Investigations of ultrafast nuclear response induced by ... · PDF fileInvestigations of ultrafast nuclear response induced by resonant ... advantage of the semiclassical approach

JOURNAL OF CHEMICAL PHYSICS VOLUME 114, NUMBER 15 15 APRIL 2001

Investigations of ultrafast nuclear response induced by resonantand nonresonant laser pulses

Anand T. N. Kumar, Florin Rosca, Allan Widom, and Paul M. Championa)

Department of Physics and Center for Interdisciplinary Research on Complex Systems,Northeastern University, Boston, Massachusetts 02115

~Received 13 November 2000; accepted 25 January 2001!

We analyze the nonstationary vibrational states prepared by ultrashort laser pulses interacting witha two electronic level molecular system. Fully quantum mechanical expressions are derived for allthe moments of the coordinate and momentum operators for the vibrational density matricesassociated with the ground and excited electronic states. The analysis presented here provides keyinformation concerning the temperature and carrier frequency dependence of the moments, andrelates the moments to equilibrium absorption and dispersion line shapes in a manner analogous tothe ‘‘transform methods’’ previously used to describe resonance Raman scattering. Particularattention is focused on the first two moments, for which simple analytical expressions are obtainedthat are computationally easy to implement. The behavior of the first two moments with respect tovarious parameters such as the pulse carrier~center! frequency, pulse width, mode frequency,electron-nuclear coupling strength, and temperature is investigated in detail. Using rigorousanalytical formulas, we also discuss the laser pulse induced squeezing of the nuclear distributions aswell as the pulse induced vibrational heating/cooling in the ground and excited states. The momentanalysis of the pump induced state presented here offers a convenient starting point for the analysisof signals measured in pump–probe spectroscopy. The moment analysis can also be used, ingeneral, to better understand the material response following ultrashort laser pulse excitation.© 2001 American Institute of Physics.@DOI: 10.1063/1.1356011#

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I. INTRODUCTION

Femtosecond laser pulses, with durations shorter ttypical nuclear vibrations, are widely used to prepare specmolecular states and study their evolution in real time.1–22

An example is pump–probe spectroscopy, or femtoseccoherence spectroscopy, where an ultrashort pump lpulse is used to excite the sample of interest. The subseqnonstationary response of the medium is monitored by alayed probe pulse. The effect of a short laser pulse incidon a two electronic level molecular system is to induce nstationary vibrational states~vibrational coherence! in boththe ground and excited electronic levels. Ground state coence, usually ascribed to impulsive stimulated Raman stering processes,4–6 is the dominant contribution in systemwith short-lived excited states and for off-resonaexcitation.2,10,11,14,16 Vibrational coherence in the excitestate is, however, the dominant contribution for systemshave long-lived excited states.1,7–9,12,13,19,22

A common description of pump–probe spectroscopybased on the third-order susceptibility (x3) formalism, whichprovides a unified view of four-wave mixingspectroscopies.23–26 The state of the molecular system ibetween the pump and probe pulse interactions is, howenot explicit in thex3 formalism.27 It is often the case that thmost interesting part of a pump–probe signal is whenpump and probe pulses are well separated in time, a lusually termed the well separated pulse~WSP! approxima-

a!Electronic mail: [email protected]

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tion. A physical model that treats the pump and probe eveseparately is therefore very useful. An example isdoorway-window picture,26 which can be used to represepump and probe events in terms of Wigner phase space wpackets, and readily enables a semiclassical interpretatiopump–probe experiments.28–30 Another approach in thewell-separated pulse limit is based on a description ofpump induced medium using nonstationary effective linresponse functions.10,25,31–33We have recently demonstratethe application of an effective linear response approachpump–probe spectroscopy using a displaced thermal srepresentation of the pump induced~doorway! densitymatrix.27 A key requirement for this representation isknowledge of the moments of position (Q) and momentum(P) in the doorway state. A moment analysis of the doorwstate is well justified by the fact that the vibrational staprepared by short pulses are highly localized in (Q,P) phasespace. We thus expect the doorway states to be descradequately by the first few moments. In addition, whenfirst moments of the doorway state are incorporated intoeffective linear response functions, the resulting pumprobe signals are in excellent agreement with the predictiof the third-order response approach,27 which implicitly in-corporates all the moments of the doorway density matrixaddition to providing an accurate description of pump–prosignals in the well-separated pulse limit, a knowledge ofmoments allows a clear physical interpretation of the amtude and phase profiles observed in pump–probe expments of nonreactive samples.27,34

5 © 2001 American Institute of Physics

IP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

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6796 J. Chem. Phys., Vol. 114, No. 15, 15 April 2001 Kumar et al.

Apart from pump–probe experiments, a fully quantittive understanding of the matter states prepared by shorser pulses can by itself serve as a fruitful exercise. Forstance, there has been an increasing interest in using sand multiple laser pulses to control molecular dynamicsto generate optimal vibrational motion.9,35–38Numerous the-oretical studies have addressed impulsive preparation oflecular states in detail and are usually based on a semisical treatment of the laser pulse interaction.36–40 Theadvantage of the semiclassical approach is that it provsimple analytical expressions that are valid in the short pulimit,36–38 and enables several key insights into the genetion and detection of coherent wave packet motion.39,40 Afully quantum treatment, on the other hand, has usuallytailed a direct numerical solution of the Schro¨dingerequation.35,37,41

One of the primary motivations of this work was thinterpretation of the observed amplitude and phase profilefemtosecond pump–probe experiments of myoglobin.27,34Asmentioned earlier, a fully quantum mechanical evaluationthe moments of the doorway state is key to the effectlinear response approach, which substitutes for the comptionally more intensive third-order response approach.27 It isalso clear that a precise determination of the first and higmoments of the nonstationary wave packets induced by spulse excitation is useful in the above-mentionapplications.9,35–37 In the present work, we consider a twelectronic level system with a multimode set of linearly dplaced harmonic oscillators, and obtain general expressfor the nth moments ofQ and P in the doorway densitymatrix. The first two moments are studied in detail, anddirectly connected to equilibrium absorption and dispersline shape functions. Because of this connection, absoscale calculations of the moments are made possible evecomplex multimode systems, with only a knowledge of tmeasured equilibrium absorption cross sections. Thisproach is analogous to ‘‘transform’’ calculations in resnance Raman scattering.42–46In addition, the temperature dependence of the moments is clearly revealed incalculations presented here, and the results are shownin good agreement with earlier treatments based on thepulsive ~short pulse! approximation.36–40

The general outline of the paper is as follows. In Sec.we briefly discuss the background for the present workdiscuss the basic expressions involved in a perturbative trment of the laser pulse interaction. In Sec. III, we calculthe first and second moments ofQ andP in the ground statedensity matrix using the general expressions for thenth mo-ments presented in Appendix A. In Sec. IV, we carry ousimilar analysis of the first two moments of the excited stdensity matrix. In Sec. V, we present simulations to studybehavior of the first two moments ofQ andP as a function ofthe laser pulse carrier frequency, pulse width, modequency, coupling strength, and temperature.

II. BACKGROUND

We first briefly review the basic expressions involvin a perturbative treatment of the pump induced density mtrix. Consider the interaction of a two electronic level systewith the pump laser pulse whose electric field is represen


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as E(t). We write the total Hamiltonian asH(t)5H0

1HI(t), with the free HamiltonianH0 and the interactionHamiltonian HI having the following typical form~in theinteraction picture! for a molecular system with two electronic levels:

H05S He1\Vv 0

0 HgD ,

~1!

HI~ t !52m~ t !•E~ t !52S 0 mge~ t !•E~ t !

meg~ t !•E~ t ! 0 D ,

where Hg and He5 Hg1V are, respectively, the adiabatBorn–Oppenheimer Hamiltonians for the ground and excielectronic states.V is defined as the difference potential thspecifies the electron nuclear coupling,\Vv is the verticalelectronic energy gap at the equilibrium position of tground state, andm(t) is the electric dipole moment operatoevolving according to the free HamiltonianH0 . Let r~2`!denote the initial density matrix of the system before tarrival of the pump fields. The evolution of the density mtrix due to the laser pulse interaction can be described byLiouville equation:47

i\]r~ t !

]t5@HI~ t !,r~ t !#. ~2!

Equation ~2! can be solved forr(t) iteratively to variousorders in the interactionHI(t). For pump–probe calculations, the second-order term is of relevance since it islowest order in the pump interaction that can contributethe signals.48,49From Eq.~2!, we obtain27 the following formfor the density matrix to second-order in the pump intertion:

r~ t !5 r~2`!1S i

\ D 2E2`

t

dt2E2`

t2dt1@mk~ t2!,

@m l~ t1!,r~2`!##Ek~ t2!El~ t1!. ~3!

Note that we have omitted the first-order term as it is nrelevant for a pump–probe calculation. In Eq.~3!, the indicesk and l refer to the vector components and summation orepeated indices is implied. Note that the second-orderturbative term is also referred to in the literature as the dsity matrix jump50 and the doorway function.28,30,51For well-separated pump and probe pulses, one is interested indensity matrix in Eq.~3! after the pump fieldsE(t) haveceased to evolve. Ast→` in Eq. ~3!, the interaction picturedensity matrix becomes time independent, and the timeguments will be dropped in the following development.

If we project the density matrixr onto the electronicbasis, we obtain the nuclear sub-density matrices forground and excited electronic states. Noting that the secoorder density matrix in Eq.~3! has no electronic coherencedue to the even number of dipole interactions, we may w

r5 reue&êu1 rgug&^gu5S re 0

0 rgD . ~4!


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6797J. Chem. Phys., Vol. 114, No. 15, 15 April 2001 Ultrafast nuclear response

In general, the nuclear subdensity matricesrg and re maycontain off-diagonal elements~vibrational coherence! in thenumber state representation due to the broad frequency strum of an ultrashort laser pulse. The presence of odiagonal terms reflects the nonstationary nature ofr, whichin general does not commute withH0 . The vibrational co-herence translates into time-dependent wave packetssemiclassical phase space (Q,P) Wigner representation othe density matrix.26,40,52,53On the other hand, the highllocalized nature~in Q andP) of impulsively driven nonsta-tionary states suggests that we calculate their moments instateu ~which representsg or e) using

Xu~ t !5Tr@ûuX~ t !uu&ru#/Tr@ ru#. ~5!

With a(a†) as the mode annihilation~creation! operators,Xrepresents Q5(a1a†)/A2, P5 i (a†2a)/A2, or theirhigher powers. The timet in Eq. ~5! is larger than the pulseduration so that the density matrixrs is time independentHowever, thet50 values54 of the first moments ofQ andP,i.e., Qu(0) andPu(0), respectively~for u5g or u5e), de-termine the effective initial conditions for the subsequenuclear dynamics on the potential surfaceu. Since Qu(0)and Pu(0) also denote shifts from thermal equilibrium, wmay represent the nonstationary nuclear density matrixthe electronic state ‘‘u’’ as a coherent-thermal state:

ru5D~lu!rT(u)D†~lu!. ~6!

Here, D(lu) is the quantum mechanical displacemeoperator55

D~lu!5exp~lua†2lu* a!, ~7!

with lu5(Qu(0)1 i Pu(0))/A2. rT(u) is the equilibrium ther-

mal density matrix corresponding to the nuclear Hamiltonof the electronic levelu:

rT(u)5Z21exp~2Hu /kBT!. ~8!

The displaced thermal state representation in Eq.~6! hasbeen shown27 to provide an accurate and computationaefficient approach to calculating pump–probe signals. Tpresents us with the primary motivation for a rigorous mment analysis of the pump induced density matrix. Aswill see in the following, the laser pulse can induce highmoment changes~also termed squeezing! in the vibrationaldistributions in addition to merely displacing them. Equati~6! is only an approximation to the full second-order densmatrix in Eq.~3!. The higher moments ofru calculated herecan be incorporated in a manner analogous to Eq.~6! using adisplaced and squeezed state representation.

In Appendix A, we derive general expressions for tnth moments ofQ(t) andP(t) for the pump induced nucleadensity matricesrg and re for a two level system, with asingle linearly coupled~undamped! mode. In what follows,we discuss the effect of the pump pulse interaction onvibrational populations and the first two moments ofQ andP in the ground and excited states.


