# Exciton recurrence motion in aggregate systems in the presence of quantized optical fields.

**ABSTRACT** The exciton dynamics of model aggregate systems, dimer, trimer, and pentamer, composed of two-state monomers is computationally investigated in the presence of three types of quantized optical fields, i.e., coherent, amplitude-squeezed, and phase-squeezed fields, in comparison with the case of classical laser fields. The constituent monomers are assumed to interact with each other by the dipole-dipole interaction, and the two-exciton model, which takes into account both the one- and two-exciton generations, is employed. As shown in previous studies, near-degenerate exciton states in the presence of a (near) resonant classical laser field create quantum superposition states and thus cause the spatial exciton recurrence motion after cutting the applied field. In contrast, continuously applied quantized optical fields turn out to induce similar exciton recurrence motions in the quiescent region between the collapse and revival behaviors of Rabi oscillation. The spatial features of exciton recurrence motions are shown to depend on the architecture of aggregates. It is also found that the coherent and amplitude-squeezed fields tend to induce longer-term exciton recurrence behavior than the phase-squeezed field. These features have a possibility for opening up a novel creation and control scheme of exciton recurrence motions in aggregate systems under the quantized optical fields.

**0**Bookmarks

**·**

**60**Views

- [Show abstract] [Hide abstract]

**ABSTRACT:**Absorption properties of molecular trimers are studied within a model including a single monomer internal vibrational degree of freedom. Upon photoabsorption, three excited electronic states which are coupled excitonically are accessed. Band shapes resulting from different electronic coupling strengths and geometries are analyzed. It is shown that geometric information can be extracted from the band intensities. Taking data recorded for perylene bisimide aggregates as an example, the spectra for monomer, dimer, and trimer systems are compared.The Journal of Chemical Physics 05/2007; 126(16):164308. · 3.12 Impact Factor - SourceAvailable from: Masayoshi Nakano
##### Article: Theoretical study on exciton dynamics in dendritic systems: exciton recurrence and migration.

[Show abstract] [Hide abstract]

**ABSTRACT:**The optical functionalities such as exciton recurrence and migration for dendritic systems, e.g., dendrimers, are investigated using the quantum master equation (QME) approach based on the ab initio molecular orbital configuration interaction (MOCI) method, which can treat both the coherent and incoherent exciton dynamics at the first principle level. Two types of phenylacetylene dendrimers, Cayley-tree dendrimer and nanostar dendrimer with anthracene core, are examined to elucidate the features of excion recurrence and migration motions in relation to their structural dependences. It is found that the nanostar dendrimer exhibits faster exciton migration from the periphery to the core than Cayley-tree dendrimer, which alternatively exhibits exciton recurrence motion among dendron parts in case of small relaxation parameters. Such strong structural dependence of exciton dynamics demonstrates the advantage of dendritic molecular systems for future applications in nano-optical and light-harvesting devices.Molecules 01/2009; 14(9):3700-18. · 2.43 Impact Factor - SourceAvailable from: Masayoshi NakanoMasayoshi Nakano, Ryohei Kishi, Takuya Minami, Hitoshi Fukui, Hiroshi Nagai, Kyohei Yoneda, Hideaki Takahashi[Show abstract] [Hide abstract]

**ABSTRACT:**We propose a novel dynamic exciton expression based on the quantum master equation approach using the ab initio molecular orbital (MO) singly excited configuration interaction (CI) method developed in our previous paper [M. Nakano, M. Takahata, S. Yamada, R. Kishi, T. Nitta, K. Yamaguchi, J. Chem. Phys. 120 (2004) 2359]. This expression is derived from the partition of polarization density in the configuration basis into the electron and hole contributions, and can describe both the coherent and incoherent dynamics of electron and hole density distributions, e.g., dynamic electric polarization, exciton recurrence and exciton migration.Chemical Physics Letters 01/2008; 460:370-374. · 2.15 Impact Factor

Page 1

Exciton recurrence motion in aggregate systems in the presence of quantized optical

fields

Masayoshi Nakano, Suguru Ohta, Ryohei Kishi, Masahito Nate, Hideaki Takahashi, Shin-Ichi Furukawa, Hiroya

Nitta, and Kizashi Yamaguchi

Citation: The Journal of Chemical Physics 125, 234707 (2006); doi: 10.1063/1.2390695

View online: http://dx.doi.org/10.1063/1.2390695

View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/125/23?ver=pdfcov

Published by the AIP Publishing

Articles you may be interested in

Oscillatory and rotatory exciton recurrence motions in double-ring molecular aggregates controlled by two-mode

circular-polarized laser field

AIP Conf. Proc. 1504, 691 (2012); 10.1063/1.4771789

Quantum Master Equation Approach to Exciton Recurrence Motion in a RingShaped Aggregate Complex

Induced by CircularPolarized Laser Field

AIP Conf. Proc. 1148, 330 (2009); 10.1063/1.3225307

Rapid motion capture of mode-specific quantum wave packets selectively generated by phase-controlled optical

pulses

J. Chem. Phys. 127, 054104 (2007); 10.1063/1.2753834

A nonMarkovian optical signature for detecting entanglement in coupled excitonic qubits

AIP Conf. Proc. 893, 1081 (2007); 10.1063/1.2730272

Determination of the Wigner function of an optical field using the atomic Talbot effect

AIP Conf. Proc. 461, 243 (1999); 10.1063/1.57892

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 133.1.24.40

On: Mon, 08 Dec 2014 11:40:30

Page 2

Exciton recurrence motion in aggregate systems in the presence

of quantized optical fields

Masayoshi Nakano,a?Suguru Ohta, Ryohei Kishi, Masahito Nate,

Hideaki Takahashi, and Shin-Ichi Furukawa

Department of Materials Engineering Science, Graduate School of Engineering Science,

Osaka University, Toyonaka, Osaka 560-8531, Japan

Hiroya Nitta and Kizashi Yamaguchi

Department of Chemistry, Graduate School of Science, Osaka University, Toyonaka,

Osaka 560-0043, Japan

?Received 17 July 2006; accepted 12 October 2006; published online 21 December 2006?

The exciton dynamics of model aggregate systems, dimer, trimer, and pentamer, composed of

two-state monomers is computationally investigated in the presence of three types of quantized

optical fields, i.e., coherent, amplitude-squeezed, and phase-squeezed fields, in comparison with the

case of classical laser fields. The constituent monomers are assumed to interact with each other by

the dipole-dipole interaction, and the two-exciton model, which takes into account both the one- and

two-exciton generations, is employed. As shown in previous studies, near-degenerate exciton states

in the presence of a ?near? resonant classical laser field create quantum superposition states and thus

cause the spatial exciton recurrence motion after cutting the applied field. In contrast, continuously

applied quantized optical fields turn out to induce similar exciton recurrence motions in the

quiescent region between the collapse and revival behaviors of Rabi oscillation. The spatial features

of exciton recurrence motions are shown to depend on the architecture of aggregates. It is also found

that the coherent and amplitude-squeezed fields tend to induce longer-term exciton recurrence

behavior than the phase-squeezed field. These features have a possibility for opening up a novel

creation and control scheme of exciton recurrence motions in aggregate systems under the quantized

optical fields. © 2006 American Institute of Physics. ?DOI: 10.1063/1.2390695?

