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Gaussian–Markovian quantum Fokker–Planck approach to nonlinearspectroscopy of a displaced Morse potentials system: Dissociation,predissociation, and optical Stark effects
Yoshitaka Tanimura and Yutaka MaruyamaInstitute for Molecular Science and The Graduate University for Advanced Studies, Myodaiji,Okazaki 444, Japan
~Received 27 March 1997; accepted 6 May 1997!
Quantum coherence and its dephasing by coupling to a dissipative environment play an importantrole in time-resolved nonlinear optical response as well as nonadiabatic transitions in the condensedphase. We have discussed nonlinear optical processes on a multi-state one-dimensional system withMorse potential surfaces in a dissipative environment. This was based on a numerical study usingthe multi-state quantum Fokker–Planck equation for a colored Gaussian–Markovian noise bath,which was expressed as a hierarchy of kinetic equations. This equation can treat strong system-bathinteractions at a low temperature heat bath, where quantum effects play a major role. The approachapplies to linear absorption measurements as well as four-wave mixing including pump-probespectroscopy. Laser induced photodissociation and predissociation have been studied for thepotential surfaces of Cs2 . We have calculated nuclear wave packets in Wigner representation andtheir monitoring by femtosecond pump-probe spectroscopy for various displacements of potentialsand heat-bath parameters. Numerical calculations of probe absorption spectra for strong pump pulseare also presented and discussed. The results show dynamical Stark splitting, but, in contrast to theBloch equations which contain an infinite-temperature dephasing, we find that at finite temperaturetheir peaks have different heights even when the pump pulse is on resonance. ©1997 AmericanInstitute of Physics.@S0021-9606~97!52230-6#
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I. INTRODUCTION
Femtosecond spectroscopy, such as impulsive Ramoptical Kerr, and pump-probe spectroscopy, provides a dimeans for studying nuclear dynamics in the condenphase.1–6 The understanding of such highly resolved mesurements demands theoretical descriptions which go faryond simple models. Tremendous insight has been gainecomparing qualitative arguments,7 quantitative analyticalcalculations8–11 and numerical studies12–19 with experiment.The response function approach,20 which is based on a perturbative expansion of the optical polarization in powersthe laser fields, has been successfully applied to stfour-wave21,22 and six-wave mixing experiments.23,11 Calcu-lation of the response functions involves integration overnuclear degrees of freedom. Thus, one could obtain thesponse function only for a system with harmonic potensurfaces. It is possible to include a non-Condon dipole inaction or a weak anharmonicity into theNth-order responsefunction by using a nonequilibrium generating functionwhich is obtained by the path-integral approach.10,24,11 Ap-plicability of this approach is, however, still limited.
Alternatively, optical processes can be calculated usindirect integration of the equations of motion in the preseof the fields. By calculating the relevant wave function25 ordensity matrix elements26–33 it becomes possible to exploroptical processes for a system with arbitrary potential sfaces. A difficulty with this approach is the proper treatmeof dephasing processes induced by a heat bath. These cincorporated using equations of motion for a reduced denmatrix, such as the quantum master equation or the quan
J. Chem. Phys. 107 (6), 8 August 1997 0021-9606/97/107(6)/17
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Fokker–Planck equation. Effects of the bath are then tainto account by introducing a damping operator, which cbe obtained by assuming Gaussian-white noise fluctuatand a bilinear system-bath interaction expressed asHSB
5RScnxn5Scn(a11a2)(bn11bn
2), wherea6 andbn6 are
the creation and annihilation operators corresponding tosystem and bath coordinates, respectively. We should nothat the reduced density matrices equation with the bilininteraction can be applied only for the high temperature stem, i.e.,\vc /kBT!1, wherevc is the characteristic fre-quency of the system. If one applies these equations beythis limitation, then one obtains unphysical results suchthe negative probability of density matrix elements. For tmaster equation, this phenomenon is known as breakindynamic positivity,34 which is the limitation of the reducedequation of motion approach. If one modifies the interactin the resonant form~or the rotating wave approximatioform!, i.e., HSB8 5Scn(a1bn
21a2bn1) then this temperature
limitation can be relaxed. We should notice, however, tthis modification of the Hamiltonian alters the dynamics dscribed by the original Hamiltonian, though the obtainequation of motion can be applied to the low temperatsystem.
We can relax this temperature limitation without modfying HSB by employing the colored Gaussian–Markovianoise bath instead of the Gaussian-white noise bath.time correlation function of noise fluctuation,V(t), in theGaussian-white noise is expressed as^V(t)V(t8)&5d(t2t8), whereas ^V(t)V(t8)&5exp@2g(t2t8)# in theGaussian–Markovian case. If the characteristic time scal
177979/15/$10.00 © 1997 American Institute of Physics
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1780 Y. Tanimura and Y. Maruyama: Nonlinear spectroscopy of a Morse potentials system
the system, 1/vc , is much longer than the correlation timenoise,t51/g, then one may regard the noise as thed func-tion in t. In the present case of femtosecond experimehowever, the noise must be treated as a finitely correlafunction of time. Thus, the generalization to the GaussiaMarkovian is also a requirement of describing a systemthe realistic condition.