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III. NONEQUILIBRIUM GROUND STATE DENSITYMATRIX

For simplicity, we restrict the discussion here to scafields assuming that the medium is isotropic. We makeCondon approximation that neglects the nuclear coordindependence of the dipole operator and writem5mgeug&êu1megue&^gu. The system is assumed to be initially in thground electronic state with an equilibrium Boltzmann dtribution for the vibrational states, i.e.,r(2`)5ug&rT

(g)^gu,whererT

(g) is of the form in Eq.~8! with Hs replaced byHg .The ground state density matrix to second order in the pufields is evaluated using Eq.~3! as rg5^gurug&. We find

rg5 rT~g!2

umgeu2

\2 E2`

`

dt8E0

`

ds E~ t8!E~ t82s!e2Gegusu

3H e2 iVvsexpS 2i

\Et82s

t8ds8V~s8! D

1

rT~g!1h.c.J , ~9!

where the subscript~1! denotes time ordering and h.c. dnotes Hermitian conjugate. We have defineds5t82t9 forconvenience.V(s) evolves in time viaHg in the interactionpicture. We have also introduced homogeneous broadeusing the factore2Gegusu to account for electronic dephasinprocesses. It is evident from Eq.~9! that the action of theelectron-nuclear coupling force~which appears through thtime-ordered exponential! on the equilibrium staterT

(g) re-sembles a square wave pulse interaction that turns on anat times separated by an intervals. The total effect of thepulse on the ground state is obtained by a superpositiosquare wave interactions for all possible time intervwithin the duration of the pump pulse.

Equation ~9! is valid for arbitrary ground and excitestate nuclear Hamiltonians. In what follows we considemodel of linearly displaced harmonic oscillators for thground and excited states and takeV52(\/mv0)1/2f Q5

2\v0DQ, with dimensionlessQ and relative displacemenD. The electron-nuclear coupling forcef is expressed asf5D(mv0

3\)1/2 where m and v0 are, respectively, the reduced mass and frequency of the mode. With these detions, the ground and excited state Hamiltonians takeform

Hg5\v0

2@Q21 P2#, ~10a!

He5\v0

2@~Q2D!21 P2#2

\v0D2

2. ~10b!

A. Populations

The pump pulse induces a net change in the groundexcited electronic state populations, which is reflected inzeroth moment, i.e., the trace of the respective nuclear dsity matrices. The electronic population in the ground stafter the pump pulse interaction is calculated asNg

5Tr@ rg#. Using Eq.~9!, we find,


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Ng512umgeu2

\2 E2`

`

dt8E0

`

ds E~ t8!E~ t82s!

3$e2 iV00se2g(s)e2Gegusu1c.c.%, ~11!

whereV005Vv2v0D2/2 is the zero-zero electronic transtion frequency andg(s) is the harmonic oscillator correlatiofunction. For a single undamped mode,

g~s!5~D2/2!@~ n11!~12e2 iv0s!1n~12eiv0s!#, ~12!

with n5(exp(\v0 /kBT)21)21. Analytic expressions forg(s)for the damped harmonic oscillator case at arbitrary tempture have been derived earlier56 and can be incorporated ithe expressions derived in this work. In terms of the FoutransformE(v) of the electric field, Eq.~11! takes the form

Ng512umgeu2

p\2 E2`

`

dv uE~v!u2F I~v!, ~13!

whereF I(v) is the imaginary part of the complex equilibrium line shape function defined as

F~v!5 i E0

`

ds ei (v2V00)se2Gegusue2g(s). ~14!

The imaginary part of the line shape function is direcrelated to the absorption cross section assA(v)5(8pumgeu2/3\c)vF I(v). We take the electric field to beof the form E(t)5E0G(t)cos(vct) whereG(t) is a dimen-sionless Gaussian envelope function. The corresponFourier transform is thenE(v)5E0@G(v2vc)1G(v1vc)#/2 where G(v) is the Fourier transform ofG(t).When this definition is used in Eq.~13!, four terms resultfrom the expansion ofE2(v). Equation~13! contains contri-butions from the line shape function at negative frequenci.e., F I(v,0). It is clear that the integrand in Eq.~14! ishighly nonresonant forv,0, so that the line shape functiomakes a vanishing contribution at negative frequencThus, we neglect those terms in Eq.~13! that contain thenegative frequency part of the pulse envelope spectrum,G(v1vc) and take only the positive frequency part of tintegral in Eq.~13!. This is a valid~‘‘rotating wave’’! ap-proximation, given that we are concerned with pulses thave optical carrier frequencies. We then arrive at thelowing result:

Ng512umgeu2E0

2

4p\2 E0

`

dv G2~v2vc!F I~v!. ~15!

The depletion in the net ground electronic state popution thus depends on the convolution of the laser intenspectrum with the absorption line shape as one would expSince the total number of molecules must be conserved~ne-glecting decay through other nonradiative channels duthe pump pulse!, the corresponding number of molecultransported to the excited state by the pump is simply giby Ne512Ng . It should be remembered that Eq.~15! hasbeen derived in the weak field perturbative limit and tnumber of molecules depleted from the ground state is vsmall ~i.e., Ng'1).


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The line shape functionF~v! plays a central rolethroughout the calculations presented in this work. Whileelectronic populations depend only on the imaginary parF~v!, we will see in the following that the moments ofQ

and P in the pump induced nonequilibrium density matrinvolve both the absorptive and dispersive line shapesF I(v)andFR(v).

B. Vibrational dynamics

The interaction with the laser pulse sets up both eltronic and vibrational coherence in the system. However,electronic coherence vanishes for a second-order interacwith the pump laser field, owing to the even dipole interations in the second-order perturbative expression for the dsity matrix in Eq. ~3!. We therefore focus attention on thvibrational coherence induced in the electronic states.

1. First moment

The first moment ofQ(t) in the ground state is obtaineby lettingn51 in the general expression Eq.~A10!. We find

Qg~ t !51

2iNg(l 50

`

(m50

`

d1,2l 1m

~21! l 1m

l !m!Dm~2n11! l

3@Cm~ t !1~21!mCm* ~ t !#, ~16!

whereCm(t) is given by Eq.~A11!. Due to the Kronecker’sdelta, we must havel 50 andm51, so that

Qg~ t !52~D/Ng!Im@C1~ t !#. ~17!

Using Eqs.~A11! and ~A9!, we find

C1~ t !5umgeu2

2p\2E2`

`

dv E~v!$~ nT11!e2 iv0tE* ~v2v0!

3@F~v!2F~v2v0!#2nTeiv0tE* ~v1v0!

3@F~v!2F~v1v0!#%. ~18!

Equation~18! again involves nonresonant contributions frothe line shape function at negative frequencies, i.e.,F~v,0!.For pulses with optical carrier frequencies, the contributof the negative frequency components is negligible andbe dropped from the above integral. We then find, after ming a change of variablesv→v2v0 in the second part ofthe above integral and using the definition ofE(v) followingEq. ~12!,

Qg~ t !5uA1gucos~v0t1w1g!, ~19!

where A1g5Qg(0)1 i Pg(0)5uA1guexp(2iw1g) is the com-plex amplitude for first moment dynamics in the groustate, with the effective ‘‘initial’’ position and momentumQg(0) andPg(0) given by:

Qg~0!52umgeu2E0

2~2n11!D

8p\2Ng

3E0

`

dv Gp~v2vc ,2v0!DF I~v!, ~20a!


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Pg~0!5umgeu2E0

2D

8p\2NgE

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`

dv Gp~v2vc ,2v0!DFR~v!.

~20b!

Here, we have introduced the product spectral functGp(v,nv0):

Gp~v,nv0!5G~v!G~v1nv0!, ~21!

and the difference operatorD ~not to be confused with thedimensionless couplingD) whose action is to generate diferences as46

D F I~v!5F I~v!2F I~v2v0!, ~22a!

D2 F I~v!5D @F I~v!2F I~v2v0!#

5F I~v!22F I~v2v0!1F I~v22v0!,

~22b!

and so on for the higher powers ofD.Equations~20a! and~20b! are fully quantum mechanica

expressions for the first moments and are valid for arbitrpulse widths, temperature, and electron-nuclear displacemD. It is clear that the mean initial position and momentuimparted by the pump pulse to the ground state wave paare related roughly to the derivatives of the imaginary areal parts of the equilibrium line shape function, resptively. In Appendix B, we discuss the impulsive limit of thabove-mentioned results and also present a closed formpression for the mean position assuming a Gaussian~semi-classical! form for F I(v). The single mode expressionsEqs. ~20a! and ~20b! can be readily generalized to the mutimode case, with the same equations applying for a gimode with parametersv i , D i , and ni . The line shapes canbe obtained using the multimode form of the equilibriucorrelation function in Eq.~12!. In fact, Eqs.~20a! and~20b!~as well as the rest of the moment expressions derived inwork! are quite general multimode expressions so long asmode of interest is linearly coupled andF(v) is obtainedfrom the experimental line shape. In this case, the remainmultimode subspace resides in the experimental line shfunction, unrestricted by approximations such as linear cpling and mode mixing.

2. Second moment

In addition to the first moments of the nuclear positiand momentum, it is also of interest to calculate the puinduced changes in the variances~or the uncertainties! of theposition and momentum distributions. Before the interactwith the laser fields, the position and momentum uncertaties ~defined in terms of the variancessA

25A22A2, with A5Q,P) have equal values at temperatureT:

sQ5sP5An11/2. ~23!

The effect of the pump pulse is to distort the equilibriunuclear distributions and modify the above-given variancTo see what is involved, we expressQ and P in terms ofaanda† using the definitions following Eq.~5! and obtain thesecond moment ofQ(t) in the pump induced ground state a


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s.

Qg2~ t !5~1/2Ng!Tr@ rg~ a2~ t !1a†2~ t !1a†a1aa†!#

5ng11/21uA2gucos~2v0t1w2g!, ~24!

where we have also useda(t)5ae2 iv0t. The quantitiesng

andA2g5uA2guexp(2iw2g) are, respectively, the mean valuof the number operatorn5a†a and the mean value ofa2, forthe state described byrg . It is clear from Eq.~24! thatthe second moment in the pump induced state consistboth vibrational coherence at frequency 2v0 and the meanoccupation number of the ground state. The amplitudethe second moment dynamics at frequency 2v0 is given

by uA2gu. If Qg2(t)2Qg(t)2 is time-dependent, the nuclea

distribution has been squeezed by the laser pu

interaction.57–60 In what follows, we evaluateQg2(t) using

the general result Eq.~A10!, which assumes the second-ordperturbative expression forrg given in Eq.~9!. A direct com-parison of the results with the identity in Eq.~24! subse-quently leads to expressions forng andA2g to second-orderin the fields.

If we setn52 in Eq.~A10!, the allowed sets of values o( l ,m) are (1,0) and (0,2) and we find

Qg2~ t !5

2n11

2Ng~112 Re@C0~ t !# !2

D2

2NgRe@C2~ t !#. ~25!

Using Eqs.~A11! and ~15! we have Re@C0(t)#5(Ng21)/2.The quantityC2(t) is also readily evaluated using Eq.~A11!.On comparing the final result with Eq.~24! and ignoringnonresonant contributions, the second-order expressionsmean occupation number and the oscillatory amplitude ofsecond moment are found as

ng5n2umgeu2E0

2D2n~ n11!

8p\2Ng

3E0

`

dv G2~v2vc!D2F I~v1v0!, ~26!

A2g52umgeu2E0

2D2

8p\2NgE

0

`

dv Gp~v2vc ,22v0!

3@~ n21n11/2!D2F I~v!2 i ~ n11/2!D2FR~v!#.

~27!

As discussed in Appendix A, the second moment of

momentum operator is obtained asPg2(t)5Qg

2(t1p/2v0),so that

Pg2~ t !5ng11/22uA2gucos~2v0t1w2g!. ~28!

The variances ofQ and P in the pump induced ground statcan be directly obtained using Eqs.~24! and~28! along withthe first momentsQg(t) andPg(t) given by Eq.~19!. For thetime-dependent uncertainty product, we find

sQg~ t !sPg~ t !>ng11

2~12uA1gu2!2

uA2gu2

~2ng11!