I. INTRODUCTION

The investigation of interaction dynamics between

atomic/molecular aggregate systems and laser fields has at-

tracted much attention because of the fundamental under-

standing of electronic excitations in aggregates and nonlinear

optical responses as well as their potential applications in

photonics and optoelectronics.1–13In particular, the coherent

processes of electronic excitations in aggregate systems by

incident laser fields become one of the most attractive recent

topics in relation to the development of quantum atomic/

molecular devices.11–29For example, the coherent processes

can create quantum superposition states composed of plural

electronic excited states by irradiating laser fields, leading to

the spatial recurrence motion of excitation, i.e., exciton,

among constituent monomer units in aggregate systems.23,24

The exciton transfer between constituent units has been

detected experimentally and been analyzed theoretically for

dimers in solution. For example, the coherent excitation

energy transferbetween

?2,2?-binaphthyl ?BN?? in solution has been observed as a

damped oscillation corresponding to exciton recurrence as-

sociated with the interaction energy ?41 cm−1? and dephasing

timeof0.2 psbyHochstrasser

Yamazaki et al. have observed that various anthracene

two-identicalchromophores

andco-workers.25,26

dimers, e.g., dianthrylbenzene ?DAB? and dithiaanthra-

cenophane ?DTA? ?see Fig. 1?, exhibit the oscillatory behav-

ior of the fluorescence anisotropy decay in solution, originat-

ing in the recurrence motion of an exciton between two

anthracenes.27–29Although such oscillatory signals are gen-

erally difficult to detect during long time due to the rapid

dephasing by solvent molecules, relatively rigid molecular

structure such as DAB and DTA turned out to be an impor-

tant factor of realizing the detection of the recurrence motion

of exciton.23,24Indeed, the fact that the dephasing time for

DTA ?1.0 ps? is longer than that for BN ?0.2 ps? is explained

by the less flexible and more rigid conformation for DTA

than for BN. In our previous papers,23,24we have applied the

quantum master equation approach to the exciton recurrence

motions in aggregate systems, i.e., a dimer model and den-

dritic aggregate models composed of dipole coupled two-

state monomers. A generation of one-exciton state by an ex-

ternal classical laser field and subsequent exciton dynamics

after cutting the laser field turned out to cause the recurrence

motion of exciton. It has also been found that the configura-

tion of each dipole unit significantly affects the coherent

exciton motions, which can be controlled by tuning the

frequency of an external field.

In the present study, we investigate the effects of quan-

tized optical fields, i.e., coherent and squeezed fields with

small average number of photons, on the exciton dynamics

of aggregate systems. As is well known, the dynamics of

a?Author to whom correspondence should be addressed. Electronic mail:

mnaka@cheng.es.osaka-u.ac.jp

THE JOURNAL OF CHEMICAL PHYSICS 125, 234707 ?2006?

0021-9606/2006/125?23?/234707/14/$23.00© 2006 American Institute of Physics

125, 234707-1

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 133.1.24.40

On: Mon, 08 Dec 2014 11:40:30

Page 3

atomic/molecular systems interacting with quantized optical

fields provides various attractive influences on the dynamics,

i.e., collapses, quiescence, and revival behavior of Rabi

oscillation.30–33There have been lots of studies on such dy-

namics for monomer and aggregate systems using the

Jaynes-Cummings ?JC? model30–33and numerical approaches

based on more complicated models.19–22On the other hand,

the dynamics of superposition state composed of plural ex-

citon states in the presence of the quantized optical field has

not been clarified though such investigation is intriguing and

important in view of the novel control scheme of coherent

processes of excitons in aggregate systems. The nonrelax-

ation case is examined here because we focus on the genera-

tion and dynamical behavior of exciton motions purely

caused by the quantized optical field and, in particular, illu-

minate the difference in the behavior among the classical and

various quantized optical fields. Three types of quantized

optical fields, i.e., coherent, amplitude-squeezed, and phase-

squeezed fields, are examined. These fields are known to

provide several different features in the coherent processes,

i.e., revival-collapse behavior of Rabi oscillations of atomic/

molecular populations, originating in the different quantum

statistics of photon distributions.31–33The features of exciton

motion and coherency between exciton states are analyzed

using the dynamics of off-diagonal exciton density matrices

in addition to the exciton population ?diagonal density ma-

trix? dynamics. We also examine the effect of aggregate ar-

chitecture on the exciton dynamics using dimer, trimer, and

pentamer models. On the basis of the present results, we

clarify the feature of generation of exciton recurrence motion

and its time-evolution behavior caused by various quantized

optical fields, and discuss a possibility of controlling the ex-

citon recurrence motion in aggregate systems using the quan-

tized optical fields.

The present paper is organized as follows. In Sec. II, we

explain our model aggregate systems and the calculation pro-

cedure of aggregate-quantized optical field dynamics. In Sec.

III, we present the results for dimer, trimer, and pentamer

models interacting with three kinds of quantized optical

fields,i.e.,coherent, amplitude-squeezed,

squeezed fields, as well as those with classical laser fields.

The features of dynamical behavior of exciton motions are

analyzed using the diagonal exciton density matrices by fo-

cusing on the effects of the initial quantum statistics of quan-

tized optical fields and the architecture of aggregate systems.

This is followed by a conclusion in Sec. IV.

andphase-

II. METHODOLOGY

A. Construction of exciton state models of aggregate

systems

The dynamics of aggregate systems in the presence of a

quantized optical field is performed in a manner of two-step

procedure: an exciton state model for an aggregate is con-

structed at the first step, and the time evolution of the aggre-

gate exciton state model interacting with a quantized optical

field is carried out at the second step. The aggregate is com-

posed of two-state monomers, which are approximated to be

dipoles. The kth monomer possesses a transition energy,

E21

tween a dipole k?l? and a line drawn from the dipole site k to

l is ?kl??lk?. This dipole approximation is considered to be

acceptable if the intermolecular distance ?Rkl? is larger than

the size of a monomer. A general form of Hamiltonian for an

arbitrary aggregate model composed of two-state monomers

?N, the number of monomers? is presented by14

k??E2

k−E1

k?, and a transition moment, ?12

k. The angle be-

Hagg= H0+ Vint

=?

k

N

?

ik

2

Eik

kaik

†aik+1

2?

k,l

N

?

ik,ik?

il,il?

2

?ikik?

k?ilil?

l

??cos??kt− ?tk?