We could obtain a hierarchy of kinetic equations for rduced density matrices which can describe the system inacting with the colored Gaussian–Markovian noise bat35
Physically, one can think of this hierarchy of equationsdealing with a set of density matrices modeling the varionumbers of phonon excited states in very special way. Tequation was originally obtained for a discrete two-level stem, and can be regarded as a generalization of the quamaster equation or the generalized quantum master equaWe then showed that a similar hierarchy of kinetic equatiocould be obtained for a system in the coordinate represetion, which can be regarded as a generalization of the qutum Fokker–Planck equation.36,37 In principle, we canchoose any representation to describe quantum dynamica system. Practically, however, the coordinate representahas some advantages for studying a system with anharmpotential surfaces. First, the coordinate description allowsto make direct interpretations of the dynamics. Thus, we measily discuss the classical and the quantum systems onsame basis. Second, we can suppress the open boundaryditions, where the wave packet can go out from the edgepotential. In the discrete state representation, the eigensbecome coetaneous for an open boundary system, wmakes it impossible to integrate the equation of motiThus, if one has to deal with the problem on an open bouary such as the problem of photo dissociation, one needadapt the coordinate space representation. Third, calculaare easier. One has to calculate a number of eigenstateseigenenergies to describe a system in the discrete statesresentation. Various interactions, such as laser interactand the system-bath interactions are then expressed as mces in this basis. Such calculations are computationallytensive except for a system with harmonic potential surfacIn the coordinate representations, we can avoid such calations for any shape of potentials and interactions.
The quantum Fokker–Planck equation was originaaiming to study a single potential surface system. Bysimple and straightforward generalization, then, we canrive the multi-state quantum Fokker–Planck equation toply to a system with multi-potential surfaces.38 In this paper,we present a comprehensive study of the various regimeoptical transition for a Morse potentials system usingmulti-state Fokker–Planck equation for a GaussiaMarkovian noise bath. The present model permits thetailed study of nuclear wave packets in Wigner represetion and their monitoring by femtosecond pump-prospectroscopy for various displacements of potentialsheat-bath parameters. Special attention is paid to a largeplacement case, where laser induced photodissociationpredissociation play an important role.
The organization of this paper is as follows: We pres
J. Chem. Phys., Vol. 107
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the procedure for calculating the linear absorption andpump-probe spectrum in Sec. II. The multi-state FokkePlanck equation is presented in Sec. III. In Secs. IV andthe linear absorption and pump-probe spectra are calculfor various displacements and heat-bath parameters. InVI, numerical results of pump-probe spectra for a stroGaussian pump pulse, which show optical Stark splitting,presented and discussed. Section V is devoted to concluremarks.
II. LINEAR ABSORPTION AND PUMP-PROBESPECTROSCOPY
We consider a molecular system with electronic stadenoted byu j &. The Hamiltonian of the system is
HS0~ t !5
p2
2M1(
j(
ku j &U jk~q;t !^ku. ~2.1!
Here,q is a nuclear coordinate strongly coupled to the eltronic state andp is its conjugate momentum. The diagonelementU j j (q) is the potential surface of thej th electronicsurface, and the off-diagonal elementU jk(q) with j Þk rep-resents the diabatic coupling between thej th and thekthstates. In this paper, we study a pump-probe experimentthree-level or four-level system with Morse potential sufaces denoted byug&, ue&, ue8& and u f &. ~Fig. 1!. The transi-tion frequency betweenge and e f are denoted byvge andve f , respectively. We assume that the system is initiallythe ground equilibrium staterg5ug&rg^gu, whererg is theequilibrium distribution function of the ground potential suface. In addition, the primary nuclear coordinate is coupto a bath. The total Hamiltonian is then expressed as
HS~ t !5HS0~ t !1H8, ~2.2!
where the HamiltonianH8 describes coupling of the molecular system to a bath of nuclear degrees of freedom. At
FIG. 1. Potential surfaces of the displaced Morse oscillators system.~a! isfor the three-level system denoted byug&, ue&, and u f &, respectively. Wedisplay theue& state for three different displacements;d51 ~dashed line!,d53 ~solid line! and d57 ~dotted line!. The resonant frequency betweeug& andue&, andue& andu f & are, respectively, expressed byvge andve f . ~b!is for the system with the anti-bonding state (ue8&). In this case, we onlyprobe between theug& and ue& states.
, No. 6, 8 August 1997
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1781Y. Tanimura and Y. Maruyama: Nonlinear spectroscopy of a Morse potentials system
point, we need not specifyH8 any further. In a pump-probeexperiment, the system is subjected to two light pulsespump and a weak probe whose frequencies and wave veare denoted byV1 , k1 andV2 , k2 , respectively. We assumthat the pump laser carrier frequencyV1 is close to the elec-tronic transition frequency betweeng ande. The probe fre-quencyV2 is chosen to~i! V2'vge for a measurement betweeng ande, and~ii ! V2've f for a measurement betweee and f . The total Hamiltonian is then given by
HA~ t !5HS~ t !1E1~ t !~eik1r2 iV1tm111e2 ik1r1 iV1tm1
2!
1E2~ t !~eik2r2 iV2tm211e2 ik2r1 iV2tm2
2!, ~2.3!
where E1(t) and E2(t) are the temporal envelopes of thpump and probe pulses, andm1
15ue&^gu andm125ug&^eu are
the dipole operators of the pump. The dipole of the probchosen to bem2
15ue&^gu and m225ug&^eu for ~i! and m2
1
5u f &^eu and m225ue&^ f u for ~ii !.
The observable in optical measurements is the polartion defined by
P~r ,t 
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1782 Y. Tanimura and Y. Maruyama: Nonlinear spectroscopy of a Morse potentials system
d
dtr~ t !52
i
\@HS~ t !,r~ t !#, ~2.14!
until time t with the initial condition r(0)5m21rg and by
taking the element tr$m22r(t)%.
