can be derived from the time-correlator result which allows for a statistical collection of initial vibrational states, whose quantum averaging is denoted by av,25
So, the theory for absorption/emission from a stationary state as well as a moving wave packet is complete. Recently, femtosecond broadband stimulated Raman spectroscopy FSRS has been developed as a powerful technique that can reveal vibrational structural information of stationary or transient excited states.6,7 The technique has also been called femsosecond time-resolved Raman spectroscopy. A typical FSRS experiment can be explained with the

d 2 dsd =

3 l s 2 c 4 2

deisil+A, , , 5

where A , , is the three-time correlation function,

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Theory of dynamic stimulated Raman spectra
* * = M eih2/2/M eih1/M eih2/2/

M eih1+/av.

In deriving Eq. 4 from Eq. 5, we assume just a single initial state i and inserted the completeness relation f f f = 1 after the propagator eih1/ in Eq. 6. However, Eq. 4 provides for a simpler wave packet interpretation of spontaneous Raman scattering. For absorption or emission, the extension of the theory from an initial stationary state to a moving wave packet has been made.4 In the case of Raman scattering, the advent of FSRS necessitates an extension of the theory also from an initial stationary state to a moving wave packet. In this paper, we provide such a theory to ll the gap. In Sec. II, we provide the quantum theory for stimulated Raman scattering from a moving wave packet based on the time evolution of the third-order density operator and the corresponding thirdorder polarization. We use Feynman dual time-line diagrams to depict the relevant polarization terms. By making a transformation to a set of physically motivated time, we can sum the three termsone for resonance Raman scattering and two for hot luminescenceto give an interesting result that the third-order polarization for stimulated Raman scattering is simply the overlap of a rst-order wave packet created by the Raman pump on the upper electronic state with a secondorder wave packet on the initial electronic state that is created by the coupling of the Raman pump and probe elds acting on the molecule. In Sec. III, we outline the calculation procedure and the model potentials, and in Sec. IV we present the results and discuss the interpretation of the FSRS spectra. We conclude in Sec. V.

FIG. 2. Dual time-line diagram for resonance Raman scattering RRS, with ket evolution on the left and bra evolution on the right. At no time does the bra and ket vectors coexist on excited state 2.

aabQ,tb,

where Q denotes the nuclear coordinates and abQ , t is the vibrational density matrix associated with electronic states a and b. We assume that the system is initially in a pure vibrational state on electronic state 1 prepared by the actinic pump pulse,

= 1Q,t1Q,t.

In the presence of the Raman pump-probe pulses, the density matrix evolves according to the quantum Liouville equation

i = H0 + Vt, t t, t

with H0 = 1h1Q1 + 2h2Q2,
II. THEORY A. Third-order polarization for Raman scattering from a moving wave packet

Vt = Q ER,t,

Q = 221Q1 + c.c.,
ER,t = puE put tDexpik pu R + prE prt tDexpik pr R,
We assume that the role of the femtosecond actinic pump pulse is to prepare the nonstationary vibrational wave packet 1t on the excited electronic state 1, which is subsequently probed with a pair of Raman picosecond pump and femtosecond probe pulses which engender a polarization between the excited electronic state 1 and a second excited electronic state 2. Here, we assume that there is no signicant temporal overlap between the actinic pump and the Raman pump-probe pulses. This is the case with the experiments on stimulated Raman scattering from an excited electronic state. If the actinic and the Raman pump and probe pulses overlap signicantly, then we would need a fth-order polarization P5t to describe the stimulated Raman scatterpr ing starting from the ground electronic state 0, but since they are well separated, a third-order polarization P3t pr sufces. Our focus is on the system of two Born-Oppenheimer excited electronic states and their associated vibrational manifolds, and we can write the density matrix as
where haQ is the vibrational Hamiltonian for electronic state a; abQ is the coordinate-dependent transition dipole between electronic states a and b; and E put tD and E prt tD, peaked at tD, are the real Raman pump and probe elds which propagate in space R which can be taken to be a constant in the long wavelength approximation with wave vectors k pu and k pr, and polarizations pu and pr, respectively. We assume that E put tD and E prt tD appear after the femtosecond actinic pump pulse peaked at t = 0 has passed, and their envelopes have coincident peaks at tD, which is interpreted as the time delay between the actinic pump and the Raman pulses. For simplicity, we will assume parallel polarizations pu = pr and write ab = ab pr; and is the relaxation superoperator which describes vibronic dephasing and relaxation effects.

