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The non-Markovian master equation for stochastically perturbed systems: effect on spectral lineshape
THEO CHEM ELSEVIER
Journal of Molecular Structure (Theochem) 361 (1996) 49-56
The non-Markovian master equation for stochastically perturbed systems: effect on spectral lineshape Gautam Gangopadhyay”,
Deb Shankar Rayb>*
“S.N. Bose National Centre for Basic Sciences, DB-17, Sector-I. Salt Lake City, Calcutta-700 064. India bDepartment of Physical Chemistry, Indian Association for the Cultivation of Science, Jadavpur. Calcutta-700 032, India
Received 28 February 1995;accepted 14 March 1995
Abstract Based on the system-reservoir model in which the intermediate states of the system are perturbed by stochastic interaction with the surroundings, we have generalized the master equation for the description of quantum dissipative processes and applied it to a three-level system to calculate transient optical response in pump-probe spectroscopy. We have shown that while the two damping processes act independently in the Markovian description, the interference of stochastic and system-reservoir interactions and the resulting non-Lorentzian structure of the band-shape function appear as the essential non-Markovian features in the dissipative dynamics.
1. Introduction
The problem of dissipative dynamics of quantum systems within the framework of system-reservoir models has been the subject of wide attention over the last few decades [l-4]. The generalizations of the appropriate master equation [5] describing the dissipative dynamics for the cases where the system is non-linear [6], or driven by a superintense field [7] in the Markovian and non-Markovian regime and also for the cases where the reservoirs are of phase sensitive nature [8,9], have been the important related issues in this context. These generalizations have yielded interesting results in multiphoton dissociation [ 10,l I], pump-probe ultrafast nonlinear spectroscopy and quantum optics, in general [51. It has been recognized recently that the internal * Corresponding author.
dynamics of molecules is susceptable to stochastic processes [12-141. The experiments on transient resonance fluorescence from molecular iodine have demonstrated the importance of stochastic interactions in the intermediate states during second order optical processes. If one takes only radiation damping into account, then in the process of Raman scattering only, coherence is maintained during the evolution of the whole system from the initial to the final state. When the random modulation from the surroundings comes into play the coherence is disturbed, resulting in fluorescence in which coherence is completely lost and in a broad Raman scattering in which coherence is partially lost. The interaction of the surrounding solvent molecules can also suitably modulate the energies of the intermediate states of, say, model organic molecules so that strong Raman signals and weak fluorescence are observed. These are some of the commonly
0166-1280/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved SSDI
0166-1280(95)04301-2
G. Gangopadhyay, D.S. Ray/Journal of Molecular Srructure (Theochem)
50
occurring simple situations in which stochastic interactions with the system play an important role in destroying quantum coherence. It is therefore worthwhile generalizing the master equations describing the dissipative dynamics of the system in contact with the reservoir for the case in which the system in addition is stochastically perturbed. The emphasis here is mainly on the dynamics induced by stochastic interactions in the system in the excited states which affect quantum coherence significantly, so that the optical response of the system becomes manifestly modified. It must also be emphasized that this stochasticity has nothing to do with system-reservoir interaction, which is essentially a dynamical description and which ensures the main relaxation of the system. Thus, our purpose here is to present a generalized non-Markovian master equation for a system which is stochastically perturbed and is in contact with the thermal reservoir. Although the scope for the theoretical treatment of system-reservoir models is very broad, we confine our scheme within the weak coupling limit and Born approximation. Since the destruction of quantum coherence takes place on a time scale that corresponds to the correlation time involved in the stochastic interactions, it is necessary to go beyond the Markov approximation. However, we point out that the generalization presented here is valid for a correlation time which is short but finite. In the following, we apply the generalized master equation to a typical three-level system to calculate transient susceptibility in the pump-probe experiments [ 151. We show that while the two damping processes act independently in the Markovian description, the interference of stochastic and system-reservoir interaction processes and the resulting nonLorentzian structure of the band-shape function appear as essential non-Markovian [ 161 features of the dissipative dynamics.
