Quantum Master Equations for Non-linear Optical Response of Molecular Systems

0.18 MB·English

by  Tomas Mancal

Quantum master equations for non-linear optical response of molecular systems
Tomáš Mančal and František Šanda
Faculty of Mathematics and Physics, Charles University in Prague,
Ke Karlovu 5, CZ-121 16 Prague 2, Czech Republic
Generalized master equationsvalid for thethird orderresponse of an optically driven multi-level
electronic system are derived within Zwanzig projection formalism. Each of three time intervals of
the response function is found to require specific master equation and projection operator. Exact
cumulant response functions for the harmonic profiles of the potential energy surfaces leading to
Gaussian spectral diffusion are reproduced. The proposed method accounts for the nonequilibrium
stateofbathattheborderofintervals,andcanbeusedtoimprovecalculationsofultrafastnon-linear
spectra of energy transferring systems.
2
1
0 I. INTRODUCTION An explicite inclusion of some important environmental
2 modesinto the Hamiltonianalsoincreasesthe sizeofthe

n system beyond feasibility. Practical calculations thus re-
Non-linear response theory forms the theoretical ba-
a quire some type of reduced dynamics where only elec-
J sis for description of many important physical phenom-
tronic DOF are treated explicitly. A convenient form
enaandexperimentaltechniques,inparticularthemulti-
7 of the reduced dynamics, master equations, is realized
2 dimensionaltechniquesofnon-linearopticalspectroscopy
by deriving kinetic equations for expectation values of
[1]. Inrecentyears,two-dimensional(2D)coherentspec-
relevant quantities averaged over the DOF of the bath
] troscopy[2],developedfirstinNMR[3],hasbeenbrought
h [15,16]. Ifthesequantities(suchastransitiondipolemo-
into the near infra-red and optical [4, 5] domains. The
p ment)donotdependonthebathDOF,allrelevantinfor-
- 2D Fourier transformed spectrum completely character- mation is carried by the reduced density matrix (RDM).
t izes the third order non-linear response of a molecular
n
a ensemble in amplitude and phase [2] providing thus, the It was demonstrated that RDM master equations de-
u maximal information accessible in a three pulse optical rivedbyprojectionoperatortechniquereproducethelin-
q experiment. Sinceitsfirstexperimentalrealization,ithas ear response exactly [17]. For higher order response,
[ yieldednewinsightsintophoto-induceddynamicsofelec- however, the same approach neglects bath correlation

2 tronic excited states of small molecules [6], polymers [7], between different time intervals of the molecular photo-
v large photosynthetic aggregates [8] and even solid state induced evolution separated by interaction with laser
3 systems[9]. Forinstance,longlivingelectroniccoherence pulses (see e.g. Section 3 in Ref. [18]). In other words,
0 has been detected in energy transfer dynamics of photo- standardRDMmasterequationsdonotproperlyaccount
8 synthetic proteins [10], a find that stimulated a renewed forthe non-equilibriumstate ofthe phononbathpresent
3
interest in the properties of energy transfer in biological after the second and the third laser pulse. This neglect
.
1 systems and its arguably quantum nature. canleadtoacompletelossoftheexperimentallyobserved
1 dynamics in simulated 2D spectra, such as in the case of
0 Most 2D experiments are well described by the third
the vibrational line shape modulation of a single elec-
1 order perturbation semiclassical light-matter interaction
tronic transition [19]. This modulation interplays with
: response function theory. Simulations must include de-
v simultaneous effect of electronic coherence. Accounting
phasing of electronic coherence and electronic energy
i reliablyforbathcorrelationsamongintervalsistherefore
X transfer between levels which are both induced by inter-
of utmost importance for interpretation of the role of
r action of the electronic degrees of freedom (DOF) with
a quantum coherence in natural light-harvesting. The dif-
environment (bath). Response functions of model few
ference between the 2D lineshape calculated with exact
level systems with pure dephasing (no energy transfer
formula, and by RDM, i. e. neglecting the correlations,
betweenthelevels)canbecalculatedusingsecondcumu-
is demonstrated in Fig. 1. We employed the definitions
lant in Magnus expansion [1], and expressed in terms of
2D spectra from Ref. [19] and the response theory from
the energy gap correlation function (EGCF), C(t). For
Ref. [1].
