Exciton Recurrence Motion in Double-Ring Molecular Aggregates

J. Phys. Chem. C 2010, 114, 6067–6076

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Exciton Recurrence Motion in Double-Ring Molecular Aggregates Induced by Two-Mode Circular-Polarized Laser Field Takuya Minami, Kyohei Yoneda, Ryohei Kishi, Hideaki Takahashi, and Masayoshi Nakano* Department of Materials Engineering Science, Graduate School of Engineering Science, Osaka UniVersity, Toyonaka, Osaka 560-8531, Japan ReceiVed: August 30, 2009; ReVised Manuscript ReceiVed: February 23, 2010

Exciton recurrence motions in double-ring molecular aggregates are investigated using the quantum master equation approach. The rotatory and oscillatory recurrence behaviors are found to be controlled by changing the relative direction of the circular polarization of an incident two-mode laser field. These dynamics are understood by the relative phases among off-diagonal exciton density matrices, the feature of which is determined by the interaction between the circular-polarized field and the degenerate states of the doublering structure. The results of exciton relaxation effects caused by the exciton-phonon coupling and static disorder also demonstrate the robustness of the exciton recurrence motions in double-ring molecular aggregates. 1. Introduction Coherent and incoherent dynamics of exciton (electron-hole pair) in molecules and molecular aggregates have been intensively investigated toward a fundamental understanding of quantum dynamical phenomena in nanomaterial and biological systems and a new development of molecular-based nanoscale devices.1–26 Laser irradiation can induce the coherent oscillation of excitation, i.e., exciton recurrence motion, among chromophores in super- and supramolecules by creating superposition states composed of plural excited states.27–36 Yamazaki et al. have experimentally observed the exciton recurrence motion by detecting the damping oscillation of fluorescence anisotropy (which persists for several picoseconds) in several anthracene dimers.33–35 They have clarified the mechanism of exciton recurrence motion in molecular aggregates composed of dipolecoupled two-state monomers based on the Frenkel exciton model.37 In our previous paper, we have theoretically investigated the exciton recurrence motion in ring-shaped aggregate complexes induced by a circular-polarized laser field and have observed the rotatory exciton recurrence motions.36 The rotatory dynamics have been found to be governed by the interaction between a circular-polarized field and two pairs of degenerate exciton states, originating in three- or higher-fold rotation symmetry of the molecular aggregate. In addition, mutually counter-rotatory exciton recurrence motions have turned out to be induced simultaneously by a two-mode circular-polarized laser field when the system has several pairs of degenerate states with large energy differences and sufficiently small overlaps of exciton distributions. These results suggest that there is still plenty of room to create and control other types of exciton recurrence motions in rotationally symmetric aggregates with circularpolarized laser fields. In this paper, therefore, we investigate the exciton recurrence motion in double-ring molecular aggregates, after irradiation of a two-mode circular-polarized laser field using the quantum master equation (QME) approach.38 We here consider the dynamics of the model system under the Markov approximation for simplicity in order to extract the * Corresponding author: fax, +81 668506268; e-mail, mnaka@ cheng.es.osaka-u.ac.jp.

structure-property relationship of exciton recurrence motion in the double-ring system as well as to examine its robustness against the Markov dissipation effects. Two kinds of incident laser fields are considered, that is, a two-mode circular-polarized laser field with the same rotation direction (con-rotatingpolarization) and that with mutually opposite rotation direction (counter-rotating-polarization). From the comparison between the exciton states and exciton dynamics in different-size doublering aggregates, we discuss the features and the origin of exciton recurrence motion in the double-ring structure. We also investigate the phase relaxation effects and the static disorder effects on the exciton dynamics in the con-rotating-polarization case using several relaxation parameters and bicentric doublering structures, respectively. The present study will contribute to the investigation of spatiotemporal dynamics of exciton and to the development of a novel control scheme of excitonic coherence in nanoscale materials, which will be realized by the recent nanofabrication technology.39,40 2. Methodology 2.1. Quantum Master Equation Approach. We briefly explain the QME approach to the exciton dynamics in dipole-dipole coupled aggregate systems.30,31,36,38 This consists of a two-step procedure: (1) an exciton state model for an aggregate is constructed, and (2) the time evolution of exciton is carried out by numerically solving the QME. We consider a molecular aggregate model composed of two-state monomers with excitation energies {ωi} and the magnitudes of transition moments {µi} (i ) 1, 2, ..., N, where N is the number of monomers). The Hamiltonian HS for the exciton state is expressed by N

HS )

∑

ωi |i〉〈i| +

i

1 2

N

∑ Jij|i〉〈j|

(1)

i,j

Here, {|i〉} indicates the aggregate basis, which represents the excitation localized on monomer i, and Jij represents the dipole-dipole coupling constant between monomers i and j, given by