3cos2~2v0t1w2g!. ~29!


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In obtaining Eq.~29!, small terms involving higher powerof the electric field strength have been ignored, since wein the weak field limit. Furthermore, in typical weak copling situations,uA2gu is much smaller thanuA1gu since itscales with the square of the coupling strengthD. Thus, thetime-dependent term of Eq.~29! is very small and the uncertainty product is nearly time independent.

A direct inspection of Eqs.~26! and ~27! reveals an ap-parent contradiction. When the system is initially atT50,we haveng50, i.e., there is no change in the vibrationpopulations due to the pump–pulse interaction. The osctory amplitudesA1g and A2g that describe the ground stacoherence are, however, not necessarily vanishing atT50,as evident from Eqs.~27! and ~20a! and ~20b!. The numberbasis elements ofrg therefore obey the following relations

~ rg!0 jÞ0, ~ rg!0051, ~ rg! j j 50, ; j Þ0, ~30!

in violation of the density matrix inequality condition61

ur i j u2<r i i r j j for any given pair of vibrational levelsi and j.Thus, strictly speaking,rg is not a complete vibrationadensity matrix.40 Furthermore, we have from Eqs.~26!to ~29!, at T50,

sQg~0!sPg~0!> 12~12uA1gu2!, ~31!

which is always less than the zero point limit of 1/2. Theapparent discrepancies are due to the second-order perttive approximation to the full density matrix employed heas well as elsewhere.29,30,32,37,38,50A genuine change in thevibrational populations in the ground state can only ocdue to a fourth-order interaction, via stimulated Raman stering processes. Thus, the complete expressions for theond moments of the pump induced density matrix wonecessitate the inclusion of fourth-order terms in the perbation expansion. This is beyond the scope of the prepaper. However, we can estimate the magnitude of the ein the calculation to be roughlyuA1gu2 @from Eq.~31!#, whichgoes as the fourth power of the electric field strength antypically very small for weak fields. We will return todiscussion of this issue in Sec. V, where we also presequantitative estimate for the error made in a second-operturbative treatment for a simple model system.

IV. NONEQUILIBRIUM EXCITED STATE DENSITYMATRIX

The excited state density matrix is obtained from Eq.~3!as re5êurue&. Here, the terms with a single dipole interation on either side ofrT in the expression forr will contrib-ute. For the linear displaced Hamiltonians in Eqs.~10a! and~10b!, we find

re5umgeu2

\2 E2`

`

dt8E2`

t8dt9 E~ t8!E~ t9!

3H exp~ iVvs2Gegusu!S expS 2 iv0DE0

t8ds8 Q~s8! D D

2

3 rT~g!S expS iv0DE

0

t9ds8 Q(s8) D D

1

1h.c.J , ~32!

where the subscript2 denotes anti-time ordering.


re

-

eba-

rt-ec-

r-ntor

is

aer

A. Populations

The fraction of excited state moleculesNe is simplyNe512Ng as the sum of the ground and excited state polations must be conserved~ignoring other channels for thedecay of the excited state!. We thus have from Eq.~15!,

Ne5~ umgeu2E02/4p\2!E

0

`

dv G2~v2vc!F I~v!. ~33!

In the weak field approximation assumed here,Ne!1. Thus,only a small fraction of the ground state is transported toexcited state by the pump interaction.

B. Vibrational dynamics

1. First moment

In Appendix B, we calculate the moments of the shiftcoordinate operatorQe8(t)5 Qe(t)2D for the excited statedensity matrix~the subscripte indicates time evolution dueto He). The first moment is obtained by lettingn51 in Eq.~A21!. Only the term with (l 50,m51) contributes and wefind

Qe8~ t !52~D/Ne!Im@D1~ t !#. ~34!

From Eqs.~A22! and ~A19!, we find after neglecting nonresonant contributions,

Qe8~ t !5uA1eucos~v0t1w1e!, ~35!

whereA1e5Qe8(0)1 i Pe(0)5uA1euexp(2iw1e) is the ampli-tude for the first moment dynamics~about the equilibriumposition D) in the excited state, with the effective initiaposition and momentum given by

Qe8~0!52umgeu2E0

2D

4p\2NeE

0

`

dv Gp~v2vc ,2v0!

3@F I~v2v0!2nDF I~v!#, ~36a!

Pe~0!50. ~36b!

Thus, the excited state wave packet receives no initial mmentum. Correspondingly, the initial phasew1e can onlytake the values 0 orp. Note that Eq.~35! gives the time-dependent mean position with respect to the excited sequilibrium positionD. To revert to the ground state equilibrium as the origin, we simply write

Qe~ t !5D1uA1eucos~v0t1w1e!. ~37!

The absence of initial momentum on the excited stwave packet might be expected on intuitive grounds. Tvibrational modes that are formed in the excited electrostate are subject to the electron-nuclear coupling forceturns on ~during the pump interaction! and thereafter, re-mains a constant. Thus, the forces on excited state nuclestep function-like. In contrast, the ground state nuclei feelelectron-nuclear coupling force only for a short time durithe pulse interaction, so that the forces on the ground s


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mncndI

ioap

bs

s

c

e-

ltsna

nd

ob-

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rstlearnt,

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are impulsive in nature. It is interesting to note that Eq.~36b!is predicted by earlier treatments that use the semiclasFranck approximation,51,62 where more general potentiawere treated than the harmonic approximation made inpresent case. It is also notable from Eq.~36a! that in sharpcontrast to the ground state case, the excited state firstment involves only the imaginary part of the line shape fution F I(v). This difference reflects the fact that the groustate coherence is created by Raman-type processes.well known42,43,46 that the resonance Raman cross sectinvolves both the real and imaginary parts of the line shfunction. The excited state coherence is created solelyabsorption-like events occurring on the bra and ket sidethe density matrix as expressed in Eq.~32!.

2. Second moment

From the definitions in Eqs.~A13! and ~A12!, we haveQe8(t)5(aee

2 iv0t1ae†eiv0t)/A2 so that the following gen-

eral relation holds analogous to Eq.~24!:

Qe82~ t !5ne11/21uA2eucos~2v0t1w2e!, ~38!

where ne is the mean value of the occupation numberne

5ae†ae , andA2e5uA2eue2 iw2e is the mean value ofa2 in the

pump induced excited state. If we letn52 in the generalexpression derived in Eq.~A21!, the allowed sets of valuefor ( l ,m) are (1,0) and (0,2) and we get

Qe82~ t !5

2n11

NeRe@D0~ t !#2

D2

2NeRe@D2~ t !#. ~39!

Using Eq. ~A22! we have Re@D0(t)#5Ne/2. The quantityRe@D2(t)# is also evaluated readily using Eq.~A22!. Com-paring the final result with Eq.~38!, we find that the meanoccupation number of the pump induced excited state oslator ne is given by

ne5n2n~ n11!D21umgeu2E0

2D2

8p\2NeE

0

`

dv G2~v2vc!

3@~ n11!2F I~v2v0!1n2F I~v1v0!#. ~40!

The oscillatory amplitude is given by

A2e5~ umgeu2E02D2/8p\2Ne!

3E0

`

dv G~v2vc1v0!G~v2vc2v0!F8~v!,

~41!

where we have defined the function

F8~v!5@F I~v2v0!22nDF I~v!

1n2D2F I~v1v0!#. ~42!

The amplitudeA2e for the second moment dynamics at frquency 2v0 is real so that the phasew2e can take on only thevalues 0 orp. This is in contrast to the ground state resuderived in Sec. III B 2. Both the first and the second momein the excited state depend only on the absorptive line sh


al

e

o--

t isney

of

il-

tspe

function F I(v) whereas the ground state moments depeon bothF I(v) andFR(v).

The second moment of the momentum operator is

tained as before by lettingt→t1p/2v0 in Qe82(t). The first

moment and the second moment results can be combineobtain the uncertainties inQ and P. For the position uncer-tainty, we find

sQe~ t !5@ ne1~12A1e2 !/21~A2e2A1e

2 /2!cos~2v0t !#1/2.~43!

The momentum uncertainty is obtained assPe(t)5sQe(t1p/2v0). For the time-dependent uncertainty produwe get

sQe~ t !sPe~ t !>ne11

2~12A1e

2 !2A2e~A2e2A1e

2 !

~2ne11!

3cos2~2v0t !, ~44!

where we have neglected terms that contain powers ofA1e

andA2e larger than second, since they involve higher powof the field strengthE0 , which is assumed to be very weaEquation~44! is simpler than Eq.~29! for the ground statedue to the lack of an initial phase for the excited state fiand second moments. Once again, for weak electron-nuccoupling, the uncertainty product is nearly time independesinceA2e is typically much smaller thanA1e .

We note from the above-given expressions thatsecond-order density matrixre is not in explicit violation ofthe uncertainty principle as is the pump induced ground sdensityrg . To see this clearly, we first note from Eqs.~36a!and ~33! that at T50, uA1eu<D since Gp(v2vc ,2v0)<G(v2vc). Also, from Eq.~40!, we havene>D2/2 , pro-vided the vibrational frequency can be neglected inF I(v6v0) ~a condition known as ultrafast electrondephasing26!. It then follows from Eq.~44! that atT50,

sQe~ t !sPe~ t !*1/2, ~45!

where the error in Eq.~45! is to fourth order in the fieldstrength and is very small for weak fields. Thus, the secoorder approximation to the excited state density matrixmore accurate than for the ground state density matrix. Tdifference is not surprising, given that a change in the vibtional populations of the excited state can be effectivelyduced by a second-order interaction. This stands in contto the ground state vibrational levels, where the populatichange due to scattering processes via a fourth-order inaction.

V. SIMULATIONS AND DISCUSSION

For the simulations reported here, fast Fourier traforms were employed to calculate the complex line shafunction in Eq. ~14!, for a given set of model parameterThe precalculated line shape functions are subsequentlyin all the moment calculations, thereby reducing the comtation time significantly. The single integration in the mment expressions is computationally straightforward, asintegrands are smooth functions that are highly converg~owing to the Gaussian spectral envelope of the laser pu!.


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In this section, we investigate the behavior of the groundexcited state moments as a function of various paramesuch as the pulse carrier frequency, temperature, modequency, coupling strength, and pulse duration.

One attractive feature of the moment expressionsrived in this work is that apart from parameters specificthe mode under consideration, all other information aboutrest of the modes~and the bath! is carried through in theequilibrium line shape functionsF I andFR . These are notindependent quantities but are rather connected byKramers–Kronig relations. One can therefore obtainF I(v)directly from the measured experimental absorption crsection ~on an absolute scale!. The Kramers–Kronig rela-tions can then be used to obtainFR(v) from F I(v). Thus,given a molecule in the condensed phase whose absorspectrum is known experimentally~and is possibly broad anasymmetric!, one can use the results derived here to calcuthe moments of the pulse induced nonequilibrium state onabsolute scale, for any given mode of the system. Furtmore, since the relation between moments andF I(v),FR(v) is linear, we can also extend the main results derivin this paper to account for inhomogeneous broadeningto a distribution of electronic 0–0 transition frequencies.63

A. First moments

1. Ground state

Consider the ground state first moments ofQ and Pderived in Eqs.~20a! and ~20b!. Several interesting featureare evident from a direct inspection of these expressioWhen the carrier frequency of the pump pulsevc is tunedacross the resonant maximum (Vv), Qg(0) changes signowing to its dependence on the derivative-like functiDF I(v). It is also clear from Eq.~20a! that for detuning tothe red of the absorption band,Qg(0) has a sign opposite tthat of D, while it has the same sign asD for blue detuning.The dependence ofPg(0) on DFR(v) implies that in theregion of resonance, the momentum impulse is opposidirected with respect to the electron-nuclear coupling forcf.Also, the sign ofPg(0) changes as we tune away from resnance on either side ofVv . When the pulse carrier frequency is detuned far from the resonant transition~but ne-glect of the nonresonant terms remains valid!, DF I(v) dropsmore rapidly to zero when compared withDFR(v) so thatQg(0) is much smaller thanPg(0). Thus, for off-resonantexcitation, the ground state coherence is dominated by amentum impulse. Under off-resonant excitation, Eq.~20b!implies that Pg(0) has the same sign asD and thereforepoints in the same direction as the electron-nuclear coupforce. The well known off-resonant impulsive limit15 is thusseen to arise from the dependence of the momentum impon the derivative of the real part of the line shape function39

The fully quantum mechanical treatment further shows tthe approach to the off-resonant limit depends stronglythe temperature and mode frequency, as we will see infollowing. The distinct dependence ofQg(0) andPg(0) ontemperature (T) is also noted from Eqs.~20a!–~20b!. Whilethe T dependence ofQg(0) is determined both by the facto


drs

re-

e-

e

e

s

ion

tenr-

de

s.

ly

-

o-

g

lse

tne

(2n11) and the line shapeF I(v), Pg(0) depends onTsolely through the real part of the line shape functiFR(v).

