Disorder-Induced Quantum Beats in Two-Dimensional Spectra of

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Disorder-induced Quantum Beats in Two-dimensional Spectra of Excitonically Coupled Molecules Vytautas Butkus, Hui Dong, Graham R. Fleming, Darius Abramavicius, and Leonas Valkunas J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.5b02642 • Publication Date (Web): 31 Dec 2015 Downloaded from http://pubs.acs.org on January 4, 2016

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The Journal of Physical Chemistry Letters

Disorder-induced Quantum Beats in Two-dimensional Spectra of Excitonically Coupled Molecules Vytautas Butkus,†,‡ Hui Dong,¶,§ Graham R. Fleming,¶,§ Darius Abramavicius,† and Leonas Valkunas∗,†,‡ Department of Theoretical Physics, Faculty of Physics, Vilnius University, Sauletekio 9-III, 10222 Vilnius, Lithuania, Center for Physical Sciences and Technology, Gostauto 9, 01108 Vilnius, Lithuania, Department of Chemistry, University of California, Berkeley, California 94720, United States, and Physical Biosciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, United States E-mail: [email protected]

∗

To whom correspondence should be addressed VU ‡ FTMC ¶ UCB § LBNL †

1 ACS Paragon Plus Environment


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Abstract Quantum superposition of molecular electronic states is very fragile due to thermal energy fluctuations and the static conformational disorder induced by the intimate surrounding of constituent molecules of the system. However, the nature of the long-lived quantum beats, observed in time-resolved spectra of molecular aggregates at physiological conditions, is still being debated. Here we present our study of the conditions, when long-lived electronic quantum coherences originating from recently proposed inhomogeneous broadening mechanism are enhanced and reflected in the 2D electronic spectra of the excitonically coupled molecular dimer. We show, that depending on the amount of inhomogeneous broadening, the excitonically coupled molecular system can establish long-lived electronic coherences, caused by a disordered sub-ensemble, for which the dephasing due to static energy disorder becomes significantly reduced. On the basis of these considerations we present explanations, why the electronic or vibrational coherences were or were not observed in a range of recent experiments.

Graphical TOC Entry

Keywords static energy disorder, long-lived excitonic coherence, excitonic dimer, two-dimensional spectroscopy


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Optical transitions from electronic ground state to electronic excited state are responsible for the light absorption ability of separate molecules in gas or liquid phases. In aggregates of strongly interacting molecules, collective excitation turns to be responsible for spectroscopic behavior of the whole aggregate as a supermolecule, even if its constituents are not bound chemically. In terms of the Heitler–London approximation, this collective excitation ability can be reflected as the coherent superposition of molecular excited states, thus, establishing the excitonic spectrum 1,2 . The coherent nature of electronic excitations is very important for functioning of various molecular systems, ranging from macro-molecules of biological origin to artificial nanostructures used for design of specific molecular electronic devices. Static and dynamic disorder, induced by variability of the local environment, essentially disturbs the coherent relationship between the molecular wave functions. Distribution of the molecular transition energies and coupling strengths, that do not change during the time interval of the processes under consideration, are referred to as the diagonal static disorder and the off-diagonal static disorder, respectively. Dynamic disorder arises from the time-dependent fluctuations of system’s electronic and nuclear parameters on the time scales of the relevant ultrafast processes. Consequently, both static and dynamic disorder influences the exciton dynamics and the extent of exciton delocalization. In large molecular crystals and aggregates 3,4 , the exciton delocalization amounts to a significant influence to excitation dynamics and photochemical properties. For example, a substantial exciton delocalization was shown to be crucial for the robust, efficient and untrapped excitation transfer through a network of coupled molecules in photosynthetic complexes 5 ; the enhancement of the quantum transport 6 can be ensured by the environmental noise, driving the system to the optimal regime with respect to the exciton localization. Details of the exciton dynamics are now intensively studied using various techniques of coherent nonlinear spectroscopy, for example, by means of the two-dimensional (2D) electronic spectroscopy 7–11 . Along with many advantages regarding excitation selectivity or spectral and temporal resolution, offered by the femtosecond laser excitation/detection,