10.1021/jp908373k  2010 American Chemical Society Published on Web 03/10/2010

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1 µi µj{cos(θij - θji) - 3 cos θij cos θji} Rij3

Jij )

(2)

Rij is an intermolecular distance and θij(θji) is the angle between the transition moment of monomer i(j) and the vector drawn from monomer i to j. The one-exciton basis {|ψR〉} with excitation energies {ωR} and transition moments (µR,β ) µR,βxex + µR,βzez, where ex and ez represent the orthogonal unit vectors on the aggregate plane, i.e., z-x plane) obtained by diagonalizing the HS matrix are expressed as N

N

i

i

∑ |i〉〈i|ψR〉 ≡ ∑ |i〉CiR,

|ψR〉 )

(R ) 2, ..., M)

(3) where M is equal to N + 1 in the one-exciton model, and R ) 1 indicates the vacuum state of exciton. The time evolution of on- and off-diagonal exciton density matrices in the exciton state basis {|ψR〉} is performed using the QME in the Born-Markov approximation38

dFex RR )dt

M

∑

M

ex ΓRR;mmFmm

+ iF ·

m

∑

(µRnFex nR

-

Fex Rn µnR)

n

(4) and ex dFRβ ex ) -i(ωR - ωβ)FRβ dt

M

ex + ∑ ΓRβ;mnFmn M

∑ (µRnFnβex - FexRnµnβ)

F ) F1(ex cos ω1t ( ez sin ω1t) + F2(ex cos ω2t ( ez sin ω2t)

∑ 〈i|ψR〉〈ψR|F(t)|ψβ〉〈ψβ|i〉 ) ∑ CiRFRβex (t)C*iβ

Fiiagg(t) )

R,β

(5)

R,β

(10) 2.2. Exciton Recurrence Motion. The fundamental mechanism of exciton recurrence motion is briefly explained in this section. We assume the system after cutting off applied laser field at t0. For simplicity, the nonrelaxation case (ΓRβ;mn ) 0, in this case, eqs 4 and 5 correspond to the quantum liouvile equation) is considered. Using eqs 4, 5, and 10, the exciton population on monomer i is described as (see Appendix A for derivation)

∑ ∑C R

n

+2

∑ ∑C R>β m,n

ex iRC* iRFRR(t0)

m

ex iRC* iβ |FRβ(t0)|

[

cos (ωR - ωβ)t -

(R * β)

arctan

The relaxation parameters are represented by38 M

ΓRR:mm ) 2δRm

N

∑ ∑ |CiR| |Cik| γ(i,i)(ωm - ωk) 2

k

2

i

N

2

∑ |CiR|2|Cim|2γ(i,i)(ωm - ωR)

(6)

i

and

ΓRβ;mn )

N

k

i

∑ ∑ [δβnC*iR|Cik|2Cimγ(i,i)(ωm - ωk) + δRmC*in |Cik | 2Ciβγ(i,i)(ωn - ωk)] -

N

∑ [C*iRCimC*C in iβ{γ(i,i)(ωm - ωR) - γ(i,i)(ωn - ωβ)}]

(7)

(

ex Im[FRβ (t0)] ex Re[FRβ (t0)]

)]

(11)

The second term represents the coherent exciton oscillation due ex , where its amplitude and to off-diagonal density matrix FRβ frequency are determined by expansion coefficient products CiRC*iβ () CiRCiβ, due to real numbers) and energy difference (ωR - ωβ) between exciton states R and β. When exciton states R and β are coherently created and the products of expansion coefficients on monomers i and j satisfy the following condition

CiRCiβ > 0 M

(9)

where the left(+)- and right(-)-hand circular polarizations are defined as the clockwise and anticlockwise rotations of polarizations, respectively, when viewed along the direction of the field propagation. Equations 4 and 5 are numerically solved by the fourth-order Runge-Kutta method. The spatial dynamics of exciton distribution is obtained by the basis conversion from the delocalized exciton state basis {|ψR〉} to the localized aggregate basis {|i〉}

Fagg ii (t) )

m,n

iF ·

The form of γ(i,i)(ω) is taken to satisfy the thermal equilibrium condition:41 γ0(i,i) indicates the high-temperature limit of γ(i,i)(ω), and kB is the Boltzmann constant. We here consider a twomode circular-polarized laser field F represented by

and

CjRCjβ < 0

(12)

ex induces the oscillation of exciton population between then, FRβ monomers i and j. In this way, coherent excitation creating superposition states of exciton states induces the exciton recurrence motion among constituent monomers in super- and supramolecules.

i

where

γ(i,i)(ω) )

0 2γ(i,i) 1 + exp(-ω/kBT)

(8)

3. Results and Discussion 3.1. Molecular Aggregate Models. Figure 1 shows the double-ring molecular aggregate models composed of two-state monomers (shown by arrows) with excitation energies of 38000 cm-1 and transition moments of 10 D. The distance between adjacent monomers in the inside and outside rings is set to 15

Exciton Recurrence Motion

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Figure 1. Structures of double-ring aggregates involving eight monomers in the inside ring, while 16 (a), 20 (b), and 24 (c) monomers are in the outside rings. X ()3 or 7) represents the difference in the center position between the inside and outside rings for models (aX), and ∆R represents the smallest inter-ring distance between the inside and outside rings.