− 3 cos ?klcos ?tk?/Rkl

3?aik

†aik?ail

†ail?,

?1?

where the first and the second terms represent a noninteract-

ing Hamiltonian and the dipole-dipole interaction, respec-

tively. We use a.u. ??=m=e=1? throughout this paper. Eik

an energy of state ikfor monomer k, and ?ikik?

of a transition matrix element between states ikand ik? for

monomer k. The aik

ation and annihilation operators for the ikstate of monomer

k. By using the basis for the aggregate ???i1

the number of monomers?, which is constructed by a direct

product of a state vector for each monomer ???ik

elements of the first and the second terms in Eq. ?1? are,

respectively, represented by14

kis

k

is a magnitude

†and aik?represent, respectively, the cre-

1?i2

2¯?iN

N?? ?N is

k??, the matrix

FIG. 1. Structures of dianthrylbenzene ?DAB? ?a? and dithiaanthra-

cenophane ?DTA? ?b?.

234707-2Nakano et al.J. Chem. Phys. 125, 234707 ?2006?

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 133.1.24.40

On: Mon, 08 Dec 2014 11:40:30

Page 4

??j1

?

1¯ ?jN

N??

k

N

?

ik

2

Eik

kaik

†aik??j1?

1¯ ?jN?

N?=?

k

N

Ejk

k??

n=1

N

?jnjn??,

?2?

and

?j1

1¯ ?jk

k¯ ?jl

l¯ ?jN

N?

?

k,l

N

?

ik,ik?,

il,il?

2

?ikik??ilil?f??kl,?lk,Rkl?alk

+aik?ail

+ail??

?jl?

1¯ ?jk?

k¯ ?jl?

l¯ ?jN?

N?

=?

k,l

N

?jkjk??jljl?f??kl,?lk,Rkl?? ?

n?k,l

N

?jnjn??,

?3?

where

f??kl,?lk,Rkl? ? ?cos??kl− ?lk? − 3 cos ?klcos ?lk?/?2Rkl

3?.

?4?

By diagonalizing this Hamiltonian matrix ?Eq. ?1??, we can

obtain a new state model ?aggregate exciton state model?

witheigenenergies

?El

?l=1,...,M?, where M is the size of the basis used. We

consider the two-exciton model, in which one and two exci-

tons are taken into account and thus M is 1+NC1+NC2. The

transition dipole matrix elements ??ll?

the polarization of the applied field for this state model are

calculated by

agg?

andeigenstates

???l

agg??

agg? in the direction of

?ll?

agg= ??l

agg??agg??l?

agg? = ?

j1,...,jN

jl?,...,jN?

??l

agg??j1

1¯ ?jN

N?

???jl?

1¯ ?jN?

N??l?

agg??

k

N

?jkjk?

?k??

n=k

N

?jnjn??,

?5?

where ?jkjk?

monomer in the direction of polarization vector of the

applied field. It is noted that only the transition moments

between the ground ?1? and the one-exciton states l

?l=2,...,N+1? and those between the two-exciton states

l? ?l?=N+2,...,M? and the one-exciton states exist in the

present two-exciton model.

?k

is the jk−jk? transition matrix element of the kth

B. Time evolution of an M-state aggregate exciton

model interacting with single-mode quantized

and classical optical fields

1. Quantized optical field case

The total Hamiltonian for an M-state aggregate exciton

model in the presence of a single-mode quantized optical

field is represented by19–22

H = Hagg+ Hqfield+ Hagg-qfield

=?

l

M

El

aggbl

†bl+?c†c +1

2?? +?

l,l?

M

K?ll?

aggbl

†bl??c + c†?,

?6?

where the coupling parameter K is

K =?2??

V?

1/2

.

?7?

The first ?Hagg?, the second ?Hqfield?, and the third ?Hagg-qfield?

terms on the right-hand side of Eq. ?6? represent the unper-

turbed aggregate exciton system, the single-mode quantized

optical field, and the interaction between them, respectively.

In the first term, bl

operators for the aggregate exciton state l, respectively. In

the second term, ? indicates the frequency of the single-

mode quantized optical field in a cavity, and c†and c are the

creation and annihilation operators for that single-mode field.

In the third term, V is the volume of the cavity containing the

quantized optical field. We neglect the couplings between

aggregate exciton states and dipole vibrations ?nuclear

motions of monomers? in order to focus on the coherent

processes purely caused by the interaction between exciton

and quantized optical fields in this study. Such approxima-

tion can be acceptable for either the case where the time

intervals considered are much faster than any change of

nuclear configuration of the aggregate or the case where the

couplings between excitons and nuclear motions are much

smaller than other couplings. The matrix elements of the

above Hamiltonian are obtained using a double Hilbert space

???l

?l=1,2,...,M? and the photon-number states of the single

mode, i.e., ??n?? ?n=0,1,2,...,??. The time evolution of the

coupled system composed of excitons and a quantized opti-

cal field is described by the time-dependent Schrödinger

equation:

†and blare the creation and annihilation

agg,n?? spanned by the aggregate exciton states ???l

agg??

?

?t???t?? = − iH???t??,

?8?

where ???t?? represents the state at time t of the coupled

system. The matrix form of this equation is given by

234707-3Exciton recurrence motion in aggregate systemsJ. Chem. Phys. 125, 234707 ?2006?

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 133.1.24.40

On: Mon, 08 Dec 2014 11:40:30

Page 5

?

?t??k

agg,n???t?? = − iEk

agg??k

agg,n???t??

2???k

− i?n +1

agg,n???t??

− iK?

k,k?

M

?kk?

agg??n??k?

agg,n − 1???t??

+?n + 1??k?

agg,n + 1???t???.

?9?

In order to solve this equation, we employ the sixth-order

Runge-Kutta method with a small time step ?a period of a

quantized optical field/600?, which is found to provide suffi-

ciently converged and quantitatively correct results. The den-

sity matrix elements at time t are obtained by

?k,n;k?,n??t? ? ??k

agg,n???t???k?

= ??k

agg,n??

agg,n???t?????t???k?

agg,n??.

?10?

The initial density matrix ??f,n;g,n??t0?? can be separated

into the product of an aggregate exciton density matrix

??f,g

matrix ??n,n?

ground state at the initial time t=t0. As the initial quantized

optical field, three types of single mode fields, i.e., coherent,

amplitude-squeezed and phase-squeezed fields, are consid-

ered. The single-mode coherent field ??? is generated as fol-

lows from the vacuum field ?0? by operating displacement

operator.34–40

agg?t0?? and a single-mode quantized optical field density

qfield?t0??. The aggregate is assumed to be in the

??? = exp??c†− ?*c??0?,

?11?

where ? is the eigenvalue of photon-annihilation operator c.

The probability distribution of finding n photons in the co-

herent field is a Poisson distribution and its element of the

quantized optical field density matrix is represented by

?n,m?t0? =?n ˆ??n+m?/2e−?n ˜?

?n!m!