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these methods provide a possibility of observation of the long-lived quantum beats, witnessing the established quantum mechanical superposition of electronic, vibrational or mixed states 12 . Most recently, the non-adiabatic effects of coupling to the nuclear degrees of freedom causing the speed-up of excitation dynamics or charge transfer 9,13–17 , were revealed by the analysis of such quantum beats. However, the influence of the disorder on the coherent beats still remains unclear. The phenomenon of long-lived coherent beatings has been supported by a range of mechanisms: spatially correlated dynamic disorder 18 , non-secular quantum dynamics 19 and nuclear vibrations 20 . It was shown theoretically, that the static disorder significantly reduces the lifetime of the excited state coherences, which are of electronic nature, while the effect on the vibrational coherences is less pronounced 21 , although the lifetime of the excited state coherences can be maintained due to the lifetime borrowing in the case of strong electronic–vibrational mixing 12,20 . Recently, Dong and Fleming have theoretically demonstrated the opposite effect, how substantial static diagonal disorder can enhance the lifetime of the electronic coherences reflecting themselves via long-lived signals in the integrated two-color photon echo (PE) spectroscopy 22 . For the proposed mechanism, the overlap of the inhomogeneously broadened spectral lineshapes is necessary allowing for a part of systems in the disordered ensemble to possess full exciton delocalization and, thus, give a strong longlived signal. Evidently, this effect should be well pronounced in the two-dimensional electronic spectroscopy, the signals of which are directly related to the PE signal through the projection–slice theorem 23 . In this paper, we study the competition between static and dynamic disorder, both responsible for the decay of the quantum beats in the 2D spectrum, and demonstrate that their lifetime and the amplitude might be significantly enhanced. Also, we reveal new properties of 2D spectra of disordered systems, such as waiting time-dependent change of observed coherence frequency and electronic coherences with frequencies smaller than the excitonic energy gap, not available in the two-color PE spectra.


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Let us consider the molecular dimer in the theoretical frame of the Frenkel excitonic model 1,2,4,10 . If the diagonal or off-diagonal static disorder is neglected, the excitonic splitting ∆0e is fixed and depends on the difference of the site energies of each molecule δ0 ≡ ¯2 − ¯1 and the resonant coupling constant J as ∆0e ≡

q δ02 + 4J 2 .

(1)

The uncorrelated static disorder of molecular transition energies affects the excitonic splitting. We characterize the energy fluctuations of both molecules by the Gaussian distribution function with the same standard deviation σD : 1 2 −2 2πσD

P (i ) =

e

−

(i −¯ i )2 2σ 2 D

,

(2)

where i is the energy of the i-th (i = 1, 2) molecule and ¯i is its mean value. The statistical distribution of the excitonic energy gap ∆e can then be derived 22 and yields:

−

P (∆e ) =

e

√

δ0 +

2 ∆2 e −4J

4σ 2 D

2

−

√

δ0 −

2 ∆2 e −4J 2 4σ D

+e p √ −1 ∆e 2 πσD ∆2e − 4J 2

2

.

(3)

This distribution function is shown in Fig. 1 for chosen parameters (|J| = 150 cm−1 , δ0 = 400 cm−1 ), which are typical for many photosynthetic pigment–protein complexes 4 , and for a few different values of the static diagonal disorder: σD = 40 cm−1 , 80 cm−1 and 160 cm−1 . The distribution has a singularity at ∆e = 2J, which becomes significant at larger values of the static disorder (see blue curve in Fig. 1). Such a distribution has two wellexpressed parts: a narrow peak at ∆e = 2J and a wide distribution around ∆e = ∆0e . These two contributions represent two sub-ensembles of the pool of dimers in the sample and have very different coherent properties. The narrow part of the distribution at ∆e = 2J represents a sub-ensemble of homo-dimers, for which the site energy gap δ0 is zero. In homo-dimers, excitons are completely delocalized over the two sites. This means, that their energy correlation functions are identical and the dephasing of the 5 ACS Paragon Plus Environment


corresponding superposition state is significantly slower than that of systems with the large site energy gap 22,24 . The wide part of the distribution represents a sub-ensemble of hetero-dimers. The widely distributed excitonic energy gap values will result in a negligible contribution to the beatings of the total spectral signal due to dephasing, caused by the averaging over the coherent oscillatory signals of different frequencies, represented by the distribution. From Fig. 1 it is evident, that increasing the disorder results in a relative enhancement of the total signal, stemming from the sub-ensemble of the homo-dimers. ∆0e

2J 0.020

P(∆e )

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0.015

σD = 40 cm−1

0.010

80 cm−1 160 cm−1

0.005 0.000 300

350

400

450

500

Excitonic splitting ∆e

550 600

(cm−1 )

Figure 1: Excitonic splitting energy distribution P (∆e ) for different values of static disorder σD .