In order to illustrate the above-mentioned aspects ofground state first moments, we consider two model lshapes. In the first example, shown in the left panel of Figa low frequency mode@v0550 cm21, S5(D2/2)50.1] iscoupled to a homogeneously broadened (Geg5300 cm21)two level system. Figure 1~a! shows the imaginary part oline shape function at two different temperatures. The cosponding mean position and momentum for the 50 cm21

mode are shown in the panels directly below, for a rangevc values across the resonance maximum. The pulse widchosen to be 50 fs. The second example, shown in the rpanel of Fig. 1, consists of a line shape that is strongly teperature dependent due to the presence of a strongly couoverdamped bath mode in addition to the 50 cm21 mode.Vibrational damping was incorporated64 using a rigoroustheory presented previously.56

FIG. 1. Temperature and carrier frequency dependence of the mean nuposition and momentum in the pump induced ground state.~a! Absorptionline shape functionF I(v) for a homogeneously broadened system (Geg

5300 cm21) consisting of a single linearly coupled mode (v0550cm21,S50.1). The solid line showsF I(v) for T50 K and the dashed linefor T5300 K. ~b! Same as in~a! but with an additional overdamped batmode (vb510 cm21,Sb520,gb550 cm21) included in the line shape.~c!and~d! The pump pulse induced effective initial position and momentumthe ground state~in dimensionless units! for the 50 cm21 mode, for both lineshapes in~a!. ~e! and~f! The mean position and momentum for the strongtemperature-dependent line shape in~b!. For all the simulations, a Gaussiawith full width at half maximum of 50 fs was used for the electric field, athe total pulse energy was fixed at 1 nJ.


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to


From the simulations, it is seen that some of the genfeatures evident from a direct inspection of Eqs.~20a! and~20b! are borne out; note in particular, the behavior of tposition and momentum as roughly the derivatives ofabsorptive and dispersive line shapes. Thus, the momenincrement is peaked at absorption maximumVv and is di-rected opposite to the excited state equilibrium shift, whin the present case is positive. The position incremenpeaked at roughly the full width at half maximum of thabsorption, and changes sign as the carrier frequency is tacrossVv . The difference in the temperature dependencethe position and momentum increments is also clear. Iseen that for the weakly temperature-dependent line shthe momentum increment is nearly unaffected by tempeture changes, while the position increment increases dramcally owing to the Bose–Einstein factor. The situation is dferent for the strongly temperature-dependent line shapewhich case, it is seen that the momentum imparted towave packet changes significantly with temperature, witdecreased value at theVv @corresponding to the fact thaFR(v) is broader at higher temperature, which in turn implies that the derivative is smaller#. It is also interesting tonote the behavior asvc is tuned away from resonance, wheit is seen thatQg(0) approaches zero whilePg(0) still has anonzero value.

The relative magnitudes ofQg(0) and Pg(0) are bestunderstood by plotting the amplitude and phase of the wpacket motion defined in Eq.~19!. In Fig. 2, we plot theinitial amplitude and phase of the pump induced wave pacas a function of the pulse carrier frequency for both exampconsidered in Fig. 1. It is seen from the phase plot~lower

FIG. 2. The first moment amplitudeuA1gu and phasew1g for the 50 cm21

mode in the pump induced ground state, plotted for both line shapessidered in Fig. 1.~a! and~b! uA1gu andw1g for the Lorentzian line shape inFig. 1~a! and ~c! and ~d! for the phonon broadened line shape in Fig. 1~b!.The solid lines correspond toT51 K and the dashed lines correspondT5300 K.


al

em

his

edf

ise,

a-ti-

-inea

-

e

ets

panels! that at high temperatures~larger n) the phase re-mains close to 0 orp. When the temperature is lowered, thphase varies more continuously over a range of 2p. Thiscorresponds to the fact that the position increment@real partof the complex displacement,A1g5Qg(0)1 i Pg(0)] domi-nates the momentum increment~imaginary part! at high tem-peratures, except at resonant maximumVv . Furthermore,when the pump is tuned toVv , the purely impulsive(Qg(0)50⇒wg5p/2) nature of the ground state coherenis evident. As the pump carrier frequency is tuned towardoff-resonance limit, the phase is again seen to approachimpulsive limit (wg52p/2), and this approach occurs mucmore rapidly at lower temperatures than high. This is agowing to the fact that the high temperature case is dominaby the position increment. Note that the momentum imparto the ground state wave packet by resonant and nonresopulses are opposite in direction, with the momentum oresonantly induced wave packet directed opposite toelectron nuclear coupling force.

Several of the above-mentioned aspects of detuningpendence ofQg(0) and Pg(0) have been noted in earliesemiclassical treatments.36–40 The results for the excitationfrequency dependence of the ground state first momentsin qualitative agreement with the predictions of Cina aco-workers,37 who have used semiclassical pulse propagato study the impulsive preparation of ground state nuclmotion due to single and multiple laser pulse excitatioTheir calculations suggest that excitation in the preresonregion induces much larger increments in the nuclear ption than excitation directly on resonance. From the preswork, this aspect directly follows from the dependence ofposition increment on the derivative of the absorption lishape function as expressed in Eq.~20a!. We also note thatsemiclassical models have been used to show that the grstate momentum impulse depends on the derivative ofreal part of polarizability.39 The present fully quantum mechanical treatment clearly exposes the distinct temperadependence of the pulse induced position and momentumthe ground state. While the temperature dependence isincluded explicitly in prior treatments, more complicated ptential surfaces have been analyzed. But the general natuthe pump induced position and momentum predicted hereharmonic potentials remains valid.

2. Excited state

Turning to the excited state first moment, recall froSec. IV that the wave packet created in the excited sreceives no initial momentum. As mentioned earlier, the laof initial momentum for the excited state wave packet canattributed to the step function-like nature of the forces feltthe nuclei of excited state molecules. This argument ispected to be valid for more complicated potentials thanharmonic model considered here. This is indeed the casshown by earlier treatments.51 On the other hand, the grounstate nuclei are subject to an impulsive square wave foThis results in a nonzero average initial momentum ofground state wave packet.

n-


at

-

at

fungth

enInt

e

fofr

re

. I5s

alat

tcais

ptioc

y

,qub

-ce

siretech

ketncyaxi-rier

first

i-

ex-rac-

dif-nant

ound


Consider the initial displacement of the excited st

wave packetQe8(0) in Eq. ~36a!. The temperature depen

dence ofQe8(0) is mainly determined by the factorn, which

appears as the coefficient of the derivative-likeDF I(v). At

zero temperature, or for temperatures low enough so thn!1, all the terms in the integrand of Eq.~36a! are positive.

Thus, the overall sign ofQe8(0) is always opposite to that oD, irrespective of other parameters such as the laser pwidth and carrier frequency. This leads us to the followiinteresting conclusion: when the laser pulse interacts wisystem that is initially atT50 ~or with negligible thermal

populationn), the wave packet created in the excited statealways placed initially on the side of the excited state pottial well that lies toward the ground state equilibrium.other words, the initial phasew1e of the excited state firsmoment dynamics is always fixed at 0 orp, depending onlyon the sign potential displacementD. As the temperature is

increased, the difference termDF I(v) that appears as th

coefficient ofn in Eq. ~36a! begins to remove the positivityof the integrand, since the differenceF I(v)2F I(v2v0)can take on either positive or negative values. However,line shapes that are broad compared to the vibrational

quency,DF I(v) is still quite small compared toF I(v) andthe derivative term dominates only at very high temperatu

~when n@1).We take the model line shape treated in Fig. 1~a! to

illustrate key aspects of the excited state first momentFig. 3, we plot the excited state first moments for thecm21 mode. In addition to the amplitude of oscillation

uA1eu5uQe8(0)u about the excited state equilibrium, the initivalue of the first moment with respect to the ground stequilibrium Qe(0) @see Eq.~37!# is also plotted for clarity.We note that in contrast to the ground state [email protected]~a!#, the sign of the temperature-dependent changes inexcited state amplitude depends on the direction of therier frequency detuning from the absorption maximum. Itseen that for red detuning from absorption maximum,uA1eudecreases with increase in temperature, whereas for bluetuning, the amplitude increases with temperature. This opsite behavior on the red and blue sides of the absorpmaximum can be understood from the ‘‘particle-like’’ aspeof the excited state coherence, which is complementarthe ‘‘holelike’’ nature of the ground state coherence.40,52Wewill return to this point briefly. As is clear from Fig. 3Qe(0) changes its sign with respect to the ground state elibrium position asvc is tuned across resonance, but its asolute magnitude remains less thanuDu. As the temperatureis increased, the derivative line shape in Eq.~36a! becomesmore significant due to largern. The antisymmetry of theamplitude profile with respect toVv becomes more pronounced so that excitation toward the blue side indularger displacements than red excitation.

We note from Figs. 1~c! and 3~a! that the mean positionfor the ground and excited state are oppositely signed wrespect to the ground state equilibrium. This behavior islated to the particle- versus hole-like nature of the exciand ground state nuclear wave packets. In Fig. 4, we s


e

lse

a

is-

re-

s

n0

e

her-

de-o-n

tto

i--

s

th-

de-

matically depict the impulsively excited nuclear wave pacin the ground and excited states, for the carrier frequetuned to three different values across the absorption mmum Vv . Consider the case when the laser pulse carfrequency is

FIG. 3. Temperature and carrier frequency dependence of excited statemoment for the 50 cm21 mode of the example in Fig. 1~a!. The effective

initial ( t50) mean positionQe(0)5D1Qe8(0) is plotted in~a!. The mo-mentum in the excited state is vanishing and is not shown.~b! The ampli-

tude uA1eu5uQe8(0)u of the oscillatory motion about the excited state equlibrium positionD.

FIG. 4. Schematic of the initial conditions prepared in the ground andcited states of a two-electronic level system, by an impulsive pump intetion. The ground and excited state wave packets are depicted for threeferent values of the pump carrier frequency as it is tuned across the resomaximumVv and for off-resonant excitation (vNR). The dotted line indi-cates the carrier frequency of the laser pulse. The arrows above the gr

state wave packets indicate the direction of the initial momentumPg(0).


giThga

redthkv

f tpr

i,p

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dmthc

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w-tionvethe

Forere

-ns

it is

romem-

ithtate. 4.it a

osi-and

ndss


tuned toward the red side of the absorption maximum (vR).The pump creates a hole in the ground state near the reof resonance, which is toward positive displacements.nuclear distribution is therefore effectively centered at netive [email protected]., Qg(0),0] with respect to equilib-rium. The excited state nuclear distribution is, however, cated directly near the region of resonance, and is centerepositive displacements. It should be remembered thatnormalized mean position of the excited state wave pac@Eq. ~36a!# is much bigger than that of the ground state wapacket @Eq. ~24!#, since Ne512Ng!1. The schematicdrawn in Fig. 4 merely expresses the opposite signage oground and excited state first moments, but does not resent their magnitudes.