Figure 2. Excitation energies, ωR, and the magnitude of transition moments, |µ1R|, between the ground (1) and exciton states (R) for models a, b, c, and aX (X ) 3, 7).

au (∼7.94 Å). Models a, b, and c are the concentric doublering models with different-size outside rings, while models (aX) represent the bicentric models with a difference in the center position X () 3 or 7 au) between the inside and outside rings, the sizes of which are identical to those in model a, respectively. Figure 2 shows the excitation energies ωR and the magnitudes of transition moments from the ground (1) to exciton (R) states, x 2 z 2 1/2 µ1R () ((µ1R ) + (µ1R ) ) ), for models a, b, c, and aX. Models a, b, and c, which are the concentric double-ring models, possess

only two pairs of degenerate states with nonzero transition moments due to the rotation symmetry of the inside and outside rings. On the other hand, models (aX), which have bicentric double-ring structures, possess several pairs of near-degenerate states with nonzero transition moments due to their asymmetric structures. In the following sections, we discuss the coherent exciton dynamics for model a and compare it with the dynamics for other models. Accordingly, it is meaningful to discuss the feature of the products of expansion coefficients CiRCiβ for these

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Figure 3. Products of expansion coefficients CiR for states 4, 5, 8, and 9 (states 4, 5, 9, and 10 for model c). The white and black circles represent positive and negative signs of CiRCiβ, respectively, and the sizes represent the magnitudes.

models (see Figure 3). The magnitude and sign of CiRCiβ ex primarily determine the contributions of FRβ to the recurrence amplitudes and phases of exciton distributions among constituent monomers, respectively (see eq 11) in optically allowed exciton states, e.g., states 4, 5, 8, and 9 for model a. The size and color of the circle represent the magnitude and sign of CiRCiβ for monomer i, respectively. The other combinations of CiRCiβ, e.g., Ci4Ci5 for model a, are excluded due to the nonoscillatory behavior of the corresponding Fex Rβ composed of the superposition states of a degenerate pair. As shown in Figure 3a, the exciton distribution patterns for Ci4Ci8 and Ci5Ci9 predict the inter-ring exciton recurrence motion due to the phase separation (node line) between the inside and outside rings, whereas those for Ci4Ci9 and Ci5Ci8 predict the intraring exciton recurrence motion due to the phase separation (node lines) within each ring. Similarly, for models b and c, the inter- and intraring recurrence motions are predicted from the exciton distribution patterns of {Ci4Ci8 and Ci5Ci9 (Ci5Ci11)} and {Ci4Ci9 and Ci5Ci8 (Ci5Ci10)}, respectively (see Figure 3, panels b and c). Similar intraring and inter-ring phase separations are predicted to be reproduced even in other analogous double-ring molecular aggregates. This is because the distributions and signs of CiRCiβ depend on the structural symmetry and features of molecular aggregates through the intermonomer interaction. However, the amplitude of CiRCiβ tends to be small as the inter-ring interaction becomes weak (see Figure 3a-c). This corresponds to the exciton localization on each ring when the inter-ring interaction is weak.

In contrast to models a-c, models (a3) and (a7) do not preserve the distinct inter- and/or intraring exciton phase distribution patterns (see Figure 3a3 and Figure 3a7) because nonuniform interactions among monomers cause asymmetric distributions of expansion coefficients and their products. It is therefore predicted that a similar phase distribution pattern is observed among models a, b, and c, originating in the concentric doublering structure, though the static disorder, e.g., slight inconformity of the centers between the inside and outside rings, might somewhat attenuate the recurrence dynamics. 3.2. Exciton Dynamics in Double-Ring Molecular Aggregate. Model a is coherently excited by a two-mode circularpolarized laser field with frequencies of (ω1, ω2) ) (35909 cm-1, 36714 cm-1), resonating to exciton states (4, 5) and (8, 9). The field intensity and irradiating cycles are set to I1 ) I2 ) 10 MW/cm2 and 400 optical cycles (∼0.367 ps), respectively. Two kinds of polarizations are examined, that is, both of modes 1 and 2 have left-hand circular polarizations [con-rotatingpolarization case (LL)], and modes 1 and 2 have right- and left-hand circular polarizations, respectively [counter-rotatingpolarization case (RL)]. We set the temperature and hightemperature limit of relaxation parameter to T ) 300 K and 0 ) 10 cm-1, respectively, the value of which is employed γ(i,i) to reproduce the similar dephasing time of several picoseconds for 2,15-dithia[3,3](1,5)anthracenophane (DTA) in THF solution.33 Figure 4 shows the time-evolution of exciton population dynamics between the inside and outside rings (II), real parts