2. The master equation 2.1. Thegeneralization for a stochastically perturbed system
We consider an N-level system coupled to a
361 (1996) 49-56
reservoir and undergoing stochastic perturbation. The total Hamiltonian can be written as HT=H,+R+
V,(t)+
vz
(1)
where Hs and R refer to the Hamiltonian for the system and the reservoir respectively, and V, represents the interaction between the system and the reservoir. The system in addition is stochastically perturbed by the interaction V,(t), which is assumed to be of the following form:
(2) where Si is any system operator of the level population type. gi(t) is the external stochastic field which is, for example, represented by an exponentially correlated gaussian noise with properties
and (3b) Here, I is the inverse of correlation time and I’@/2 is the variance of gi(t). By ((. .)) we mean stochastic averaging. The system reservoir interaction term V, can be represented by a sum of products of the form P’z =hCQiFi
(4)
where Qi and Fi are the functions of the system and reservoir operators respectively. The joint density operator x for the system and the reservoir obeys the Liouville-von Neumann equation in the interaction picture dx - -gqt):x] at-
(5)
Here, V(t) denotes the sum of the two interaction terms Vi and I’, in the interaction picture. We formally integrate Eq. (5) and iterate twice. Taking a trace over the reservoir variables as well as after stochastic averaging and proceeding as in Ref. [6], we obtain an equation for “coarsegrained” time evolution for the reduced density
G. Gangopadhyay, D.S. Ray/Journal of Molecular Structure (Theochem)
operator s for the system in the interaction picture: i(t) = - C
’
sj(’
-
7)S(f
-
s(t) = -C{SiSiS-
- s(t - T)Sj(t- T)Si(t)l((gj(t - T)gi(t)))) + UQiP)Qj(t - T)s(~ - T) -
Qj(t
-
~)f(t
- lQi(t)s(t -
S(t -
-
-
T)Qi(t)I(Fi(t)Fj(t
T)Qj(t
T)Qj(t
-
-
-
-
2SiSSi +SSjSi}Ws
- {SiSlS - 2SiSS; + SSiSi}Gs
- C{[(QiQjs
T))R
- QjsQi)
f+'iT
ij
T)
T)Qi(t)I(Q(r
2ioT
i.e. r is short but finite. Eq. (8) then reduces to the following form:
T)si(t)l((gi(t)g,(t - T))) - [$(t)s(t- +$(t - 7) -
is to write
s(t - T) 25 s(t) -
1’ dr{ [Si(t)Sj(t - r)~(t - r)
ij
approximation
51
361 (1996) 49-56
-
(QiSQj
-
J(QiQjS
-
(Q;iQj
T)Fi(t))R}
-
-
SQj Qi)]
qi
QjSQi)G$
(6)
where all operators are in the interaction picture. Since Si - s and Qi - s are population type and excitation (de-excitation) type system operators, respectively, we write QfP(t) = ei"~'Q~P
(7)
and
-
SQ,Qi)G,TI)fi(wi
wj)
(10)
where II’S = 1 dr((gi(t)gi(t s
- r)))
I,drrjki(t)gi(t-
GS =
T)))
f ' dTe-i~‘(Fj(t)J$(t
wif =
J0 S!P I = SF I
+
' 0 .I
wj; =
Eq. (6) then reduces to the following form:
d7e-i‘+i(q(t
- 7))a
- r)Fj(t))R Gl; =
s(t) = - C{Si(t)Si(t)s(t
- r) - 2Sj(t)s(t - T)Si(t)
i + S(t -
x
I
(11) T)Si(t)Si(t)}
i dT((gi(f)gi(t
ESij
1 dT{
-
IQiQjdt
Note that WS and Gs are due to stochastic interaction. The terms containing s on the right hand side of Eq. (10) are the non-Markovian contributions. In the Markovian limit, Eq. (10) may be further simplified to the following form:
~1))
-
~1 -
QjJ(t
-
T)Qil
s(t) = - C{SjSiS
- 2SiSSi + SSjSj} WS
X eCiqT(Fi(t)F,(t - T))R - [Qis(t
-
T)Qj
-
~(f -
T)QjQil
X e-'q'(F,(t - ')Fi(t))R}ei'"'+w/'r
(8)
Here, we make two approximations. The first is the secular approximation which is a good approximation in the weak coupling limit, and the second
-
(QisQj - sQjQi)IW'i}S(wi
+wj)
(12)
It is immediately apparent that both the stochastic and the reservoir induced dissipative processes act independently in the Markovian limit.
52
G. Gangopadhyay. D.S. Ray/Journal of Molecular Structure (Theochem)
2.2. The two-levelsystem and the harmonic oscillator heat bath
Before going over to a non-Markovian scheme it is better to be more specific about the system. For simplicity, instead of a general N-level system we take the case of a two-level atom and identify the level population type operator Si as si=+02)(21-11)(11)
where j 1) and 12) refer to the two levels of the atom, or equivalently as Si = CJ=in terms of the Pauli operator. Further, if we replace S10terms on the right hand side of Eq. (10) by a “zeroeth order” part (i.e. the Markovian part of Eq. (lo)), we obtain i(t) = -(c7pzs - 2czZsaz+ sa,az) ws
-
C{[(QiQjs-QjsQi)%T
-(QiSP,
The first two lines on the right hand side of Eq. (13) denote the Markovian part and the rest is the non-Markovian contribution, It is important to emphasize that the Gs and Gi$ terms are the new elements of the present non-Markovian theory which are due to the finite but short correlation time. These terms illustrate the frequency dependence of spectral density functions W,f as follows:
(15) We now explicitly consider a two-level system coupled to a heat bath of harmonic oscillators characterized by creation and annihilation operators, b\ and bj respectively. The reservoir Hamiltonian and its interaction with the two-level system are described by R = C
- SQiQi)] Wjy}S(Ui -t wj)
361 (1996) 49-56
tiw,b,ibj
(16) respectively, SOwe identify Qi and Fi in Eq. (4) as
QI= P)(ll = g+
Q2 = /I)(21 = C- = Qi
and
Fl =
C gjb, j
X
Gi:
QiGG
- Qj k (S - ~u~,scJ~) Ws + C L,,
inn
-Qi
mn
+ ;(s -41T,SCJws + c L, mn X
QjGi;
k (S - 4~~s~~)Ws + C L,, i
6(Wi+
Wj)
1
1
QjQiGj; )
(13)
F2 = cg;b; i
= F/
(17)
One can also calculate the spectral density functions Wit and the corresponding G,$ terms using thermal field partition functions in the usual way. This reduces the expression for Lij (Eq. 14) to the following form:
c =$$ Lij
(1 + n(W))
Ii
x (a+a_ S - 2u_ su+ + su+u_) Y(W)- -+w)(u_u+s
- 2u+su_ +X-u+)
where Li, is defined as (18)
Lij =
{[(QiQjs-QjsQi)Wij+ Here, y and fi are defined as the natural damping - (QisQj -sQjQi)I~F)S(wi+~j) (14) rate y = 27rD(w)Ig(w))’ and the thermal average
G. Gangopadhyay,
D.S. Ray/Journal
of Molecular
excitation number n = [exp(hu/kT) - l]-‘, where D(w) denotes the density of states of the harmonic oscillator bath modes. The y terms without ii represent the loss of energy from the system to the reservoir, while yfi terms represent the diffusion of fluctuations in the reservoir into the system mode. In the limit as fi -+ 0 (i.e. the thermally induced effects are small), one can reduce Eq. (13) to the following form: jr
-(~+~)(s-40zso,) -
Structure (Theochem)
361 (1996) 49-56
53
as a relatively slow process remains unaffected. These interference terms which make their presence felt beyond the Markovian regime are a clear signature of non-Markovian characteristics of the dissipative process. In the next section, we show that these decay characteristics modify the band-shape of the transient susceptibility functions in the pump-probe experiment dealing with ultrafast optical processes.