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Assuming that the pump pulse leads to excitation and the probe pulse leads to deexcitation on either the bra or ket interaction, the perturbative response of the density matrix to third order is
* dte11tteih1tt/21E* t tD pr 1* 21 t21E prt tDeih1tt/ ,

13 dte21tteih2tt/

21E put
11 t 22 t21E prt tDeih1tt/ ,
dte21tteih2tt/21E put tD 14 15
11 where ta is a time after the actinic pump pulse is over, but before the Raman pump and probe pulses appear,

teih1tt/.

The third-order polarization is given by

* 3 P3t = Tr2121 t. pr

dte22tteih2tt/
Now, let us assume that the damping constants ab are given by the simple form
1* 1 * 21 t 21 t21E* t tDeih2tt/ , pu

ab = a + b/2

with 1 = 0 and 2 = .
Using Eqs. 1115, the third-order polarization yields four terms:

3 PCARSt;tD =

3 t t1
dt3E put1 tDE* t2 tDE put3 tD pr 17
* * 1taeih1tta/21eih2/2tt1/21eih1t1t2/21eih2/2t2t3/21eih1t3ta/1ta, 3 PRRSt;tD =
dt3E put1 tDE prt2 tDE* t3 tD pu 18

3 PHLIt;tD =

ih1t3ta/ t1
* * 21eih2/2t2t3/21eih1tt2/21eih2/2tt1/21eih1t1ta/1ta, t2
dt3E prt1 tDE put2 tDE* t3 tD pu 19

1tae and

3 PHLIIt;tD = 3 t

ih1t3ta/

* * 21eih2/2t1t3/21eih1tt1/21eih2/2tt2/21eih1t2ta/1ta,
dt3E prt1 tDE* t2 tDE put3 tD pu 20
* * 1taeih1t2ta/21eih2/2t1t2/21eih1tt1/21eih2/2tt3/21eih1t3ta/1ta.
Here, 1ta is the initial wave packet prepared on electronic state 1 by the ultrashort actinic pump. Equation 17 describes coherent anti-Stokes Raman scattering CARS, while Eqs. 1820 describe Raman scattering with three terms: resonance Raman scattering RRS where there is no intermediate population on electronic state 2, and hot luminescence terms HLI and HLII where there is population on electronic state 2 for the time period t2 , t3. The detected signal for CARS has a wave vector, kCARS = k pu k pr + k pu = 2k pu k pr, which is usually not in the direction of k pr, whereas for RRS, HLI, and HLII, the detected signal is in
the direction of k pr. Our interest here is in the Raman scattering terms arising from a moving wave packet on electronic state 1, with the detected signal in the direction k pr of the Raman probe, and we can therefore omit the CARS term. We can use Feynman dual time-line diagrams to depict the three Raman scattering terms: Fig. 2 for RRS, Eq. 18, and Fig. 3 for the two HL terms, Eqs. 19 and 20. In off-resonance Raman scattering, the excited electronic state 2 would be a virtual state to all three termsRRS, HLI, and HLIIand so they would make similar contributions to the nal result, but for on-resonance Raman scat-

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tering, the excited electronic state 2 is real, and the HLI and HLII terms which give rise to population on electronic state 2 would dominate over the RRS term. In any case, we have to sum all three terms to obtain the nal result for Raman scattering.
B. Transformation to physically motivated time , , results
A physically more useful set of time variables is obtained by taking appropriate differences in the set t3 , t2 , t1 , t
to obtain a new set , , , where a and correspond to the times spent by the bra and ket vectors, respectively, in the excited electronic state 2, b is the time the bra vector has returned to the electronic state 1 before closure with the ket vector to give the third-order polarization. This new set of time variables is shown in Fig. 4, and it turns out to be identical to the set used in Eqs. 5 and 6, so we now have an interpretation for the variables there. With the new set of time variables, the three terms, Eqs. 1820, share a common integrand and we can sum them up,
3 P3t = PRRSt + PHLIt + PHLIIt
E* t tDE prt tDE put tD pu
* * 1taeih1tta/21eih2/2/21eih1/21eih2/2/21eih1tta/1ta
dE* t tDE prt tDE put tD pu
* * * 1taeih1tta21eih2/2/21eih121eih2/2/21eih1tta/1ta = 2t211t, 1 2

where 1t = 2

dE put tDeih2/2/ 22
equality in Eq. 21 leads to Eqs. 5 and 6. The change in spectral intensity of the probe pulse, I pr, after transmission through an optically thin sample of thickness l is given by30 I pr = 20cE prl, 2 E pr2 = 82lC ImE* P3, pr 3n 24

