Correlated driving and dissipation in two-dimensional spectroscopyJian Xu, Hou-Dao Zhang, Rui-Xue Xu, and YiJing Yan Citation: J. Chem. Phys. 138, 024106 (2013); doi: 10.1063/1.4773472 View online: http://dx.doi.org/10.1063/1.4773472 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v138/i2 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

THE JOURNAL OF CHEMICAL PHYSICS 138, 024106 (2013)

Correlated driving and dissipation in two-dimensional spectroscopyJian Xu,1,a) Hou-Dao Zhang,1 Rui-Xue Xu,2,b) and YiJing Yan1,2

1Department of Chemistry, Hong Kong University of Science and Technology, Kowloon, Hong Kong2Hefei National Laboratory for Physical Sciences at the Microscale, University of Science and Technologyof China, Hefei, Anhui 230026, China

(Received 11 October 2012; accepted 13 December 2012; published online 10 January 2013)

The correlation between coherent driving and non-Markovian dissipation plays a vital role in opticalprocesses. To exhibit its effect on the simulation of optical spectroscopy, we explore the correlateddriving-dissipation equation (CODDE) [R. X. Xu and Y. J. Yan, J. Chem. Phys. 116, 9196 (2002)],which modifies the conventional Redfield theory with the inclusion of correlated driving-dissipationeffect at the second-order system–bath coupling level. With an exciton model mimicking the Fenna–Matthews–Olson pigment-protein complex, we compare between the Redfield theory, CODDE, andexact hierarchical dynamics, for their results on linear absorption and coherent two-dimensionalspectroscopy. We clarify that the failure of Redfield approach originates mainly from the neglect ofdriving–dissipation correlation, rather than its second-order nature. We further propose a dynami-cal inhomogeneity parameter to quantify the applicable range of CODDE. Our results indicate thatCODDE is an efficient and quantifiable theory for many light-harvesting complexes of interest. Tofacilitate the evaluation of multi-dimensional spectroscopy, we also develop the mixed Heisenberg–Schrödinger picture scheme that is valid for any dynamics implementation on nonlinear responsefunctions. © 2013 American Institute of Physics. [http://dx.doi.org/10.1063/1.4773472]

I. INTRODUCTION

Long-lived quantum coherence in photosynthetic sys-tems probed by ultrafast two-dimensional (2D) electronicspectroscopy has attracted much attentions recently, both inexperiment and theory.1–7 Many studies suggest that quan-tum coherence may contribute to promoting the efficiency ofexcitation energy transfer processes,1–11 by which the cap-tured solar energy is transferred through the antenna sys-tems to the reaction center. The detailed understanding forsuch complex systems relies on a close interplay between the-ory and experiments, especially the simulation on 2D spec-troscopic signals.12–14 The hierarchical equations of motion(HEOM) formalism15–20 has been recently applied to thesimulation of the 2D spectra of the Fenna–Matthews–Olson(FMO) pigment-protein complexes,21–25 assuming a simpleelectronic exciton model.1, 26, 27 Since the HEOM theory isexact, the discrepancy between simulation and experimentwould be due to the simplification of model, which neglectssuch as the vibration nature of the FMO molecular system.Approximate but reliable methods are called for to simulatecomplex systems, especially their 2D spectra, with realis-tic molecular models that are often too expensive for exactHEOM evaluations. The existing approximate evaluations aremainly based on second-order formulations. The most oftenused one is the Redfield theory. For the aforementioned FMOmodel system, the Redfield theory results in a largely under-estimated quantum coherence time.27 It would suggest thatsecond-order treatments be inadequate for the case of mod-erate system–bath coupling strength with moderate memory.

a)xujian@ust.hk.b)rxxu@ustc.edu.cn.

However, the conventional Redfield theory is an incompletesecond-order theory, which neglects the correlated drivingand dissipation dynamics.

In this work, we will show that the correlated driving-dissipation equation (CODDE)28–30 would be the choiceof second-order theory for the case of moderate system–bath coupling strength and non-Markovianicity. In particular,CODDE would be a powerful theoretical tool in the study ofphotosynthetic light-harvesting systems. To address this issue,we propose a dynamical inhomogeneity parameter to the mea-sure of the validity range of CODDE.

As complete second-order quantum master equation the-ories are concerned, CODDE is a variation of time-localformulation. The latter is characterized by a local time-dependent dissipation superoperator R(t) and often suf-fers numerical stability problem, due to the underlyingnonlinearity.28, 29 The construction of CODDE involves theseparation of R(t) into the field-free and field-dependentparts, R(t) ≡ Rs + Rsf(t). The field-free Rs is the sameas the conventional Redfield Markovian theory, while thefield-dependent part Rsf(t) is treated in a time-nonlocal (ormemory) resum scheme.28, 29 This is consistent with the factthat memory is physically clocked by the time-dependentexternal field action. CODDE resolves this memory effectinto a set of linear equations. As a result, CODDE is nu-merically efficient due to both the second-order treatmentand the linear construction. Note that the conventional time-nonlocal quantum master equation theory also assumes a lin-ear construction,28, 29 but leads often to artifact resonances inthe simulated spectrum.31

We will demonstrate the intrinsic advantages of CODDEformulation through its evaluations of the FMO model sys-tems. The resulting 2D spectrum, which is even comparable

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024106-2 Xu et al. J. Chem. Phys. 138, 024106 (2013)

to that from HEOM, shows a significantly longer quantumcoherence than the Redfield theory evaluation. This fact an-nounces that CODDE is a theory for the case of moder-ate non-Markovian system–bath coupling. The linearity ofCODDE further supports the mixed Heisenberg–Schrödingerpicture scheme for efficient evaluation of nonlinear responsefunction. Considered in this work are both CODDE andHEOM evaluations of third-order optical response functions,in which the detection time t3-propagation is carried out inthe Heisenberg picture, while the excitation t1-propagationand waiting time t2-propagation remain in the Schrödingerpicture. Thus, the three-dimensional time-domain evolutionreduces to two-dimensional plus one-dimensional dynamics.The mixed Heisenberg–Schrödinger scheme resembles thedoorway-window picture of third-order nonlinear responsefunctions.12, 32 We also employ the block-matrix dynamics ofboth CODDE and HEOM to further facilitate the evaluationsof optical response functions.

The paper is organized as follows. We present theCODDE formalism, its deviation, and the HEOM formal-ism in Sec. II, Appendixes A and B, respectively, togetherwith the underlying generalized Liouville-space algebra. InSec. III, we discuss the mixed Heisenberg–Schrödinger pic-ture prescription of third-order optical response functions,in conjunction with the block-matrix dynamics in a generalLiouville space. Numerical demonstrations are given in Sec.IV on a model FMO system at both 77 K and 298 K. Both thelinear absorption and 2D electronic spectroscopy signals aresimulated, with comparisons between the CODDE, Redfieldapproximation, and HEOM evaluations, to clarify unambigu-ously the importance of the driving–dissipation correlation.Proposed is also a non-Markovianian bath influence measureto quantify the CODDE validity range. Finally, we summarizethe paper in Sec. V.

II. NON-MARKOVIAN QUANTUM DISSIPATIVEDYNAMICS: PERTURBATIVE VERSUSNON-PERTURBATIVE THEORIES

A. Prelude

Consider the reduced dynamics of a molecular system,driven by external fields and also subject to dissipation due tothe presence of bath environment. The reduced system den-sity operator is defined as ρ(t) ≡ TrBρtotal(t), i.e., the trace ofthe total density operator over the environment bath subspace.The total Hamiltonian, Htotal(t) = Hs + Hsf(t) + hB + HSB,comprises the reduced system, system–field coupling, bath,and system–bath interaction components. Let Ls · = [Hs, · ]and Lsf(t) · = [Hsf(t), · ]. The reduced system Liouvillian inthe presence of external classical field is L(t) = Ls + Lsf(t).Set hereafter ¯ = 1.