Figure 4. Time evolutions of exciton populations between the inside and outside rings (II), off-diagonal density matrices (III), and snapshots of exciton density distributions in one recurrence cycle (shown in dotted square) (I) after irradiation of a two-mode circular-polarized laser field with con-rotating-polarization (ILL-IIILL) and counter-rotating-polarization (IRL-IIIRL) for model a. Applied laser field resonates to the exciton states 4, 5, 8, and 9, having (ω1, ω2) ) (35909 cm-1, 36714 cm-1) with intensities of I1 ) I2 ) 10 MW/cm2 and cutoff time of 400 optical cycle (∼0.367 0 ps). Temperature and high-temperature limit of relaxation parameters are set to be T ) 300 K and γ(i,i) ) 10 cm-1, respectively.

of the off-diagonal density matrices in the exciton basis (III), and snapshots of exciton density distributions (I) (the size of the blue circle represents the magnitude of Fagg ii ) in one recurrence cycle shown in dotted square (see II or III). Con- and counterrotating-polarization cases are shown in ILL-IIILL of Figure 4 and IRL-4IIIRL of Figure 4, respectively. ILL and 4IILL of Figure 4 show the inter-ring exciton recurrence motion between the inside and outside rings, whereas IRL and IIRL of Figure 4 show the intraring exciton rotation in the clockwise direction. We ex between compare the time evolution of the real parts of FRβ con- and counter-rotating-polarization cases in order to grasp the difference between these recurrence motions. In the conex ex and F59 oscillate with identical rotating-polarization case, F48 ex ex phases, while F49 and F58 do with the opposite phases (see Figure 4IIILL). In contrast, in the counter-rotating-polarization case, Fex 49 ex ex ex and F58 oscillate with same phases, while F48 and F59 do with opposite phases (see Figure 4IIIRL). Taking into account CiRCiβ distributions (see Figure 3a) together with relative phase among ex , the con-rotating-polarization case intensifies the inter-ring FRβ phase distribution pattern (Ci4Ci8 and Ci5Ci9), while it almost completely cancels out the intraring pattern (Ci4Ci9 and Ci5Ci8). In contrast, the counter-rotating-polarization case cancels out the inter-ring phase distribution pattern (Ci4Ci8 and Ci5Ci9), while it intensifies the intraring phase distribution pattern (Ci4Ci9 and ex Ci5Ci8). It is thus found that the relative phases among FRβ determine the feature of exciton recurrence motion. We next ex illustrate the origin of the relative phase differences in FRβ between con- and counter-rotating-polarization cases. As shown in ref 42, a circular-polarized laser field in resonance with the degenerate states induces the superposition state composed of these states with relative phase difference of π/2

|ψ((t)〉 )

1 (|ψx〉 ( i|ψz〉)e-iωt √2

(13)

where |ψk〉 indicates the exciton state with k-polarized transition moment and |ψ( 〉 indicates the exciton states induced by

left(+)- and right(-)-hand circular-polarized laser fields, respectively. In the same way, the coherent excitation of two pairs of degenerate states (R and β) creates the superposition state

|Ψ(t)〉 )

{

1 1 (|ψRx〉 ( i|ψRz〉)e-iωRt + √2 √2 1 (|ψβx〉 ( i|ψβz〉)e-iωβt √2

}

(14)

where ωβ > ωR is assumed. The signs of the second and the fourth terms depend on the polarization direction of the resonant laser field. The relationships of the real parts of off-diagonal density matrices in the con-rotating-polarization case are expressed as (see Appendix B for derivation) ex ex ex ex (Re[FRxβx ], Re[FRzβz ], Re[FRxβz ], Re[FRzβx ]) ∝ (cos ∆ωt, cos ∆ωt, sin ∆ωt, -sin ∆ωt) (15) ex where ∆ω ) ωβ - ωR. This indicates that Re[Fex Rxβx] and Re[FRzβz] ex ex exhibit in-phase time-evolution, while Re[FRxβz] and Re[FRzβx] do antiphase evolution. On the other hand, in the counterrotating-polarization case, we obtain (see Appendix B for derivation)

ex ex ex ex (Re[FRxβx ], Re[FRzβz ], Re[FRxβz ], Re[FRzβx ]) ∝ (cos ∆ωt, -cos ∆ωt, sin ∆ωt, sin ∆ωt) (16)

ex which shows that Re[Fex Rxβx] and Re[FRzβz] exhibit antiphase time ex ex evolution, while Re[FRxβz] and Re[FRzβx] do in-phase evolution. Namely, this analysis indicates that the in- and antiphase ex ] are switched between the con- and combinations of Re[FRβ counter-rotating-polarization cases as shown in model a. It is thus generally predicted that the exciton recurrence motion, driven by two pairs of degenerate states, can be controlled by