,

?12?

where ?n ˆ? is the mean number of photons in the coherent

field. On the other hand, the single-mode ideally squeezed

field can be generated from the vacuum field ?0? by operating

squeezing and displacement operators:37–40

??,?? = exp??c†− ?*c?exp???*c2− ?c†2?/2??0?,

?13?

where ? can be expressed by ?=rei?using real modulus r

and argument ?. The r and ?/2 represent squeezing intensity

and direction, respectively. The direction of ? is taken to be

aligned with Re??? axis in the complex ? plane. The

squeezed field is generated by a number of nonlinear optical

processes including optical parametric oscillation and four-

wave mixing. This field state exhibits the property that the

variance of the quadrature operator x ˆ1?x ˆ1? less than the value

of 1/2 for the vacuum and the coherent field states. From the

Heisenberg uncertainty relation between x ˆ1and x ˆ2, the vari-

ance of another quadrature operator x ˆ2?x ˆ1? exceeds 1/2. If

?=0 in Eq. ?13?, the squeezed field has a phase uncertainty

higher than that of a coherent field of the same average pho-

ton number and a narrower photon-number distribution. This

is referred to as an amplitude-squeezed field. If ?=? in Eq.

?13?, the squeezed field has an amplitude uncertainty higher

than that of a coherent field and a broader photon-number

distribution. This is referred to as a phase-squeezed field.

In this study, we consider these two types of squeezed fields

??=0 and ?? with a relatively weak squeezing intensity r

=0.5. The elements of the squeezed field density matrix are

represented by37–40

?n,m?t0? =

1

????n!m!?

? exp?1

*?

?

2??

n/2?

?*

2?*?

?*??2?

?2???,

m/2

exp?− ????2?

2

?*

???2+1

?2???Hn?

2

?

?Hm

??

??

?14?

where ?=cosh r, ?=ei?sinh r, and ??=??+??*, and Hnis

the Hermite function.

The aggregate reduced density matrix elements are ob-

tained by

FIG. 2. Structures of dimer ?D?, trimer ?T?, and pentamer ?P?, which are

composed of identical two-state monomers ?shown by arrows and symbols

a, b, and c? with an excitation energy E21

transition moment ?12

tersection of axes that run along the transition dipoles at right angles to one

another is fixed to be 30 a.u.

k??E2

k−E1

k=38 000 cm−1? and a

k?=10 D?. The distance between monomer and the in-

234707-4 Nakano et al.J. Chem. Phys. 125, 234707 ?2006?

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 133.1.24.40

On: Mon, 08 Dec 2014 11:40:30

Page 6

?f,g

agg?t? =?

n

?f,n;g,n?t?.

?15?

In

???i1

ments at time t are expressed by

the

2¯?iN

representation

N??, the aggregate exciton density matrix ele-

usingthe aggregate basis

1?i2

?i1,i2,...,iN;i1?,i2?,...,iN?

agg

?t? =?

l,l?

M

??i1

1¯ ?iN

N??i

agg??ll?

agg?t?

???l?

agg??i1?

1¯ ?iN?

N?.

?16?

2. Classical laser field case

In the case of classical laser field, the total Hamiltonian,

Eq. ?6?, is changed to the following form:

H = Hagg+ Hagg-cfield

=?

l

M

El

aggbl

†bl−?

l,m

M

?lm?F cos ?t?bl

†bm,

?17?

where Hagg-cfieldrepresents the interaction between the aggre-

gate exciton and an external classical laser field with fre-

quency ? and amplitude F, which propagates in the direction

perpendicular to the aggregate plane. The ?lmis the compo-

nent of the transition moment between exciton states l and m

in the same direction with the polarization of the classical

laser field. It is noted in contrast to Eq. ?6? that there is no

field Hamiltonian because the classical laser field only plays

a role of a driving force. Therefore, the matrix elements of

the above Hamiltonian are described using a single Hilbert

space ???l

citon states. Similar to the case of quantized optical field, the

time evolution of the aggregate exciton is obtained by nu-

merically solving the time-dependent Schrödinger equation,

Eq. ?8?, with a time step ?a period of a classical laser field/

80?, and the aggregate density matrix ?f,g

culated by ??f

the representation of the aggregate basis are also expressed

by Eq. ?16?.

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 133.1.24.40

On: Mon, 08 Dec 2014 11:40:30

agg?? ?l=1,2,...,M?, representing the aggregate ex-

agg?t? is directly cal-

agg?. The matrix elements in

agg???t?????t???g

III. RESULTS AND DISCUSSION

A. Exciton states of dimer, trimer, and pentamer

models

Figure 2 shows the structures of dimer ?D?, trimer ?T?,

and pentamer ?P? models, which are composed of identical

two-state monomers ?shown by arrows? with an excitation

energy E21

?12

is orthogonally oriented to each other on a plane, is known to

be the simplest case to observe the exciton recurrence motion

between the monomers.24The distance between monomer

and the intersection of axes that run along the transition di-

poles at right angles to one another is fixed to be 30 a.u. The

trimer ?T? and pentamer ?P? models are symmetrically con-

structed from the dimer ?D? units in order to realize exciton

recurrence motions. The one- and two-exciton states of these

aggregates are calculated by diagonalizing aggregate Hamil-

tonian Hagg?Eq. ?1??. The exciton states are shown with ex-

citation energies and transition moments in Fig. 3. For dimer

model ?D?, state 1 is a nonexciton ?ground? state, states 2 and

3 are one-exciton states, and state 4 is a two-exciton state,

as seen from the configurations involved in these wave

functions. Indeed, the excitation energy of state 4 ?E4

=76 000 cm−1? is about twice as those of states 2 ?E2

=37 966.6 cm−1? and 3 ?E3

nearly equal to the excitation energy of the monomer

?38 000 cm−1?. For trimer ?T? ?pentamer ?P?? model, states

from 2 to 4 ?2 to 6? are classified into one-exciton states and

the remaining states are done into two-exciton states.

Figure 4 schematically shows the exciton wave functions

of one-exciton states with significant transition moments

from the nonexciton ?ground? state by plotting the expansion

coefficients ???i1

monomersiterepresented

???i1

the positive and negative coefficients, and the size of circles

represents the amplitudes of coefficients. For dimer ?D?,

states 2 ?Fig. 4?D-2?? and 3 ?Fig. 4?D-3?? turn out to have

in-phase and out-of-phase exciton wave functions with the

same amplitude on two monomer sites a and b ?0.7071?,

respectively. These exciton states are nearly degenerate ?en-

k??E2

k−E1

k=38 000 cm−1? and a transition moment

k?=10 D?. The dimer model ?D?, each monomer of which

agg

agg

agg=38 033.4 cm−1?, which are

1¯?iN

N??l

agg?? of one-exciton states on each

by the

N??. The white and black circles correspond to

aggregatebasis

1?i2

2¯?iN

FIG. 3. One- and two-exciton states for dimer ?D?, tri-

mer ?T?, and pentamer ?P?. Exciton state energies

?cm−1? and the amplitudes of transition moments ?ij

?D? are also shown.

234707-5Exciton recurrence motion in aggregate systems J. Chem. Phys. 125, 234707 ?2006?