The fluctuating system–bath interaction, described interms of HSB(t) = eihBtHSBe−ihBt , has the general form ofHSB(t) = −∑

QmFm(t), with system operators {Qm} spec-ifying the dissipative modes. The stochastic bath operators{Fm(t)} are set to be of 〈Fm(t)〉B = 0 and assumed to fol-low the Gaussian statistics in thermal equilibrium bath en-sembles. For clarity, we treat explicitly the single dissipative

mode case of HSB(t) = −QF (t). The Gaussian bath influ-ence is fully characterized by the bath correlation functionC(t) ≡ 〈F (t)F (0)〉B. It is related to the bath spectral densityJ(ω) via the fluctuation-dissipation theorem.29, 33 For the con-structions of both CODDE and HEOM, we expand C(t) intoan exponential series, based on such as the Padé spectrumdecomposition schemes.34, 35 In this work, we focus on theDrude model of the interacting bath spectral density function

J (ω) = 2λγω

ω2 + γ 2. (1)

Here, λ and γ denote the reorganization energy and the de-phasing rate, respectively. In contact with the optimal HEOMconstruction for Drude dissipation,36–39 we adopt the [N/N]Padé spectrum decomposition scheme.35 It leads to the Drudebath correlation function the form of

C(t) ≈ C[N/N](t) =N∑

k=0

cke−γkt + 2�Nδ(t). (2)

The k = 0 term with γ 0 ≡ γ is the Drude pole contribu-tion. The other N contributions are from the [N/N] Bose func-tion poles, where {γ k > 0} are all positive and can be read-ily determined.35 For Drude dissipation the [N/N] schemealso leaves automatically a δ-function, which amounts tothe white-noise residue resum.40 It is the last term in Eq.(2), where �N = λγ/(2kBT )

(N+1)(2N+3) , with kB being the Boltzmannconstant and T the temperature.36–38 While the white-noiseresidue resum renders a convenient accuracy control crite-rion for an exact HEOM evaluation,36–39 we will see soonthat it plays no roles in CODDE, due to the underlying non-Markovian treatment. Note that bath correlation function C(t)in an exponential form as Eq. (2), with complex exponents ingeneral, dictates the CODDE and HEOM constructions; seeAppendixes A and B, respectively.

B. The CODDE approach

1. General comments

CODDE is a second-order quantum master equationformalism.28, 29 It differs from the conventional time-local for-malism by its memory resummation treatment of the field-dressed dissipation, while the field-free dissipation superop-erator Rs is intact. The latter is Markovian and can be writtenvia28, 29

RsO ≡ [Q, QO − OQ†], with Q ≡ C(−Ls)Q. (3)

Here, C(−Ls) ≡ [∫ ∞

0 dt eiωtC(t)]ω=−Ls , evaluated at thefield-free system Liouvillian Ls and with the true or non-approximated bath correlation function C(t). The evaluationof Q is, therefore, often made in the Hs-representation thatdiagonalizes the field-free system Hamiltonian.

In Appendix A, we present the derivation of the CODDEformalism,28, 29 including its integro-differential form inEq. (A10). With the aid of the exponential expansion of bathcorrelation function in Eq. (2), we arrive at the differential

024106-3 Xu et al. J. Chem. Phys. 138, 024106 (2013)

form of CODDE,

ρ = −[iL(t) + Rs]ρ −N∑

k=0

Q[ckρ(−)k − c∗

kρ(+)k ], (4a)

ρ(−)k = −[iL(t) + γk]ρ(−)

k − �sfk (t)ρ,

ρ(+)k = −[iL(t) + γ ∗

k ]ρ(+)k − ρ [�sf

k (t)]†,(4b)

with QO ≡ [Q, O] and

�sfk (t) ≡ iLsf(t)(iLs + γk)−1Q. (5)

Here the reduced system ρ(t) is not just dictated by the Red-field dissipation superoperator Rs, but also coupled with theauxiliary density operators (ADOs), {ρ(±)

k (t)}. Remarkably,these ADOs account for the non-Markovianicity from the cor-relation between driving and dissipation.28, 29 Involved there-fore are only the finite-{γ k} components of the bath cor-relation function. The δ-function (white-noise) residue termin Eq. (2) has no contribution to the non-Markovian ADOdynamics. Note also that ρ(t) is Hermitian, while ρ

(+)k (t)

= [ρ(−)k (t)]†. In Subsection II B 2, we will extend the underly-

ing CODDE propagator for non-Hermitian cases, as involvedin the evaluation of individual Liouville-space pathway con-tribution to the third-order optical response function.

The non-Markovianicity to the reduced system ρ(t) dy-namics [cf. Eq. (4a)] arises from the ADOs {ρ(±)

k (t)}. Ne-glecting them reduces to the Redfield theory, involving onlyfield-free dissipation superoperator Rs. In fact, the CODDEconstruction highlights the nature of the correlated driving-dissipation non-Markovianicity. This feature is evident byEq. (4b), in which ρ

(±)k (t) depends on ρ(t) through �sf

k (t)of Eq. (5). We elaborate the underlying correlated non-Markovianicity with the following observations: (i) Not justHsf(t) = 0 but also 1/γ k = 0 leads to �sf

k (t) = 0 [cf. Eq. (5)].The resultant independent evolution of ρ

(±)k (t) is also subject

to the decay rate of γ k. This decay parameter is just the spec-ified exponent of the bath correlation function C(t) in the ex-ponential series expansion of Eq. (2). (ii) As a result, the δ(t)component of Eq. (2), with an effective 1/γ N + 1 = 0, resultsin a zero ADO. It confirms that the Markovian componentof bath correlation function plays no role in the correlateddriving-dissipation non-Markovianicity.

Moreover, the natural initial conditions to Eq. (4), priorto the external field action, where Hsf(t0) = 0 and thus�sf

k (t0) = 0, with t0 → −∞, are the steady-state (ther-mal equilibrium) solutions to Eq. (4) themselves. We havetherefore ρ

(±)k (−∞) = 0 via Eq. (4b), followed by ρ(− ∞)

= ρeq(T) via Eq. (4a), with the thermal equilibrium reducedsystem density matrix being evaluated via (iLs + Rs)ρeq(T )= 0, together with normalization.41, 42 Non-Markovianicity(i.e., ρ

(±)k = 0), originated from finite memory components

of bath correlation function, emerges only upon the time-dependent external field takes action. When external fieldis over, �sf

k (t) = 0 and ρ(±)k (t) evolve independently, as de-

scribed in (i) above, with the overall decay rate of γ k, towardsthe steady-state of ρ

(±)k (t → ∞) = 0.

2. The CODDE approach to nonlinear response theory

For efficient evaluation of various Liouville-space path-way contributions to the third-order response function, wepropose the CODDE-space dynamics be implemented in amixed Heisenberg–Schrödinger interaction picture scheme.This is the main formulation development of this work andthe final results will be presented in Sec. III.

To proceed, we recall the linearity of the CODDE formal-ism [Eq. (4)], which reads in the matrix-vector form as ρ(t)= −i L(t)ρ(t), with the CODDE-space state operators beingarranged in a column vector, ρ(t) = {ρ(t); ρ(−)

k (t), ρ(+)k (t)},

and the CODDE dynamics generator L(t) a matrix be-ing specified with Eq. (4). It is additive, L(t) = Ls +Lsf(t), with respect to its field-free and field-dressed com-

ponents. The latter has the form of Lsf(t) = − Dεin(t),where εin(t) represents the classical incoming field, whileD is the CODDE-space analogue of the transition

dipole commutator of D (·) = [ μ, (·) ] and will be spec-ified later. Treating the field-dressed component as aperturbation, the interaction picture technique leads toδρ(t) ≡ ρ(t) − ρeq(T ), for example, the first-order expressionof

δρ(1)(t) = i

∫ t

−∞dτ

[Gs(t − τ ) Dρeq(T )

]εin(τ ). (6)