It is important to recognize that the schematic shownFig. 4 is strictly validonly in the impulsive approximationi.e., when the pulses are much shorter than a vibrationalriod. In this case, the nuclei of the ground and excited smolecules do not have the time to evolve in their respecsurfaces within the duration of the pulse. Thus, one canture the particle in the excited state as being instantaneocreated on the same side of the well where the corresponhole is created in the ground state. However, when the puhave a duration that is not too short compared to the vibtional period, the wave packets evolve during the pulseteraction. The centroids of the ground and excited state wpackets are thus no longer expected to be located on osites sides of the ground state equilibrium. In order to qutify the impulsive limit, it is useful to calculate the population weighted sum of the ground and excited state fimoments. Using Eqs.~19! and ~37!, we arrive at

NeQe~0!1NgQg~0!52umgeu2E0

2D

8p\2 E0

`

dv F I~v!

3G~v2vc!~D2G~v2vc1v0!!.

~46!

The right-hand side of Eq.~46! is an integral over the secondifference of the pulse spectrum. It can be taken to be a squantity in the impulsive limit, where the spectral bandwidof the pulse is broad compared to the vibrational frequenThus, we may set the right-hand side of Eq.~46! to zeroin the impulsive limit. This in turn implies thatQe(0)}2Qg(0); i.e., the mean position of the ground and excitstate nuclear wave packets are situated on opposite sidthe ground state equilibrium position. This situation hoonly for modes with vibrational period much longer than tpulse duration. For longer pulses, the right-hand side of~46! becomes appreciable. In Fig. 5, we plot the ground aexcited state first moments for the 50 cm21 mode of theprevious example, and the sum defined in Eq.~46!, for threedifferent pulse widths. It is seen that the ground and excstate first moments are oppositely signed for the impulsexcitation. However,Qe(0) andQg(0) are no longer sym-metric when the pulse duration approaches three quartethe vibrational period. In this case,Qe(0) is positive, i.e., the


one-

-ate

ete

hee-

n

e-teec-ly

nges--

veo--

t

all

y.

ofs

q.d

de

of

excited state wave packet is situated to the right ofground state equilibrium position for the entire range of crier frequencies plotted.

The relationshipQe(0)}2Qg(0), which is valid inthe impulsive limit, can be used to understand the increin the asymmetry of the excited state amplitude profile wtemperature, as observed in Fig. 3~b!. As we have seenearlier, Qg(0) approaches zero as the temperature is loered. This means that according to the inverse relaQe(0)}2Qg(0), thedisplacement of the excited state wapacket also approaches 0, but from the opposite side ofequilibrium position as the ground state wave packet.example, consider the schematic depicted in Fig. 4, whQe(0).0 for excitation toward the red (vc5vR). The cor-responding approach ofQe(0) toward zero at low temperatures implies an increase in the amplitude of oscillatioabout the shifted equilibrium positionD. This is shown bythe dashed wave packet in Fig. 4. On the other hand,clear that for excitation toward the blue (vc5vB), the cen-troid of the excited state wave packet approaches zero fthe negative side of the ground state equilibrium as the tperature is lowered. This corresponds to a wave packet wdecreased amplitude of oscillations about the excited sequilibrium, as shown by the dashed wave packet in FigThus the ground and excited state first moments exhib

FIG. 5. Study of the symmetry of the ground and excited state mean ptions as a function of the pulse width. The population weighted ground

the excited state first momentsNgQg(0) and NeQe(0) for the 50 cm21

mode~with periodTvib;666 fs! of the example in Fig. 1~a! are plotted alongwith their sum, for three different pulse widths. The top panel correspoto a pulse widthtp550 fs, middle paneltp5200 fs, and the bottom panel iwith tp5500 fs.


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in

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thatp

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sllythatde

nales-en-

ccu-ofari-ns.galthe

statep–sun-

is

r


striking contrast in the temperature dependence that cantentially be used to assign the origin of vibrational coherein pump–probe spectroscopy.27

It is clear from the above-mentioned illustrative eamples that the analytic expressions for the first momepresented in Eqs.~20a! and~20b! and~36a! expose the tem-perature and carrier frequency dependence of the firstments. They also offer a physically intuitive picture of thinitial conditions prepared by the pump pulse. The accurof the first moment expressions is verified by calculationsthe pump–probe signals using these expressions in theplaced state representation of the doorway state in Eq.~3!.The resulting signals agree well with those calculated usthe full third-order response approach.27 We also note thatthe depiction of pump induced wave packets in Fig. 4 ctradicts prior predictions arising from time-dependent wapacket pictures of impulsive stimulated light scatterinThese pictures suggest that the ground state wave packalways created toward decreasing energy gaps.10,65,66How-ever, the analysis presented here as well as in oworks37,38,62 shows that the ground state wave packetduced by impulsive excitation is strongly sensitive to tcarrier frequency of the laser pulse.

3. Multimode case

The above-considered examples illustrate some ofkey aspects of impulsively driven ground and excited stcoherence for simple systems with a bath and a single ocally coupled mode. As discussed earlier, the expressderived for the single-mode case can be readily extendemultimode systems. As an example, we consider a two-msystem consisting of a low frequency mode at 50 cm21 and ahigh frequency mode at 220 cm21 coupled to a homogeneously broadened two electronic level system. An ovdamped low frequency bath mode is also included to broathe line shapes. The absorption line shape atT5300 K isplotted in Fig. 6~a!, and is nearly a Gaussian owing to thsemiclassical limit of strong coupling and high temperatuIn the lower panels of Fig. 6, we plot the first moments of t50 and 220 cm21 modes in the ground state, assumingpulse width of 50 fs. Along with the fully quantum mechancal results for the mean position and momentum, amplitand phase for the two modes, we also plot~for comparison!the analytic result for the mean position in Eq.~B7!, valid inthe impulsive and semiclassical~Gaussian! approximations.

From the simulations, we once again see the role plaby thermal factor (n) in determining the amplitude and phabehavior. The high frequency mode~smaller n) exhibits alarger proportion of the momentum than the low frequenmode~larger n). Note that for the chosen pulse duration50 fs, the low frequency oscillator~period;666 fs! is drivenby the laser pulse in the impulsive limit. The high frequenmode~period;150 fs! is, however, dynamic even during thpulse interaction and is far from being impulsively driveAs would be expected, the agreement of the analytic refor impulsive excitation in Eq.~B7! with the fully quantumexpression in Eq.~20a! is much better for the low frequenc50 cm21 mode than for the higher frequency 220 cm21


o-e

ts

o-

yfis-

g

-e.t is

er-

eeti-nstoe

r-n

.

e

d

y

.lt

mode. While much of the earlier work on thisubject36,38,67,68has been carried out assuming vibrationaabrupt pulses, we see that a more complete treatmentaccounts for the finite pulse width is essential in a multimosituation.

B. Second moments

Apart from inducing vibrational coherence~off-diagonalterms!, the laser pulse also induces changes in the vibratiopopulations. In Secs. III B 2 and IV B 2, we presented exprsions for the second moments of the coordinate and momtum, as well as the pulse induced change in the mean opation number in the ground and excited states. It isinterest to calculate the pulse induced changes in the vances of the ground and excited state nuclear distributioThe variances ofQ andP reflect the pulse induced squeezinof the vibrational wave packet, as well as the vibrationheating and cooling in the ground and excited states bylaser pulse. Squeezing of the ground state and excitedwave packets can contribute to overtone signals in pumprobe spectroscopy.27 In the present section, we discussome aspects of pulse induced position and momentumcertainties for a simple model system.

FIG. 6. ~a! Absorption line shape atT5300 K for a three-mode systemconsisting of a strongly coupled overdamped bath mode (vb510 cm21,Sb

520,gb550 cm21) and two other modes with parameters (v1550 cm21,S150.1) and (v25220 cm21, S250.1). The homogeneous dampingGeg510 cm21. ~b! and~c! The pulse induced mean position~solid line! andthe mean momentum~dashed line!, along with the analytic expression fomean position calculated in the impulsive approximation in Eq.~B7!~circles!. ~d! and ~e! The amplitude and phase of the 50 and 220 cm21

modes.


. A

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a-

sncomu

iem

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al-

pec-

ve

ics

e to

the

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tumfthe

thetiesm-

n.

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We first consider the ground state second momentsdiscussed following Eq.~24!, the second moments ofQ andP involve both the diagonal and off-diagonal density matelements. The diagonal elements~which are time independent as long as there is no coupling of the mode to an exnal bath! give rise to a constant width in the coordinate amomentum distributions. The off-diagonal terms give risesqueezing dynamics at twice the vibrational frequency. FrEq. ~27!, we note that the real and imaginary parts of tsecond moment amplitudeA2g are related roughly to thesecond derivative of the absorptive and dispersive line shfunctionsD2F I(v) and D2FR(v). The temperature dependence of the real and imaginary parts ofA2g are mainlygoverned by the factorsn21n11/2 and n11/2, respec-tively. Thus, the phasew2g of the second moment dynamicis strongly dependent on temperature and carrier frequeAt zero temperature, the real and imaginary parts are cparable in magnitude, whereas for high temperatures sthat n2@n, the real part ofA2g dominates. This situation issimilar to the first moment, where the mean position~realpart! dominates the momentum~imaginary part! at high tem-peratures.

In Fig. 7, we consider the 50 cm21 mode of the modelsystem in Fig. 1~a!, and study the temperature and carrfrequency dependence of the pulse induced position and

FIG. 7. Temperature and carrier frequency dependence of the positionmomentum uncertainties in the pulse induced ground state vibrationaltribution. The pulse induced changes~from the thermal equilibrium values!in the position uncertaintydsQ , momentum uncertaintydsPg , and theirproductd(sQgsPg) are plotted at timest50 ~solid line!, t5p/4v0 ~dottedline!, andt5p/2v0 ~dashed line!. ~a!–~c! dsQg , dsPg , andd(sQgsPg) atT50 K. ~d!–~f! The corresponding quantities atT5300 K.


s

r-

om

pe

y.-

ch

ro-

mentum uncertainties. Since the pulse induced changes invariances are very small, it is convenient to plot the diffeences of various quantities from the thermal equilibrium v

ues, i.e.,dsQg5sQg2An11/2, dsPg5sPg2An11/2 and

d(sQgsPg)5sQgsPg2(n11/2). We plotdsQg and dsPg

for a range of carrier frequencies across the absorption strum, at three different times during a half-periodp/2v0 ofthe second moment dynamics at frequency 2v0 . We see thatat t50, dsQg mimics the second derivative of the absorpti

line shape owing to its dependence onD2F I(v). At t5p/4v0 ~which is a quarter period of the overtone dynam

at 2v0) the dispersive termD2FR(v) dominates. The ab-sorptive and dispersive line shapes thus act in quadraturdetermine the overall dynamics ofsQg(t), which is thusstrongly sensitive to the carrier frequency detuning fromabsorption maximum. According to the relationsPg(t)5sQg(t1p/2v0), the momentum uncertainty simpl‘‘lags’’ the position uncertainty by 1/4 vibrational period, ashown in Fig. 7~b!. The position momentum uncertaintproduct is plotted in Fig. 7~c! and is independent of timeNote the small dip of the uncertainty product below tvacuum product of one-half, which arises from the teuA1gu2 in Eq. ~29!. As discussed following Eq.~29!, thisunphysical behavior is attributed to the neglect of fourorder interactions in the present treatment. While the incsion of higher order interactions is beyond the scope ofpresent paper, we can roughly say that the magnitude ofdip, given by uA1gu2, effectively estimates the error due tthe second-order perturbative approximation made in Eq.~9!.

At higher temperature, the mean occupation numberng

dominates the expression for the position and momenuncertainties. As discussed previously, the variances oQand P at high temperatures are mainly determined by

absorptive functionD2F I(v), which is the real part ofA2g .This is evident from Figs. 7~d! and 7~e!, where the roleplayed by the dispersive term is insignificant compared toabsorptive part. Also, note the increase of the uncertainby nearly two orders of magnitude relative to the low teperature case. The uncertainty product, shown in Fig. 7~f!,once again remains constant with time. The error termuA1gu2

in Eq. ~29! is far less significant compared tong at high

temperature, and we havesQgsPg>ng11/2. Indeed, Fig.

7~f! simply reflects the behavior ofdng5ng2n calculated inEq. ~26!, which is also shown in the figure for comparisoThere is a spread in bothsQg and sPg near the absorptionmaximum, corresponding to pulse induced heating. Nearwings of the absorption spectrum, however, bothsQg andsPg are narrowed from their equilibrium values, corresponing to a laser pulse induced cooling of the ground state.