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Figure 5. Time evolutions of off-diagonal density matrices with snapshots of exciton density distribution in one recurrence cycle (shown in dotted square) for models b and c in the con-rotating-polarization case. Applied laser field resonates to exciton states 4, 5, 8 (10), and 9 (11), having (ω1, ω2) ) (35831 (35771) cm-1, 36681 (36672) cm-1) with intensities of I1 ) I2 ) 10 MW/cm2 and cutoff time of 350 (320) optical cycles for model b (for model c). Temperature and hightemperature limit of relaxation parameters are set to be T ) 300 K and 0 γ(i,i) ) 10 cm-1, respectively.

tuning the polarized direction of two-mode circular-polarized laser fields such as con- and counter-rotating-polarization cases. We also examine the exciton dynamics induced by two-mode circular-polarized laser field in the con-rotating-polarization case for models b and c to compare with the corresponding dynamics in model a. The frequencies, intensities, and irradiating cycles of the incident laser field are set to be (ω1, ω2) ) (35831, 36681 cm-1), I1 ) I2 ) 10 MW/cm2 and 350 optical cycles (∼0.322 ps), respectively, for model b, and those are set to be (ω1, ω2) ) (35771, 36672 cm-1), I1 ) I2 ) 10 MW/cm2 and 320 optical cycles (∼0.295 ps), respectively, for model c. Theses frequencies are chosen to resonate with exciton states 4, 5, 8, and 9 for model b, and states 4, 5, 10, and 11 for model c. Parts b and c ex with snapshots of of Figure 5 show the time-evolutions of FRβ exciton density distributions in one recurrence cycle shown in dotted square. The inter-ring exciton recurrences with smaller amplitudes compared to model a are observed for both models b and c though the amplitudes of Fex Rβ are almost the same. These differences in recurrence amplitude are simply derived from the magnitudes of CiRCiβ explicitly shown in parts a-c of Figure 3. Similarly, it is easy to foresee that the counter-rotatingpolarization case also reproduces the exciton rotations, though they have smaller amplitudes, for models b and b. As discussed in section 3.1, the spatial distributions and signs of CiRCiβ, which determine the oscillating pattern of exciton recurrence motion, are predicted to be similar among the molecular aggregates with analogous structure. Their amplitudes, however, depend on the intensity of intermonomer interaction. We therefore conclude that the double-ring molecular aggregate inherently possesses the common oscillating pattern of exciton recurrence motion, characterized by intra- or inter-ring oscillation. In addition, twomode circular polarized laser field will induce the inter-ring oscillation or intraring rotation of exciton in the double-ring ex . However, system by controlling the relative phases among FRβ

Minami et al.

Figure 6. Time evolutions of exciton populations in the con-rotatingpolarization case between the inside and outside rings with relaxation 0 parameters: γ(i,i) ) 10 cm-1 (I), 50 cm-1 (II), and 200 cm-1 (III) for model (a). Temperature is set to be T ) 300 K. See the legends of Figure 4 for further explanation.

sufficient intensity of inter-ring interaction is required for inducing such exciton recurrence motion. 3.3. Phase Relaxation Effect. Next, we examine the exciton dynamics using several relaxation parameters in order to evaluate the robustness of exciton recurrence motion in the double-ring molecular aggregate against phase relaxation. Figure 6 shows the time evolutions of exciton populations calculated under the condition of room temperature (T ) 300 K) and using 0 ) 10, a high-temperature limit of relaxation parameters, γ(i,i) -1 50, 200 cm , for model a in the con-rotating-polarization case. 0 ) 10 and 200 cm-1 are chosen so as to The values of γ(i,i) reproduce the oscillation decay time of ∼3 ps in the fluorescent anisotropy experiment for DTA33 and the fast photon energy harvesting period in phenylacetylene nanostar dendrimer43,44 at room temperature, respectively. The frequencies (ω1, ω2) ) (35909 cm-1, 36714 cm-1), intensities I1 ) I2 ) 10 MW/cm2 and irradiating cycles (400 optical cycles (∼0.367 ps)) of the incident laser field are identical to those employed in Figure 4a, respectively. The dotted line inserted in Figure 6 represents the cutoff time of the laser field. It is found that the recurrence amplitude during irradiating laser field becomes smaller with increasing the relaxation parameter. Especially for Figure 6III, we can hardly observe the population oscillation because the 0 ) 200 cm-1) immediately large relaxation parameter (γ(i,i) destroys the coherence between the exciton states. These results show that relaxation parameter is closely related not only to the dephasing time but also to the recurrence amplitude. For model a in γ0(i,i) ) 50 cm-1 (Figure 6I), the recurrence amplitude is large enough to be observed, the duration time of which is ∼200 fs after cutting of laser field. Considering the dependence of recurrence amplitude on the inter-ring interaction (see section 3.2), if sufficient inter-ring interaction exists like model a, the exciton recurrence motion can emerge even in the case of the 0 value (50 cm-1, Figure 6II), corresponding to intermediate γ(i,i) the oscillation decay time of several hundreds of femtoseconds. 3.4. Static Disorder Effect. In this section, we discuss the static disorder effect originating in the disagreement of center position between inside and outside rings. The exciton recurrence motion is examined and compared among the bicentric