For simulations of the 2D spectra, we used the third-order system response function formalism 10,25,26 . The dynamic disorder was accounted for by treating the dynamics of the bath as the overdamped Brownian oscillator and using the Drude–Lorentz spectral −1

density C 00 (ω) = 2λω (ω 2 + γ 2 ) . The reorganization energy λ and bath relaxation rate γ were both set to 50 cm−1 . The secular Redfield theory was used to account for the excited state lifetime and population transfer at 77 K temperature. As the excitation transfer rates are not directly related to the discussed effect, the limitations of the secular Redfield theory 27 are not crucial; therefore using more sophisticated non-secular theories 10,27 or system response functions 28 would not change here reported conclusions qualitatively. The spectra calculated for particular realizations from the chosen distribution were averaged over 1000 realizations of diagonal disorder.


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The 2D spectra, simulated using the same values of the static disorder as in Fig. 1, together with the corresponding absorption spectra and the waiting time t2 traces of the lower cross-peak are shown in Fig. 2a-c. For all these cases the spectral beats show significant changes with the disorder: the lifetime of coherent beatings when σD = 80 cm−1 (Fig. 2b) is significantly shorter than for σD = 40 cm−1 (Fig. 2a), however, increasing the static disorder up to σD = 160 cm−1 results in re-emergence of much longer coherence lifetime (Fig. 2c). (b)

12.5 12.5

σD = 40 cm−1

(c)

σD = 80 cm−1

5.0

3.1

12.0

6.3

σD = 160 cm−1

12.5

ω3

(103 cm−1 )

Abs. (a.u.)

(a)

11.0

Amplitude

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


3 2 1

-6.3

−ω1

12.0

(103 cm−1 )

13.0

-5.0 11.0

3 2 1

−ω1

12.0

(103 cm−1 )

13.0 3 2 1

0

0

0

-1 -2 -3

-1 -2 -3

-1 -2 -3

0

100

200

300

Waiting time t2

400 500 (fs)

0

100

-3.1 11.0

200

300

Waiting time t2

400 500 (fs)

0

−ω1

100

12.0

(103 cm−1 )

200

300

Waiting time t2

13.0

400 500 (fs)

Figure 2: Absorption spectra (upper panel), 2D spectra at delay time t2 = 0 fs (middle panel) and the lower cross-peak (indicated by ’X’ symbol) time dependencies (lower panel) in the case of static disorder σD equal to 40 cm−1 (a), 80 cm−1 (b) and 160 cm−1 (c). If the pure dephasing was zero and population/coherence transfer could be neglected, the lifetime of electronic coherent beats would eventually be determined by the excitonic gap distribution given in Eq. 3 alone. In order to characterize the beats at different system’s parameters in our model, we performed the Fourier transform over the waiting time t2 dynamics of each point in the spectrum, calculated by taking into account only the coherent contributions (coherent double-sided Feynman diagrams 10,25 ). The Fourier amplitude, integrated over all excitation/detection frequencies, for the fixed coherence fre7 ACS Paragon Plus Environment


∆0e

2J σD = 160 cm−1 (a.u.)

40 cm−1 80 cm−1

Anorm

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0

200

400

Frequency ω2

600

(cm−1 )