Here, Gs(t) ≡ exp(−i Lst) denotes the field-free CODDE-space propagator. The initial thermal equilibrium CODDE-space state is just ρeq(T ) ≡ {ρeq(T ); 0, 0}, as analyzed earlier.The third-order δρ(3)(t) can be obtained similarly. It involvesthree D actions, associating with three sequential incomingfields, εin(τ 1), εin(τ 2), and εin(τ 3), and the subsequent field-free propagations of Gs(τ2 − τ1), Gs(τ3−τ2), and Gs(t−τ3),in between and after.12, 43 The third-order nonlinear opticalresponse function can then be formulated accordingly. Incontact with the sequential pump-probe optical configura-tion, the above three field-free propagations are also referredto the excitation time (t1 ≡ τ 2 − τ 1), waiting time (t2 ≡τ 3 − τ 2), and detection time (t3 = t − τ 3) propagations,respectively.12, 32

We shall be interested in the CODDE dynamics eval-uation on various Liouville-space pathway contributions tothe third-order optical response function.12, 43 To that end,

we denote D ≡→D −

←D, in analogy to writing the transi-

tion dipole commutator as D ≡→D −

←D, with

→DO ≡ μO and

←DO ≡ Oμ that are used in specifying the Liouville-space

pathways.12, 43 The CODDE-space analogues of→D and

←D can

be specified readily. The final results, in terms of their actionson an arbitrary CODDE-space state, ρ ≡ {ρ; ρ

(−)k , ρ

(+)k }, are

obtained as30

→Dρ = {μρ; μρ

(−)k , μρ

(+)k } + {0; [μ, �k] ρ, 0 },

←Dρ = {ρμ; ρ

(−)k μ, ρ

(+)k μ} + {0; 0, ρ [�†

k, μ] }(7)

with [cf. Eq. (5)]

�k ≡ (iLs + γk)−1Q. (8)

024106-4 Xu et al. J. Chem. Phys. 138, 024106 (2013)

The resulting ρ(0) =→Dρ or

←Dρ serves as the initial door-

way state for the subsequent field-free propagation, ρ(t)= Gs(t)ρ(0) or ρ(t) = −i Lsρ(t) that reads in thematrix-vector form as

∂

∂t

⎡⎢⎢⎣ρ

ρ(−)k

ρ(+)k

⎤⎥⎥⎦ = −

⎡⎢⎢⎣iLs+Rs ckQ −c∗

kQ

0 iLs+γk 0

0 0 iLs+γ ∗k

⎤⎥⎥⎦⎡⎢⎢⎣

ρ

ρ(−)k

ρ(+)k

⎤⎥⎥⎦.

(9)

This is just Eq. (4) but with Lsf(t) = 0. The dummy sumover k as the last term in Eq. (4) is also implied here. Now,Eq. (9) propagates the CODDE-space state ρ(t) that is non-Hermitian: ρ = ρ† and/or ρ

(+)k = (ρ(−)

k )†. Apparently, Eq. (7)is also valid for a non-Hermitian ρ. In relation to the third-order optical response functions to be detailed in Sec. III,one starts with the thermal equilibrium state, ρ = ρeq(T ), and

evaluates the initial doorway state ρ(0) =→Dρ or

←Dρ via

Eq. (7), followed by the t1-propagation via Eq. (9). The re-sulting ρ(t1) is set to be the new ρ, by which ρ(t2 = 0; t1)

=→Dρ or

←Dρ serves as the new initial condition for the t2-

propagation. The t3-propagation can be repeated in the samemanner.

For the completeness of formalism and also for the lateruse, we identify the CODDE’s Heisenberg picture A(t), cor-responding to an arbitrary system dynamical variable A(t), asfollows. Start with the expectation value

A(t) = tr[Aρ(t)] ≡ 〈〈A|ρ(t)〉〉 ≡ 〈〈A|ρ(t)〉〉 . (10)

The last identity is the CODDE-space correspondence, whereA ≡ {A; A(−)

k , A(+)k } = {A; 0, 0} and

〈〈A|ρ〉〉 ≡ 〈〈A|ρ〉〉 +∑

k

{〈〈A(−)k |ρ(−)

k 〉〉 + 〈〈A(+)k |ρ(+)

k 〉〉}. (11)

Moreover, the last identity of Eq. (10) can be recast as A(t)= 〈〈A| Gs(t)|ρ(0)〉〉. The Heisenberg picture of the CODDEdynamics is then A(t) ≡ A(0) Gs(t), with A(0) = A = {A;0, 0}, as specified earlier. We have therefore A(t)= −i A(t) Ls, for the row vector of A(t) ≡ {A(t); A(−)

k (t),A

(+)k (t)}, associated with the same matrix i Ls in Eq. (9).

We immediately arrive at the CODDE-space dynamics in theHeisenberg picture44

˙A(t) = −A(t)(iLs + Rs), (12a)

˙A

(−)k (t) = −A

(−)k (t)(iLs + γk) − ckA(t)Q,

˙A

(+)k (t) = −A

(+)k (t)(iLs + γ ∗

k ) + c∗k A(t)Q.

(12b)

Here the superoperators take actions from the right sideon an arbitrary operator O. They can readily be iden-tified as ORs = [O, Q]Q − Q†[O, Q] that correspondsto Eq. (3), while OLs = [O,Hs] and OQ = [O, Q],respectively.

In Appendix B, we present the HEOM formalism in boththe Schrödinger and the Heisenberg pictures. Considered inSec. III is the mixed Heisenberg–Schrödinger interaction pic-ture for efficient evaluation of third-order optical response

functions. It is to have the Schrödinger dynamics of Gs(t)ρ(0)[Eq. (9) or Eq. (B1) with field-free Liouvillian] for the t1-and t2-propagations, while implement in parallel the Heisen-berg dynamics of A Gs(t3) [Eq. (12) or Eq. (B4)] for the t3-propagation.

III. EFFICIENT EVALUATION OF NONLINEARRESPONSE FUNCTIONS

In this section we elucidate an efficient evaluation ofthird-order optical response function, accomplished by us-ing the mixed Heisenberg–Schrödinger scheme, together withthe block-matrix dynamics implementation. The proposedmethod is rather general, applicable to generalized Liouville-space dynamics, such as CODDE and HEOM considered inthis work.

A. The CODDE in block-matrix dynamics

Let us start with the block-matrix dynamics implemen-tation. It actually goes with distinct optical processes thatare further separated into different Liouville-space path-way contributions and visualized with double-sided Feyn-man diagrams.12 We will come back to them later; seeFig. 1 and Eq. (19). The block-matrix dynamics emergeshere in virtue of the Born-Oppenheimer principle. Consider,for example, a system of three adiabatic electronic mani-folds, the initial ground |g〉, the excited |e〉, and doubly-excited |f〉. For a molecular aggregate of size M, the elec-tronic levels in these three manifolds are Ng = 1, Ne

= M, and Nf = M(M − 1)/2, respectively. In general,an adiabatic system Hamiltonian assumes the form of Hs

= ∑uHu|u〉〈u|, which is block-diagonal in the Hilbert space,

due to the fact of Huv = Huδuv as inferred above. The adi-abatic Hu here is an Nu × Nu matrix. The correspondingsystem Liouvillian is block-diagonal in the Liouville space,i.e., Ls = ∑

uv Luv|uv〉〉〈〈uv| or Luv,u′v′ = Luvδuu′δvv′ . In thiswork, we assume that each individual system dissipativemode is also block-diagonal and can be cast in the formof Q = ∑

u Qu|u〉〈u|. As a result, the field-free dissipativepropagator is diagonal in the generalized Liouville space andreads Gs(t) = ∑

uv Guv(t)|uv〉〉〈〈uv|. In the CODDE space,this amounts to not just Rs = ∑

uv Ruv|uv〉〉〈〈uv|, but also �k

= ∑u �k,u|u〉〈u|. Apparently, �k,u = (iLuu + γk)−1Qu [cf.

Eq. (8)]. We have also RuvOuv ≡ Quv(QuOuv − OuvQ†v),

with Qu ≡ C(−Luu)Qu [cf. Eq. (3)] and QuvOuv = QuOuv

− OuvQv . Here Ouv is an arbitrary Nu × Nv matrix.Note that the block-diagonalized dissipative modes al-low excitation energy relaxation within a single excitonicmanifold, but induce only dephasing between differentmanifolds.