J. Phys. Chem. C, Vol. 114, No. 13, 2010 6073 TABLE 1: Excitation Energy Differences between States 4, 5, 8, and 9 for Models a and aX excitation energy difference ∆ωRβ () ωβ - ωR) (cm-1) model a model a3 model a7

Figure 7. Time evolutions of off-diagonal density matrices for models aX (X ) 3, 7) in the con-rotating-polarization case with snapshots of exciton density distributions in one recurrence cycle (shown in dotted square). See the legend of Figure 4 for further explanation.

double-ring molecular aggregate models (aX) and the concentric double ring model (a). The parameters of incident laser field are equal to those shown in the legends of Figure 4, that is, the frequencies, intensities, and irradiating cycles are (ω1, ω2) ) (35909 cm-1, 36714 cm-1), I1 ) I2 ) 10 MW/cm2, and 400 optical cycles (∼0.367 ps), respectively. The temperature and high temperature limit of relaxation parameter are set to be T 0 ) 10 cm-1, respectively. Figure 7 shows the ) 300 K and γ(i,i) ex for 0.5 time evolutions of the off-diagonal density matrices FRβ ps with snapshots in one recurrence cycle for models a3 and a7 after irradiating the two-mode circular-polarized laser field with the same rotating direction. It is noted that model a3 shows the inter-ring exciton recurrence motion and characteristic phase dynamics of Fex Rβ similar to model a, while the clockwise exciton rotation also appears in the outside ring (see Figure 7a3 and Figure 4 in the con-rotating-polarization case). This additional intraring rotation of exciton originates in the asymmetric interactions among the constituent monomers, which are characterized by CiRCiβ distributions in the outside ring (see Figure 3a3). Considering the CiRCiβ distributions and relative ex , the intraring recurrence patterns of Ci4Ci9 phases among FRβ and Ci5Ci8 are not completely canceled, especially in the outside ring in contrast to model a. This result shows that a small deviation in center positions of the double rings hardly changes the inter-ring recurrence inherent in model a, while it induces an additional exciton rotation in the outside ring. Although model a7 also shows the asymmetric exciton distribution (see Figure 3a7), no distinct inter- and intraring rotations of exciton are observed in contrast to model a3 (see Figure 7a7). This is understood by the disorder in relative phase among Fex Rβ. In fact, ex is no longer negligible at the relative phase disorder for FRβ ex ex (blue line with colored square) and F58 (green ∼300 fs, e.g., F49 line with colored rhombus) do not have mutually opposite phase completely. It is, however, noted that the ordered relative phases are observed in earlier time region. This time dependence of relative phase disorder explicitly denotes the existence of ex , derived from the energy frequency difference among FRβ splitting between near-degenerate states. Table 1 lists the excitation energy differences ∆ωRβ () ωβ - ωR) among exciton states 4, 5, 8, and 9, which correspond to the frequencies of

∆ω45

∆ω48

∆ω49

∆ω58

∆ω59

∆ω89

0 1 15

804 823 886

804 825 894

804 822 871

804 824 879

0 2 8

ex ex . For a completely concentric model a, all the induced FRβ FRβ ex ex except for F45 and F89 (the superposition of the same degenerate pairs) develop with the same frequencies and thus persist their relative phases. In contrast, for bicentric model aX, exciton states 4 and 5 (8 and 9) are no longer degenerate and thus lead to the relative phase modulation among Fex Rβ according to the frequency differences among them. The subsequent discussion focuses on ex the relative phase modulation between Fex 49 and F58, the frequency difference of which is the largest among other combinations of ex ex ex ex , F49 , F58 , and F59 . For model a7, the ∆ω49 - ∆ω58 ) 23 F48 -1 ex and cm , leading to the relative phase modulation between F49 ex in the period of ∼1.45 ps. This is rapid enough to affect the F58 ex with a period of subpicoseconds. In relative phases among FRβ contrast, for model a3, the relative phase modulation is insignificant just after cutting off the laser field due to the slight frequency difference (∆ω49 - ∆ω58 ) 3 cm-1), corresponding to the period of ∼11.1 ps. These results indicate that the relative phase modulation with the period of ∼10 ps is allowed to retain the exciton recurrence motion with a period of subpicoseconds inherent in the double-ring structure. In order to clarify the ex on the center dependence of relative phase disorder among FRβ deviation in the double rings, we also examine other bicentric double-ring models bX and cX (not shown in the figure), which correspond to models b and c, respectively, with center deviations of X a.u. between the inside and outside rings. Figure 8 shows the time period giving relative phase ex ex ex ex and F58 (F4,11 and F5,10 ) calculated modulation between F49 from ∆ω49 + ∆ω58 (∆ω4,11 + ∆ω5,10) for models aX and bX (for model cX), as a function of the smallest inter-ring distance ∆R (see Figure 1aX). It is noted that phase modulation periods attain 10 ps at ∆R ∼17 au for all the models, regardless the center deviation X. It is presumable that these energy splittings become significant when asymmetric inter-ring interactions compete with symmetric intraring interactions. In addition, steep change in energy difference, derived from the 1/R3 dependence of intermonomer interaction (see eq 2), assures that the slight center deviation hardly affects the ex . It is conserelative phase modulation among induced FRβ quently found that the inter-ring recurrence motion inherent in the double-ring aggregates is robust for slight center