B. Pump-probe spectroscopy for an arbitrary shapeand strength of pump pulses
Vibrational wave packets have proven to be an effectprobe of interatomic potentials.42 In the pump-probe spectroscopy, the pump transfers a small fraction of the groustate distribution to the excited state, thereby creating a ‘‘pticle’’ in the excited state and a ‘‘hole’’ in the ground statThe particle and the hole then evolve during the delay pet, which are detected by the probe absorption signal. It wbe convenient in the following calculations to expressspectrum using expectation values rather than a correlafunction. This can be done as follows. Let us considerevolution of the system subject only to the pump field. THamiltonian is given by Eq.~2.12! and the correspondingsolution of the Liouville equation is denotedr0(t):
d
dtr0~ t !52
i
\@HA
0~ t !,r0~ t !#. ~2.15!
We next introduce a modified Hamiltonian which includonly the negative frequency component ofE2
HA8 ~ t 
1783Y. Tanimura and Y. Maruyama: Nonlinear spectroscopy of a Morse potentials system
R4~ t,t8,t8,t9!5trH m22 exp← S 2
i
\ Et8
t
dtHSD m21 exp← S 2
i
\ Et8
t8dtHSD m1
2 exp← S 2
i
\ Et9
t8dtHSD m1
1
3 exp← S 2
i
\ E2`
t9dtHSD rg
exp→ S i
\ E2`
t
dtHSD J .
Equation~2.21! is a commonly used description of the response functions for a two-level system.21
In case~ii !, we have
tr$@m22~ t !,m2
1~ t8!#rg%5E0
t
dt8E0
t8dt9E1~t8!E1~t9!e2 iV1~t82t9!(
j 53
4
Rj8~t8,t9,t8,t !, ~2.22!
where
R38~ t,t8,t8,t9!5trH m22 exp← S 2
i
\ Et8
t
dtHSD m21 exp← S 2
i
\ Et9
t8dtHSD m1
1
3 exp← S 2
i
\ E2`
t9dtHSD rg
exp→ S i
\ E2`
t8dtHSD m1
2 exp→ S i
\ Et8
t
dtHSD J ,
~2.23!
R48~ t,t8,t8,t9!5trH m22 exp← S 2
i
\ Et8
t
dtHSD m21 exp← S 2
i
\ Et8
t8dtHSD m1
1
3 exp← S 2
i
\ E2`
t8dtHSD rg
exp→ S i
\ E2`
t9dtHSD m1
2 exp→ S i
\ Et9
t
dtHSD J .
arefo
q.
umurem
-
om
In the impulsive limit, the pump and the probe pulsesshort compared with the dynamical time scales of the solvand solute nuclear degrees of freedom. We can theremake the following assumption,
E1~ t !5u1d~ t !, E2~ t !5u2d~ t2t!, ~2.24!
where u1 and u2 are their areas and we takeu15u251.Then Eq.~2.9! with Eqs.~2.21! or ~2.23! reduces to
Pk2~ t !5ImH eiV2t(
j 51
4
Ri~ t,t!J ~2.25!
with
R1~ t,t!5R2~ t,t!5trH m22 exp← S 2
i
\ Et
t
dtHSD re~t!m21
3 exp→ S i
\ Et
t8dtHSD J , ~2.26!
R3~ t,t!5R4~ t,t!5trH m22 exp← S 2
i
\ Et
t
dtHSDm21rg
3 exp→ S i
\ Et
t
dtHSD J , ~2.27!
or
Pk2~ t !5ImH eiV2t(
i 53
4
Ri8~ t,t!J ~2.28!
with
J. Chem. Phys., Vol. 107
entre
R38~ t,t!5R48~ t,t!5trH m22 exp← S 2
i
\ Et
t
dtHSDm21re~t!
3 exp→ S i
\ Et
t
dtHSD J , ~2.29!
respectively, where
re~t
q.
ta
ediasth-e
re
eor
e
ctdon
a
ndhc
ianise
nacequa-
re-
ath,
he
1784 Y. Tanimura and Y. Maruyama: Nonlinear spectroscopy of a Morse potentials system
III. QUANTUM FOKKER–PLANCK EQUATION FOR AMULTI-STATE SYSTEM INTERACTING WITH AGAUSSIAN–MARKOVIAN NOISE BATH
The multi-state density matrix for the Hamiltonian E~2.1! may be expanded in the electronic basis set as
r~ t !5(j ,k
u j &r jk~q,q8;t !^ku. ~3.1!
Here,r jk(q,q8;t) is expressed in the coordinate represention. Alternatively, we can switch to the Wigner~phasespace! representation
Wjk~P,R;t ![1
2p\E2`
`
dreiPr /\r jk~R2r /2,R1r /2;t !,
~3.2!
and the density matrix may then be written as
W~ t !5(j ,k
u j &Wjk~P,R;t !^ku. ~3.3!
The Wigner representation has the following advantagfirst it allows us to compare the quantum density matrixrectly with its classical counterpart. Second, using phspace distribution functions, we can further easily imposenecessary boundary conditions~e.g., periodic or open boundary conditions!, where particles can move in and out of thsystem. This is much more difficult in the coordinate repsentation.
We now specify the heat-bath Hamiltonian. We considan environment consisting of a set of harmonic oscillatwith coordinatesxn and momentapn . The interaction be-tween the system and thenth oscillator is assumed to blinear with a coupling strengthcn . The total Hamiltonian isthen given by
HA~ t !5HA0~ t !1H8, ~3.4!
where
H85(n
F pn2
2mn1
mnvn2
2 S xn2cnq
mnvn2D 2G . ~3.5!