Turning to the second moments of the excited state wpacket, we recall from Sec. IV that the excited state momeonly involve the absorptive line shape functionF I(v). FromEq. ~43! we see that the position uncertainty oscillates~at

frequency 2v0) between (ne11/21A2e2A1e2 )1/2 at t50

and (ne11/22A2e)1/2 at t5p/2v0 . The momentum uncer

tainty lags the position uncertainty byp/2v0 . In Fig. 8, wedepict the detailed behavior of the position and moment

ndis-


rootig

ng

he

he

tth

atthlintioe

a-dthu-veob-isnaleic

msula-lya-mpasnic

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ll.

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onsdedesss.

ted

eorp-

rm,

ad


uncertainties of the excited state wave packet and their puct, for three different times stretching over a half periodthe second moment dynamics. The uncertainty produconce again practically independent of time, especially at htemperatures. This is in accord with Eq.~44!, which showsthat the oscillatory term diminishes rapidly with increasitemperatures and the product attains the limitne1(12A1e

2 )/2.It is interesting to note from Eq.~40! for ne that, if we

neglect the vibrational frequency in comparison with telectronic dephasing rate constantGeg so thatF I(v6v0)5F I(v), we have, using Eq.~33!, ne>n1D2/2. This isprecisely the mean occupation number for the displaced tmal state as expressed in Eq.~6!. We will see in the follow-ing that the same limit forne is obtained in the ultrashorpulse limit. In contrast, the mean occupation number inground stateng given by Eq.~26! does not attain a simplelimit for ultrafast electronic dephasing.

The simulations of the present section serve to illustrthe application of simple yet rigorous expressions forsecond moments and their connection to the equilibriumshapes. It is clear that the detailed behavior of the posiand momentum uncertainties can also be evaluated ovwider range of pulse widths and temperature.

FIG. 8. Temperature and carrier frequency dependence of the positionmomentum uncertainties in the pulse induced excited state vibrationaltribution. The pulse induced changes~from thermal equilibrium values! inthe position uncertaintydsQe , momentum uncertaintydsPe , and theirproductd(sQesPe) are plotted at timest50 ~solid line!, t5p/4v0 ~dottedline!, andt5p/2v0 ~dashed line!. ~a!–~c! dsQe , dsPe , andd(sQesPe) atT50 K. ~d!–~f! The corresponding quantities atT5300 K.


d-fish

r-

e

eeen

r a

C. Dependence on pulse width

A key ingredient for the creation of nonstationary vibrtional states is an ultrashort laser pulse with a bandwisufficiently large to excite several vibrational levels. The sperposition of many such levels results in a localized wapacket. Since several time scales are involved in the prlem, we first define some useful limits. The impulsive limitwhen the pulse duration is much shorter than the vibratioperiod, i.e.,tp!v0

21, with the electronic dephasing timscale remaining arbitrary. If, in addition, the electrondephasing is much more rapid than the pulse, i.e.,Geg

21!tp

!v021 , we obtain the ‘‘snapshot limit.’’26 While the condi-

tion Geg21!tp is justified for many condensed phase syste

with broad and featureless absorption spectra, the simtions in Fig. 6 show that the impulsive limit is not generalapplicable for all modes in a multimode system with vibrtional frequencies spanning the bandwidth of the pupulse. A more extreme ultrashort pulse limit is definedthat when the pulse duration is shorter than both electroand vibrational time scales, i.e.,tp!Geg

21!v021 . This is an

interesting limit to consider from a theoretical point of viewIt has generally been recognized that coherent vib

tional motion in the ground state vanishes both in thetrashort pulse limit and in the very long pulse limit.25,40,52,65

It is obvious that a very long pulse does not have the sucient bandwidth to excite coherent motion. The vanishingthe ground state coherence in the opposite limit of very shpulses arises due to the fact that the broad spectrum oflaser pulse bleaches all the nuclear coordinates to an eextent. Thus, a moving hole cannot be induced in the grostate.40 Both these limits are clearly obtained from thground state first moments in Eqs.~20a! and ~20b! and thesecond moment amplitude in Eq.~27!. For an infinitely long

pulse, the pulse envelope spectrumG(v2vc) approaches a

delta function and the product spectral functionGp(v2vc ,nv0) in the moment expressions is vanishingly sma

In the ultrashort pulse limit, we may setGp(v2vc ,nv0)

>G2(v2vc) and remove it outside the integrals in Eq

~20a!, ~20b! and Eq. ~27!, since G2(v2vc) varies muchmore slowly than the line shape functions. The expressithen reduce to integrals over the differences of bounfunctionsF I(v) andFR(v) and hence vanish. It should bnoted this occurs only in the ultrashort pulse limit; in the lestringent impulsive limit, the moments are given by Eq~B2a! and ~B2b!.

Turning to the pulse width dependence of the exci

state moments, we note thatQe8(0) also vanishes for long

pulses because the product spectral functionGp(v2vc ,v0) in Eq. ~36a! is negligible. However, the integral inEq. ~36a! does not vanish when the pulse is very short. If wneglect the variation of the pulse spectrum over the abstion line shape, the second term of Eq.~36a! is simply anintegral over the derivative of the bounded functionF I(v)and therefore vanishes. We are then left with the first te

which reduces to the following result if we letGp(v

2vc ,v0)>G2(v2vc) and use Eq.~33!:

ndis-


m

um

tend

ve

t

enh

shormtath

hahex

sei

pt

anu

qundreal

ean

bcleo

ooe

ineob-Mbe

forea

theen

omofionre-

athellyof

-

slysorp-

ment


Qe8~ t !52Dcos~v0t ! ~tp→0!. ~47!

By making similar approximations in Eqs.~40! and~41!, weobtain the following expressions for mean occupation nu

ber ne and the second momentQe2(t) in the ultrashort pulse

limit:

ne5n1D2/2 , ~48a!

Qe82~ t !5n11/21D2cos2~v0t !. ~48b!

The moments ofP(t) are simply obtained by substitutingt→t1p/2v0 in Eqs. ~48a! and ~48b!. The position and mo-mentum uncertainties become time independent in thetrashort pulse limit and approach their thermal equilibriuvalues given by Eq.~23!:

sQe~ t !5sPe~ t !5A~ n11/2!. ~49!

From Eqs.~47! and~37!, we haveQe(0)50. The centroid ofthe excited state nuclear wave packet is thus initially locavertically above the ground state equilibrium position aoscillates with an amplitudeD. Equations~47!–~49! togetherimply that for infinitely short pulses, the excited state wapacket is simply the thermal staterT , initially displacedfrom equilibrium as in Eq.~6! with a coherent displacemenle5D/A2. Note thatuleu25D2/25S is the quantity usuallycalled the electron nuclear coupling strength, and represthe mean number of phonons in a coherent state thatbeen displaced byle from the vacuum (n50).

Since the pulse induced ground state coherence vaniboth in the limit of pulses that are too long and too shcompared to vibrational and electronic dephasing tiscales, we would expect the amplitude of the ground scoherent motion to peak at some intermediate value ofpulse width. We have also seen that the amplitude and pof the induced vibrational motion is strongly sensitive to tdetuning of the laser frequency from the absorption mamum. It is therefore important to consider both the lapulse width and the carrier frequency if, for instance, oneinterested in determining the conditions for generating omal displacements in the ground state.36–38 The expressionsderived here for the ground and excited state momentsamenable to fast numerical computation. The computatioadvantage offered by the analytic expressions enables mdimensional plots that capture the behavior of the nonelibrium moments over an entire manifold of pulse widths acarrier frequencies simultaneously. Furthermore, the expsions derived here allow us to incorporate the experimentmeasured absorption line shape~and its Kramers–Kronigtransform, the dispersion line shape!. This enables absolutscale calculations of the pulse induced moments forgiven mode in a complex multimode system.

As an example, we consider the heme protein myoglo~Mb!, which is an oxygen storage protein found in muscells. Mb possesses a highly asymmetric and broad abstion spectrum ~Soret band! in its ligand-free, high spin~deoxy, S52! state. In previous studies, the Soret bandMb was modeled using a non-Gaussian inhomogenedistribution63 of electronic energy levels, ascribed to disord


-

l-

d

tsas

esteteese

i-rsi-

reallti-i-

s-ly

y

in

rp-

fusr

in the position of the central iron atom of the porphyrring.69,70 In Fig. 9~a!, we plot the simulated absorption linshape with mode coupling strengths and frequenciestained from previous resonance Raman studies of theSoret line shape.69,71 Of the numerous modes coupled to thSoret transition, we consider two modes at 50 and 220 cm21

and study the pump induced initial amplitude and phasethese modes. In Fig. 9~b!, we plot the amplitude and phasfor the 220 cm21 mode in the ground electronic state, asfunction of carrier frequencyvc and pulse widthtp . Theapproach of the ground state amplitude to zero in bothshort and long pulse limits is clear from Fig. 9. It is also sethat the laser pulse width for whichuA1gu is maximum issensitive to the detuning of the pulse center frequency frVv . The initial amplitude for excitation on the red sidethe absorption maximum is much larger than for excitattoward the blue side of the absorption maximum. Thisflects the asymmetry of the Mb absorption spectrum withmuch larger slope on the red side than on the blue side ofabsorption maximum. Although there are numerous opticacoupled modes in the model, the mode specific natureEqs. ~20a! and ~20b! allows us to calculate the nonequilib

FIG. 9. ~a! Absorption line shape for an asymmetric inhomogeneoubroadened system, with parameters chosen to mimic the deoxyMb abtion spectrum at room temperature.~b! Three-dimensional view depictingthe laser pulse width and carrier frequency dependence of the first moamplitudeuA1gu and phasew1g , for the 220 cm21 mode~coupling strengthS50.05) in the ground electronic state.


iththm.ith

40iot.e

teesentnd

ci-

ery

ith

heb-umnalde-si-rierhiscalsta-

ofasern-

nditu

s

ate

del


rium moments for a specific vibrational mode of choice, wthe effect of the remaining modes carried through inequilibrium line shape functions. In Fig. 10, we plot the aplitude and phase of the 50 cm21 mode in the ground stateNote that the optimal amplitude of the ground state motionattained for pulses nearly as long as 100 fs in contrast to220 cm21 mode which attains an optimal amplitude nearfs. The optimal pulse widths are thus not in direct proportto the vibrational frequencies as one might naively expec

We plot the initial position of the excited state wavpacketQe(0) for the 50 and 220 cm21 mode in the excitedstate in Fig. 11. The initial position approaches the excistate equilibrium in the limit of long pulses implying that thoscillations vanish. For very short pulses, the limit expresin Eq. ~47! is clearly obtained, with the position incremeapproaching zero. As discussed previously, this correspoto the fact that the excited state wave packet is placedrectly above the ground state equilibrium position and oslates with an amplitudeD. Also, note that the detuning de

FIG. 10. ~a! Three-dimensional plot depicting the laser pulse width acarrier frequency dependence of the pulse induced first moment amplprofile ~a! and phase profile~b! for the 50 cm21 mode (S50.1) in theground state. The mode is coupled to the inhomogeneously broadenedtem considered in Fig. 9~a!.


e-

se

n

d

d

dsi-

l-

pendence of the excited state position increment is vstrongly sensitive to the laser pulse width~see Fig. 5!. Forvery short pulses, the mean position changes its sign wrespect to the ground state equilibrium asvc is tuned acrossVv . As the pulse width becomes longer, the profile of tfirst moment slowly drifts toward the excited state equilirium and stays on one side of the ground state equilibriposition for long enough pulses. The three-dimensiosimulations presented in Figs. 9–11 capture the highlytailed behavior of the pump induced first moments in potion and momentum as a function of pulse width and carfrequency. The simple analytical formulas presented in twork thus enable us to make fully quantum mechaniquantitative predictions for the laser pulse induced nontionary vibrational states in complex multimode systems.