ex ex Figure 8. Time periods of phase modulation between Fex 49 and F58 (F4,11 and Fex 5,10) calculated from ∆ω49 + ∆ω58 (∆ω4,11 + ∆ω5,10) for models aX and bX (for model cX), as a function of the smallest inter-ring distance ∆R. Inserted dotted line shows a time period of 10 ps.

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deviations of the double rings though that causes an additional intraring exciton rotation in the outside ring. 4. Concluding Remarks We have investigated the inherent exciton recurrence motion in molecular aggregate systems with double-ring structure and the controllability of exciton recurrence motion using a twomode circular-polarized laser field, as well as the phase relaxation in the Markov approximation and static disorder effects. It is found that the inter-ring exciton recurrence motion is induced by irradiation of laser field with con-rotatingpolarization, while the intraring exciton rotation is induced by irradiation of laser field with counter-rotating-polarization. These differences between con- and counter-rotating-polarization cases are explained by the difference in relative phases among excitonic superposition states, originating in the interaction between circular-polarization and degenerate exciton states. We have predicted that such a control scheme of exciton recurrence motion using a two-mode circular-polarized laser field can be applied to other types of molecular aggregates with rotation symmetry. From the comparison of the results of different-size double-ring aggregate models, we have found the strong dependence of the recurrence amplitude on the inter-ring interaction. The exciton recurrence motions using several relaxation parameters in QME approach reveal the robustness of such exciton recurrence motion against vibrational relaxation (phase relaxation) when sufficient inter-ring interaction exists. On the other hand, the static disorder, originating from the deviation in center positions of the double rings, insignificantly affects the exciton recurrence behavior inherent in the concentric double-ring aggregate, as long as the intraring interaction is sufficiently stronger than the inter-ring interaction. These results predict that the strong inter-ring interaction induces the robust exciton recurrence motion against vibrational relaxation, while too strong an inter-ring interaction tends to cause the relaxation of the recurrence motion due to the static disorder such as large deviations in center positions of the double rings. The present results could be observed on structure-controlled molecular aggregate systems composed of quantum dots45 or artificial ring-shaped aggregates46,47 by using two-dimensional spectroscopy, which can unravel the excitonic coherence in complex molecular systems such as photosynthetic systems.8,9,48 It is also interesting to experimentally investigate how the control of relative phase differences among excitonic coherences predicted in our study impacts on the energy transfer process in artificial and real molecular systems. Although, in the present study, we have examined the exciton dynamics under the Markov approximation by assuming that the phonon-bath is consistently in equilibrium, one may consider that a nonMarkovian approach is preferable for ultrafast exciton dynamics when the phonon correlation time is comparable to or larger than the period of dynamical behavior of exciton, e.g., exciton systems in condensed phase.49,50 The non-Markovian effects on the exciton recurrence motion predicted in this study will be investigated at the next stage. Acknowledgment. This work is supported by Grant-in-Aid for Scientific Research (Nos. 21350011 and 20655003) from Japan Society for the Promotion of Science (JSPS), Grantin-Aid for Scientific Research on Priority Areas (No. 18066010) from the Ministry of Education, Science, Sports and Culture of Japan, and the global COE (center of excellence) program “Global Education and Research Center for Bio-Environmental Chemistry” of Osaka University.