The aforementioned electronic adiabatic basis set leadsto the partial expansion of the CODDE-space state op-erator as ρ = {ρ; ρ(−)

k , ρ(+)k } = ∑

uv ρuv|u〉〈v|, where ρuv

= {ρuv; ρ(−)k,uv, ρ

(+)k,uv}. The field-free CODDE propagation of

ρ(t) = Gs(t)ρ(0) follows then the adiabatic block-matrix dy-namics of ρuv(t) = Guv(t)ρuv(0). Its equation of motion can

024106-5 Xu et al. J. Chem. Phys. 138, 024106 (2013)

FIG. 1. Eight Liouville-space pathways (upper) and double-sided Feynman diagrams (lower) for two-dimensional spectroscopy in the rotating-wave approxima-tion. Each Liouville-space pathway starts from the upper-left circle, while the double-sided Feynman diagram starts from the bottom, following the conventionof Ref. 12.

be deduced from Eq. (9) and reads explicitly as

ρuv = −(iLuv + Ruv)ρuv −N∑

k=0

Quv[ckρ(−)k,uv − c∗

kρ(+)k,uv],

ρ(−)k,uv = −(iLuv + γk)ρ(−)

k,uv, (13)

ρ(+)k,uv = −(iLuv + γ ∗

k )ρ(+)k,uv.

Similarly, for the dynamics operator A in theCODDE space, we have A = ∑

uv Avu|v〉〈u|, withAvu = {Avu; A(−)

k,vu, A(+)k,vu}. It follows the Heisenberg picture

of Avu(t) = Avu(0) Guv(t). The corresponding equation ofmotion reads [cf. Eq. (12)]

˙Avu(t) = −Avu(t)(iLuv + Ruv),

˙A

(−)k,vu(t) = −A

(−)k,vu(t)(iLuv + γk) − ckAvu(t)Quv, (14)

˙A

(+)k,vu(t) = −A

(+)k,vu(t)(iLuv + γ ∗

k ) + c∗k Avu(t)Quv.

Here, the Redfield tensor takes action on to the left,OvuRuv ≡ (OvuQuv)Qu − Q†

v(OvuQuv), whereas OvuQuv

= OvuQu − QvOvu.The system is initially at the thermal equilibrium in

ground-state manifold, i.e., ρeq(T ) = ρeqgg(T )|gg〉〉. Transi-

tions between different electronic manifolds are allowed viaoptical transition dipoles μuv or μ = ∑

u =v μuv|u〉〈v|, whereμuu = 0 and μuv assumes an Nu × Nv matrix form. TheCODDE-space analogue of such transitions defined in Eq. (7)now is needed to be recast into a block-matrix form. It readsexplicitly as

→Duv′ ρv′v = {

μuv′ ρv′v; μuv′ ρ(−)k,v′v, μuv′ ρ

(+)k,v′v

}+ {

0; [μ, �k]uv′ ρv′v, 0}, (15a)

←Du′vρuu′ = {

ρuu′μu′v; ρ(−)k,uu′μu′v, ρ

(+)k,uu′μu′v

}+{

0; 0, ρuu′[�†k, μ]u′v

}, (15b)

with [μ, �k]uv = μuv�k,v − �k,uμuv and [�†k, μ]u′v

= �†k,u′μu′v − μu′v�

†k,v . The resulting

→Duv′ ρv′v or

←Du′vρuu′

resembles ρuv that consists of a set of Nu × Nv matrices.

B. Mixed Heisenberg–Schrödinger dynamicsin nonlinear optical response functions

Next, we proceed to elucidate the mixed Heisenberg–Schrödinger scheme implementation of third-order optical re-sponse function, starting from its generic form in the general-ized Liouville space

R(3)(t3, t2, t1) = i3〈〈μks| Gs(t3) Dk3

× Gs(t2) Dk2 Gs(t1) Dk1 |ρeq(T )〉〉. (16)

Here, the tetradic bracket notation is adopted as Eq. (11) inthe CODDE space or its counterpart in the HEOM space (cf.Appendix B), together with the corresponding initial thermalequilibrium state ρeq(T ), field-free propagator Gs(t), transi-tion dipole operator μ and its commutator D, in the spec-ified linear space. The subscripts k1, k2, and k3 denote thewavevectors of three time-ordered incoming pulsed fields;each of them is associated with a transition dipole commuta-tor D. The signal polarization field is collected along one ofthe four-wave-mixing phase-matched directions of ks = ±k3

± k2 ± k1.It is well known that for the above three-manifolds sys-

tem there are eight Liouville-space pathways (α = 1, . . . , 8)and their Hermitian conjugates, contributing to the total third-order optical response,12 i.e.,

R(3)(t3, t2, t1) = i38∑

α=1

[Rα(t3, t2, t1) − c.c.]. (17)

Figure 1 depicts these eight pathways, together with the cor-responding double-sided Feynman diagrams for the underly-ing optical transitions in the rotating-wave approximation.12

These pathway contributions remain their block-matrix ex-

024106-6 Xu et al. J. Chem. Phys. 138, 024106 (2013)

pressions in the generalized Liouville space, e.g.,

R1(t3, t2, t1) = 〈〈μge| Geg(t3)←Deg

× Gee(t2)←Dge Geg(t1)

→Deg|ρeq

gg(T )〉〉. (18)

The underlying mixed Heisenberg–Schrödinger dynam-ics are as follows. We start with the thermal equi-librium in the ground-state manifold, which readsρ

eqgg(T ) = {ρeq(T ); 0, 0} in the CODDE space, with

ρeq(T) = 1 when Ng = 1. In the R1-pathway, the first(k1) incoming laser pulse excites the system to the (eg)-

coherence, i.e., ρeg(0) =→Degρ

eqgg(T ) [Eq. (15a)], followed

by the field-free t1-propagation, ρeg(t1) = Geg(t1)ρeg(0),then the second pulsed field excitation, ρee(0; t1)

=←Dgeρeg(t1) [Eq. (15b)], and further the t2-propagation,

ρee(t2; t1) ≡ Gee(t2)ρee(0; t1). We can thus recast Eq. (18)

as R1(t3, t2, t1) = 〈〈μge| Geg(t3)←Deg|ρee(t2; t1)〉〉. Having

the t1- and t2-propagations performed in the Schrödingerpicture, we implement in parallel the t3-propagation with theHeisenberg dynamics, involving μge(t3) = μge Geg(t3) in theR1-R4 and R8 pathways, while μef (t3) = μef Gf e(t3)in the R5-R7 components. Identify also ρgg(t2; t1)

= Ggg(t2)[←Degρge(t1)], where ρge(t1)= Gge(t1)[

←Dgeρ

eqgg(T )]

= ρ†eg(t1), and further ρfg(t2; t1) = Gfg(t2)[

→Df eρeg(t1)].

The mixed Heisenberg–Schrödinger dynamics for theeight pathway contributions can then be summarized as45

R1(t3, t2, t1) = 〈〈μge(t3)|←Deg|ρee(t2; t1)〉〉,

R2(t3, t2, t1) = 〈〈μge(t3)|←Deg|ρ†

ee(t2; t1)〉〉,R3(t3, t2, t1) = 〈〈μge(t3)|

→Deg|ρ†

gg(t2; t1)〉〉,

R4(t3, t2, t1) = 〈〈μge(t3)|→Deg|ρgg(t2; t1)〉〉,

R5(t3, t2, t1) = −〈〈μef (t3)|→Df e|ρ†

ee(t2; t1)〉〉,R6(t3, t2, t1) = −〈〈μef (t3)|

→Df e|ρee(t2; t1)〉〉,

R7(t3, t2, t1) = −〈〈μef (t3)|←Dge|ρfg(t2; t1)〉〉,

R8(t3, t2, t1) = 〈〈μge(t3)|→Def |ρfg(t2; t1)〉〉.