The character of the heat bath is specified by the spedistribution: All information about the bath which is requirefor a reduced description of the system dynamics, is ctained in its initial temperature and its spectral density
J~v![v(n
S cn2
4mnvn2D ~d~v2vn!1d~v1vn!!. ~3.6!
J(v) is related to the symmetric correlation function ofcollective bath coordinate (X5Scnxn),
1
2^X~ t !X1XX~ t !&5\E dvJ~v!cothS b\v
2 D cos~vt !,
~3.7!
whereb51/kBT is the inverse temperature of the bath, athe time evolution ofX is determined by the pure batHamiltonian @Eq.~3.5! with q50]. We assume an Ohmidissipation with the Lorentzian cutoff,
J. Chem. Phys., Vol. 107
-
s;-ee
-
rs
ral
-
J~v!5Mz
2p
vg2
g21v2 . ~3.8!
With the assumption of the high temperature bathb\g<1,this spectral density represents a Gaussian—Markovnoise where the symmetric correlation function of the noinduced by the heat bath, is given by
1
2^X~ t !X1XX~ t !&5
Mzg
be2gt. ~3.9!
Thus,z and g correspond to the friction and the relaxatiotime of the noise, respectively. In this case, one can trover the heat-bath degrees of freedom and obtain the etion of motion in the hierarchy form.35–37 The importantpoint is that the restriction does not involve the system fquencies~which can be small or large compared tob21), butonly a high temperature requirement with respect to the bwhich is much easier to meet. For thenth member of hier-archy,Wjk
(n) , wherej andk represent nonadiabatic states, tequation of motion is expressed as38
]
]tWjk
~0!~P,R;t !
52P
M
]
]RWjk
~0!~P,R;t !21
\E dP8
2p\
3(m
@Xjm~P2P8,R;t !Wmk~0!~P8,R;t !
1Xmk* ~P2P8,R;t !Wjm~0!~P8,R;t !#1
]
]PWjk
~1!~P,R;t !,
~3.10!
]
]tWjk
~1!~P,R;t !
52P
M
]
]RWjk
~1!~P,R;t !21
\E dP8
2p\
3(m
@Xjm~P2P8,R;t !Wmk~1!~P8,R;t !
1Xmk* ~P2P8,R;t !Wjm~1!~P8,R;t !#2gWjk
~1!~P,R;t !
1]
]PWjk
~2!~P,R;t !1zgS P1M
b
]
]PDWjk~0!~P,R;t !,
~3.11!
and
, No. 6, 8 August 1997
tictlg
onseitorc-
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1785Y. Tanimura and Y. Maruyama: Nonlinear spectroscopy of a Morse potentials system
]
]tWjk
~n!~P,R;t !
52P
M
]
]RWjk
~n!~P,R;t !21
\E dP8
2p\
3(m
@Xjm~P2P8,R;t !Wmk~n!~P8,R;t !
1Xmk* ~P2P8,R;t !Wjm~n!~P8,R;t !#2ngWjk
~n!~P,R;t !
1]
]PWjk
~n11!~P,R;t !
1nzgS P1M
b
]
]PDWjk~n21!~P,R;t !. ~3.12!
Here,z is the friction constant and
Xi j ~P,R;t !5 i E2`
`
dr exp~ iPr /\!Ui j ~R2r /2;t !,
~3.13!
Xi j* ~P,R;t !52 i E2`
`
dr exp~ iPr /\!Ui j ~R1r /2;t !,
are the Fourier transform representation of the potenterms which are convenient for studying the quantum effeThe hierarchy elementsWjk
(n) are defined in the path integraform.36,37 The equation of motion is derived by performintime derivative of these hierarchy elements. Physically,can think of this hierarchy of equations as dealing with aof Wigner functions, modeling the states of the system wvarious number of phonons excited in the bath. In this fmulation,Wjk
(0) includes all order of the system-bath interation and is the exact solution for the Hamiltonian Eq.~3.4!.Then Wjk
(1) ,Wjk(2) , . . . ,Wjk
(n) describe the distribution functions with a smaller set of the system-bath interaction, cresponding to the complete set of the system-bath intetions minus 1st, 2nd, . . . ,nth order of the system-batinteraction, respectively. Thus, one can think that this formlation takes the opposite direction to the conventional perbative expansion approaches, where the 0th member doeinclude any system-bath interaction, then the first, secothird, etc, members gradually take into account the highorder interactions and approach to the exact solution.shall be interested only in the 0th member of the hierarWjk
(0) which is identical toWjk defined in Eq.~3.3!. The otherelementsnÞ0 are not directory relate to the physical obseable and introduced for computational purposes. For dhierarchyNg@vc wherevc is the characteristic frequencof the system such as the frequency of the harmonic potial, the above hierarchy can be terminated by36
]
]tWjk
~N!~P,R;t !
52P
M
]
]RWjk
~N!~P,R;t !21
\E dP8
2p\
J. Chem. Phys., Vol. 107
als.
et
h-
r-c-
-r-notd,r-ey
-p
n-
3(m
@Xjm~P2P8,R;t !Wmk~N!~P8,R;t !
1Xmk* ~P2P8,R;t !Wjm~N!~P8,R;t !#2NgWjk
~N!~P,R;t !
1GWjk~N!~P,R;t !1NzgS P1
M
b
]
]PDWjk~N21!~P,R;t !,
~3.14!
where
G[z]
]PS P1M
b
]
]PD . ~3.15!