21eih1tta//1ta, and

i 2t = 1

dE* t tD pu

E prt tD1taeih1tta/

* 21eih2/2/21eih1/.

where C is the number of molecules per unit volume; n is the refractive index; E pr, the spectrum of the incoming Raman probe pulse, is given by E pr = 1 2
We obtain the interesting result that the third-order polarization is given by an overlap between a rst-order wave packet 1t on electronic state 2 created by the Raman pump 2 pulse acting on the moving wave packet on electronic state 1 and a second-order wave packet 2t on electronic 1 state 1 created by the simultaneous coupling of the Raman pump and probe elds acting on the moving wave packet on electronic state 1. This second-order wave packet 2t 1 can also be viewed simply as the Raman probe pulse acting on the rst-order wave packet 1t. Clearly, summing the 2 three termsRRS, HLI, and HLIIleads to a result where the ket rst order and bra second order evolution in the dual time-line diagrams can be separated to give the results in Eqs. 22 and 23, respectively. It can be shown that for scattering from a stationary state with cw lasers, the third

eitE prtdt;

and P3 is given by the Fourier transform of the polarization, Eq. 21, P3 = 1 2

eit P3tdt.

In the calculation, we rst choose a set of potentials and pulses. We then nd the wave packet 1ta created by the actinic pump pulse. This is followed by calculating 1t 2 based on Eq. 22, and 2t based on the adjoint of 1 Eq. 23; the overlap between them gives the time-dependent polarization P3t, based on Eq. 21. We can then calculate P3 by Eq. 26 and E pr by Eq. 25, which are then used in Eq. 24 to nd the Raman gain.

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FIG. 4. Physically motivated set of time variables , , for the dual time-line diagrams.
FIG. 3. Dual time-line diagram for hot luminescence HLI and HLII. The difference between the two diagrams lies in the interaction time of the Raman pump on the ket evolution line. In both diagrams, the bra and ket vectors coexist on excited state 2 in the time interval t1 , t2; we say that there is population on excited state 2.
We assume that there are three electronic states, 0, 1, and 2, and associated potential energy surfaces, as illustrated in Fig. 1. The initial wave packet in electronic state 1 is created by an ultrashort actinic pump pulse Eact acting on an initial vibrational wavefunction or wave packet 0t in the ground electronic state 0, which can be written within rst-order perturbation theory as 1t = i
III. NUMERICAL PROCEDURE AND MODEL POTENTIALS A. Numerical procedure

eih1tt/10Eact0tdt.

From the quantum theory of FSRS above, we can see that the stimulated Raman spectrum from a moving wave packet can be calculated using the time-dependent wave packet method. In the calculations below, we use the split operator method and associated fast-Fourier transform technique for wave packet propagation.15 The numerical method used here is effective and accurate for one-dimensional problem and can be extended to multidimensional problems. We take the pump or probe electric eld to have the form E pu/prt tD = E0,pu/prg pu/prt tDeipu/prt , 27
This can be solved iteratively for small time steps as 1t + t = eih1t/1t i + t10Eact + t0t + t + Ot2. 30 Similarly, the rst-order wave packet 1t on electronic 2 state 2 created by the Raman pump pulse, Eq. 22, is solved iteratively by 1t + t = eih2/2t/1t i + t21E put + t1t + t + Otand the second-order wave packet 2t on electronic state created by the coupling of the Raman pump and probe elds with the medium, Eq. 23, is solved iteratively by
where E0,pu/pr is a constant indicating the peak intensity of the laser elds, and the Gaussian envelope g pu/prt tD has the form g pu/prt tD = exp 2 ln 2

t tD2 , 2

giving a laser pulse intensity proportional to g pu/prt tD2 with full width at half maximum FWHM of.