(19)

They can be classified as the excited-state emission (R1,R2), ground-state bleaching (R3, R4), excited-state ab-sorption (R5, R6), and double-excitation absorption (R7,R8) contributions.46–48 The underlying mixed Heisenberg–Schrödinger scheme of efficient evaluations on them is alsoself-evident in Eq. (19).

There are several configurations of 2D signals due to dif-ferent four-wave-mixing phase-matched direction. In partic-ular, those at kI = k3 + k2 − k1 and kII = k3 − k2 + k1

are called the rephasing (photon-echo) and the non-rephasingsignals, respectively. In the impulsive limit, these two typesof 2D spectra can be evaluated via

SkI/II (ω3, t2, ω1) = Re∫ ∞

0dt3

∫ ∞

0dt1e

i(ω3t3∓ω1t1)

× RkI/II (t3, t2, t1), (20)

and related, respectively, to

RkI = R2 + R3 + R5 (rephasing),

RkII = R1 + R4 + R6 (non-rephasing).(21)

The opposite sign in the phase factor argument ω1t1 inEq. (20) indicates the distinct rephasing and free-inductiondecay processes. In experiment, the time interval t1 oftenscans from the negative to positive region, yielding the 2Dsignal of SkI+kII = SkI + SkII . This is in fact the pump-probeabsorption configuration, involving all the six, R1 to R6, path-ways. In the following, we adopt this spectrum as our numer-ical demonstrations.

IV. NUMERICAL DEMONSTRATIONS WITHQUANTIFICATION OF THE ACCURACY

A. The FMO model and simulation setup

We will examine the effect of the driving-dissipation cor-relation on the coherent optical spectroscopy of FMO com-plex, followed by the quantification of the applicability rangeof the CODDE approximation. The FMO complex of thephotosynthetic green sulfur bacteria Chlorobium tepidum hasbeen regarded as one of most important prototypes for inves-tigating the efficient photosynthetic excitation energy trans-fer process because of relatively small size. It consists of atrimer; however, the couplings between different monomersare rather weak, so that we can focus on the monomer caseinstead. We adopt a previously studied exciton model,1, 21, 27

in which each FMO monomer is treated as an excitonic ag-gregate of seven bacteriochlorophyll molecules (BChls), withthe following excitonic (|e〉-manifold) Hamiltonian (in unit

FIG. 2. Linear absorption spectra of the FMO complex calculated at T= 77 K using different methods, with the molecular |e〉-manifolds Hamil-tonian of Eq. (22). Drude dissipation parameters for each on-site excitationenergy fluctuation are λ = 35 cm−1 and γ −1 = 100 fs.

024106-7 Xu et al. J. Chem. Phys. 138, 024106 (2013)

of cm−1):

He =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

280 −106 8 −5 6 −8 −4−106 420 28 6 2 13 1

8 28 0 −62 −1 −9 17−5 6 −62 175 −70 −19 −576 2 −1 −70 320 40 −2

−8 13 −9 −19 40 360 32−4 1 17 −57 −2 32 260

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦.

(22)

Here, the site energies along the diagonal are the relative en-ergies with respect to the BChl 3 electronic excitation en-ergy of 12120 cm−1. Involved in 2D spectrum simulation isalso the doubly-excited |f〉-manifold, which consists of 21states, {|jk〉; k > j = 1, . . . , 7}, with Hf = ∑

jk εojk|jk〉〈jk|

+ ∑jk =j ′k′ Jjk,j ′k′ |jk〉〈j ′k′|, where k > j and k′ > j′. We ne-

glect the Coulomb interactions, so that the on-site bi-excitonenergy is from the diagonal terms of Eq. (22), assumingεojk = εo

j + εok , while Jjk,j ′k′ is from off-diagonal terms de-

scribing the inter-site couplings.26 The ground |g〉-manifoldconsists of one level (|0〉). The on-site energy fluctuations en-ter via the individual dissipative modes of Qj = B

†j Bj , where

Bj = |0〉〈j | + ∑k |k〉〈jk|. The optical transition dipole vec-

tors in μ = ∑j μj (Bj + B

†j ) are extracted from the crystal

structure of C. tepidum (PDB code: 3ENI), with { μj } for allseven BChls being assumed to have the same magnitude butdistinct directions defined in the molecular frame. Referred tothe crystal structure, each BChl dipole vector is assumed tobe pointing from its NB to ND atom. To account for the orien-tation disorder effect, we take the average results over 1000samples. The orientation of each sample is tuned by rotatingthe molecular frame with respect to the laser fields of a ran-dom Euler angle. The static on-site energy disorders arisingfrom different local environments and contributing to inho-mogeneous broadening in spectroscopy are not included. Asa result, individual peaks in the simulated spectra will be lesscongested, as the main interest here is the dynamical aspectsof the correlated driving and dissipation.

The bath spectral density associated with each indepen-dent on-site excitation energy fluctuation assumes the Drudeform of Eq. (1), with all the same λ = 35 cm−1 and γ −1

= 100 fs, in accord with previous studies.1, 21, 27 We will eval-uate the spectroscopic signals for the model FMO system atboth T = 77 K and 298 K. Based on the established accu-racy control criterion on Drude dissipation,36–39 we exploitthe [1/1]– and [0/0]–PSD schemes to expand the bath correla-tion function in the form of Eq. (2), at these two temperatures,respectively.

For the spectroscopic simulations to be presented soon,we compare explicitly the results between CODDE, Redfieldtheory, HEOM-2, and HEOM-4. The latter two refer to theHEOM dynamics truncated at the tier ntrun = 2 and 4 lev-els, respectively, with the leading fourth- and eighth-ordertreatments of system-bath interaction; see Appendix B. Forthe model system specified above at both 77 K and 298 K,the HEOM-4 evaluations are rather accurate and used, there-fore, as references to assess other approaches exploited inthis work. For the second-order approaches, we focus on the

CODDE versus the Redfield theory evaluations. The conven-tional time-nonlocal quantum master equation (equivalent toHEOM-1) results in optical signals that suffer badly from spu-rious peaks31 and thus will not be shown explicitly below.

B. Optical electronic spectra at 77 K

Figure 2 displays the linear absorption spectra of theFMO model system at 77 K, calculated by HEOM-4, HEOM-2, CODDE, and the Redfield methods. Each simulated spec-trum contains six absorption peaks, which can be identified22

to the delocalized excitons, except the exciton 4, as it has adiminishing oscillator strength. For later use, we denote thedelocalized exciton energy as εm in ascending order. Refer-ring to the accurate HEOM-4 evaluation, the Redfield methodexhibits an overall peak shift towards lower energies. It isknown that the absorption peak position is sensitive to thebath correlation time γ −1, with an effective blueshift boundedby the reorganization energy λ, in the inhomogeneous(γ −1 → ∞) limit.12 The insufficient blueshifts of the Red-field theory evaluation shown in Fig. 2 are ascribed to itsMarkovian nature, assuming an effective γ −1

eff ≈ 0. Remark-ably, CODDE appears even better than HEOM-2 that is a cer-tain fourth-order theory.19 It gives an overall excellent agree-ment with the HEOM-4 reference, for the moderate dephasingtime parameter (γ −1 = 100 fs) considered here.

Figure 3 shows the absorptive 2D signals, SkI + SkII [cf.Eq. (20)], at different waiting times, evaluated with the afore-mentioned four approaches. Here we concern about the 2Dspectral shapes and the cross peak amplitude oscillations, asthey reveal the pigment-protein conformations and the dy-namical coherence underlying the excitation energy trans-fer processes, respectively. In 2D spectroscopy, the diagonalpeaks (ω1 = ω3) amplitudes represent the excitonic popula-tions, while the anti-diagonal cross peaks (ω1 + ω3 = const.)reflect the correlations between different excitons. Adoptingthe convention, the cross peak centered at (ω1 = εm, ω3 = εn)is labeled by CPmn, while the diagonal one at (ω1 = ω3 = εn)by DPn.