Using this hierarchal structure we may deal with strosystem-bath interactions in addition to a colored noise. Inwhite noise limitg@vc we may terminate the hierarchy oEqs.~3.10!–~3.14! by settingN50, obtaining the multi-statequantum Fokker–Planck equation for a Gaussian-white nbath:
]
]tWjk
~0!~P,R;t !52P
M
]
]RWjk
~0!~P,R;t !
21
\E dP8
2p\(m
@Xjm~P2P8,R;t !
3Wmk~0!~P8,R;t !1Xmk* ~P2P8,R;t !
3Wjm~0!~P8,R;t !#1GWjk
~0!~P,R;t !.
~3.16!
If we consider a system with a single potential surface, ththe above equation further reduces to the quantum FokkPlanck equation that was obtained by Caldeira and Legge44
Since we have assumedb\g<1, the temperature requirement of the Gaussian-white case is more stringent thanGaussian–Markovian case.
IV. NUMERICAL CALCULATIONS OF LINEARABSORPTION SPECTRUM
We consider the displaced Morse potentials systemfined by ~see Fig. 1!;
Ugg~R!5Ee$12e2a~R2D1!%2,
Uee~R!5Ee$12e2a~R2D2!%21\vge ,~4.1!
U f f~R!5Ee$12e2a~R2D1!%21\~vge1ve f!,
whereEe , a, andD j are the dissociation energy, the curvture of the potential, and the displacement, respectively.the end of the next section, we will also include the anbonding state,e8 and the diabatic coupling betweene ande8 described by
Ue8e8~R!5Eee22a8~R2D2!1\vge ,
Uee8~R!5Ae2D~R2D3!2. ~4.2!
, No. 6, 8 August 1997
a
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o
-
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ese
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1786 Y. Tanimura and Y. Maruyama: Nonlinear spectroscopy of a Morse potentials system
Hereafter, we employed the dimensionless coordinatemomentum defined byr[RAMv0 /\ and p[PA1/M\v0,respectively, wherev0[AUgg9 (R)/M . The displacemenand curvature of the potential,D1 , D2 , a, etc. are also measured in this unit. We setEe53649.5 @cm21#, a50.6361,and D1540.598 ~4.64788 @Å#! as the ground state of thCs2 molecule,45,46 which has been studied by a varietyspectroscopic techniques.47–50The fundamental frequency ithen given byv0538.7 @cm21#. We calculate linear absorption and pump-probe spectra for various displacemend[D22D1 . For the anti-bonding state, the parameters wchosen to bea850.6361, D51.0, A5300 @cm21#, and d8[D32D1511.09, respectively. We have used two valuesfriction z50.16@cm21# ~weak! andz547.8@cm21# ~strong!and have chosen the inverse correlation timeg54.8@cm21#,and the initial temperatureT5300 K, which satisfiesthe conditionb\g[\g/kBT50.023!1.
We first calculate the initial equilibrium state by integrating the equation of motion from timet52t i to t50with the temporally initial condition,
Wgg~0!~p,r ;2t i !)5exp@2b~p21Ugg~r !!#,
Wgg~n!~p,r ;2t i !50. ~4.3!
Note that Eq.~4.3! is the equilibrium state of the systemitself, but, it is not the equilibrium state of the total systesince it neglects the system-bath interaction. In the preformalism, such interaction can be taken into account bynonzero hierarchy elements, i.e.,Wjk
(n)(p,r ;t)Þ0. By inte-grating the equation of motion from timet52t i to t50, thedensity matrix comes to the ‘‘true’’ equilibrium state dscribed by the full set of hierarchyWgg
(n)(p,r ;t50), if we setut i u for a sufficiently longer time than the characteristic timof the system. In the following, we use the calculated fullof hierarchyWgg
(n)(p,r ;t50) as the true initial condition.The numerical integrations of these kinetic equatio
were performed by using second-order Runge–Kutta metfor finite difference expressions of the momentum andcoordinate space. The size of mesh was chosen to b3231-13031601 in the mesh range210,p,10 and 34,r ,57–234,p,34 and 34,r ,106. On the mesh, thekinetic operatorp]W/]r is approximated by a left-hand difference,pi(W(pi ,r j )2W(pi ,r j 21))/Dr for pi.0 and by aright-hand differencepi(W(pi ,r j 11)2W(pi ,r j ))/Dr for pi
,0.51 The discrete Fourier expression is used for the pottial kernel Eq. ~3.13!. We have taken into account abo11–24 hierarchy elements forW(n). The accuracy of the calculations was checked by changing the mesh size andnumber of terms in the hierarchy.
After obtaining the equilibrium state, we calculate tlinear absorption by integrating the equation of motion E~3.10!–~3.14! instead of the Liouville equation Eq.~2.14!following the procedure explained in Sec. II.
In Fig. 2 we present the linear absorption spectratween theg and e state for different displacements:~a! thesmall d51; ~b! the intermediated53; and ~c! the larged
J. Chem. Phys., Vol. 107
nd
e
f
,nte
t
sd
e30
-
he
.