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probe pulses, both peaking at t = tD, was used to produce the dynamic Raman spectrum. In the off-resonance case, the Raman pump pulse has an 826 nm central wavelength and a Gaussian envelope of 500 fs FWHM, while the broadband Raman probe pulse has a 900.37 nm central wavelength and 10 fs FWHM. In the resonance case, the Raman pump pulse has a 400 nm central wavelength and 500 fs FWHM, while the Raman probe pulse has a 416.67 nm central wavelength and 10 fs FWHM. For simplicity, the lifetime in state 1 is taken to be innite, while for state 2 it is nite with = 200 cm1. The Condon approximation was used, with all transition dipoles set to unity, ab = 1. In order for the 500 fs Raman pump pulse to come into play as much as possible, the third-order polarization for the time-dependent FSRS was calculated after seven periods of motion in state 1, where each period is 133.33 fs, and for time delay tD ranging over nearly one period of time from 933 to 1073 fs, in steps of 10 fs.
FIG. 5. Potential energy surfaces used in the calculation: 500 cm1 harmonic ground electronic state 0 potential minimum energy at 0 eV and zero dimensionless unit, 250 cm1 harmonic rst excited electronic state 1 potential minimum energy at 3.0 eV and displacement of two dimensionless units, and 500 cm1 harmonic second excited electronic state 2 potential minimum energy at 5.0 eV and displacement of four dimensionless units. A moving wave packet on state 1 is produced at time t = 0 by a delta function actinic pulse acting on the v = 0 state of the ground electronic potential. Femtosecond stimulated Raman scattering FSRS is obtained from the moving wave packet on state 1 using a pair of Raman pump and probe pulses which has the same peak time tD. For resonance FSRS, the Raman pump pulse has a 400 nm central wavelength and a Gaussian envelope of 500 fs FWHM, while the broadband Raman probe pulse has a 416.67 nm central wavelength and 10 fs FWHM. In the off-resonance case, the Raman pump pulse has an 826 nm central wavelength and 500 fs FWHM, and the Raman probe pulse has a 900.37 nm central wavelength and 10 fs FWHM.
IV. RESULTS AND DISCUSSION
The rst step in the FSRS calculation is nding the rstorder wave packet 1t created by the Raman pump pulse 2 on the state 2 surface. To visualize this wave packet, it can be expanded in terms of the eigenstates 2 of the state 2 surface,

1 = C2 t2.

i * 2t + t = eih1t/2t + t21E* t + t pr 1t + t + Ot2. 2 From these, we obtain the third-order polarization
* P3t + t = 2t + t211t + t. 3 3
Since we need to take the overlap of 1t with the second2 order wave packet 2t on the state 1 surface, it may also 1 be useful to project 1t onto the eigenstates 1 of the 2 state 1 surface,

1 1t = C1 t1. 2

Similarly, the second-order wave function 2t can be ex1 panded as
The Fourier transform of P t yields P which is used in Eq. 24 to nd the Raman gain.
B. Model potentials and laser elds used

1 C1 t,

2 = C1 t1.
The potentials used in the numerical calculations are shown in Fig. 5. The model consists of 500 cm1 harmonic ground 0 electronic state and second excited 2 electronic state potentials, but a 250 cm1 harmonic rst excited 1 electronic state potential. The minima of the 1 state and the 2 state potentials are displaced relative to the ground potential minimum by two and four dimensionless units, respectively. The minimum energies of the 1 state and 2 state are at 3.0 and 5.0 eV, respectively, above that of the ground potential minimum. In our FSRS calculation, we used a delta function actinic pulse at t = 0 to prepare the initial moving wave packet on state 1 from an initial v = 0 vibrational state on the ground potential. A delayed pair of Gaussian Raman pump and
1 The contour plots of the time dependence of C2 t, 2 and C1 t are shown in Figs. 6a6c, respectively, for the off-resonance case826 nm, 500 fs FWHM Raman pump; 900.37 nm, 10 fs FWHM Raman probewith time delay or peak times of the Raman pump and probe pulses at 933 fs. Both the C2 t and C1 t distributions 1 peak at low vibrational quantum numbers. The C1 t distribution has an envelope governed by the Gaussian Raman pump pulse and is periodic, with a period of about 133 fs corresponding to the vibrational period in state 1. In this 1 off-resonance case, the peaks of the C1 t distribution are attained when the wave packet in state 1 is at the right turning point, e.g., at delay times of 1 ps, 1.133 ps, etc., where the energy mismatch between the central energy of the Raman pump pulse and the energy gap between surfaces and 2 is at a minimum. As expected, the C1 t distribu-

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FIG. 6. Color Off-resonance case826 nm, 500 fs FWHM Raman pump; 900.37 nm, 10 fs FWHM Raman probewhere the peaks of the Raman pump and probe pulses are at 933 fs and lifetime on state 2 is given by = 200 cm1: a The time-dependent vibrational distribution C1t of the 2 rst-order wave packet 1t over the eigenstates of surface 2. b The time-dependent vibrational distribution C1 t of the same wave packet 1t over the eigenstates of surface 1. c The time-dependent vibrational 2 distribution C2t of the second-order wave packet 2t on the vibrational eigenstates of surface 1. d The modulus of the third-order polarization, 3 P t from the product of the coefcients in b and c. e The Raman gain spectrum.