Examine the diagonal peaks first. The HEOM-4,CODDE, and HEOM-2 results, plotted in Figs. 3(a)–3(c),respectively, show a similar trend with the increasingt2: the prominent diagonal elongations (non-Markovcharacteristics)5 at early waiting times, followed by thegradual change to symmetric shapes (Markov behavior).However, the Redfield theory results in Fig. 3(d) show the“star” shapes of diagonal peaks in the whole range of t2.The “star” shape produced as 2D Lorentzian49 comes as nosurprise in the Redfield Markovian evaluation on the under-lying Drude dissipative dynamics during the time intervalsof t1 and t3. The above observations are also consistent withthe fact that we do not include the static inhomogeneityin the excitation energy distributions. Similar to the caseof linear absorption spectra, good agreements are achievedbetween the three non-Markovian approaches. We have thusdemonstrated again, in particular, that CODDE may supporta reliable evaluation on 2D spectrum of light-harvesting

024106-8 Xu et al. J. Chem. Phys. 138, 024106 (2013)

ω1(104 cm−1)

ω3(1

04 cm

−1 )

0

1.2 1.22 1.24 1.261.2

1.22

1.24

1.26

0

200

400

ω1(104 cm−1)

200 fs

1.2 1.22 1.24 1.26

0

100

200

300

ω1(104 cm−1)

400 fs

1.2 1.22 1.24 1.26

0

100

200

300

(a) HEOM−4 T = 77 K

t2 =

ω1(104 cm−1)

ω3(1

04 cm

−1 )

0

1.2 1.22 1.24 1.261.2

1.22

1.24

1.26

0

200

400

ω1(104 cm−1)

200 fs

1.2 1.22 1.24 1.26

0

100

200

300

ω1(104 cm−1)

400 fs

1.2 1.22 1.24 1.26

0

100

200

(b) CODDE

t2 =

ω1(104 cm−1)

ω3(1

04 cm

−1 )

0

1.2 1.22 1.24 1.261.2

1.22

1.24

1.26

0

200

400

ω1(104 cm−1)

200 fs

1.2 1.22 1.24 1.26

0

100

200

300

ω1(104 cm−1)

400 fs

1.2 1.22 1.24 1.26

0

100

200

300

(c) HEOM−2

t2 =

ω1(104 cm−1)

ω3(1

04 cm

−1 )

0

1.2 1.22 1.24 1.261.2

1.22

1.24

1.26

0

100

200

ω1(104 cm−1)

200 fs

1.2 1.22 1.24 1.26

0

50

100

150

ω1(104 cm−1)

400 fs

1.2 1.22 1.24 1.26

0

50

100

(d) REDFIELD

t2 =

FIG. 3. Absorptive 2D spectra evaluated with different methods for the same FMO model system at 77 K (a)–(d). Involved are also the 21 bi-exciton levels.

pigment-protein complexes, where the system-bath couplingstrength and non-Markovianicity are often both moderate.50

To examine the quantum coherent behavior in thewaiting-time-resolved 2D spectra, Fig. 4 depicts the evolu-tions of amplitudes of eight cross peaks, which are groupedinto four pairs, CP12 vs. CP21, CP13 vs. CP31, CP15 vs.

CP51, and CP23 vs. CP32, as shown from the top pair panelsto the bottom pair there. The depicted values are collected byintegrating the absolute magnitude over the region of 50 cm−1

× 50 cm−1, centered around the corresponding peak, with-out additional scaling. As a result, the y-ranges of the in-dividual frames in Fig. 4 are (arbitrary unit): (a) [20, 70],

024106-9 Xu et al. J. Chem. Phys. 138, 024106 (2013)

FIG. 4. Amplitude oscillations of eight cross peaks extracted from thewaiting-time-resolved 2D spectra evaluated at 77 K, with “CPmn” indicat-ing the cross peak at (ω1 = εm, ω3 = εn).

(b) [5, 20], (c) [14, 25], (d) [2, 10], (e) [10, 30], (f) [5, 15],(g) [40, 150], and (h) [30, 200]. While the energy gap be-tween excitons determines the oscillation period, the coher-ence time of great interest is closely related to the dissipa-tive dynamics. Clearly seen from Fig. 4, the coherent am-plitude oscillations predicted by Redfield theory can onlysurvive within 200 fs, while those from the three non-Marvovian approaches persist for about 400 fs. To confirmthe underestimation in quantum coherence by the Marko-vian nature of the Redfield theory, we reduce the dephas-ing time parameter to γ −1 = 50 fs, and the Redfield dynam-ics can provide about the same coherence time as the ex-act ones (results not shown here). The Markovian simplifi-cation of the Redfield dynamics does underestimate the quan-tum coherence time in photosynthetic complexes.27 In con-trast, CODDE provides the quantitatively accurate predictionson the quantum coherence time, as compared to the HEOMevaluations. CODDE is the viable second-order theory to

FIG. 5. Linear absorption spectra for the same FMO model system calcu-lated at T = 298 K using different methods.

the optical properties of photosynthetic pigment-protein com-plexes, not just the linear absorption but also the 2D spec-tral shapes and the underlying coherent energy transfertime.

C. Optical electronic spectra at 298 K

The individual approximation deteriorates when the tem-perature raises to 298 K, due to the enhanced effective non-Markovianicity that will be analyzed in Sec. IV D. Fig-ure 5 reports the evaluated linear absorption signals via thefour approaches. Apparently, all simulated spectra are broad-ened in comparison with their 77 K counterparts in Fig. 2.But CODDE and HEOM-2 underestimates the broadening,whereas the Redfield theory does the opposite and fails com-pletely to reveal any spectral information, at the room temper-ature. The simulated 2D spectra in Fig. 6 have the same trendsof discrepancy in different methods: CODDE and HEOM-2are over-featured, while Redfield theory evaluations are muchtoo congested.

Figure 7 reports the simulated quantum beating behaviorsusing different methods at T = 298 K. The coherent beatingslast significantly shorter than their 77 K counterparts, withalso overall smaller amplitudes. The y-ranges of individualframes in Fig. 7 are (same unit as Fig. 4): (a) [5, 60], (b) [5,25], (c) [5, 30], (d) [3, 13], (e) [5, 20], (f) [3, 13], (g) [5, 80],and (h) [5, 80]. In contrast to the Redfield approach that re-sults in almost no coherence, with the cross peak amplitudesdecreased monotonically, either CODDE or HEOM-2 repro-duces at least qualitatively the quantum beating behaviors ofexperimental observations at ambient condition.7 The quan-tum coherence calculated with either CODDE or HEOM-2survives at least 200 fs, about the same (by order of mag-nitude) as that of the HEOM-4 evaluation, despite their 2Dspectral shapes appear quite different.

D. Towards the applicability measure on CODDE

To quantify the applicability range of CODDE, let usstart with the benchmark HEOM-4 results, for the underly-ing mechanisms as implied in the above evaluations at twotemperatures. Observed from the accurate HEOM-4 calcula-tions of 2D spectrum is a substantial diagonal elongation inFig. 6(a) at 298 K, compared with Fig. 3(a) at 77 K. Notethat we did not include static inhomogeneity in the modelcalculations. Therefore, the observed 2D spectral diagonalelongation is concerned about the dynamical inhomogeneity,which increases as temperature raises from 77 K to 298 K.Proposed here is the following dynamical inhomogeneityparameter

α = max{∣∣ 1

2ck/γ2k

∣∣; k = 0, 1, · · · }. (23)

As inferred from Eq. (2), this parameter measures the non-Markovianicity of system-bath coupling. Consequently, itmay also be used to quantify the applicability range (α � 1)of CODDE, assuming that the system-bath coupling strength(not specified here) is also moderate. For the present FMOsystem in study, we have α = 0.7 at 77 K and α = 2.6 at

024106-10 Xu et al. J. Chem. Phys. 138, 024106 (2013)

ω1(104 cm−1)

ω3(1

04 cm

−1 )

0

1.2 1.22 1.24 1.261.2

1.22

1.24

1.26

0

20

40

60

80

ω1(104 cm−1)

200 fs

1.2 1.22 1.24 1.26

0

10

20

30

40

ω1(104 cm−1)