Gaussian stochastic bath, this result is exact, and the
EGCF of the electronic transitions thus determines fully In this Letter, we show that the exact third order re-
boththe linearandthe non-linear(transient)absorption sponse function of a multilevel molecular system with
spectra. Forenergytransferringsystems,suchasFrenkel pure dephasing can be calculated by certain generaliza-
excitons in photosynthetic complexes [11], the exact re- tion of RDM master equations. We introduce special
sponsefunctions cannotbe constructedbysecondcumu- parametric projection operators that enable us to derive
lant. Photosynthetic complexes are relatively large, and distinct equation of motion for each time interval of the
the proper methods to simulate finite timescale stochas- response function. These parametric projectors offer a
tic fluctuations at finite temperatures [12] carry a sub- systematic approach to improvement of current master
stantial numerical cost. Stochastic simulations can thus equation method by incorporating bath correlation ef-
be readily implemented only for small systems [13, 14]. fectsintothecalculationofnon-linearresponseinenergy
2
numerical calculations. Resonance coupling
A B
M
Hres = X Jij|¯iih¯j| (2)
i6=j=1
3
ω
betweentheexcitedstatescauseselectronicenergytrans-
ferbetweenmolecules. DiagonalizationofaverageHamil-
correlated no correlation tonian hHi is required to identify the positions of peaks
in absorption spectrum, which correspond to transition
ω ω
frequencies ω = (ǫ −ǫ )/~ between electronic eigen-
1 1 ig i g
states |ii with eigenenergies ǫ , i = 1,...,M and the
i
groundstate. Intheverysamespirit,2Dspectraarecon-
FIG. 1: Two dimensional spectra of a two-level electronic
venientlyinterpretedintermsofenergytransferbetween
system interacting with bath of harmonic oscillators at pop-
theseenergeticeigenstates. Whenresonancecouplingbe-
ulation time T = 0, left panel: calculated by exact cu-
tween molecules in the aggregate vanishes J → 0 the
mulant Eq. (8), right: by applying RDM for optical co- ij
eigenstates |ii coincide with excited states |¯ii, and so do
herences (Eq. (25)). Frequency-frequency correlation spec-
the eigenenergies ǫ and energies ε .
trum at T = 0 defined in Ref. [19] is shown for line- i i
broadening function of overdamped Brownian oscillator [1]
g(t)=(λk Tτ2 ~−1−iλτ )[e−t/τcorr −1+t/τ ] where
B corr corr corr
λ=100cm−1,τ =100fs,andthetemperatureisT =300 III. LINEAR RESPONSE
corr
K.
For harmonic PES and J → 0 the calculation of re-
ij
sponse functions of all orders can be carried out exactly
transfer systems. by second order cumulant. The linear response function
I(t)=X|di|2ρig(t)+c.c. (3)
i
II. MODEL HAMILTONIAN
is generated by time evolution of coherence elements of
the RDM ρig(t)=e−iωigt−gii(t), which can be differenti-
We model a molecular aggregate (e.g. photosynthetic ated into the convolutionless master equation [17, 20]
antenna) as an assembly from N electronic two-level
molecules coupled by resonance interaction. Third order ∂
ρ =[−iω −g˙ (t)]ρ (t). (4)
opticalexperimentsonsuchsystemsaddressonlythecol- ∂t ig ig ii ig
lective electronic groundstate |gi and the states |¯ii with
InEq. (3),d arethe transitiondipolemomentsbetween
one or two excited molecules of the aggregate. Thus the i
eigenstate |ii and ground state |gi, the line broadening
N-molecule aggregatehas M =N +(N −1)×N/2 rele-
function
vant excited states. The free evolution of the system be-
tween the times of interaction with external light pulses 1 t t′
is described by Frenkel exciton Hamiltonian g (t)= dt′ dt′′C (t′′) (5)
ij ~2 ˆ ˆ ij
0 0
M is related to the EGCF,
Hˆ =Zˆ+Vˆg+εg|gihg|+X[εi+∆Φˆi]|¯iih¯i|+Hres. (1)
i=1 C (t)=tr {e(it/~)HˆB∆Φˆ e−(it/~)HˆB∆Φˆ Wˆ }, (6)
ij B i j eq
Here, Zˆ represents kinetic energy of the bath DOF, εg in the ground state equilibrium, and g˙ij ≡ (∂gij/∂t).