800

Figure 3: Averaged Fourier amplitude, extracted from the 2D spectra in cases of static disorder σD = 40 cm−1 , 80 cm−1 and 160 cm−1 . quency ω2 is shown in Fig. 3. The width, center frequency and height of these Lorentzian peaks in the dependency represent the lifetime, the observed frequency and the intensity of the beats, respectively. The Fourier analysis, thus, illustrates, that the observed lifetime and frequency of coherent beatings non-trivially depend on the disorder. In order to study a broad parameter space, we chose to fix the resonance coupling constant at J = −150 cm−1 and perform the Fourier analysis of the phase space with respect to the site energy gap δ0 and the standard deviation of the molecular site energy distribution σD . The resulting coherence intensity, lifetime and frequency dependencies are shown in Fig. 4a-c. As it can be seen in Fig. 4a, the amplitude of the quantum beats decrease both with the disorder and the site energy gap. However, for high values of δ0 , the coherence amplitude has its minimum at σD 6= 0 (see black line in Fig. 4a). The lifetime dependence (Fig. 4b) shows, that the longest coherence lifetime of ∼ 1.4 ps is achieved for just slightly disordered homo-dimers (σD → 0, δ0 → 0). Changing the disorder and the energy gap parameters gives a complicated pattern of the coherence lifetime dependence (note the solid line marking the longest lifetime in Fig. 4b). First of all, for the large disorder, the lifetime of beats weakly depends on the site energy gap δ0 and is around 0.8 − 1.0 ps. The shortest lifetime of coherent beats is for those systems, which are slightly disordered, but the monomer site energy difference is large (bottom right part of Fig. 4b).


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(a) Amplitude (a. u.) 200 180 160 140 120 100 80 60 40 20

(b) Lifetime (ps) 1.0

1.4 1.2 1.0 0.8 0.6 0.4 0.2

Static disorder σD (cm−1 )

200 180 160 140 120 100 80 60 40 20

Static disorder σD

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(cm−1 )

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0.0

0.0

(c) Frequency (cm−1 )

(d) Error (%)

∆0e

0.35

500

0.30 0.25 0.20 0.15 0.10 0.05

450 400 350 0

40

300 80 120 160 200 2J 0

Site energy gap δ0

−1

(cm

40

80 120 160 200 0.00

Site energy gap δ0

)

(cm−1 )

Figure 4: Dependencies of (a) normalized amplitude, (b) frequency and (c) lifetime of coherences on site energy difference δ0 and static disorder σD and for J = −150 cm−1 . Results are obtained from simulations with δ0 and σD steps of 10 cm−1 and then interpolated in figures. The error—mean square deviation of the Lorentzian fit—is shown in (d) as the ratio to the normalized amplitude of coherences (a). Solid black lines in (a) and (c) trace minimal amplitude and maximal lifetime, respectively. A remarkable result is shown in the coherence frequency dependence in Fig. 4c. It follows, that in most cases the observed oscillation frequency does not correspond to an excitonic energy gap ∆0e , but instead the double value of the resonance coupling constant, 2J (the same effect is conveyed in Fig. 3). This is due to the inhomogeneous distribution of ∆e , which for such parameters is wide enough and the coherence properties are dominated by the sub-ensemble of homo-dimers, maintaining δ0 ≈ 0. Interestingly, the area in the phase space which signifies an oscillation frequency, distinct from 2J, (bottom right portion of Fig. 4c) closely matches the parameters, which lead to shortlived (< 300fs) quantum beats in Fig. 4b. The properties of coherent beats, illustrated by the phase space analysis in Fig. 4, gives a clear picture of interplay between the inhomogeneous and homogeneous dephasing in the molecular dimer. As the homogeneous dephasing is fixed by choosing a particular spectral density function, the variation of amount of static disorder provides the possibility to demonstrate the relative effect of both broadening mechanisms. The longest coherence lifetimes are obtained for dimers, where the effect of static disorder is neg9 ACS Paragon Plus Environment