(;+;GS)
3. Application: calculation of transient optical susceptibility
- z~‘y[a+cTU+0_s - sg+0_0+0_]
(19)
where f’=&?(w) Eq. (19) describes the non-Markovian dissipative dynamics of a two-level system in contact with a harmonic oscillator heat bath and perturbed by an external stochastic field. The Markovian and non-Markovian decay terms due to both of these interactions become more transparent if one calculates the evolution of population inversion and polarizations of the two-level system. Thus, the decay dynamics is as follows: (6+) = (2if’Ws + if’y)(ff+) - (Ws + WsGs +;+;Gs)(g+, (6-) = -(2if’Ws - (Ws+ (&A = -YClP
The decay of quantum coherence or dephasing takes place on a time scale where the nonMarkovian approximation generally used in the theoretical treatment of relaxation processes is invalidated. It has also been argued that nonMarkovian features might be probed through the transient optical absorption rather than any steady state measurement. With this end in view, we now calculate transient optical susceptibility which takes into account the non-Markovian features described in Eq. (20). We consider a model three-level system initially in the ground state. The pump pulse, which is ideally a S-pulse, selectively excites the system from the ground state to an excited state 1 in the presence of a continuous probe wave which monitors the probe absorption, resulting in transition of the system from state 1 to state 2 (see Fig. 1). Levels 1 and 2 thus effectively comprise a two-level system. Since the probe absorption is weak, we
s2
+ if’y)(a_)
Ep (t)
W,G,+;+;G,)(o_) + (a,))
1 (20)
It is immediately apparent that in addition to the usual independent damping terms corresponding to stochastic and system-reservoir interactions, there are terms like f’ W, and y/2Gs which illustrate the interference of these two interactions in the dephasing processes; the decay of population
t
6/t 1 t
4
Fig. 1. The pump pulse in a three-level scheme which is described by a &pulse excitation from the ground state g to 1. The probe field 4(t) induces transition from level 1 to 2.
G. Gangopadhyay, D.S. Ray/Journal of Molecular Structure (Theochem)
54
assume that pll >> p22, i.e. the perturbation approximation can be conveniently used. Thus, the effective two-level system, which is in contact with the heat bath and whose intermediate states 1 and 2 are perturbed by stochastic interaction and interaction with the probe wave, can be described by the following set of Bloch equations in the rotating wave approximation: (a,) = i(wO+ 2f’ Ws +f’y)(a+)
- islo e’Wf(aZ)
- (Ws + WsGs +;+;Gs)(d (b_) = -i(q
is further defined as
~~b(Q-4= P21 (th2
P(t) =
P21 =
b+)
PI2
P(t) =
bJ+)P12
L-l
-
- (a_) e’“‘) - y( l/2 + (a,))
Here, the weak probe field Rabi frequency a0 is the expressed through relation cl(t) = (Ro/2)(e-“’ + eiWf); w,, refers to the frequency of the effective two-level system. The damping processes in Eq. (21) have been incorporated according to the prescription of the last section. Making use of the following slowly varying envelope approximation and the abbreviations (c)
= S_ eP
(a,) = SZ
A=w,-w
+
h-)P21
s+(0) [P -
iRoLml
(22)
we can rewrite Eq. (21) as 3, = ifs, S_ = -ifs_
- is2& - ~$3, + iR$,
[ p - (if - 4
(ff+(t))
= eiW’L-’ [p _si;‘yya)]
x L-l
3, [ P - (if - x)
’ x(t - t’)E(t’)dt’
s0
- 2ip21EOeiw’
1
Here, the Laplace transform of SZ(t) is defined by S, = Jo” e -P’SZ(t) dt G ,9,(r). The response function x(w) in the frequency domain is then expressed as
3, [ P - (if - 7,)
1
Identifying the imaginary part of X(W) as the required transient optical susceptibility, we obtain after explicit evaluation of the inverse Laplace transform in Eq. (26) I
x”(w) = $p2112Re
d~e[if--Yu]'(uz(t 0
(24)
where E(t) represents the external probe field. P(t)
-
T))} (27)
(23)
To calculate the susceptibility, the linear response term in the expression for polarization is considered as follows: P(t) =
(25)
I
and
- y&
3, = -iRo(S+ - s-) - y(1/2 + S,)
1
3,
x(w) = -2i 1p2112C1
Y Y WsGs+-+-&=~a 2 2
of S,, (a,) can be
(if - “ia)
A+2f’Ws+f’y=f W,+
(0-j
or
S+(t) =
(21)
(o+) = S+ eiWt
=