Page 7

ergy gap between states 2 and 3: ?23=67 cm−1? and possess

almost the same transition moment ??21

from the ground state. For trimer ?T?, states 2 ?Fig. 4?T-2??

and 4 ?Fig. 4?T-4?? have similar transition moments ??21

=10.08 D and ?31

the near-degenerate excitation energies ??24=94 cm−1?. The

exciton wave function of state 2 exhibits an in-phase distri-

bution, which has the largest amplitude on the central site

b ?0.7019? together with the identical smaller distributions on

the both-end sites a and c ?0.5037?. In contrast, the wave

function of state 4, which has two node lines between sites a

and b, and between sites b and c, has the largest amplitude of

the central site b ?−0.7123? with smaller identical amplitudes

on both-end sites a and c ?0.4953?. For pentamer ?P?, there

are three one-exciton states, 2 ?Fig. 4?P-2??, 4 ?Fig. 4?P-4??,

and 6 ?Fig. 4?P-6??, with significant transition moments from

the ground state ??21

=10.5 D?.Thesestates are

??24=53 cm−1and ?46=62 cm−1?. The exciton wave func-

tion of state 2 has an in-phase distribution, in which the

amplitude becomes larger as going from site c ?0.3007? to

site a ?0.6279? through site b ?0.4609?. State 4 has positive

and negative distributions with large amplitudes on sites

a ?0.5571? and c ?−0.5872?, respectively. In state 6, the dis-

tributions on sites a ?−0.5435? and c ?−0.2545? have an op-

posite sign to those on site b ?0.5362? though the variation in

amplitude on each site is similar to that for state 2. Although

the distributions of two-exciton states are not shown, they

tend to be more delocalized, i.e., distributed over neighbor-

ing sites, than those of one-exciton states.23

agg???31

agg?=7.07 D?

agg

agg=9.92 D? from the ground state as well as

agg=12.3 D, ?41

agg=6.17 D, and ?61

also nearly

agg

degenerate

B. Exciton recurrence motion in the case of classical

laser field

As shown in previous papers,23,24near-degenerate exci-

ton states create a superposition state by the irradiation of

nearly resonant laser field. Firstly, we clarify the origin of

exciton recurrence behavior using a dimer model with a pair

of near-degenerate exciton states 2 and 3. Let us consider the

situation that the near-degenerate one-exciton states, ??m

and ??n

constructed from the two aggregate bases, ??????ia,ib?

=?10?? and ??????ia?,ib??=?01??, which represent the distribu-

tion of one exciton on sites a and b, respectively:

agg?

agg?, with energies Em

aggand En

aggare approximately

??m

agg? = C?m??? + C?n???

and

??n

agg? = C?n??? + C?n???.

?18?

The time-dependent wave function ???t?? ?Eq. ?8?? under the

external field in near resonance to the near-degenerate exci-

ton states is represented by

???t?? = Am?t?e−iEm

aggt??m

agg? + An?t?e−iEn

aggt??n

agg?.

?19?

We examine the time evolution of exciton distribution in the

aggregate basis after cutting off the external laser field at t

=t0. Note that Am?t0? becomes a constant value after t=t0.

The density matrix elements in the aggregate basis ????? are,

therefore, expressed as23

????agg??? = ?Am?t0??2?C?m?2+ ?An?t0??2?C?n?2+ 2 Re?C?mC?n

+ 2 Im?C?mC?n

= ?Am?t0??2?C?m?2+ ?An?t0??2?C?n?2+ 2?P2+ Q2cos??Em

*Am?t0?An

*?t0??cos?Em

agg− En

agg?t

*Am?t0?An

*?t0??sin?Em

agg− En

agg?t

agg− En

agg?t − arctan?Q/P??,

?20?

where

P ? Re?C?mC?n

*Am?t0?An

*?t0??

and

Q ? Im?C?mC?n

*Am?t0?An

*?t0??.

?21?

If there are only two site ?? and ?? configurations with real

coefficients and the following condition is satisfied:

C?m= C?nand C?m= − C?n,

then

FIG. 4. Exciton wave functions of one-exciton states

with significant transition moments from the nonexciton

?ground ?1?? state by plotting expansion coefficients

???i1

mer site for dimer ?D-n?, trimer ?T-n?, and pentamer

?P-n? ?n indicates the state number?. The white and

black circles indicate the positive and negative coeffi-

cients, respectively, and the size of circles represents

the magnitudes of coefficients.

1¯?iN

N??l

agg?? of one-exciton states on each mono-

234707-6 Nakano et al.J. Chem. Phys. 125, 234707 ?2006?

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 133.1.24.40

On: Mon, 08 Dec 2014 11:40:30

Page 8

????agg??? − ????agg???

= 2??P2+ Q2+?P?2+ Q?2?cos??Em

− arctan?Q/P??,

agg− En

agg?t

?22?

where P? and Q? represent Eq. ?21? for C?mand C?n. Equa-

tion ?22? indicates that the exciton distribution on site b os-

cillates with the opposite phase to that on site a. Namely, the

exciton distribution oscillates between two sites a and b with

frequency ?mn?=?Em

?D? interacting with an applied electric field ?classical laser

field?, which isresonant

=38 033.4 cm−1? and has a power of 5 MW/cm2. Because

this is relatively weak, the population of two-exciton state is

predicted to be negligible. The applied laser field is cut off at

the 1000 optical cycle ??36 260 a.u.=876.8 fs?. Figure 5

shows the variation of exciton population ?a?, population in

the aggregate basis ?b?, population on sites ?c?, and on- and

off-diagonal density matrices ??22

dimer model ?D?. As shown in Fig. 5?a?, the variation of

population between states 1 ?ground state? and 3 is dominant,

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 133.1.24.40

On: Mon, 08 Dec 2014 11:40:30

agg−En

agg??. We here consider dimer model

with excitonstate3

??

agg?t? and ??23

agg?t??? ?d? for

while only the slight variation is detected in state 2 until

1000 optical cycle ?see Fig. 5?d??. This indicates a superpo-

sition of states 2 and 3 though it is very small due to the

weak coupling between the aggregate and the applied field in

resonance with state 3. After cutting off the applied field, the

population of each state is shown to be constant. Using Eq.

?16?, the population in the representation of aggregate basis

??ia,ib?? is calculated ?Fig. 5?b?? and then the population on

sites a ??aa

tively, by

agg?t?? and b ??bb

agg?t?? is obtained ?Fig. 5?c??, respec-

?aa

agg?t? = ?10;10

agg?t? + ?11;11

agg?t?

and

?bb

agg?t? = ?01;01

agg?t? + ?11;11

agg?t?.

?23?

As expected from Eqs. ?20? and ?22?, the superposition of

states 2 and 3 is shown to be created and to give the coherent

oscillation of exciton population between states ?10? and ?01?,

i.e., sites a and b, after the cutoff time ?1000 optical cycle?.

As seen from Eq. ?20?, the origin of this oscillation after the

cutoff time is a constant finite value of ??23

exhibits a coherent oscillation:

agg?t??, where ?23

agg?t?