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ligible; as discussed above, such a regime is achieved in two cases: when the dimer is composed of identical molecules of similar site energies and the distribution of the site energies is narrow, or, on the contrary, when the disorder value is high, resulting in signal, dominated by the oscillatory contribution of the sub-ensemble of homo-dimers. This gives the possibility to generalize the same conclusion to any molecular aggregates since in the case of the large values of disorder the selected dimers will be responsible for the long-lived coherence of the electronic origin. If we define the inhomogeneous dephasing as the process of decay of quantum beats in the spectrum, caused by the static disorder in the ensemble, and decoherence as the disruption of the coherent exciton, caused by the dissipative system–bath coupling, the concurrence of dynamic and static disorder translates into interplay of decoherence and spectral dephasing, respectively. Thus we suggest, that the situation, when the static energy disorder is effectively excluded and long-lived quantum beats are observed, can be referred to as the “decoherence-limited” regime of quantum coherences. Depending on the system parameters, our proposed mechanism of decoherence-limited electronic quantum coherence beats might be very important regarding experimental observables. Coherence beats of electronic origin in principle could have been observed in a range of recent experiments of 2D electronic spectroscopy 14,15,29–34 . For example, long-lived quantum beats of electronic nature were successfully captured in 2D spectra of circular nanoring composed of six identical porphyrins 34 . With the static energy disorder around σD = 100 cm−1 and resonant couplings up to ∼ 400 cm−1 , the conditions for the decoherence-limited quantum beats were ensured and electronic quantum coherences of 80 cm−1 surviving longer than 0.5 ps were reported. The conditions for the decoherence-limited quantum beats (σD = 500 cm−1 ) were fulfilled for the biscyanine homo-dimer, studied using the 2D electronic spectroscopy 32 . However, the analysis of spectral region of experimental data, where the expected electronic coherence of 2J ≈ 1600 cm−1 could have been extracted, was not performed and only beatings, assigned to strong vibrational mode of 1400 cm−1 , were reported.


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A natural (almost) dimeric system that could be addressed in the perspective of the considered effect is the bacteriochlorophyll–bacteriopheophytin dimer in the photosynthetic bacterial reaction center. The resonant coupling constant and the amount of static disorder were suggested to be J = 202 cm−1 and σD = 55 cm−1 35 and the site energy gap δ0 = 650 cm−1 36 . With such parameters, our scheme produces only short-lived electronic quantum beats 30,31 . The conditions of decoherence-limited regime were not met in the system of weakly coupled dimers in the water soluble chlorophyll-binding protein, what explains, why the quantum electronic coherences of 305 cm−1 and 610 cm−1 witnessing quantum superpositions between known energy states were not extracted 33 . The quantum coherences were theoretically shown to be very significant for the charge separation in the photosystem II reaction center. Here strong electronic–vibrational mixing with vibrational mode of 250 cm−1 results in mixed quantum beats 14,15 . The presented effect of elimination of the static disorder contribution to the quantum beats in the 2D spectrum is easily explained for the simple system of the electronic dimer. However, it still remains to be shown, how it would extend to the spectral signals of multi-chromophore systems. For example, it would be very useful to evaluate this effect on the spectral signals of the FMO complex 29 , where the quantum beats were observed first. The influence of the inhomogeneous broadening to the spectral evolution of the FMO complex must be noticeable since its 2D spectrum is dominated by the significantly elongated and overlapping spectral features. As the resonance coupling constants in FMO are less than 100 cm−1 and the standard deviation of the Gaussian distribution of the molecular energies is around 60 cm−1 37 , the disorder induced beats could arise for the low frequency (< 200 cm−1 ) quantum beats. In conclusion, the effect of disorder induced electronic quantum coherences has many implications and complications regarding the experimental observation of coherent beats. Firstly, the frequencies of the detected beats do not necessarily match the frequencies of the corresponding excitonic energy gaps, which can be estimated, for example, as differences between resolved peaks in the absorption spectrum. Secondly, these beats in the 2D spectrum at initial waiting times might still be overwhelmed by the short-lived



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quantum beats of frequencies, corresponding to the excitonic energy gaps, making the observed oscillation frequency dependent on the waiting time. Exciton delocalization is a decisive phenomenon, controlling the amplitude and lifetimes of coherences, affected both by dephasing and decoherence. If a complete exciton delocalization is not ensured by the system’s parameters and static disorder is negligible, electronic quantum coherences will be very weak and/or short-lived, and difficult to be captured experimentally. However, a certain amount of static disorder can drive the system to the decoherence-limited regime, where the long-lived electronic quantum coherences will be observed in time-resolved coherent spectroscopy.

Acknowledgement This work was supported by the Research Council of Lithuania (LMT grant no. MIP080/2015), U.S. Department of Energy under contract DE-AC02-05CH11231 and the Division of Chemical Sciences, Geosciences and Biosciences Division, Office of Basic Energy Sciences through grant DE-AC03-76SF000098 (at LBNL and U.C. Berkeley). V. B. acknowledges support by project “Promotion of Student Scientific Activities” (VP1-3.1ŠMM-01-V-02-003) from the Research Council of Lithuania.

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Disorder-Induced Quantum Beats in Two-Dimensional Spectra of

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