Using the Laplace transform written as
_ (W, + WsGs +;+;Gs)(o-) eP
+ P12(t)P21
where p(t) and p denote the density matrix and the dipole moment operator respectively. In our notation, we express
+ 2f ‘W, +f ‘y)(a+) - iRo e -‘w’(a,)
(bz) = -iQ,((c+)
361 (1996) 49-56
Since probe absorption is sufficiently weak, we may write (uz(t - 7-)) M -l/2. Eq. (27) then reduces to the form
x”(w) =
&1
Incorporating
p21
1’Re the
{ eCi$“‘,lr
1)
approximate
(28) frequency
G. Gangopadhyay. D.S. Ray/Journal of Molecular Structure (Theochem)
dependence of y as
and for a time that is long in the relevant time scale, we obtain
(1
+Gs)
H’s+++?)
X
(A+2FWs+,y,,)2+(l+Gs)2
361 (1996) 49-56
55
shown that the two dissipative processes interfere in the non-Markovian description, This interference is manifested in the expression for transient optical response functions in ultrafast optical pump-probe spectroscopy. Although the treatment that we have presented here is valid for a correlation time which is short but finite, we hope that an understanding of the ultrafast processes in pumpprobe spectroscopy particularly in the non-linear regime, in general, will be amenable in terms of such a generalized master equation approach.
I (29)
Here, ~rl~wl,-,, = F and A = w- wo, and we have omitted the terms containing higher powers of F. x”(w) exhibits a lineshape which is different from the usual Lorentzian structure. It is also important to note than the influence of both the dissipative processes and the non-Markovian nature of its dynamics for short but finite correlation time has become prominent in the shape and width of the lineshape function. Also, because of the F-terms, the resonance in the transient absorption is shifted. We note in passing that the asymmetry in the lineshape function is a typical feature well-known in experimental ultrafast pump-probe spectroscopy.
4. Summary and conclusions Based on the system-reservoir model, in which the intermediate states of the system are perturbed by the external stochastic interactions with the surroundings, we have developed a generalized non-Markovian master equation for the description of the quantum dissipative processes. The dissipation concerns two decay processes, one of which is the system-reservoir interaction whereas the other is the stochastic interaction. While the former description depicting the usual relaxation processes is completely dynamical in nature, the latter in the absence of any knowledge of correlation functions derived from dynamics, the interaction is phenomenological in character and is responsible for dephasing processes. We have
Acknowledgements One of us (DSR) would like to thank the Department of Science and Technology, India, for financial support under the “laser facility scheme”.
References [I] P. Ullersma, Physica, 32 (1966) 27; 32 (1966) 56; 32 (1966) 74; 32 (1966) 90. [2] M. Lax, Phys. Rev. 129 (1963) 2342; 145 (1966) 109. G.S. Agarwal, Phys. Rev. A, 2 (1970) 2038; 3 (1971) 1783. [3] W.H. Louisell, Quantum Statistical Properties of Radiation, Wiley, New York, 1973. [4] S.A. Adelman and J.D. Doll, J. Chem. Phys., 61 (1974) 4242; 64 (1976) 2375. [S] G. Gangopadhyay and D.S. Ray, in S.H. Lin, A.A. Villaeys and Y. Fujimura (Eds.), Advances in Multiphoton Processes and spectroscopy, Vol. 8, World Scientific, Singapore, 1993. [6] G. Gangopadhyay and D.S. Ray, Phys. Rev. A, 46 (1992) 1507. [7] G. Gangopadhyay and D.S. Ray, Phys. Rev. A, 43 (1991) 6424. [8] C.W. Gardiner and M.J. Collet, Phys. Rev. A, 31 (1985) 3761. [9] N. Lu and M. Bergou, Phys. Rev. A, 40 (1989) 237. [lo] J. Lin, M. Hutchison and T.F. George, in S.H. Lin, A.A. Villaeys and Y. Fujimura (Eds.), Advances in Multiphoton Processes and Spectroscopy, Vol. 1, World Scientific, Singapore, 1984. [I l] G. Gangopadhyay and D.S. Ray, J. Chem. Phys., 96 (1992) 4693; 97 (1992) 4104. [12] T. Takagahara, E. Hanamura and R. Kubo, J. Phys. Sot. Jpn., 43 (1977) 802; 43 (1977) 811; 43 (1977) 1522; 44 (1978) 742. [13] Y. Tanimuraand R. Kubo, J. Phys. Sot. Jpn., 58 (1989) 58; Y. Tanimura, T. Suzuki and R. Kubo, J. Phys. Sot. Jpn., 58 (1989) 1850.