?23

agg?t? = 2 Re?A2?t0?A3

*?t0?e−i?E3

agg−E2

agg?t?.

?24?

Indeed, Fig. 5?d? shows a finite constant value of ??23

which implies a preservation of the coherency between states

2 and 3.

Next, we examine the exciton dynamics of dimer model

?D? in the presence of a continuous wave laser ?with fre-

quency ?=38 0.33.4 cm−1and a power of 5 MW/cm2?.

From Fig. 6?a?, we observe a primary Rabi oscillation be-

tween the population of states 1 and 3, while the population

of state 2 exhibits slight oscillatory amplitudes. The ampli-

tude of the variation in population of state 4 ?two-exciton

state? is much smaller than others. In contrast to the case of

cutoff field shown in Figs. 5?b? and 5?c?, we cannot observe

a distinct coherent recurrence motion of exciton between

populations, ?01;01

a and b ?Fig. 6?c??, but observe only slight oscillatory am-

plitudes in the envelopes of the Rabi oscillations. As a result,

the exciton population exhibits almost the same Rabi oscil-

latory behavior between ?01;01

sites a and b, which is caused by the driving laser field. This

is ascribed to the dominant oscillatory behavior of diagonal

terms in Eq. ?20?, ?A3?t??2?C?3?2?note that A3?t? varies de-

pending on t in this case and ???=?01? and ?10??, whose

amplitudes are much larger than other terms. This is also

exemplified by the fact that the amplitudes of oscillatory

behavior of ??23

that of ?33

Similar behavior is predicted to be observed for trimer

?T? and pentamer ?P? models because near-degenerate exci-

ton states ?states 2 and 4 for trimer ?T? and states 2, 4, and 6

for pentamer ?P?, see Fig. 3? involve mutually in-phase and

out-of-phase wave functions on different sites ?sites a?c? and

b for trimer ?T? and sites a and c for pentamer ?P?, see Fig.

4?. For trimer ?T?, the relative distribution on sites a?c? and b

between states 2 and 4 shows the same relation as that on

agg?t??,

agg?t? and ?10;10

agg?t? ?Fig. 6?b?? or between sites

agg?t? and ?10;10

agg?t? or between

agg?t?? and ?22

agg?t? ?see Figs. 6?a? and 6?d??.

agg?t? are significantly smaller than

FIG. 5. Variation of exciton population ?diagonal density matrix? ?a?, popu-

lation in the aggregate basis ?b?, population on sites ?c?, and on- and off-

diagonal density matrices in the exciton state basis ??22

for dimer model ?D? in the case of cutoff field. The applied electric field

?classical laser field? is resonant with exciton state 3 ??=38 033.4 cm−1? for

dimer ?D? ?see Fig. 4?D-3?? and has a power of 5 MW/cm2. The electric

field is cut off at the 1000 optical cycle ??36 260 a.u.=876.8 fs?.

agg?t? and ??23

agg?t??? ?d?

234707-7Exciton recurrence motion in aggregate systems J. Chem. Phys. 125, 234707 ?2006?

Page 9

sites a and b between states 2 and 3 of dimer ?D? ?see Figs.

4?D-2?, 4?D-3?, 4?T-2?, and 4?T-4??. For pentamer ?P?, we

observe in-phase ?states 2 and 6? and out-of-phase ?state 4?

relations on sites a and c ?see Figs. 4?P-2?, 4?P-4?, and 4?P-

6??. These features are expected to cause the exciton recur-

rence motion between sites a?c? and b for trimer ?T? and

between a and c for pentamer ?P? if these degenerate states

come into a superposition state by a ?near? resonant electric

field. For trimer ?T?, we apply an electric field with a reso-

nant frequency to state 4 ?38 046.5 cm−1? and a power of

5 MW/cm2, while for pentamer ?P?, we apply an electric

field with a resonant frequency to state 4 ?38 002.6 cm−1?

and a power of 1 MW/cm2. Similar to the dimer ?D? case,

the cutoff field and continuous field cases are examined. In

the cutoff field case, the applied field is cut off at 1000 op-

tical cycle for trimer ?T? ??36 250 a.u.=876.5 fs? and at

3000 optical cycle for pentamer ?P?

=2632.7 fs?. Figure 7 shows the exciton population on sites

?Figs. 7?a? and 7?b?? and some on- and off-diagonal density

matrices ??22

?66

Figs. 7?c? and 7?d?? in the cutoff field case for trimer ?T? and

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 133.1.24.40

On: Mon, 08 Dec 2014 11:40:30

??108 880 a.u.

agg?t? and ??24

agg?t??, ??26

agg?t?? for trimer ?T?, and ?22

agg?t??, and ??46

agg?t?,

agg?t?, ??24

agg?t?? for pentamer ?P?, see

pentamer ?P? models. As expected, we observe an exciton

recurrence motion between sites a?c? and b for trimer ?T?,

and that between sites a and c for pentamer ?P?, in which the

population on two sites c is about twice as that of site a as

predicted from Fig. 4?P-4?. Similar to the dimer ?D? case,

these recurrence oscillations are shown to originate in the

finite value of ??24

??46

field cases for trimer ?T? and pentamer ?P? models. Similar

to the dimer ?D? case ?Fig. 6?, the exciton population on a

pair of sites, between which the exciton recurrence motion is

observed in the cutoff field case, shows mutually in-phase

coherent Rabi oscillation driven by the continuous laser field,

while the exciton recurrence motion between these sites is

not observed distinctively but is only detected as a carrier

wave with slight amplitudes, which originates in the slight

finite values of ??24

and ??46

agg?t?? for trimer ?T?, and ??24

agg?t?? and

agg?t?? for pentamer ?P?. Figure 8 shows the continuous

agg?t?? for trimer ?T? ?Fig. 8?c?? and ??24

agg?t?? for pentamer ?P? ?Fig. 8?d??. These are ascribed

agg?t??

FIG. 6. Variation of exciton population ?diagonal density matrix? ?a?, popu-

lation in the aggregate basis ?b?, population on sites ?c?, and some on- and

off-diagonal density matrices in the exciton state basis ??22

?d? for dimer model ?D? in the presence of a continuous wave laser ?with

frequency ?==38 033.4 cm−1and a power of 5 MW/cm2?. See Fig. 5 for

further legends.

agg?t? and ??23

agg?t???

FIG. 7. Variation in population on sites ??a? for trimer ?T? and ?b? for

pentamer ?P?? and the amplitudes of some on- and off-diagonal density

matrices ?in the exciton state basis? ??c? for trimer ?T? and ?d? for pentamer

?P?? in the cutoff field case ?see Fig. 5?. For trimer ?T? case, an electric field

with a resonant frequency to state 4 ?38 046.5 cm−1? and a power of

5 MW/cm2is applied, while for pentamer ?P? case, an electric field with a

resonant frequency to state 4 ?38 002.6 cm−1? and a power of 1 MW/cm2is

applied. The applied field is cut off at 1000 optical cycle for trimer

?T? ??36 250 a.u.=876.5 fs? and at 3000 optical cycle for pentamer

?P? ??108 880 a.u.=2632.7 fs?.