-
57. The linear absorption with and without the anti-bondistate (e8 state! give the same result. In each figure, we hacalculated two cases of frictionz50.16 @cm21# ~weak! andz547.8 @cm21# ~strong!, respectively. Since we assumethat the probe pulse connects only between theg and estates, the contribution of the linear absorption is only froWeg(p,r ;t). Figure 2a is for small displacement. Each pearepresent the transitions between the vibrational levels ofground and the excited states. Since the Morse potentiathe vicinity of potential minimum is well approximated bthe harmonic surface, in this small displacement case,absorption spectra resemble those from the displacedmonic oscillator system with the fundamental frequencyv0
538.7@cm21#. Ford53, transitions to the higher vibrationalevels in thee state can take part in. Thus, we observe mapeaks in the weak damping case. Due to the anharmonof the potential, the interval of vibronic lines decreasesfrequency increases. In the strong damping case, eachbronic line is broadened and we simply observe the envelof the spectrum. Because the resonant frequency betwee
FIG. 2. Linear absorption spectra for different displacement. In each figwe display spectra for the weak damping case~solid line! and the strongdamping case~dashed line!.
, No. 6, 8 August 1997
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ti
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et tocket
forflcu-
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rent
1787Y. Tanimura and Y. Maruyama: Nonlinear spectroscopy of a Morse potentials system
ground and excited state is not linear function of coordin~see Fig. 1a! such like a displaced harmonic oscillators sytem, the envelope of peaks is not symmetric Gaussian suway that the blue side of the spectrum is amplified atexpense of the red side. Ford57, the transition mainly oc-curs between the ground state and continuum dissociastates and the spectrum is widely spread out.52 The shape ofspectra in the weak and strong damping cases are alidentical and overlapped. This is because, in this large
FIG. 3. Impulsive pump-probe spectra of the two-level system for differdisplacement in the weak damping case. Here, we probe between thug&and ue& states.
J. Chem. Phys., Vol. 107
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ae
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placement case, a laser excitation brings the wave packthe continuous dissociation states, where the wave pacannot show coherent oscillations.
V. NUMERICAL CALCULATIONS OF IMPULSIVEPUMP-PROBE SPECTRUM
Next we present the impulsive pump-probe spectravarious displacement betweeng ande states as a function othe frequency and the time. We have carried out the ca
tFIG. 4. Impulsive pump-probe spectra of the three-level system for diffedisplacement in the weak damping case. Here, we probe between theue& andu f & states.
, No. 6, 8 August 1997
1788 Y. Tanimura and Y. Maruyama: Nonlinear spectroscopy of a Morse potentials system
FIG. 5. The time-evolution of the wave packet of theue& state for the displacementd53 in the weak damping case.
thest
t
xs
e,
nicbeos-ally-3bce-
theosethe
e-en
lation for the weak and strong damping, however, sincedifference of them is mostly the existence of vibronic linas seen in Fig. 2, hereafter we present the results forweak damping (z50.16 @cm21#! only. We show the resulfor ~i! probe absorption betweeng ande in Fig. 3 and for~ii !the probe absorption betweene and f in Fig. 4.
We calculated the signal following the procedure eplained in Sec. III by integrating the equation of motion Eq~3.10!–~3.14! instead of the Liouville equation Eq.~2.14!. Incase ~i!, both the particle@Eq. ~2.26!# and the hole@Eq.~2.27!# contributes to the signal. Figure 3a shows the sptrum for small displacementd51. As mentioned in Sec. IV
J. Chem. Phys., Vol. 107
e
he
-.
c-
the system is well approximated by the displaced harmooscillator, if the displacement is small, and the pump-prospectrum is therefore similar to the displaced harmoniccillators case. The height of each peak changes periodicwith T51/v05861 @fs# corresponding to the coherent motion of the particle created by the pump-pulse. Figureshows the pump-probe spectrum for intermediate displamentd53.0. The small peaks in the figure correspond tovibronic bands as observed in Fig. 2b. The envelope of thsmall peaks reflects the shape of excited wave packet andpeak of the envelope shows oscillating motion with the priod about 1000@fs#. Since the resonant frequency betwe
FIG. 6. The time-evolution of the wave packet of theue& state for the displacementd57 in the weak damping case.
, No. 6, 8 August 1997
ef
-ves
tualntt iscan
gh-in
1789Y. Tanimura and Y. Maruyama: Nonlinear spectroscopy of a Morse potentials system
FIG. 7. Impulsive pump-probe spectrum of the system with anti-bondstate for the displacementd53 in the weak damping case~see Fig. 1b!.Here, we probe between theug& and ue& states.
also
J. Chem. Phys., Vol. 107
the ground and excited state (dv5Uee(r )2Ugg(r )2veg) isnot a linear function ofr , the shape of the envelope as thfunction of dv is quite different from the original shape othe wave packet. For instancedv is a rapidly decreasingfunction of r in the ranger ,a, wherea is about 50 ford53, but gradually increases forr .a after it has attained itsminimum (dv52380 @cm21# for d53) at r'a. Thus, ifthe wave packet is in the area ofr ,50, the envelope corresponding to the wave packet is broadened and moquickly, but if the wave packet is inr .50, the envelopebecomes sharp and moves slowly, compared with its acshape and speed. Figure 3c is for the large displacemed57. In this case, the kinetic energy of the wave packelarger than the dissociation energy and the wave packetescape from the potential. Compared with Fig. 3b, the hiest peak shifts from2380 @cm21# to 2880@cm21#, since theminimum of dv now becomes2880 at r'a554 for d57. Corresponding to the dissociation processes, wehave a new peak about 0@cm21#, which agrees with the
g
FIG. 8. The time-evolution of the wave packet of theue& state~the bonding state! andue8& state~the anti-bonding state! for the displacementd53 in the weakdamping case. In each figure, the upper one is forue& whereas the lower one is forue8&.