56
G. Gangopadhyay, D.S. Ray/Journal of Molecular Strucrure (Theochem)
1141 A.A. Villaeys, A. Boeglin and S.H. Lin, Phys. Rev. A, 44
(1991) 4660; 44 (1991) 4671. [15] P.C. Brecker, H.L. Fragnito, J.Y. Bigot, C.H. Britocruz and C.V. Shank, Phys. Rev. Lett., 63 (1990) 505. W. Vogel, D.G. Welsch and W. Wilhelmi, Phys. Rev. A, 37 (1988) 3825.
361 (1996) 49-56
E.T.J. Nibbering, K. Duppen and D.A. Wiersma, J. Chem. Phys., 93 (1990) 5477. [16] S. Mukamel, Phys. Rep., 93 (1982) 1; A.A. Villaeys, J.C. Vallet and S.H. Lin, Phys. Rev. A, 43 (1991) 5030.
Journal of Molecular Structure (Theochem) 361 (1996) 49-56
The non-Markovian master equation for stochastically perturbed systems: effect on spectral lineshape Gautam Gangopadhyay”,
Deb Shankar Rayb>*
“S.N. Bose National Centre for Basic Sciences, DB-17, Sector-I. Salt Lake City, Calcutta-700 064. India bDepartment of Physical Chemistry, Indian Association for the Cultivation of Science, Jadavpur. Calcutta-700 032, India
Received 28 February 1995;accepted 14 March 1995
Abstract Based on the system-reservoir model in which the intermediate states of the system are perturbed by stochastic interaction with the surroundings, we have generalized the master equation for the description of quantum dissipative processes and applied it to a three-level system to calculate transient optical response in pump-probe spectroscopy. We have shown that while the two damping processes act independently in the Markovian description, the interference of stochastic and system-reservoir interactions and the resulting non-Lorentzian structure of the band-shape function appear as the essential non-Markovian features in the dissipative dynamics.
1. Introduction
The problem of dissipative dynamics of quantum systems within the framework of system-reservoir models has been the subject of wide attention over the last few decades [l-4]. The generalizations of the appropriate master equation [5] describing the dissipative dynamics for the cases where the system is non-linear [6], or driven by a superintense field [7] in the Markovian and non-Markovian regime and also for the cases where the reservoirs are of phase sensitive nature [8,9], have been the important related issues in this context. These generalizations have yielded interesting results in multiphoton dissociation [ 10,l I], pump-probe ultrafast nonlinear spectroscopy and quantum optics, in general [51. It has been recognized recently that the internal * Corresponding author.
dynamics of molecules is susceptable to stochastic processes [12-141. The experiments on transient resonance fluorescence from molecular iodine have demonstrated the importance of stochastic interactions in the intermediate states during second order optical processes. If one takes only radiation damping into account, then in the process of Raman scattering only, coherence is maintained during the evolution of the whole system from the initial to the final state. When the random modulation from the surroundings comes into play the coherence is disturbed, resulting in fluorescence in which coherence is completely lost and in a broad Raman scattering in which coherence is partially lost. The interaction of the surrounding solvent molecules can also suitably modulate the energies of the intermediate states of, say, model organic molecules so that strong Raman signals and weak fluorescence are observed. These are some of the commonly
0166-1280/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved SSDI
0166-1280(95)04301-2
G. Gangopadhyay, D.S. Ray/Journal of Molecular Srructure (Theochem)
50
occurring simple situations in which stochastic interactions with the system play an important role in destroying quantum coherence. It is therefore worthwhile generalizing the master equations describing the dissipative dynamics of the system in contact with the reservoir for the case in which the system in addition is stochastically perturbed. The emphasis here is mainly on the dynamics induced by stochastic interactions in the system in the excited states which affect quantum coherence significantly, so that the optical response of the system becomes manifestly modified. It must also be emphasized that this stochasticity has nothing to do with system-reservoir interaction, which is essentially a dynamical description and which ensures the main relaxation of the system. Thus, our purpose here is to present a generalized non-Markovian master equation for a system which is stochastically perturbed and is in contact with the thermal reservoir. Although the scope for the theoretical treatment of system-reservoir models is very broad, we confine our scheme within the weak coupling limit and Born approximation. Since the destruction of quantum coherence takes place on a time scale that corresponds to the correlation time involved in the stochastic interactions, it is necessary to go beyond the Markov approximation. However, we point out that the generalization presented here is valid for a correlation time which is short but finite. In the following, we apply the generalized master equation to a typical three-level system to calculate transient susceptibility in the pump-probe experiments [ 151. We show that while the two damping processes act independently in the Markovian description, the interference of stochastic and system-reservoir interaction processes and the resulting nonLorentzian structure of the band-shape function appear as essential non-Markovian [ 161 features of the dissipative dynamics.