234707-8Nakano et al.J. Chem. Phys. 125, 234707 ?2006?

Page 10

to the dominant oscillatory behavior of diagonal density ma-

trix, i.e., population of state 4 for trimer ?T? and pentamer

?P?, whose amplitudes are much larger than other diagonal

densities, i.e., population of state 2 for trimer ?T? and that of

states 2 and 6 for pentamer ?P?, in the present case ?a weak

aggregate-field coupling with the resonant field frequency to

state 4?.

In summary, for the classical laser field, the exciton re-

currence motion is observed on the sites relating to near-

degenerate exciton states after cutting off the external field,

but is not observed distinctively in the case of continuous

external laser field due to the dominant Rabi oscillation of

exciton population of the resonant state.

C. Exciton recurrence motion in the case of quantized

optical field

1. Coherent field case

Firstly, we consider the quantum dynamics of dimer ?D?

interacting with a coherent field with an initial average pho-

ton number ?n?=8 and a resonant frequency to state 3

?E3

?109Å3?Eq. ?7??, which corresponds to relatively weak

aggregate-field coupling. The number of photon basis ??n?? is

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 133.1.24.40

On: Mon, 08 Dec 2014 11:40:30

agg=38 033.4 cm−1? in a cavity with a volume V=7.0

taken to be 49, which turns out to be sufficient for obtaining

quantitative resutls in the present study. The variation in ex-

cition population ?diagonal density matrix? in each exciton

state of dimer ?D? in the presence of the coherent field is

shown in Fig. 9?a?. The Rabi oscillation is observed between

states 1 ?nonexciton state? and 3 ?resonant state?, while the

amplitudes of exciton population of state 2, which is near

degenerate with state 3, and that of state 4 ?two-exciton state?

are much smaller than those of states 1 and 3. This situation

is similar to the case of classical laser field ?see Fig. 6?a??. As

is well known, the Rabi oscillation between states 1 and 3

exhibits collapse ?I–II, see Fig. 9?a?? and revival behaviors

?III–IV?, i.e., decreasing and increasing amplitudes, caused

by the dephasing and rephasing processes originating in the

quantum property of relatively small finite quantum

systems.30–33The stationary region, i.e., quiescent region30–33

II–III, is observed between the first collapse and revival be-

havior. It has been found that the amplitudes of off-diagonal

density matrices in the exciton state basis, which represent

the degree of coherency between exciton states, show oscil-

latory behaviors in the quiescent region after the first col-

lapse process though the populations ?diagonal density ma-

FIG. 8. Variation in population on sites ??a? for trimer ?T? and ?b? for

pentamer ?P?? and the amplitudes of some on- and off-diagonal density

matrices in the exciton state basis ??c? for trimer ?T? and ?d? for pentamer

?P?? in the case of continuous classical laser field ?see Fig. 6?.

FIG. 9. Variation in exciton population ?diagonal density matrix? ?a?, popu-

lation in aggregate basis ?b?, population on sites ?c?, and amplitudes of on-

and off-diagonal density matrices ?d? for dimer ?D? in the presence of a

coherent field with an initial average photon number ?n?=8 and a resonant

frequency with state 3 ?E3

V=7.0?109Å3.

agg=38 033.4 cm−1? in a cavity with a volume of

234707-9Exciton recurrence motion in aggregate systemsJ. Chem. Phys. 125, 234707 ?2006?

Page 11

trices? of exciton states maintain almost constant values in

that region.37–39In the present dimer ?D? case, this leads to

the fact that ?33

constant, while ?23

its an oscillatory behavior in the quiescent region. Appar-

ently, this feature is similar to the case of cutoff classical

laser field ?Eqs. ?20?–?22??, in which the diagonal density

matrices are constant after cutting off fields, while the off-

diagonal densities continue to oscillate. As a result, the quan-

tized coherent field is predicted to induce the exciton recur-

rence motion between sites a and b in the quiescent region

II–III since the Rabi oscillation driven by the quantized op-

tical field almost vanishes in the region ?see Fig. 9?a??. Note

that this is contrast to the case of continuous classical laser

field ?see Fig. 6?a??, in which the Rabi oscillations of exciton

populations ?diagonal density matrices? with large ampli-

tudes driven by the continuous classical laser field conceal

the recurrence oscillations with slight amplitudes. Indeed,

Figs. 9?b? and 9?c? show the exciton recurrence motion be-

tween aggregate bases ?01? and ?10?, or sites a and b in the

quiescent region II–III. As seen from Eq. ?22?, the amplitude

of recurrence motion is proportional to the amplitude of off-

diagonal density matrices ??23

to be larger than ?22

We apply the quantized coherent field in resonance with

state 4 for timer ?T? and pentamer ?P? models and employ

the cavity with volumes of 1.0?1010Å3for trimer ?T? and

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 133.1.24.40

On: Mon, 08 Dec 2014 11:40:30

agg?t? ?=A3?t?A3

agg?t??=2 Re?A2?t?A3

*?t?? ?see Eq. ?19?? is almost

*?t?e−i?E3

agg−E2

agg??? exhib-

agg?t??, which is, indeed, shown

aggin the quiescent region ?see Fig. 9?d??.

5.0?1010Å3for pentamer ?P?, both of which correspond to

the case of relatively weak aggregate-field coupling. Figure

10 shows the variation in populations of one-exciton states

??a? for trimer ?T? and ?d? for pentamer ?P??, the variation of

exciton population on sites ??b? for trimer ?T? and ?e? for

pentamer ?P??, and some on- and off-diagonal density matri-

ces in the exciton state basis ??c? for trimer ?T? and ?f? for

pentamer ?P??. The periods II?–III? and II?–III? indicate the

quiescent region for trimer ?T? and pentamer ?P?, respec-

tively. Similar to the dimer ?D? case, it is found from Figs.

10?a? and 10?d? that there are primary collapse-revival be-

haviors between states 1 and 4, while there is a slight super-

position from near-degenerate states ?state 2 for trimer ?T?

and states 2 and 6 for pentamer ?P??. These superposition

and stationary ?constant? population of states 1 and 4 in the

quiescent region are expected to cause the exciton recurrence

behavior. Indeed, trimer ?T? shows the exciton recurrence

motion between sites a?c? and b in the quiescent region

II?–III?, while pentamer ?P? shows that between sites a and

c in the quiescent region II?–III?. The reason of the recur-

rence motion between a and c, instead of a and b or b and c,

for pentamer ?P? originates in the dominant exciton distribu-

tion on sites a and c of the resonant state 4 for pentamer ?P?