and ε are the electronic energies of the ground- and the Further on in this work, we assume that the electronic
exciteidstates,respectively,∆Φˆi ≡Φˆi−Φˆg−hΦˆi−Φˆgiis energy gap operators ∆Φˆi are nonzero.
theenergygapoperator,andΦˆ andΦˆ arepotential en-
g i
ergysurfaces(PES)intheelectronicground-andexcited
IV. MASTER EQUATIONS FOR NON-LINEAR
states, respectively. Harmonic PES represents Gaussian
RESPONSE
fluctuations of excitons. The energy gap operator is de-
fined so that it vanishes h∆Φˆ i ≡ tr {∆Φˆ Wˆ } = 0 in
i B i eq
the ground state equilibrium Wˆeq =e−kB1THˆB/z0. Here, byItnhnreoens-lhinoertarlaesxeprepruimlseesnts,epmaorlaetceudlabrystyismteeminitserpvraolbseτd,
the ground state bath Hamiltonian reads HˆB =Zˆ+Φˆg. T,andits responseis monitoredafter time t. Non-linear
The operators ∆Φˆ describe a macroscopic number of responsefunction is conveniently dissectedinto Liouville
i
DOF so that one cannot easily treat them explicitely in space pathways [1], each representing one combination
3
differentiation of R(ji)(0,T,τ)
2
t
|ji hg| P(i) Wˆ (T +τ)=ρ (T +τ) ∂ ρ (T;τ)=[−i(ω −ω )+K(ji)(T,τ)]ρ (T;τ),
T+τ jg jg ∂T ji jg ig 2 ji
T ×Uj†(T)Ug†(τ)WˆeqUi(τ +T)βT(j+i)τ (11)
|ji hi| P Wˆ (τ) with relaxation tensor
τ ji
|gi hi| τ =ρji(τ)Ug†(τ)WˆeqUi(τ)βτ(i) K2(ji)(T,τ)=g˙ij(T)−g˙i∗i(T +τ)
P W (0)=ρ (0)Wˆ
|gi hg| 0 gi gi eq
−g˙ (T)+g˙∗(T +τ). (12)
jj ij
FIG.2: DoublesidedFeynmandiagramoftheLiouvillepath-
way R2ji for a system with three levels |gi, |ii and |ji. Inter- andinitialconditionρjg(0;τ)=R2(ji)(0,0,τ). Inthefirst
actionswithexternalfieldaredenotedbystraightarrows,the interval we find an analogicalmaster equation for ρgi(τ)
wavy arrow corresponds to emitted field. The projectors P , which is (up to a complex conjugation) identical to Eq.
0
P and P(i) are shown explicitly with point of their appli- (4). Wehavethusdemonstratedthatevolutionequations
τ T+τ
cation in theresponse function. canbeconstructedwiththestructureoftheconvolution-
less master equation, and that they yield exact third or-
der response function. Their forms, however, have now
of indices of RDM during the periods τ, T, and t. For tobe specifictoeachintervalandtoeachLiouvillespace
ourdiscussionweselectapathwayusuallydenotedasR pathway.