2. The master equation 2.1. Thegeneralization for a stochastically perturbed system
We consider an N-level system coupled to a
361 (1996) 49-56
reservoir and undergoing stochastic perturbation. The total Hamiltonian can be written as HT=H,+R+
V,(t)+
vz
(1)
where Hs and R refer to the Hamiltonian for the system and the reservoir respectively, and V, represents the interaction between the system and the reservoir. The system in addition is stochastically perturbed by the interaction V,(t), which is assumed to be of the following form:
(2) where Si is any system operator of the level population type. gi(t) is the external stochastic field which is, for example, represented by an exponentially correlated gaussian noise with properties
and (3b) Here, I is the inverse of correlation time and I’@/2 is the variance of gi(t). By ((. .)) we mean stochastic averaging. The system reservoir interaction term V, can be represented by a sum of products of the form P’z =hCQiFi
(4)
where Qi and Fi are the functions of the system and reservoir operators respectively. The joint density operator x for the system and the reservoir obeys the Liouville-von Neumann equation in the interaction picture dx - -gqt):x] at-
(5)
Here, V(t) denotes the sum of the two interaction terms Vi and I’, in the interaction picture. We formally integrate Eq. (5) and iterate twice. Taking a trace over the reservoir variables as well as after stochastic averaging and proceeding as in Ref. [6], we obtain an equation for “coarsegrained” time evolution for the reduced density
G. Gangopadhyay, D.S. Ray/Journal of Molecular Structure (Theochem)
operator s for the system in the interaction picture: i(t) = - C
’
sj(’
-
7)S(f
-
s(t) = -C{SiSiS-
- s(t - T)Sj(t- T)Si(t)l((gj(t - T)gi(t)))) + UQiP)Qj(t - T)s(~ - T) -
Qj(t
-
~)f(t
- lQi(t)s(t -
S(t -
-
-
T)Qi(t)I(Fi(t)Fj(t
T)Qj(t
T)Qj(t
-
-
-
-
2SiSSi +SSjSi}Ws
- {SiSlS - 2SiSS; + SSiSi}Gs
- C{[(QiQjs
T))R
- QjsQi)
f+'iT
ij
T)
T)Qi(t)I(Q(r
2ioT
i.e. r is short but finite. Eq. (8) then reduces to the following form:
T)si(t)l((gi(t)g,(t - T))) - [$(t)s(t- +$(t - 7) -
is to write
s(t - T) 25 s(t) -
1’ dr{ [Si(t)Sj(t - r)~(t - r)
ij
approximation
51
361 (1996) 49-56
-
(QiSQj
-
J(QiQjS
-
(Q;iQj
T)Fi(t))R}
-
-
SQj Qi)]
qi
QjSQi)G$
(6)
where all operators are in the interaction picture. Since Si - s and Qi - s are population type and excitation (de-excitation) type system operators, respectively, we write QfP(t) = ei"~'Q~P
(7)
and
-
SQ,Qi)G,TI)fi(wi
wj)
(10)
where II’S = 1 dr((gi(t)gi(t s
- r)))
I,drrjki(t)gi(t-
GS =
T)))
f ' dTe-i~‘(Fj(t)J$(t
wif =
J0 S!P I = SF I
+
' 0 .I
wj; =
Eq. (6) then reduces to the following form:
d7e-i‘+i(q(t
- 7))a
- r)Fj(t))R Gl; =
s(t) = - C{Si(t)Si(t)s(t
- r) - 2Sj(t)s(t - T)Si(t)
i + S(t -
x
I
(11) T)Si(t)Si(t)}
i dT((gi(f)gi(t
ESij
1 dT{
-
IQiQjdt
Note that WS and Gs are due to stochastic interaction. The terms containing s on the right hand side of Eq. (10) are the non-Markovian contributions. In the Markovian limit, Eq. (10) may be further simplified to the following form:
~1))
-
~1 -
QjJ(t
-
T)Qil
s(t) = - C{SjSiS
- 2SiSSi + SSjSj} WS
X eCiqT(Fi(t)F,(t - T))R - [Qis(t
-
T)Qj
-
~(f -
T)QjQil
X e-'q'(F,(t - ')Fi(t))R}ei'"'+w/'r
(8)
Here, we make two approximations. The first is the secular approximation which is a good approximation in the weak coupling limit, and the second
-
(QisQj - sQjQi)IW'i}S(wi
+wj)
(12)
It is immediately apparent that both the stochastic and the reservoir induced dissipative processes act independently in the Markovian limit.