Minami et al. Appendix A: Derivation of Equation 11. We here assume the nonrelaxation case (ΓRβ;mn ) 0). After the laser field is irradiated (cutoff time is represented as t0, when t g t0, F ) 0), eq 4 is expressed as

dFex RR )0 dt

(A1)

ex Fex RR(t) ) FRR(t0)

(A2)

Thus

Similarly, after the laser field is cut off, eq 5 (nonrelaxation case) is expressed as ex dFRβ ex ) -i(ωR - ωβ)FRβ dt

(A3)

Therefore, we obtain ex ex FRβ (t) ) FRβ (t0) exp[-i(ωR - ωβ)(t - t0)]

(A4)

From eqs A2, A4, and 10, the diagonal density matrix in aggregate basis after irradiating laser field is represented by

∑ CiRC*iRFexRR(t0) + ∑ CiRC*iβFRβex (t0)

Fiiagg(t) )

R

×

R*β

exp[-i(ωR - ωβ)(t - t0)] (A5) The first and second terms of the right-hand side of (A5) represent the time evolution of the population and coherence of exciton states, respectively. The second term is expressed as

∑ CiRC*iβFRβex (t0) exp[-i(ωR - ωβ)(t - t0)] )

R*β

∑ CiRC*iβ(Re[FRβex (t0)] + iIm[FRβex (t0)])(cos[(ωR - ωβ) ×

R*β

(t - t0)] - i sin[(ωR - ωβ)(t - t0)]) ) 2

∑ CiRC*iβ(Re[FRβex (t0)] cos[(ωR - ωβ)(t - t0)] +

R>β

ex Im[FRβ (t0)] sin[(ωR - ωβ)(t - t0)]) )

2

∑ CiRC*iβRe[FRβex (t0)]2 + Im[FRβex (t0)]2cos

R>β

[

(

(ωR - ωβ)(t - t0) - arctan


From these results and ex ex ex (t0)]2 + Im[FRβ (t0)]2 |FRβ | ) √Re[FRβ

Fagg ii (t) is expressed as

×

)]

(A6)


Fiiagg(t) )


∑ CiRC*iRFexRR(t0) + 2 ∑ CiRC*iβ|FRβex (t0)| cos

[

R

R>β

(

(ωR - ωβ)(t - t0) - arctan


)]

(A7)

Appendix B: Derivation of Equations 15 and 16. In the con-rotating-polarization case, both modes 1 and 2 are left-hand circular-polarized laser field, and then, from eq 14

|Ψ(t)〉 )

{

1 1 1 (|ψRx〉 + i|ψRz〉)e-iωRt + (|ψβx〉 + √2 √2 √2 i|ψβz〉)e-iωβt

}

1 ex Re[FRzβx ] ) - sin ∆ωt 4

×

(B1)

All these terms oscillate with identical amplitude and frequency. The relationship of relative phases among these terms is represented by ex ex ex ex (Re[FRxβx ], Re[FRzβz ], Re[FRxβz ], Re[FRzβx ]) ∝ (cos ∆ωt, cos ∆ωt, sin ∆ωt, -sin ∆ωt) (B11)

On the other hand, in the counter-rotating-polarization case, modes 1 and 2 are right- and left-hand circular-polarized laser fields, respectively. In this case, eq 14 is expressed as

|Ψ(t)〉 )

The density matrix is therefore expressed as

(〈ψβx | - i〈ψβz |)e

iωβt

{

1 1 1 (|ψRx〉 - i|ψRz〉)e-iωRt + (|ψβx〉 + √2 √2 √2 i|ψβz〉)e-iωβt

1 F(t) ) |Ψ(t)〉〈Ψ(t)| ) {(|ψRx〉 + i|ψRz〉)e-iωRt + 4 (|ψβx〉 + i|ψβz〉)e-iωβt}{(〈ψRx | - i〈ψRz |)eiωRt + } (B2)

(B10)

}

(B12)

This is different from eq B1 in terms of the sign of the second term. Similar to the con-rotating-polarization case, the relationship of relative phases among real parts of off-diagonal density matrices is represented by

which provides the following four oscillatory terms 1 1 ex FRxβx (t) ) 〈ψRx |F(t)|ψβx〉 ) ei∆ωt ) (cos ∆ωt + i sin ∆ωt) 4 4

(B3) 1 1 ex FRzβz (t) ) 〈ψRz |F(t)|ψβz〉 ) ei∆ωt ) (cos ∆ωt + i sin ∆ωt) 4 4

(B4) i ex FRxβz (t) ) 〈ψRx |F(t)|ψβz〉 ) - ei∆ωt ) 4 1 (-i cos ∆ωt + sin ∆ωt) (B5) 4 and i 1 ex FRzβx (t) ) 〈ψRz |F(t)|ψβx〉 ) ei∆ωt ) (i cos ∆ωt - sin ∆ωt) 4 4

(B6) where we define ∆ω ) ωβ - ωR. The real parts of these terms are expressed as

and

ex Re[FRxβx (t)] )

1 cos ∆ωt 4

(B7)

ex Re[FRzβz (t)] )

1 cos ∆ωt 4

(B8)

ex Re[FRxβz (t)] )

1 sin ∆ωt 4

(B9)

ex ex ex ex (Re[FRxβx ], Re[FRzβz ], Re[FRxβz ], Re[FRzβx ]) ∝ (cos ∆ωt, -cos ∆ωt, sin ∆ωt, sin ∆ωt) (B13)

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JP908373K

Exciton Recurrence Motion in Double-Ring Molecular Aggregates

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