FIG. 7. Off-resonance case: a The modulus of the third-order polarization, P3t, for peak times of the Raman pump and probe pulses ranging over one period on the state 1 surface from 933 to 1073 fs, in steps of 10 fs. b The corresponding Raman gain spectra with the 250 cm1 Stokes v v + 1, 0 cm1 Rayleigh v v, and 250 cm1 anti-Stokes v v 1 lines dominating.
tion is constant in time and concentrated at the low vibrational states after the ultrashort Raman probe pulse has passed, and this is true for all time delays. The third-order 1 2* polarization arises from the overlap of the C1 and C1 t distributions, and the modulus is shown in Fig. 6d. It has the periodicity of the C2 t or C1 t distribution, and each of the oscillations is relatively broad. The Raman gain spectrum is shown in Fig. 6e, and the Fourier transform relation applies in understanding the inverse relationship between the structures in the energy frame and the timedependent polarization in Fig. 6d. In this off-resonance case, only the 250 cm1 Stokes v v + 1, 0 cm1 Rayleigh v v, and 250 cm1 anti-Stokes v v 1 lines are seen because of the relatively broad oscillations in the thirdorder polarization. The spacings are exactly the inverse of the periodicity in the third-order polarization, and the narrow width, about 40 cm1, of each of the lines is determined by the inverse time scale of the polarization about 800 fs since E pr is very broad, as given by Eq. 24, thus affording high frequency resolution. The moduli of the third-order polarization for various delay times, determined by the ultrashort Raman probe pulse, are shown in Fig. 7a for this off-resonance case, and the corresponding Raman gain spectra are shown in Fig. 7b. There is high frequency resolution in all the FSRS spectra, with high time resolution determined by the ultrashort Raman probe pulse. The third-order polarization as a
function of delay time in this off-resonance case, Fig. 7a, is just phase shifted, with the phase of the 1063 fs polarization being about 2 off from the 933 fs polarization, which accounts for the nearly identical Raman gain spectra. The 1003 fs polarization, however, is phase shifted by about from the 933 fs polarization, so the Stokes and anti-Stokes Raman loss i.e., negative lines at 933 fs become a gain i.e., positive. In between, we have a combination of Gaussian and dispersive Stokes and anti-Stokes lines, with the dispersive lines dominating at phase shifts of / 2, at about 963 fs, and 3 / 2, at about 1033 fs. Dispersive line shapes have been observed in FSRS experiments.3135 Similar to Fig. 6, the contour plots of the time depen2 dence of C2 t, C1 t, and C1 t are shown in Figs. 8a8c, respectively, for the resonance case400 nm, 500 fs FWHM Raman pump; 416.67 nm, 10 fs FWHM Raman probewith time delay also at 933 fs. In this resonance case, the C2 t and C1 t distributions stretch to rela1 tively high vibrational quantum numbers. The C2 t and 1 C1 t distributions have a similar period of 133 fs, corresponding to the wave packet motion on surface 1, as in the off-resonance case, but their decay with time in the resonance case is determined by the lifetime on state 2, given by = 200 cm1, because of population created on state 2. 1 The maxima of the C1 t distribution are attained when the wave packet in state 1 is near the left turning point, e.g., at about 933 fs, 1.066 ps, etc., where the energy mismatch between the central energy of the Raman pump pulse and the energy gap between surfaces 1 and 2 is at a minimum. The 2 C1 t distribution is constant in time after the ultrashort Raman probe pulse has passed, but this constant distribution