52
G. Gangopadhyay. D.S. Ray/Journal of Molecular Structure (Theochem)
2.2. The two-levelsystem and the harmonic oscillator heat bath
Before going over to a non-Markovian scheme it is better to be more specific about the system. For simplicity, instead of a general N-level system we take the case of a two-level atom and identify the level population type operator Si as si=+02)(21-11)(11)
where j 1) and 12) refer to the two levels of the atom, or equivalently as Si = CJ=in terms of the Pauli operator. Further, if we replace S10terms on the right hand side of Eq. (10) by a “zeroeth order” part (i.e. the Markovian part of Eq. (lo)), we obtain i(t) = -(c7pzs - 2czZsaz+ sa,az) ws
-
C{[(QiQjs-QjsQi)%T
-(QiSP,
The first two lines on the right hand side of Eq. (13) denote the Markovian part and the rest is the non-Markovian contribution, It is important to emphasize that the Gs and Gi$ terms are the new elements of the present non-Markovian theory which are due to the finite but short correlation time. These terms illustrate the frequency dependence of spectral density functions W,f as follows:
(15) We now explicitly consider a two-level system coupled to a heat bath of harmonic oscillators characterized by creation and annihilation operators, b\ and bj respectively. The reservoir Hamiltonian and its interaction with the two-level system are described by R = C
- SQiQi)] Wjy}S(Ui -t wj)
361 (1996) 49-56
tiw,b,ibj
(16) respectively, SOwe identify Qi and Fi in Eq. (4) as
QI= P)(ll = g+
Q2 = /I)(21 = C- = Qi
and
Fl =
C gjb, j
X
Gi:
QiGG
- Qj k (S - ~u~,scJ~) Ws + C L,,
inn
-Qi
mn
+ ;(s -41T,SCJws + c L, mn X
QjGi;
k (S - 4~~s~~)Ws + C L,, i
6(Wi+
Wj)
1
1
QjQiGj; )
(13)
F2 = cg;b; i
= F/
(17)
One can also calculate the spectral density functions Wit and the corresponding G,$ terms using thermal field partition functions in the usual way. This reduces the expression for Lij (Eq. 14) to the following form:
c =$$ Lij
(1 + n(W))
Ii
x (a+a_ S - 2u_ su+ + su+u_) Y(W)- -+w)(u_u+s
- 2u+su_ +X-u+)
where Li, is defined as (18)
Lij =
{[(QiQjs-QjsQi)Wij+ Here, y and fi are defined as the natural damping - (QisQj -sQjQi)I~F)S(wi+~j) (14) rate y = 27rD(w)Ig(w))’ and the thermal average
G. Gangopadhyay,
D.S. Ray/Journal
of Molecular
excitation number n = [exp(hu/kT) - l]-‘, where D(w) denotes the density of states of the harmonic oscillator bath modes. The y terms without ii represent the loss of energy from the system to the reservoir, while yfi terms represent the diffusion of fluctuations in the reservoir into the system mode. In the limit as fi -+ 0 (i.e. the thermally induced effects are small), one can reduce Eq. (13) to the following form: jr
-(~+~)(s-40zso,) -
Structure (Theochem)
361 (1996) 49-56
53
as a relatively slow process remains unaffected. These interference terms which make their presence felt beyond the Markovian regime are a clear signature of non-Markovian characteristics of the dissipative process. In the next section, we show that these decay characteristics modify the band-shape of the transient susceptibility functions in the pump-probe experiment dealing with ultrafast optical processes.
(;+;GS)
3. Application: calculation of transient optical susceptibility
- z~‘y[a+cTU+0_s - sg+0_0+0_]
(19)
where f’=&?(w) Eq. (19) describes the non-Markovian dissipative dynamics of a two-level system in contact with a harmonic oscillator heat bath and perturbed by an external stochastic field. The Markovian and non-Markovian decay terms due to both of these interactions become more transparent if one calculates the evolution of population inversion and polarizations of the two-level system. Thus, the decay dynamics is as follows: (6+) = (2if’Ws + if’y)(ff+) - (Ws + WsGs +;+;Gs)(g+, (6-) = -(2if’Ws - (Ws+ (&A = -YClP
The decay of quantum coherence or dephasing takes place on a time scale where the nonMarkovian approximation generally used in the theoretical treatment of relaxation processes is invalidated. It has also been argued that nonMarkovian features might be probed through the transient optical absorption rather than any steady state measurement. With this end in view, we now calculate transient optical susceptibility which takes into account the non-Markovian features described in Eq. (20). We consider a model three-level system initially in the ground state. The pump pulse, which is ideally a S-pulse, selectively excites the system from the ground state to an excited state 1 in the presence of a continuous probe wave which monitors the probe absorption, resulting in transition of the system from state 1 to state 2 (see Fig. 1). Levels 1 and 2 thus effectively comprise a two-level system. Since the probe absorption is weak, we
s2
+ if’y)(a_)
Ep (t)
W,G,+;+;G,)(o_) + (a,))
1 (20)
It is immediately apparent that in addition to the usual independent damping terms corresponding to stochastic and system-reservoir interactions, there are terms like f’ W, and y/2Gs which illustrate the interference of these two interactions in the dephasing processes; the decay of population
t
6/t 1 t
4
Fig. 1. The pump pulse in a three-level scheme which is described by a &pulse excitation from the ground state g to 1. The probe field 4(t) induces transition from level 1 to 2.