2
(see Fig. 2 for the corresponding double sided Feynman
diagram), i.e. the pathway which is responsible for pho-
ton echo signal, and which contains excited state evolu- V. CONSTRUCTING MASTER EQUATIONS
tionduringthe secondinterval(waitingtime)T [1]. The BY PARAMETRIC PROJECTORS
R(ji) response function involving excited states |ii and
2
|ji reads In Ref. [17] it was pointed out that Eq. (4) agrees
withconvolutionless master equation (CLME)ofHashit-
R(ji)(t,T,τ)=|d |2|d |2 sume et al. [22], when it is evaluated to second or-
2 i j der in ∆Φˆ. Similar microscopic derivation of Eqs. (9)
to (12) by using projection techniques would open a
×tr {U (t)U (T)U (τ)W }. (7) way to constructing improved master equations for the
B jg ji gi eq
electronic energy transfer in the cases where no exact
Here, Uji(t) ≡ Ujiji(t) are abbreviated matrix elements cumulant solution is known. The Zwanzig projection
of the evolution superoperator U(t) which is the Green’s technique requires to define projection superoperator by
functionsolutionoftheLiouville-vonNeumannequation its action on a arbitrary operator Aˆ. In general form
with Hamiltonian, Eq. (1). Between each two consecu- PAˆ= Wˆ tr {ha|Aˆ|bi}, where the normalized bath
Pab ab B
tive Green’s functions in the trace of Eq. (7), the action matrix (tr {Wˆ }=1) canbe chosento achievespecific
B ab
of the external light causes transition between electronic
goals. Masterequationforthelinearresponse,Eq. (4),is
levels |gi, |ii and |ji. For a harmonic PES the second homogenousanditcorrespondstothechoiceWˆ ≡Wˆ ,
ab eq
order cumulant is exact, and it yields [21]
when QWˆ(0) = 0 (Q ≡ 1−P), and thus the inhomo-
geneouspartofthe CLME vanishes [17]for uncorrelated
R(ji)(t,T,τ) = |d |2|d |2e−i(ωjgt+(ωjg−ωig)T−ωigτ)
2 i j initial state Wˆ(0) = ρ(0)W . In the second and third
×e−gi∗i(τ+T)−gjj(T+t)+gi∗j(t+T+τ) interval the choice Wˆ = Weˆq will not yield Eqs. (9)
ab eq
×e−gi∗j(t)−gi∗j(τ)+gij(T). (8) and (11), since the inhomogeneous part of the CLME
does not vanish. In fact, the inhomogeneous terms can-
BydifferentiatingEq. (8)withrespecttotwefindthe notbeeliminatedinallthreeintervalssimultaneouslyby
response functions to be solutions of evolution (master) anysingleprojector. Thisisamicroscopiccounterpartof
equations the fact that non-linearopticalresponse function cannot
be exactly evaluated by simulating the reduced dynam-
∂
ρ (t;T,τ)=[−iω +K(ji)(t,T,τ)]ρ (t;T,τ) (9) ics of the system by a single master equation. Indeed,
∂t jg jg 3 jg the homogeneouspart of the CLME correspondsto only
oneofthefour(nonzero)contributionsofthethirdorder
with the relaxation tensor
response function
K(ji)(t,T,τ)=−g˙∗(t)+g˙∗(t+T+τ)−g˙ (t+T). (10)
3 ij ij jj R(ji)(t,T,τ)=tr {U (t)
2 B jg
(ji)
and initial condition ρ (0;T,τ)=R (0,T,τ). Master
jg 2
equationcanbesimilarlyfoundinthesecondintervalby ×(P +Q)U (T)(P+Q)U (τ)PWˆ }. (13)
ji gi eq
4
Themasterequations(4)to(12)canbederivedwhenwe function, Eq. (7), in terms of the reduced dynamics of
allowdifferent projectionoperatorto be defined for each the system. We can write
timeintervaloftheresponsefunction. Lettheprojection
operatorsP0,Pτ andPT(i+)τ inthe respectiveintervalsbe R2(ji)(t,T,τ)=|dj|2|di|2
chosensothattheyeliminatetheinhomogeneoustermsin
the CLME master equation, i.e. so that (1−P0)Wˆ(0)= ×U¯j(gi)(t;T,τ)U¯ji(T;τ)U¯gi(τ)ρ0, (17)
0;(1−P )Wˆ(τ)=0,etc(seeFig. 2). Thecorresponding
τ
where the three reduced evolution superoperators de-
projectorsreadas follows. The first coherence projection
noted by U¯ are solutions of three different master equa-
superoperator
tionsfollowingfromtheprojectors,Eqs. (14)to(16). To
P0Wˆ(t)=WˆeqtrB{Wˆ(t)}=Wˆeqρˆ(t), (14) simplifythematters,wewillconcentrateontheresponse
function where i = j, i.e. on R(ii). Since we deal with
is the widely usedArgyres-Kellyprojector[23]. The sec- 2
a multi-level electronic system with adiabatically sepa-
ondintervalcanbetreatedwiththepopulationprojection
rated states (J = 0), we can assume that U¯ (T;τ)= 1
superoperator ij ii
and thus
PτWˆ(t)=Xρkl(t)Ugl(τ)Wˆeqβτ(l)|kihl|, (15) R(ii)(t,T,τ)=|d |4U¯(i)(t;T,τ)U¯ (τ)ρ . (18)
kl 2 i ig gi 0
which yields a single master equation for all RDM ele- Note that all T-dependence in Eq. (18) is due to the
ments in the second interval. Projector, Eq. (15), to be second coherence parametric projector.