400 fs

1.2 1.22 1.24 1.26

0

10

20

30

(a) HEOM−4 T = 298 K

t2 =

ω1(104 cm−1)

ω3(1

04 cm

−1 )

0

1.2 1.22 1.24 1.261.2

1.22

1.24

1.26

0

50

100

ω1(104 cm−1)

200 fs

1.2 1.22 1.24 1.26

0

20

40

60

ω1(104 cm−1)

400 fs

1.2 1.22 1.24 1.26

0

20

40

60

(b) CODDE

t2 =

ω1(104 cm−1)

ω3(1

04 cm

−1 )

0

1.2 1.22 1.24 1.261.2

1.22

1.24

1.26

0

50

100

ω1(104 cm−1)

200 fs

1.2 1.22 1.24 1.26

0

20

40

60

ω1(104 cm−1)

400 fs

1.2 1.22 1.24 1.26

0

10

20

30

40

(c) HEOM−2

t2 =

ω1(104 cm−1)

ω3(1

04 cm

−1 )

0

1.2 1.22 1.24 1.261.2

1.22

1.24

1.26

5

10

15

ω1(104 cm−1)

200 fs

1.2 1.22 1.24 1.26

2

4

6

8

10

ω1(104 cm−1)

400 fs

1.2 1.22 1.24 1.26

2

4

6

8

10

(d) REDFIELD

t2 =

FIG. 6. (a)–(d) Absorptive 2D spectra evaluated with different methods for the same FMO model system at 298 K.

298 K, respectively. It is interesting to notice that the applica-bility range of CODDE is at least the same as that of HEOM-2. Note also that when the temperature is not too low, the k= 0 or Drude (γ 0 = γ D) term from Eq. (2) often dictates themaximum of the right-hand side (rhs) of Eq. (23), leading to

the increase of α as temperature raises. However, in the lowtemperature regime, the maximum may go by the k = 1 termof the rhs of Eq. (23), considering, for example, the Matsubarafrequency of γ 1 = 2πkBT, resulting in the opposite tempera-ture dependent behavior of the parameter α.

024106-11 Xu et al. J. Chem. Phys. 138, 024106 (2013)

FIG. 7. Amplitude oscillations of eight cross peaks extracted from thewaiting-time-resolved 2D spectra evaluated at 298 K.

V. CONCLUDING REMARKS

In conclusion, we have demonstrated that CODDE wouldbe a viable second-order quantum dissipation theory for thestudy of quantum coherence in photosynthetic complexes.The advantage of CODDE is its numerical efficiency andaccuracy predictability, applicable to where the system-bathcoupling strength and non-Markovianicity are both moderate.This scenario is rather common in photosynthetic pigment-protein molecular complex systems.5, 50 Thus, it is anticipatedthat CODDE and its further development could be a practicaltool in this field of study.

The CODDE formalism gives the most transparent de-scription on the correlated driving and dissipation dynamics,with the lowest-order of system–bath coupling being treatedproperly in both the dynamic evolution and the initial equi-librium state distribution. Apparently, this correlated dynam-ics is also included in HEOM, at any level of truncation, andin the modified Redfield approximation. The latter was de-veloped by Mukamel and co-workers51, 52 on the basis of thecumulant expansion of nonlinear optical response functions,where the system-bath coupling is treated nonperturbatively.

It is also noticed that the exact HEOM method hasbecome popular recently in the studies of photosyntheticsystems.21–25 The proposed mixed Heisenberg–Schrödingerpicture scheme has greatly improved the efficiency of HEOMevaluations on 2D spectroscopy. This advanced numericaltechnique is rather general and applicable to any dynamicsimplementation on nonlinear response functions. The result-ing CPU times on individual sample calculations in Fig. 3are (in minute) 0.4 ∼ 0.8 for CODDE, 4 ∼ 13 for HEOM-2,and 22 ∼ 76 for HEOM-4, respectively, on a single Intel(R)Xeon(R) processor X5660 @2.80 GHz. Highlighted here isthat, with about the same level of accuracy, CODDE is about10 times more efficient than HEOM-2, and both are applicableat moderate non-Markovianicity or dynamical inhomogeneity(α � 1); see Sec. IV D. The above observation suggests theimportance of HEOM truncation scheme and further the pos-sibility of developing the CODDE-like hierarchy termination

for challenging cases such as the 2D spectroscopy evaluationat 298 K as studied here. It is anticipated that the aforemen-tioned advancement also facilitates more realistic simulations,taking such as the coherent vibrational motion into account.

ACKNOWLEDGMENTS

Support from the Hong Kong RGC (605012) andUGC (AoE/P-04/08-2), the National Natural Science Foun-dation (NNSF) of China (21033008 and 21073169 andQing Nian Grant No. 61203061), Strategic Priority Re-search Program (B) of the Chinese Academy of Sciences(XDB01000000), and the National Basic Research Programof China (2010CB923300 and 2011CB921400) is gratefullyacknowledged.

APPENDIX A: THE CODDE FORMALISM: DERIVATION

This Appendix contains a brief derivation of the CODDEformalism. Let us start with the conventional time-local quan-tum master equation

ρ(t) = −iL(t)ρ(t) − R(t)ρ(t). (A1)

The involving time-local dissipation superoperator, R(t), isgiven via

R(t)ρ(t) = [Q, Q(t)ρ(t)] + H.c. (A2)

with

Q(t) =∫ t

−∞dτ C(t − τ )G(t, τ )Q. (A3)

Here, G(t, τ ) is the dissipation-free but field-dressedLiouville-space propagator, satisfying the Liouville-von Neu-mann equation, ∂tG(t, τ ) = −iL(t)G(t, τ ). To investigate thecorrelated driving-dissipation effects involved here, we makeuse of the identity L(t) = Ls + Lsf(t) and the Dyson equation

G(t, τ ) = Gs(t − τ ) − i

∫ t

τ

dτ ′ G(t, τ ′)Lsf(τ′)Gs(τ

′ − τ ).

Thus, Eq. (A3) becomes

Q(t) = Qs + Qsf(t) (A4)

with

Qsf(t) = −i

∫ t

−∞dτ

∫ τ

−∞dτ ′ C(t − τ ′)

× G(t, τ )Lsf(τ )Gs(τ − τ ′)Q. (A5)

In writing this equation, we have used the identity of∫ t

−∞dτ∫ t

τdτ ′ = ∫ t

−∞dτ ′ ∫ τ ′

−∞dτ , followed by swapping theintegration variables τ and τ ′. The time-local Eq. (A1) cannow be recast as

ρ(t) = −[iL(t) + Rs]ρ(t) − {[Q, Qsf(t)ρ(t)] + H.c.}.(A6)

For bookkeeping below, we introduce the operator

�sf(τ ; t) ≡ iLsf(τ )∫ τ

−∞dτ ′ C(t − τ ′)Gs(τ − τ ′)Q, (A7)

024106-12 Xu et al. J. Chem. Phys. 138, 024106 (2013)

which leads to Eq. (A5) the compact form of

Qsf(t) = −∫ t

−∞dτ G(t, τ )�sf(τ ; t). (A8)

To obtain the CODDE formalism, we exploit the followingsecond-order expression:[

G(t, τ )�sf(τ ; t)]ρ(t) ≈ [

G(t, τ )�sf(τ ; t)][G(t, τ )ρ(τ )

].