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FIG. 8. Color Resonance case400 nm, 500 fs FWHM Raman pump; 416.67 nm, 10 fs FWHM Raman probewhere the peaks of the Raman pump and probe pulses are at 933 fs and lifetime on state 2 is given by = 200 cm1: a The time-dependent vibrational distribution C1t of the 2 rst-order wave packet 1t over the eigenstates of surface 2. b The time-dependent vibrational distribution C1 t of the same wave packet 1t over the eigenstates of surface 1. c The time-dependent vibrational 2 distribution C2t of the second-order wave packet 2t on the vibrational eigenstates of surface 1. d The modulus of third-order polarization P3 t. e The Raman gain spectrum.
FIG. 9. Resonance case. a The modulus of the third-order polarization, P3t, for peak times of the Raman pump and probe pulses ranging over one period on the state 1 surface from 933 to 1073 fs, in steps of 10 fs. b The corresponding Raman gain spectra with well-resolved, long Raman progressions.
depends on the delay time when the Raman probe pulse is applied. At the full periods of 933 fs, 1.066 ps, etc., the 2 C1 t distribution peaks at the lower vibrational quantum numbers. At the half periods of 1 ps, 1.133 ps, etc., the 2 C1 t distribution is low in magnitude and spread out over a broad range of vibrational quantum numbers, and this gives rise to a different pattern for the the third-order polarization. The third-order polarization, whose modulus is shown in Fig. 8d, for a time delay of 933 fs, has recurring sharp lines with a period of 133 fs, Fig. 8b. In contrast to the offresonance case, these sharp lines in the third-order polarization lead to a long Raman progression going out to the v v + 9 transition in the Raman gain spectrum, as shown in Fig. 8e. Again, as in the off-resonance case, the narrow width of each of the lines in the Raman gain spectrum is determined by the inverse time scale of the polarization about 800 fs here, again affording high frequency resolution. The moduli of the third-order polarization for various delay times, in this resonance case, are shown in Fig. 9a, and the corresponding Raman gain spectra are shown in Fig. 9b. At the full periods of the delay time at 933 fs, 1.066 ps, etc., the third-order polarization has recurring sharp lines that lead to long Raman progressions, but at the half periods of the delay time at 1 ps, 1.133 ps, etc., the low magnitude and 2 spread out C1 t distribution leads to a weaker third-order polarization with broad recurring features which translate to less prominent, rapidly damped Raman progressions. In
short, at the delay time when the wave packet on surface 1 is at the point where the energy mismatch between the Raman pump pulse central energy and the energy gap between surfaces 1 and 2 is very small, we can expect to see a long Raman progression, but when the energy mismatch is large, the Raman progression will be very much muted. As in the off-resonance case, there is high frequency resolution in all the FSRS spectra, with high time resolution determined by the ultrashort Raman probe pulse. We have also calculated the dynamic absorption spectra4 of the moving wave packet from state 1 to state 2 using a delta function pump pulse at similar delay times to compare with the resonance FSRS spectra. This is shown in Fig. 10 over about one period from 933 to 1073 fs, in steps of 10 fs. As explained by Pollard et al.,4 the mean energies of the dynamic absorption spectra correspond to the energy gap between surfaces 1 and 2 at the positions where the moving wave packet on surface 1 is excited to surface 2. In resonance FSRS, Fig. 9b, the wave packet motion on surface 1 is reected in the changing envelope of the Raman progressions, which depends on the energy mismatch between the Raman pump pulse central energy and the energy gap between the two surfaces where the wave packet is at the particular delay time. Comparing Fig. 10 with Figs. 7b and 9b, it is clear that high time resolution is present in both dynamic absorption and FSRS spectra, but only FSRS can afford high frequency resolution as well, as shown by the sharp peaks in the FSRS spectra, Figs. 7b and 9b.

V. CONCLUSION

We have derived the quantum theory for femtosecond stimulated Raman spectroscopy FSRS from a moving wave

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line shapes and Raman gain and loss for the Stokes and anti-Stokes lines are shown to be present in FSRS.
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FIG. 10. Dynamic absorption spectra, at different delay times of the delta function pump pulse, over about one period on state 1 from 933 to 1073 fs, in steps of 10 fs. The envelopes of the dynamic absorption spectra reect the motion of the wave packet on state 1.
packet. It is shown that the Raman gain is determined by the time-dependent third-order polarization given simply by the overlap between a rst-order wave packet prepared on the excited state surface by the relatively long Raman pump pulse and a second-order wave packet on the initial state surface created by the ultrashort Raman probe pulse acting on the rst-order wave packet. The physics of high time resolution and high frequency resolution in the FSRS spectrum are explained in terms of the wave packet dynamics. The differences between the spectra in resonance and offresonance FSRS are also explained in terms of the wave packet dynamics and the third-order polarization. Dispersive