applied to a single RDM element as in Eq. (11), reads Nowweconstructthemasterequationsforthefirstand
P(ji)Wˆ (t) = ρˆ (t)U (τ)Wˆ β(i). In the third interval secondcoherenceintervals(i.e. inintervalscharacterized
τ ji ji gi eq τ
bytimeτ andt,respectively),anddemonstratethatthey
oftheresponseasetofsecondcoherenceprojectionsuper-
operators reflecting different bras (hi|) during the second lead to correct response function. First, we split the to-
tal Hamiltonian into a sum of the pure bath (denoted
interval have to be defined as
by B), pure electronic (S) and system–bath interaction
PT(i+)τWˆ(t)=Xρkg(t)Uki(T)Ugi(τ) (S-B)parts. BathHamiltonianhasalreadybeendefined
k above,the electronic Hamiltonian reads Hˆ =ǫ |gihg|+
S g
ǫ |iihi|,andthe system–bathcouplingHamiltonianis
Pi i
×WˆeqβT(k+i)τ|kihg|. (16) HˆS−B =~Pi∆Φˆ|iihi|. We define the Liouville superop-
eratorL using the corresponding Hamilton operator
The factorsβτ(k) andβT(k+i)τ arepresenttoensurethatthe as LS−BS−=B~−1[HˆS−B,...]− and switch to the interac-
superoperators have the projection property (Pτ)2 =Pτ tion picture induced by Hˆ = Hˆ +Hˆ . In the second
0 B S
and (P(i) )2 = P(i) . From these conditions it follows orderinL ,andassumingthatQ Wˆ(0)=0forx=0,
τ+T τ+T S−B x
e.g. β(ii) = β(i) = 1/tr {U (τ)Wˆ }. The factor β(i) and x = T +τ we obtain the following quantum master
τ+T τ B gi eq τ
evaluates in the second cumulant to βτ(i) =egi∗i(τ). equation
An alternative to this formalism would be to account
directly for the Q terms in Eq. (13), and to use the ∂ P W(I)(t)=−iP L (t)P W(I)(t)
∂t x x x S−B x x
timeconvolutionlessoperatoridentity[22]forthederiva-
t
tion of the non-linear response as in Ref. [26]. The two − dt′P L (t)Q L (t′)P W(I)(t). (19)
x S−B x S−B x x
approaches lead to the same exact results for a system ´0
of non-interacting molecules. For a coupled system, the
Thetime-dependenceoftheLiouvillianandtheupperin-
qualityoftheequationsofmotionderivedbythetwoap-
dex(I)ofthedensityoperatordenoteinteractionpicture
proaches has to be compared in a separate study. How- with respect to the Hamiltonian Hˆ =Hˆ +Hˆ .
ever, the central idea of the projector approach, i.e. its 0 B S
In the first coherence interval we use the projection
choice in a form which satisfied (1−P)Wˆ = 0, is gen- superoperator P . The construction of the ∆Φˆ oper-
eral, and it does not limit the present method to the ator ensures tha0t tr {∆Φˆ Wˆ } = 0, and so thie first
three projectors suggested above. Depending on physi- B i eq
term in Eq. (19) contributes with zero. The second
cal situation, we expect that specialized projectors can
termis onlynon-zerowhenno Hˆ occursonthe right
be introduced for interacting systems, possibly based on S−B
hand side of the density operator, and thus only the
the comparison with the results of Ref. [26].