VI. SUMMARY AND CONCLUSIONS

In this paper, we have presented a rigorous analysisthe nonstationary vibrational states prepared by a short lpulse interacting with a two electronic level system with li

de

ys-

FIG. 11. Dependence of the effective initial position of the excited st

wave packetQe(0) on laser pulse width and carrier frequency, for the mosystem considered in Fig. 9~a!. ~a! The mean position for the 50 cm21 modeand ~b! the mean position of the 220cm21 mode.


ino

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tetahe

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textlse in-

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ear electron-nuclear coupling. The chief motivation behthis work is to expose the nature of the pump-induced doway state, using a rigorous moment analysis. A knowleof the moments of the doorway state provides a convenstarting point for the analysis of pump–probe signals.27 Inthe present work, we have used moment generating funct

to derive general expressions for arbitrary moments ofQ and

P in the ground and excited state density matrices to secorder in the pump fields.

The fully quantum mechanical expressions for the fitwo moments of position and momentum reveal interestbehavior with respect to temperature, pump pulse carrierquency, and width. The pump pulse induced changes ofmean position and momentum in the ground state are foto depend on the derivatives of the absorptive and disperline shapes, respectively. This relationship enables onreadily obtain a qualitative understanding of the dependeof pump induced first moment changes on the carrierquency. The mean position of pump induced ground sexhibits a much stronger temperature dependence thanmean momentum. This implies a strong temperature depdence of the initial phase of the ground state wave pacand consequently the phase of the pump–probe signal isstrongly temperature dependent.27 While the ground statemoments depend on both the absorptive and dispersiveof the equilibrium line shape functions, the moments ofexcited state wave packet are shown to depend only onabsorptive line shape function. This is strongly indicativethe distinct mechanisms that induce ground and excited scoherences; namely stimulated Raman-type processes foground state, and absorptive processes for the excited sThe amplitude of the ground state oscillations decreasesformly with temperature for all pump carrier frequencies.contrast, the profile of the excited state first moment amtude exhibits a striking asymmetry with change in tempeture: For red excitation, the amplitude of the excited stwave packet decreases with increased temperature, whfor blue excitation, the amplitude increases with temperatThis contrasting behavior can be explained based onparticle- versus hole-like nature of the ground and excistate wave packets. It can also be used to experimendiscriminate between the ground and excited state coences.

An analysis of the second moments of the pump-indudoorway state reveals information regarding the squeezinthe ground and excited state nuclear distributions by the lpulse interaction. It is well known that a difference in thcurvature of the ground and excited state vibrational pottials ~quadratic coupling! can induce squeezed vibrationstates.57–60Apart from this ‘‘geometric’’ squeezing,57 the la-ser pulse can by itself induce a time-dependent variancthe vibrational distributions, which is also called ‘‘dynamsqueezing.’’57 The present analysis exposes the nature ofdynamical squeezing and its relation to the equilibriumsorption and dispersion line shapes. While the first momdynamics constitutes a major part of the wave-packet modetected by the probe,27 higher moment modulations, such asqueezing, will contribute weakly to overtone signals. Whave recently presented an analysis of the overtone sig


dr-ent

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dllyr-

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-

in

e-ntn

als

generated due to purely geometric squeezing in the conof nonradiative reactions.27 The analysis of overtone signagenerated by dynamical squeezing due to the laser pulsteraction will be the subject of future work.

We have pointed out the consequence of the secoorder perturbative approximation to the doorway density mtrix. Since a genuine change in the ground state vibratiopopulations requires a fourth-order interaction~stimulatedRaman scattering! of the pump pulse, the diagonal densimatrix elements calculated using only the second-orderproximation will be inaccurate. The neglect of higher ordfield interactions results in certain discrepancies in the calated expectation values of the higher moments. Forample, we showed that the ground state doorway~to secondorder! violates the properties of a genuine density matrand leads to a position-momentum uncertainty product thain violation of the uncertainty principle. While these aspehave been noted in earlier work,37,40 we have presented aestimate of the error due to truncation at second order.

Perhaps the most important aspect of the present anais the direct connection made between the laser pulseduced nonstationary states and the measurable equilibproperties of the system. This connection appears throthe dependence of the pump-induced moments on the elibrium line shape functions. A direct consequence of taspect is that the pump induced moments are highly mspecific; the only explicit parameters in the moment exprsions are the mode frequency and coupling strength. Theof the parameters relevant to the system and the bathautomatically carried by the measured line shape functioOne practical consequence is a many-fold increase in cputational efficiency. In this respect, the present appromust be contrasted with earlier treatments that expressedfull second-order doorway density matrix using the mocumbersome sum over vibronic eigenstates expressions29,50

Here we have shown that the individual moments ofdoorway density matrix can be calculated efficiently usingcorrelation function based approach.

Apart from computational advantages, the direct conntion with equilibrium line shape functions can potentially bexploited in calculating the various moments on an abso~per molecule! scale. Since absolute scale measurementthe absorption cross section are possible experimentally,may use the Kramers–Kronig relations to calculate the dpersion line shape, and subsequently incorporate theshapes in the moment expressions. This would yield prevalues for moments of the nonstationary wave packetduced by the pump pulse for any given mode in a compmultimode system. This approach is analogous to transfmethods previously used to describe resonance Rascattering.42–46,72

We finally mention that the present work can be etended beyond the linearly coupled harmonic oscillamodel assumed here. Quadratic electron-nuclear coupcan be treated using standard quantum field theoretic tniques. Non-Condon effects can also be readily incorporaby assuming an exponential dependence of the dipolement on the nuclear coordinate.52,73 Furthermore, a calculation of the moments for multiple pulse excitation36–38 can


ti

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--

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ion

tonpan-’s

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c


also be carried out along the same lines as the calculapresented here. The central point is that the initial state~be-fore the pulse interactions! is a thermal density matrix. Themultitime correlation functions can therefore be evaluaexactly using a second-order cumulant expansion.

A moment analysis of the doorway state can alsoenvisaged in spectroscopies that use multiply resonant puwith different carrier frequencies. For instance, considetime domain coherent anti-Stokes Raman spectroscexperiment26,74 involving two time coincident pulses withwave vectorsk1 and k2 , which are followed by delayedpulse with wave vectork1 . The measured signal is the totcoherent emission along the directionks52k12k2 . In thiscase, the relevant term in the third-order polarization consof contributions from three fields, one from each of the thpulses.26 The first two fields ~from the time coincidentpulses! prepare a second-order doorway state. The cosponding density matrix can be calculated by replacingtwo pump fieldsEa(t) in the second-order expression for thdoorway density matrix@given in Eq.~3!# by the fields fromthe two different pulses. One would then evaluate the varimoments for this density matrix, and subsequently makdisplaced thermal state representation as in Eq.~6!. Thethird-order polarization induced by the delayed pulse mthen be calculated using the effective linear respoapproach.27 A rigorous analysis of these experiments rquires the inclusion of the spatial dependence of the lafields and will be considered in future work.

APPENDIX A: MOMENTS OF NUCLEAR POSITIONAND MOMENTUM IN THE DOORWAY STATE

1. Ground state

The moment generating function for the coordinaQ(t), for the density matrixrg is defined as

M Q(g)~k,t !5~1/Ng!Tr@eikQ(t)rg#, ~A1!

whereNg is given by Eq.~15!. The time evolution ofQ inEq. ~A1! is governed by the ground state HamiltonianHg

and takes the following form forHg defined in Eq.~10a!:

Q~ t !5Q cos~v0t !1 P sin~v0t !. ~A2!

From a knowledge ofM Q(g)(k,t), we can calculate the time

dependent moments ofQ for the pump induced density matrix by a simple differentiation ofM Q

(g)(k,t):

Qgn~ t !5~2 i !n~]nM Q

(g)~k,t !/]kn!k50 . ~A3!

Also, since it follows from Eq.~A1! and the definition in Eq.~5! that

M Q(g)~k,t !5 (

n50

`~ ik !n

n!Qg

n~ t !, ~A4!

the moments can be directly obtained as the coefficientthe various powers ofk in the series expansion foM Q

(g)(k,t). The moments ofP(t) are trivially obtained fromthe coordinate moments by lettingt→t1p/2v0 . To see this,consider the Hamilton’s equationdQ(t)/dt5v0P(t) and the


on

d

eesapy

tse

e-e

sa

ye

-er

of

relation dQ(t)/dt5v0Q(t1p/2v0) that follows from Eq.~A2!. These two equations imply the operator relationP(t)5Q(t1p/2v0) so that thenth moments ofQ and P obey

Pgn~ t !5Qg

n~ t1p/2v0!. ~A5!

We now derive a fully quantum-mechanical expressfor M Q

(g)(k,t). Using Eq.~A1! and Eq.~9!, we have

M Q(g)~k,t !5~1/Ng!H êikQ(t)&T2

umgeu2

\2 E2`

`

dt8E0

`

ds

3E~ t8!E~ t82s!e2Gegusu

3@e2 iVvsAkg~ t2t8,s!1eiVvsBk

g~ t2t8,s!#J ,

~A6!

where we have defined

Akg~ t2t8,s!5K eikQ(t)expS iv0DE

t82s

t8ds8 Q~s8! D

1L

T

,

~A7a!

Bkg~ t2t8,s!5K expS 2 iv0DE

t82s

t8ds8 Q~s8! D

2

eikQ(t)LT

.

~A7b!

The angular brackets&T denote the average with respectthe thermal staterT

(g) @see Eq.~8!#. The thermal average cabe evaluated exactly using a second-order cumulant exsion ~which is exact for a thermal state, according to Wicktheorem75!, with the result

Akg~ t2t8,s!5e2g(s)1 iv0sD2/2e2(2n11)k2/4

3expS 2kD

2G~ t2t8,s! D , ~A8a!

Bkg~ t2t8,s!5e2g* (s)2 iv0sD2/2e2(2n11)k2/4

3expS kD

2G* ~ t2t8,s! D . ~A8b!

Here, we have defined the functionG(t,s) @which arises fromthe cross terms that result from the cumulant expansionEqs.~A7a! and ~A7b!# as

G~ t,s!52 i @~ n11!e2 iv0t~12e2 iv0s!2neiv0t~12eiv0s!#,~A9!

and g(s) is given by Eq.~12! for the undamped harmonioscillator. If we expandAk

g andBkg in powers ofk and sub-

stitute the resulting expression into Eq.~A6!, and comparethe result with Eq.~A4!, we get

Qgn~ t !5(

l 50

`

dn,2l

n! ~2n11! l

2nNg

1n!

~2i !nNg(l 50

`

(m50

`

dn,2l 1m

3~21! l 1m

l !m!Dm~2n11! l@Cm~ t !1~21!mCm* ~ t !#,

~A10!


in

lam

onve

ed

sg

te

n-

latempainandid-

ntthepe-


with

Cm~ t !52umgeu2

\2 E2`

`

dt8E0

`

ds E~ t8!E~ t82s!

3e2Gegusu2 iV00s2g(s)@G~ t2t8,s!#m. ~A11!

Here,d i , j refers to the Kronecker’s delta. The first termEq. ~A10! gives the moments of the equilibrium staterT

(g) .Equations~A10! and ~A11! along with Eqs.~A9! and ~12!provide the complete set of expressions required to calcuarbitrary moments of the coordinate operator in the puinduced density matrix. The moments ofP(t) are obtainedusing Eq.~A5!.

2. Excited state

Consider the coordinate operatorQ evolving in time dueto He defined in Eq.~10b!:

Qe~ t !5D1~Q2D!cos~v0t !1 P sin~v0t !. ~A12!

This has the expected form, with the constantD reflectingthe oscillations about the excited state equilibrium positiIn evaluating the excited state moments, it is more connient to shift the origin toD and takeQe8(t)5Qe(t)2D forthe excited state oscillator coordinate. The creation~destruc-tion! operators for the excited state oscillator with a shiftequilibrium position are defined as

Q2D5~ ae1ae†!/A2;P5 i ~ ae

†2ae!/A2. ~A13!

From Eq. ~10b!, the excited state Hamiltonian is given aHe5\v0(ae

†ae11/2)2\v0D2/2. The moment generatinfunction is then defined analogous to Eq.~A1! but with thesubstitutionQ(t)→Qe8(t) andg→ e:

M Q(e)~k,t !5~1/Ne!Tr@eikQe8(t)re#, ~A14!

whereNe is the excited state population given by Eq.~33!.Using Eq.~A2!, we also haveQe8(t)5 Q(t)2D cos(v0t) sothat

M Q(e)~k,t !5~1/Ne!Tr@eikQ(t)re#e

2 ikD cos~v0t !. ~A15!