Apparently, it can be recast as[G(t, τ )�sf(τ ; t)

]ρ(t) ≈ G(t, τ )

[�sf(τ ; t)ρ(τ )

]. (A9)

Equations (A6)–(A9) lead to the CODDE formalism theintegro-differential expression of

ρ(t) = −[iL(t) + Rs]ρ(t) +∫ t

−∞dτ K(t, τ )ρ(τ ) (A10)

with (noting that Q · ≡ [Q, · ]),

K(t, τ )ρ(τ ) = QG(t, τ )[�sf(τ ; t)ρ(τ )] + H.c. (A11)

To convert Eq. (A10) into a set of coupled equations,an exponential expansion of bath correlation function will beneeded. To proceed, let us recast Eq. (A10) as

ρ(t) = −[iL(t) + Rs]ρ(t) − Q[ρ(t) − ρ†(t)]. (A12)

Introduced here is

ρ(t) ≡ −∫ t

−∞dτ G(t, τ )[�sf(τ ; t)ρ(τ )]

=∑

k

ckρ(−)k (t). (A13)

The second identity is obtained by using the exponential ex-pansion of the bath correlation function as Eq. (2). It leads tothe involving ρk(t) the expression of

ρ(−)k (t) = −

∫ t

−∞dτ G(t, τ )[�sf

k (τ ; t)ρ(τ )] (A14)

with (noting that Gs(t) = e−iLst ),

�sfk (τ ; t) ≡ iLsf(τ )

∫ τ

−∞dτ ′ e−γk (t−τ ′)e−iLs(τ−τ ′)Q. (A15)

The time derivative on Eq. (A14) results in

ρ(−)k (t) = −[iL(t) + γk]ρ(−)

k (t) − �sfk (t)ρ(t) (A16)

with �sfk (t) ≡ �sf

k (t ; t) being identified from Eq. (A15) as

�sfk (t) = iLsf(t)

∫ t

−∞dτ ′ e−γk (t−τ ′)e−iLs(t−τ ′)Q

= iLsf(t)∫ ∞

0dτ e−γkτ e−iLsτ Q

= iLsf(t)(iLs + γk)−1Q. (A17)

Denote ρ(+)k (t) = [ρ(−)

k (t)]†, so that ρ†(t) = ∑k c∗

kρ(+)k (t).

Thus, Eqs. (A16) and (A17) are just Eqs. (4b) and (5), re-spectively. To verify that Eq. (A12) with (A13) is the sameas Eq. (4a), we should show that the δ(t)-function term inEq. (2) leads to an ADO of ρ

(±)k = 0. This result can be easily

obtained by either examining Eq. (A5) or setting γ k → ∞ inEqs. (A16) and (A17). We have thus completed the CODDEformulation of Eq. (4).

APPENDIX B: THE HEOM FORMALISM

In this work, we choose the exact HEOM approach asthe reference to calibrate the second-order evaluations of bothRedfield theory and CODDE. The explicit HEOM construc-tion is dependent on the expansion scheme of C(t). In accor-dance with the CODDE evaluations, we also adopt the formof Eq. (2). The resulting HEOM formalism in the Schrödingerpicture reads38, 53

ρn = −[iL(t) +

N∑k=0

nkγk + �NQ2

]ρn

− i

N∑k=0

√nk

|ck|(ckQρn−

k− c∗

kρn−kQ

)

− i

N∑k=0

√(nk + 1)|ck|Qρn+

k. (B1)

The ADO’s labeling index consists of a set of non-negativeintegers, i.e., n ≡ {n0, n1, . . . , nN }, specifying that ρn

≡ ρn0,n1,...,nNis of the leading orders of {nk ≥ 0} in the in-

dividual exponent terms in the bath correlation function ofEq. (2). The effect of the white-noise residue in Eq. (2) onρn is described by the �N-term of Eq. (B1). Let n = n0 + n1

+ · · · + nN and call ρn an nth-tier ADO. The zeroth-tier ADOis just the reduced system density operator, i.e., ρn=0 ≡ ρ(t).The associated ADO’s index n±

k in the last two terms ofEq. (B1) differs from n only by changing the specified nk to nk

± 1. Thus, the last two terms of Eq. (B1) specify how an nth-tier ADO depends on its associated (n − 1)th- and (n + 1)th-tier ADOs, respectively. In practice, HEOM is evaluated at acertain truncation tier level ntrun, which is done in this workby setting all higher-tier ADOs, ρn|n>ntrun = 0. The resultingdynamics treats the system-bath coupling nonperturbativelyto the (2ntrun)th-order level, as the higher order contributionsare partially included in certain resum manner.19

Similar to Sec. II B 2, we define the HEOMspace by recasting Eq. (B1) in a matrix-vector form,ρ(t) = −i L(t)ρ(t), with the HEOM-space state vector ρ

≡ {ρn=0; ρn =0} ≡ {ρ; ρn =0} consisting of all ADOs here. TheHEOM generator L(t) can also be separated into the field-free Ls and field-dressed Lsf(t) components. The latter isdiagonal in the HEOM space, having the form of Lsf(t) =−D · I εin(t), as inferred from Eq. (B1), where I denotes

the unit operator in the HEOM space, while D =→D −

←D is the

conventional transition dipole commutator. Thus, the HEOM-space correspondence to Eq. (7) reads

→Dρ = {μρ; μρn =0},

←Dρ = {ρμ; ρn =0μ}. (B2)

The resulting ρ(0) =→Dρ or

←Dρ serves as the initial door-

way state for the subsequent field-free propagation, ρ(t)= Gs(t)ρ(0) or ρ(t) = −i Lsρ(t).

In contact with the expectation value of an arbi-trary dynamical variable A in the reduced system space,A = tr(Aρ) ≡ 〈〈A|ρ〉〉 = 〈〈A|ρ〉〉, we define the correspond-ing HEOM-space inner product as 〈〈A|ρ〉〉 ≡ ∑

all n〈〈An|ρn〉〉[cf. Eq. (11)]. Here, A = {An=0; An =0}, arranged in a row

024106-13 Xu et al. J. Chem. Phys. 138, 024106 (2013)

vector, denotes the HEOM-space extension of the system dy-namics operator A. Implied here, in particular, is also theinitial conditions of An=0(t = 0) ≡ A and An =0(t = 0) ≡ 0,for the Heisenberg picture of HEOM-space dynamics to bespecified as follows. To that end, let us write down explicitlythe key ingredients in the matrix-vector form of ρ = −i Lsρ

[cf. Eq. (B1) in the absence of external field],

∂

∂t

⎡⎢⎢⎢⎢⎣×ρn−

k

ρn

ρn+k×

⎤⎥⎥⎥⎥⎦ = −i

⎡⎢⎢⎢⎢⎣× × 0 0 0× × Bn−

k0 0

0 An Ln Bn 00 0 An+

k× ×

0 0 0 × ×

⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣

×ρn−

k

ρn

ρn+k×

⎤⎥⎥⎥⎥⎦. (B3)

Here, iLn denotes the field-free quantity in the square-brackets in Eq. (B1). The superoperators An and Bn aregiven via the last two terms in Eq. (B1), respectively.In terms of the tier-up An+

kand tier-down Bn−

kcorre-

spondences, we have An+kO ≡

√nk+1|ck | (ckQO − c∗

k OQ) and

Bn−kO ≡ √

nk|ck|QO. These two associated superoperatorswill appear explicitly in the Heisenberg equation of mo-tion, as below. With the matrix Ls being specified explic-itly in Eq. (B3), we obtain A(t) = −i A(t) Ls for the rowvector of A = {×, An−

k, An, An+

k,×} the expression of ˙

An

= −iAnLn − iAn−kBn−

k− iAn+

kAn+

k, with the dummy sum

over k-index being implied in the last two terms. Therefore,the HEOM formalism in the Heisenberg picture reads explic-itly as45

˙An(t) = − An(t)

(iLs +

N∑k=0

nkγk + �NQ2

)

− i

N∑k=0

√nk + 1

|ck|[ckAn+

k(t)Q − c∗

kQAn+k(t)

]− i

N∑k=0

√nk|ck| An−

k(t)Q. (B4)

Together with the initial conditions of An=0(0) = A andAn =0(0) = 0, the above equation determines the HEOM evo-lution of A(t) = A(0) Gs(t) = A(0) exp(−i Lst) for an arbi-trary system dynamical variable A.

Note that the auxiliary operators in either Eq. (B1) orEq. (B4) are all scaled properly to support the efficient HEOMevaluation via the on-the-fly filtering algorithm.53 The filter-ing error tolerance of 2 × 10−5 is found to be sufficient fornumerically exact HEOM dynamics in both the Schrödingerand Heisenberg pictures.

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