G. Gangopadhyay, D.S. Ray/Journal of Molecular Structure (Theochem)
54
assume that pll >> p22, i.e. the perturbation approximation can be conveniently used. Thus, the effective two-level system, which is in contact with the heat bath and whose intermediate states 1 and 2 are perturbed by stochastic interaction and interaction with the probe wave, can be described by the following set of Bloch equations in the rotating wave approximation: (a,) = i(wO+ 2f’ Ws +f’y)(a+)
- islo e’Wf(aZ)
- (Ws + WsGs +;+;Gs)(d (b_) = -i(q
is further defined as
~~b(Q-4= P21 (th2
P(t) =
P21 =
b+)
PI2
P(t) =
bJ+)P12
L-l
-
- (a_) e’“‘) - y( l/2 + (a,))
Here, the weak probe field Rabi frequency a0 is the expressed through relation cl(t) = (Ro/2)(e-“’ + eiWf); w,, refers to the frequency of the effective two-level system. The damping processes in Eq. (21) have been incorporated according to the prescription of the last section. Making use of the following slowly varying envelope approximation and the abbreviations (c)
= S_ eP
(a,) = SZ
A=w,-w
+
h-)P21
s+(0) [P -
iRoLml
(22)
we can rewrite Eq. (21) as 3, = ifs, S_ = -ifs_
- is2& - ~$3, + iR$,
[ p - (if - 4
(ff+(t))
= eiW’L-’ [p _si;‘yya)]
x L-l
3, [ P - (if - x)
’ x(t - t’)E(t’)dt’
s0
- 2ip21EOeiw’
1
Here, the Laplace transform of SZ(t) is defined by S, = Jo” e -P’SZ(t) dt G ,9,(r). The response function x(w) in the frequency domain is then expressed as
3, [ P - (if - 7,)
1
Identifying the imaginary part of X(W) as the required transient optical susceptibility, we obtain after explicit evaluation of the inverse Laplace transform in Eq. (26) I
x”(w) = $p2112Re
d~e[if--Yu]'(uz(t 0
(24)
where E(t) represents the external probe field. P(t)
-
T))} (27)
(23)
To calculate the susceptibility, the linear response term in the expression for polarization is considered as follows: P(t) =
(25)
I
and
- y&
3, = -iRo(S+ - s-) - y(1/2 + S,)
1
3,
x(w) = -2i 1p2112C1
Y Y WsGs+-+-&=~a 2 2
of S,, (a,) can be
(if - “ia)
A+2f’Ws+f’y=f W,+
(0-j
or
S+(t) =
(21)
(o+) = S+ eiWt
=
Using the Laplace transform written as
_ (W, + WsGs +;+;Gs)(o-) eP
+ P12(t)P21
where p(t) and p denote the density matrix and the dipole moment operator respectively. In our notation, we express
+ 2f ‘W, +f ‘y)(a+) - iRo e -‘w’(a,)
(bz) = -iQ,((c+)
361 (1996) 49-56
Since probe absorption is sufficiently weak, we may write (uz(t - 7-)) M -l/2. Eq. (27) then reduces to the form
x”(w) =
&1
Incorporating
p21
1’Re the
{ eCi$“‘,lr
1)
approximate
(28) frequency
G. Gangopadhyay. D.S. Ray/Journal of Molecular Structure (Theochem)
dependence of y as
and for a time that is long in the relevant time scale, we obtain
(1
+Gs)
H’s+++?)
X
(A+2FWs+,y,,)2+(l+Gs)2
361 (1996) 49-56
55
shown that the two dissipative processes interfere in the non-Markovian description, This interference is manifested in the expression for transient optical response functions in ultrafast optical pump-probe spectroscopy. Although the treatment that we have presented here is valid for a correlation time which is short but finite, we hope that an understanding of the ultrafast processes in pumpprobe spectroscopy particularly in the non-linear regime, in general, will be amenable in terms of such a generalized master equation approach.
I (29)
Here, ~rl~wl,-,, = F and A = w- wo, and we have omitted the terms containing higher powers of F. x”(w) exhibits a lineshape which is different from the usual Lorentzian structure. It is also important to note than the influence of both the dissipative processes and the non-Markovian nature of its dynamics for short but finite correlation time has become prominent in the shape and width of the lineshape function. Also, because of the F-terms, the resonance in the transient absorption is shifted. We note in passing that the asymmetry in the lineshape function is a typical feature well-known in experimental ultrafast pump-probe spectroscopy.
4. Summary and conclusions Based on the system-reservoir model, in which the intermediate states of the system are perturbed by the external stochastic interactions with the surroundings, we have developed a generalized non-Markovian master equation for the description of the quantum dissipative processes. The dissipation concerns two decay processes, one of which is the system-reservoir interaction whereas the other is the stochastic interaction. While the former description depicting the usual relaxation processes is completely dynamical in nature, the latter in the absence of any knowledge of correlation functions derived from dynamics, the interaction is phenomenological in character and is responsible for dephasing processes. We have
Acknowledgements One of us (DSR) would like to thank the Department of Science and Technology, India, for financial support under the “laser facility scheme”.
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56
G. Gangopadhyay, D.S. Ray/Journal of Molecular Strucrure (Theochem)
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(1991) 4660; 44 (1991) 4671. [15] P.C. Brecker, H.L. Fragnito, J.Y. Bigot, C.H. Britocruz and C.V. Shank, Phys. Rev. Lett., 63 (1990) 505. W. Vogel, D.G. Welsch and W. Wilhelmi, Phys. Rev. A, 37 (1988) 3825.
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E.T.J. Nibbering, K. Duppen and D.A. Wiersma, J. Chem. Phys., 93 (1990) 5477. [16] S. Mukamel, Phys. Rep., 93 (1982) 1; A.A. Villaeys, J.C. Vallet and S.H. Lin, Phys. Rev. A, 43 (1991) 5030.