term P Hˆ (t)Hˆ (t′)P Wˆ(I)(t) contributes. The
0 S−B S−B 0
evaluation leads to Eq. (4) and consequently yields
VI. RECOVERING THE CUMULANT U¯gi(τ) = e−gi∗i(τ)+iωigτ. A straightforward application
EXPRESSIONS of the second coherence projection superoperator P(i)
T+τ
leads to
Letusnowdemonstratehowtheseprojectionoperators ∂
can be used for calculation of the non-linear response ∂tρ(igI)(t)=−IT+τ(t)ρ(igI)(t)−MT+τ(t)ρ(igI)(t), (20)
5
where usedtoderivecorrespondingmasterequations. Thecom-
plete responsefunction involvesa single masterequation
IT+τ(t)=itrB{∆Φˆi(t)Uii(T)Ugi(τ)Wˆeq}egi∗i(τ), (21) for the first coherenceinterval and two master equations
forthe waitinginterval. Inthe secondcoherenceinterval
and
of the response function there are N master equations
in each Liouville pathways, where N is the number of
t
M (t)= dt′tr {∆Φˆ(t)∆Φˆ(t′)U (T) singleexcitonstates. Whenenergytransferbetweensys-
T+τ B ii
´0 tem’s eigenstates is induced by non-zero J couplings,
ij
t the application of the parametric projectors represents
×Ugi(τ)Wˆeq}egi∗i(τ)− dt′IT+τ(t)IT+τ(t′). (22) an approximation whose quality depends on the previ-
´0
ous energy transfer history of the system. Nevertheless,
The coefficients I and M, Eqs. (21) and (22), can be unliketheusualmasterequationsbasedonstandardpro-
recastintothesecondcumulantbyatechniquedescribed jectors,the parametricprojectorsleadto correctlimit in
in Ref. [24]. A tedious but straightforward calculation Jij →0. The prescription(1−P)Wˆ(t)=0 for construc-
gives tion of the parametric projection operators is general,
and can be used to construct a proper projector in the
I (t)=g˙ (t+T)−g˙ (t)−g˙∗(t+T+τ)+g˙∗(t) (23) cases where P would fail.
T+τ ii ii ii ii τ+T
VII. CONCLUSIONS
and
M (t)=g˙ (t). (24)
T+τ ii In this Letter, we have outlined a general method for
calculation of non-linear response functions by quantum
Combining the results of the second order cumulant ex-
master equations. We have tailored parametric projec-
pansion of Eqs. (21) and (22) and switching back from
tion superoperators to derive master equations for each
the interaction picture we arrive at the master equation
time interval of the response function, and we demon-
Eq. (10). The solution of the master equation can be
stratedthe validityofthemethodbyreproducingawell-
easily obtained in form of the time evolution superoper-
know result exact for a multi-level system. The results
atorU¯ (t;T,τ). Finally, insertingU¯ (τ) andU¯ (t;T,τ)
ig gi ig presented here can be extended to a master equation
into Eq. (18) we obtain Eq. (8) which is the desired re-
method for calculation of a general multi-point correla-
sult. Calculationcanbe repeatedfor anyLiouville space
tion functions. This opens a new research avenue with
pathway reproducing the results of direct cumulant ex-
consequences in the theory of open systems, non-linear
pansion.
spectroscopy and potentially other branches of physics.
Note that standardchoice of ground state equilibrium
for projector would yield response function
R(ii)(t,T,τ) = |d |2|d |2e−i(ωigt−ωigτ)
2 i j
×e−gi∗i(τ)−gii(t) (25)
Comparison between 2D lineshapes simulated using Eq. Acknowledgments
(8) and Eq. (25) is made in Figure 1.
The projector technique developed above can be ex-
tended to finite resonance coupling by straightforward This work was supported by the Czech Science Foun-
abandoningoftheJ →0limit. We willshowelsewhere dation (GACR) grant 205/10/0898 and by the Min-
ij
[25]thatprovidedtheopticalcoherencesinthefirstinter- istry of Education, Youth, and Sports of the Czech
valcanbeassumedindependentofeachother(secularap- Republic through grant ME899 and the research plan
proximation), the projectors, Eqs. (14) and (15) can be MSM0021620835.
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