, No. 6, 8 August 1997
st
e-e-
-
en
eth
n-
ke
reod
t
nsusal
td
te
te
vine8
-
s
t
o
bebythe
ticalorsB,thempsian,
ne
1790 Y. Tanimura and Y. Maruyama: Nonlinear spectroscopy of a Morse potentials system
energy differences between the excited and the groundat larger .
Figure 4 shows~ii ! the probe absorption spectrum btweene and f . As explained in Sec. II, the absorption btween e and f is a particle part of~i!. Since we fixed theposition of thef state just above of thef state, the displacement between thee and f becomes2d. Thus, absorptionpeaks appear in the opposite direction ofDv compared withcase~i!. As seen from Figs. 4a, we observe the cohermotion of the envelope more clearly than the case of~i!,since the spectra of~i! involve the time-dependent particland the time-independent hole contributions, whereasspectra of~ii ! involve only the time-dependent particle cotribution.
Figure 5 shows the time-evolution of the wave pacWee(p,r ;t) for intermediate displacementd53.0 and theweak dampingz50.16 @cm21#. At time t50, the wavepacket with the shape of the ground equilibrium state is cated by the pump pulse, then it moves in the positive codinate direction. At timet50.4@ps#, the wave packet reacheto the right-hand side of a potential wall and then bouncedthe negative coordinate direction (t50.6@ps#!. Due to thestrong anharmonicity of the potential, distribution functiowith different energy have different eigen frequencies. Thanharmonic effects lead to a destruction of the initially locized wave packet as seen in figures at timet50.6,0.8 and1.0@ps#.
Figure 6 showsWee(p,r ;t) for large displacementd57.0. In this case, the wave packet is quickly broken insmall wave packets because of anharmonicity as explaineFig. 5. The small wave packets appearing at larger havelarger energy and some of them can escape from the potial. This is seen from the figure at timet50.6– 1.0@ps#.
We next show the result with the anti-bonding sta(e8 state!. Figure 7 is for the intermediate displacementd53.0 and the weak dampingz50.16 @cm21#. Comparedwith Fig. 3b, the peak at2380 @cm21# and t51.4@ps# isnoticeably small. This is because the population of wapacket in thee state decreased after passing the crosspoint due to the predissociation process. This can be sfrom the time-evolution of the wave packet shown in Fig.In each figure, the upper one is forue& ~the bonding state!whereas the lower one is forue8& ~the anti-bonding state!. Att50.0@ps#, the wave packet in thee state moves in the positive coordinate direction. The wave packet in thee state,then, reaches (t50.2@ps#! and passes (t50.4@ps#! the curvecrossing point~about r 550). The transition mainly takeplace in the vicinity of the curve crossing point, and thee8population suddenly increases when thee state wave packepasses the crossing point (t50.4@ps#!. This is because weconsidered the diabatic coupling betweene and e8 in thelocalized form @see Eq.~4.2!#. After passing the crossingpoint, the transferred wave packet starts to move in thee8state potential surface (t50.6@ps#!. Since thee8 potential isnot stable, the wave packet in thee8 state quickly moves tothe positive direction and then goes out from the edgepotential (t50.8 and 1.0@ps#!.
J. Chem. Phys., Vol. 107
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f
VI. NUMERICAL CALCULATIONS OF PROBEABSORPTION FOR STRONG PUMP PULSE: OPTICALSTARK SPECTROSCOPY
The present Fokker–Planck equation approach canapplied to a system with any shape of potentials drivenpulses of arbitrary number, shape and strength. Thus,present approach can generalize the earlier study of opStark spectroscopy for a displaced harmonic oscillatsystem.43 Following the prescription discussed in Sec. IIhere, we have calculated the pump-probe spectrum fordisplaced Morse potential system under the strong pupulse. We assume that pump and probe pulses are Gaus
E1~ t !5u1 exp@2~ t/t1!2#,
E2~ t !5u2 exp@2~ t2t!2/t22#, ~6.1!
with resonance central frequencies, i.e.,V15V25vge .Thus, we measure the transition between theug& and ue&states only. The pulse durations were taken to bet15700@fs# and t2530 @fs# and the time delay was varied betweet522.0 @ps# to t51.0 @ps#, i.e., the pump and the probpulses are overlapped. The pump intensity wasmu154.77@THz# and the probe was weakmu251.59 @GHz#. In thisstudy we have calculated spectra in the case of Fig. 1a~with-out the antibonding state! with intermediate displacementd53 for a weak couplingz50.16 @cm21#.The other param-eters were same as the impulsive case.
FIG. 9. Pump-probe spectrum for a strong excitation (mu154.77@THz#! fordifferent pulse delayst ~ps!.
, No. 6, 8 August 1997
1791Y. Tanimura and Y. Maruyama: Nonlinear spectroscopy of a Morse potentials system
FIG. 10. The time-evolution of the wave packet of theug& andue& states for strong pump excitation. In each figure, the upper one is forue& whereas the lowerone is forug&.
he
vptonantri
on
pues
one
-side
-arkvede-
ethene.by
cy
’’h
toat ofngvi-
In Fig. 9, we show the pump-probe spectrum for tstrong pulse excitation. The curve att522.0 @ps# is similarto the linear absorption spectrum~Fig. 2b!, since the pump isweak and its effects are small at this early stage. Thebronic side-band peaks are observed in the probe absorspectrum corresponding to various vibronic absorptiemission processes. Due to anharmonicity of potentialthermal effects, the vibronic transitions yield an asymmeline shape. The peak about2380 @cm21# corresponds to theabsorption atdv5Uee(r )2Ugg(r )52380@cm21# and is at-tributed to the movement of the wave packet during nimpulsive probe detection.