Analogous to Eq.~A5!, the moments of theP for the excitedstate can be obtained from the coordinate moments bysubstitutiont→t1p/2v0 . For the coordinate moments, whave from Eqs.~32! and ~A14!

M Q(e)~k,t !5

umgeu2

N e\2E2`

`

dt8E2`

t8dt9 E~ t8!E~ t9!e2Gegusu

3@eiVvsAke~ t,t8,t9!1e2 iVvsAk

e~ t,t9,t8!#,

~A16!

wheres5t82t9 and we have defined


tep

.-

he

Ake(t,t8,t9)

5exp(2 ikD cos(v0t))

3K S expS iv0DE0

t9ds8Q(s8) D D

1

3exp~ ikQ~ t !!S expS 2 iv0DE0

t8ds8Q(s8) D D

2L

T

.

~A17!

The thermal average in Eq.~A17! can be evaluated similar toEqs.~A7a! and~A7b!, using a second-order cumulant expasion. We find

Ake~ t,t8,t9!5e2g* (s)2 iv0sD2/2e2(2n11)k2/4

3expS 2kD

2Ge~ t,t8,t9! D , ~A18!

whereg(s) is defined in Eq.~12!, and

Ge~ t,t8,t9!5 i @e2 iv0t~~ n11!eiv0t82neiv0t9!

1eiv0t~~ n11!e2 iv0t92ne2 iv0t8!#. ~A19!

@Note thatGe(t,t9,t8)52Ge* (t,t8,t9).] Expansion ofAk inpowers ofk and substituting the result in Eq.~A16! gives usthe final result:

M Q(e)~k,t !5 (

n50

`~ ik !n

n!Qe8

n~ t !, ~A20!

where

Qe8n~ t !5

n!

~2i !nNe(l 50

`

(m50

`

dn,2l 1m

~21! l 1m

l !m!Dm~2n11! l

3@Dm~ t !1~21!mDm* ~ t !#, ~A21!

with

Dm~ t !5umgeu2

\2 E2`

`

dt8E2`

t8dt9 E~ t8!E~ t9!

3e2Gegusu1 iV00s2g* (s)@Ge~ t,t8,t9!#m. ~A22!

Equations~A21! and ~A22! along with Eqs.~A19! and ~12!provide the complete set of expressions required to calcuarbitrary moments of the coordinate operator for the puinduced excited state density matrix. Extension of the mresults in this section to the multimode harmonic casefor damped oscillators is straightforward, but is not consered here for simplicity.

APPENDIX B: COHERENT MOTION IN THEIMPULSIVE LIMIT

Here we consider the impulsive limit of the first momeresults obtained in Secs. III B and IV B, by assuming thatlaser pulse duration is much shorter than the vibrationalriod. Consider the ground state first moments in Eqs.~20a!


nng

As,

sn

rao-o

orfeo

tra

b

ib

igs

thiap

nal

,sultse

um,

ndad-

the

ys.

R.

es,


and ~20b!. Before we make the short pulse approximatiowe rewrite the two equations after making a simple chaof variables:

Qg~0!52umgeu2E0

2~2n11!D

8p\2NgE

0

`

dvF I~v!G~v2vc!

3@G~v2vc2v0!2G~v2vc1v0!#, ~B1a!

Pg~0!5umgeu2E0

2D

8p\2NgE

0

`

dv FR~v!G~v2vc!

3@G~v2vc2v0!2G~v2vc1v0!#. ~B1b!

If we now make the approximation,tp21@v0 , we may re-

place the expressions within square brackets in Eqs.~B1a!and ~B1b! by the first derivatives of the pulse spectrum.subsequent integration by parts then leads to final result

Qg~ Imp!~0!52

umgeu2E02~2n11!v0D

8p\2Ng

3E0

`

dv G2~v2vc!~]F I~v!/]v!, ~B2a!

Pg~ Imp!~0!5

umgeu2E02v0D

8p\2Ng

3E0

`

dv G2~v2vc!~]FR~v!/]v!. ~B2b!

The only assumption made in deriving the above resultthat the pulse duration is short compared to the vibratioperiod. The temperature,~linear! electron-nuclear couplingstrength, and all other line shape parameters remain arbitIt is seen that in the impulsive limit, the position and mmentum increments are simply given by the convolutionthe laser pulse spectrum with the derivatives of the abstion and dispersion line shapes, respectively. Also, the efof the spectral bandwidth of the laser pulse enters the abexpressions independently of the mode frequency, in conto the general expressions~20a! and ~20b!.

The impulsive limit of the excited state coherence cansimilarly obtained from Eq.~36a!. We find,

Qe(Imp)~0!52

umgeu2E02D

4p\2NeE

0

`

dv G2~v2vc!

3@F I~v!2nv0~]F I~v!/]v!#. ~B3!

A closed form solution is possible for the mean positionwe approximate the imaginary part of the line shape toGaussian, which is realized in the semiclassical limit of htemperatures or strong electron-nuclear coupling strength76

As an illustration, we obtain a closed form expression forground state by taking the following forms for the Gaussspectral function and the imaginary part of the line shafunction:


,e

isal

ry.

fp-ctvest

e

feh.ene

E~v!5E0

2~2ptp

2!1/2e2(v2vc)2tp2/2, ~B4!

F I~v!5Ap/2sT2e2(v2Vv)2/2sT

2, ~B5!

where the semiclassical linewidth is given bysT

5DAkBTv0 /\. Using these definitions in Eq.~B2a!, we get

Qg(Imp)~0!5

~2p!3/2v0umgeu2E02tp

2D~2n11!

16p\2sT3

3E0

`

dv~v2Vv!

3e2(v2vc)2tp2e2(v2Vv)2/2sT

2. ~B6!

Evaluation of the above-mentioned integral gives the firesult,

Qg(Imp)~0!5

Cv0tp4D~vc2Vv!~2n11!

~112tp2sT

2!3/2

3expF2tp

2

~112tp2sT

2!~vc2Vv!2G , ~B7!

where we have defined the constantC5pumgeu2E02/2\2.

Equation~B7!, valid in the impulsive limit of short pulsescaptures several of the features found from the general rein Eq. ~20a!. The initial amplitude is seen to change its phaacross the absorption maximumVv . The sign of the dis-placement is opposite to that of the excited state equilibrishift D for red detuning from the absorption maximumwhile it has the same sign for blue detuning. The groustate displacement vanishes for very short pulses. If, indition to the impulsive limit wherev0!tp

21 , the conditionsT@tp

21 is also satisfied~yielding the snapshot limit26!, Eq.~B7! reduces to

Qg(Imp)~0!5

Cv0tpD~vc2Vv!~2n11!

A8sT3

3exp@2~vc2Vv!2/2sT2#, ~B8!

so that the mean position has a linear dependence onpulse width.

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9N. F. Scherer, R. J. Carlson, A. Matro, M. Du, A. J. Ruggeiro, V.Rochin, J. A. Cina, G. R. Fleming, and S. A. Rice, J. Chem. Phys.95,1487 ~1991!.

10W. T. Pollard and R. A. Mathies, Annu. Rev. Phys. Chem.43, 497~1992!.11S. L. Dexheimer, Q. Wang, L. A. Peteanu, W. T. Pollard, R. A. Mathi

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ur

P

V

R.

,

tz,

nk

m

co

ue

.

.

m

.

cti

har-arestameuder

. A

ys.

tiveges;, and

sig-eri-

thealintotlex

osere-line

the

nc-to

d-

ys.

m.

hys.


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versity Press, New York, 1995!.27A. T. N. Kumar, F. Rosca, A. Widom, and P. M. Champion, J. Che

Phys.114, 701~2001!. We note a correction to Eq.~44a! of this reference,which gives the amplitude of the open band pump-probe signal. Therect form of Eq.~44a! is:

Ask5UE0

`

dvCk~v!U ,i.e., the absolute value must be taken after the integration over all freqcies.

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40D. M. Jonas, S. Bradforth, S. Passino, and G. Fleming, J. Phys. Chem99,2594 ~1995!.

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42V. Hizhnyakov and I. Tehver, Phys. Status Solidi21, 755 ~1967!.43J. Page and D. Tonks, J. Chem. Phys.75, 5694~1981!.44B. R. Stallard, P. M. Champion, P. Callis, and A. C. Albrecht, J. Che

Phys.78, 712 ~1983!.45K. T. Schomacker and P. M. Champion, J. Chem. Phys.90, 5982~1989!.46J. B. Page, inLight Scattering in Solids VI, edited by M. Cardona and G

Guntherodt~Springer, Berlin, 1991!, p. 17.47Y. R. Shen,The Principles of Nonlinear Optics~Wiley, New York, 1984!.48S. Mukamel, Phys. Rev. A28, 3480~1983!.49R. W. Boyd and S. Mukamel, Phys. Rev. A29, 1973~1984!.50B. Fain and S. Lin, J. Chem. Phys.93, 6387~1990!.51L. W. Ungar and J. A. Cina, Adv. Chem. Phys.100, 171 ~1997!.52Y. Tanimmura and S. Mukamel, J. Opt. Soc. Am. B10, 2263~1993!.53T. J. Dunn, I. A. Walmsley, and S. Mukamel, Phys. Rev. Lett.74, 884

~1995!.54Once the moments are evaluated in the well-separated limit, the effe


e

.

.

J.

,

.

r-

n-

.

ve

initial values of the moments are obtained by extrapolation back tot50.This procedure is implicit in Eq.~5!, where the density matrix is timeindependent in the WSP limit, and settingt50 directly gives the extrapo-lated initial values.

55The displacement operator is the generator of coherent states of themonic oscillator from the vaccum state. Coherent states are the neapproximation to classical states of the oscillator, and derive their ndue to their quantum statistical properties. See, for example, J. R. Klaand E. C. G. Sudarshan,Fundamentals of Quantum Optics~Benjamin,New York, 1968!.

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New York, 1977!.62Y. C. Shen and J. A. Cina, J. Chem. Phys.110, 9793~1999!.63We note that in a complete pump–probe calculation using the effec

linear response approach, the line shape functions appear in two sta27

first, in the pump induced moments, as described in the present worksecond, in the probe detection step. In other words, the pump–probenal is quadratic in the line shape functions. When incorporating expmentally measured line shapes, inhomogeneities~if present! must first bedeconvolved from the line shapes before they are employed in bothpump ~moment analysis! and probe steps of the calculation. The finpump–probe signal is obtained by convolving the inhomogeneitiesthe homogeneous signal~Ref. 27!. This procedure is analogous to thaemployed in the calculation of Raman excitation profiles using compline shape functions~Ref. 46!.

64We note that vibrational damping is neglected only for those modes whmoments are being analyzed. One can still allow for damping in themaining subspace of the system, which appear solely through theshape function. Thus, in the example considered in Fig. 1~b!, the effect ofthe overdamped 10 cm21 mode on the undamped 50 cm21 mode underanalysis is manifested solely through the line shape function. For allmodes involved in the line shape function~other than the mode of inter-est!, it is justified to use the rigorous expressions for the correlation fution g(s) derived for the damped oscillator case. When one needscalculate thepump induced moments of the damped modes, however, theundamped expressions for the correlation functionsG(t2t8,s) andGe(t,t8,t9) @Eqs.~A9! and~A19!# need to be replaced by the corresponing expressions for the damped oscillator case.

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1267 ~1986!.70P. M. Champion, J. Raman Spectrosc.23, 557 ~1992!.71V. Srajer, Ph.D. thesis, Northeastern University, 1991.72P. M. Champion and A. C. Albrecht, Annu. Rev. Phys. Chem.33, 353

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~1996!.74M. Cho, N. F. Scherer, G. R. Fleming, and S. Mukamel, J. Chem. P

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Quantum Field Theory in Statistical Physics~Dover, New York, 1975!.76C. K. Chan and J. B. Page, Chem. Phys. Lett.104, 609 ~1984!.


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