The curves att521.5 and21.0 @ps# show the dipsabout 0@cm21# caused by the unbalance between the polation and the coherent contribution of an absorption sptrum ~the coherent dips!.53 When the pump pulse becomestronger, the coherent dips are broadened. Each vibrtransition shows a Stark splitting whose magnitude is givby the proper Rabi frequencyDVnm5ADvnm
2 1(mE1(t))2,whereDVnm is the Rabi frequency between thenth vibra-tional state ofUg and mth vibrational state ofUe with the
J. Chem. Phys., Vol. 107
i-ion-d
c
-
-c-
icn
energy differenceDvnm . In the early time periods the splitting of the peaks near the center are larger than for thepeaks~see curves fort51.5 and21.0@ps# in Fig. 9!. This isbecause the corresponding Rabi frequencyDVnm changessignificantly for smallmE1(t) if Dvnm is small. The Starkpeak of the origin (Dv50), which corresponds to zero vibronic line then splits to the blue and to the red. The Stshifted peaks of the vibronic side bands can be obseroutside of the Stark peaks of the zero vibronic line. For btween t50.0 and 1.0@ps#, the vibronic mode seems to bdecoupled from the optical transition and we observespectra similar to the one from the two-level system aloThis can be explained using an argument first employedBrewer;54,55,41under strong excitation, the relevant frequenof the atomic system is notveg or v0 , but rather Rabi fre-quency DVnm'mE1(t), which represents the ‘‘dressedstates.56 For very strong excitation, this frequency is muclarger thang andv0 . Thus the oscillator cannot respondthe system and the absorption spectrum approaches ththe isolated two-level system. This decoupling at strofields can potentially be used to eliminate intramolecular
, No. 6, 8 August 1997
se-s
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ke-f
th
in
lculcb
riopencem
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isskeca
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.
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1792 Y. Tanimura and Y. Maruyama: Nonlinear spectroscopy of a Morse potentials system
brational relaxation and to enhance the selectivity of lainduced processes. Fort51.0 @ps#, the pump excitation becomes weak enough and the structure of vibronic bandrecovered.
In Fig. 9, the blue Stark peak gives an absorptive conbution, whereas the red one gives gain contribution andcomes negative aftert520.5 @ps#. In contrast, the Blochequations39,40 or the stochastic Liouville equations41 predictboth peaks to be identical in this resonant excitation caThis phenomenon had been discussed in a study of displharmonic oscillator system;43 because of the Stark effect, thsystem has two Stark shifted excited states~dressed states!. Ifthe pump field is on resonance, and the temperature ofheat bath is infinite, then the populations of these two staare the same as predicted by the Bloch or the stochaLiouville equations. However, if the temperature of the bais finite, the population of upper Stark level can relax to tlower one which gives a gain of absorption from the lowlevel to the ground state.
Figure 10 shows the time-evolution of the wave pacWgg(p,r ;t) and Wee(p,r ;t) for this case. Since we considered the Gaussian~non impulsive! excitation, the shape othe ground state wave packet is also changed. Fort520.5@ps#, the population of the excited state increases due tostrong pump pulse, then, fort50.0, 0.5 and 1.0@ps#, itslightly decreases and increases because of Rabi floppCompare with the weak excitation case~Fig. 5!, the wavepacket in the excited state seems to be bounded due toStark effects.
VII. CONCLUSIONS
In this paper we present a rigorous procedure for calating the absorption and the pump-probe spectra, and calate the spectra for a displaced Morse oscillators systemusing multi-state quantum Fokker–Planck equation foGaussian Markovian bath. In the dissociation process regwe show the correspondence between the pump-probe strum and the wave packet dynamics. For small displacemthe spectra are similar to those obtained in the displaharmonic oscillators case. When displacement becolarge, then we observe the movement of wave packet asshift of the envelope of absorption peak. Then for large dplacement, we observe the peak corresponding to the dciation. It is shown that for weak damping, the wave paccollapses into small packets with different eigen frequendue to the anharmonicity of the potential which looks liketail of the wave packet.
We have also presented numerical calculations of puprobe spectra for a strong Gaussian pump pulse. The reshow interplay between vibronic transitions and dynamiStark splitting. In contrast to the results from the convetional Bloch equations which contain an infinite-temperatdephasing, we find that at finite temperature, the Stark pemay have different heights even when the pump pulse isresonance.
In conclusion, the present multi-state quantum FokkePlanck approach provides a powerful means for the stud
J. Chem. Phys., Vol. 107
r
is
i-e-
e.ed
heestic
er
t
e
g.
the
-u-y
an,ec-t,d
eshe-o-ty
p-ltsl-eksn
–of
various chemical processes, including the coherent contromolecules, where quantum effects play a major role. Onegeneralize the present approach to study a two-dimensisystem, where the interplay between internal energy reation and chaotic dynamics with various quantum effeplays an important role. One can also use the presentproach to study off-resonant and resonant fifth-order sptroscopy, which have been the subject of receresearch.57–62,23,11 The advent of fast computers equippewith several hundred megabytes of memory make it possto study such problems with dissipation. We leave themfuture studies.
ACKNOWLEDGMENTS
We thank K. Yoshihara and S. Mukamel for useful dcussions. Financial support for this work was partially prvided by the US-Japan International Scientific Research Pgram from Japan Society of Promotion of Science~JSPS!,and by Grand-in-Aid for Scientific Research from the JapMinistry of Education, Science, Sports, and Culture.
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