?see Fig. 4?P-4??. From Fig. 10?c? and Eqs. ?20?–?22?, the

recurrence motion for trimer ?T? originates in the fact that in

the quiescent region the amplitude of off-diagonal density

matrix, ??24

agg?t??, is larger than ?22

agg?t?, and both ?44

agg?t? and

FIG. 10. Variation in population of one-exciton states ??a? for trimer ?T? and ?d? for pentamer ?P??, exciton population on sites ??b? for trimer ?T? and ?e? for

pentamer ?P??, and amplitudes of some on- and off-diagonal density matrices in the exciton state basis ??c? for trimer ?T? and ?f? for pentamer ?P??. The

applied quantized coherent field is resonant with state 4 for trimer ?T? and pentamer ?P? models, and the cavity volume is fixed to be 1.0?1010Å3for trimer

?T? and 5.0?1010Å3for pentamer ?P?.

234707-10Nakano et al.J. Chem. Phys. 125, 234707 ?2006?

Page 12

?11

that ??24

and ??26

ary in the quiescent region as shown in Figs. 10?d? and 10?f?,

is ascribed to the exciton recurrence motion between sites a

and c in the quiescent region II?–III?.

In summary, it is predicted that a quantized coherent

field, which is nearly resonant with near-degenerate exciton

states, creates a superposition of the near-degenerate states,

each of which possesses an exciton wave function on a pair

of different sites with in- and out-of-phase distributions, re-

spectively. This superposition turned out to cause exciton

recurrence motions between the different sites of aggregates

in the quiescent region in the time evolution of population in

the exciton state basis. The dimer model ?D? composed of

monomers with mutually perpendicular configuration ?Fig.

2?D? is shown to satisfy the above condition, i.e., exciton

wave functions of near-degenerate exciton states with sig-

nificant transition moments from the ground state have in-

and out-of-phase distributions on sites a and b, respectively.

As a result, dimer model ?D? exhibits an exciton recurrence

motion between sites a and b during the quiescent region in

the presence of the quantized coherent field. The aggregates

with different sizes and architectures constructed from these

dimer units ?see Figs. 2?T? and 2?P?? also exhibit similar

exciton recurrence motions between different sites, i.e., sites

a?c? and b for trimer ?T?, and sites a and c for pentamer ?P?,

in the presence of the quantized coherent field. The spatial

region of exciton recurrence motion significantly depends on

the architecture of aggregates, which affects the interaction

between monomers and thus determines the relative feature

of exciton wave function of each state.

agg?t? almost remain stationary. For pentamer ?P?, the fact

agg?t?? is larger than those of ?22

agg?t??, while ?44

agg?t?, ?66

agg?t?, ??46

agg?t??,

agg?t? and ?11

agg?t? almost remain station-

2. Squeezed field case

It is well known that the squeezed fields significantly

affect the collapse-revival behavior by changing the squeez-

ing intensity r and squeezing angle ? ?see Sec. II B 1?.41We

consider two types of squeezed fields with weak squeezing

intensity

?r=0.5?, i.e.,amplitude

??=??-squeezed fields. The cavity volumes for dimer ?D?,

trimer ?T?, and pentamer ?P? models are the same as those

for the coherent field, respectively ?see Sec. III C 1?. The

collapse time ?time period I–II for dimer ?D?? tCand revival

time tR ?time period I–IV for dimer ?D?? for resonant

squeezed and coherent fields are shown to be represented

by41

??n ˆ?

?n,

??=0?- and phase

tC??

?

?25?

and

tR?2?

???n ˆ?,

?26?

where ??n?2is a variance defined by ??n?2=?n ˆ2?−?n ˆ?2, and

? indicates the coupling between aggregate and quantized

optical field ?K?ll?

i.e., ??n?2=?n ˆ?, tConly depends on the coupling constant in

the case of resonant field case. In the same aggregate model,

aggin Eq. ?6??. For the coherent field case,

we use the same cavity volume V, namely, the same ?, and

the same initial average photon number ?n ˆ?=8, so that tC

depends on ?n, which is different between two types of

squeezed fields. The amplitude-squeezed field exhibits

the sub-Poisson distribution in the present squeezing

intensity,37–39so that ?n is smaller than that of the coherent

field with the Poisson distribution, while the phase-squeezed

field exhibits the super-Poisson distribution,37–39so that ?n

is larger than that of the coherent field. From this feature

and Eq. ?25?, the order of increasing collapse time tCis pre-

dicted as follows: phase-squeezed field?coherent field

?amplitude-squeezed field. On the other hand, tRbecomes

the same for the coherent and squeezed fields with the same

initial average photon number and aggregate-field coupling

?see Eq. ?26??. Figure 11 shows the variation of exciton

population ?Fig. 11?a? for the amplitude-squeezed field and

Fig. 11?d? for the phase-squeezed field?, the variation in

population on sites ?Fig. 11?b? for the amplitude-squeezed

field and Fig. 11?e? for the phase-squeezed field?, and the

variation in some on- and off-diagonal density matrices ?Fig.

11?c? for the amplitude-squeezed field and Fig. 11?f? for the

phase-squeezed field? for dimer model ?D?. Apparently from

Figs. 9 and 11, the features of collapse and revival times for

these quantized optical fields turn out to be in agreement

with the prediction by Eqs. ?25? and ?26?. It is also found that

the amplitudes of collapse-revivals become larger in the or-

der of the phase-squeezed field case, the coherent field case,

and the amplitude-squeezed field case. On the other hand, the

beginning time of revival becomes earlier in the order of the

amplitude-squeezed field case, the coherent field case, and

the phase-squeezed field case. In particular, the phase-

squeezed field is shown to begin the revival behavior in sig-

nificantly earlier time region ?see Fig. 11?d??. Such depen-

dences of collapse-revival behaviors on the quantum

statistics of the initial quantized optical field are explained by

the dynamics of Q function distributions for these fields.37–40

As a result, the length of quiescent region for the phase-

squeezed field becomes smaller than those for the amplitude-

squeezed and coherent fields, which are similar to each other.

From Figs. 9?c?, 11?b?, and 11?e?, the exciton recurrence mo-

tion during a longer term is observed for the amplitude-

squeezed field than for the phase-squeezed field. The length

of the exciton recurrence region for the coherent field is

shown to be similar to that for the amplitude-squeezed field.

These features are also explained by the slower dephasing

?decaying? behavior of ??23

and coherent fields than for the phase-squeezed field, as

shown in Figs. 9?d?, 11?c?, and 11?f?.

Figure 12 shows the variation in exciton population on

sites and on- and off-diagonal density matrices for trimer ?T?

and pentamer ?P? models in the presence of the amplitude-

and phase-squeezed fields. For trimer ?T?, the exciton recur-

rence motions between sites a?c? and b are observed in the

quiescent region II?–III? for the amplitude- and phase-

squeezed fields ?Figs. 12?a? and 12?c?? similar to the coherent

field case ?Fig. 10?b??. The amplitudes of recurrence oscilla-

tions are similar to each other for these fields. As expected,

the lengths of the recurrence region for the amplitude-

squeezed and coherent fields are larger than that for the

agg?t?? for the amplitude-squeezed

234707-11 Exciton recurrence motion in aggregate systems J. Chem. Phys. 125, 234707 ?2006?

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 133.1.24.40

On: Mon, 08 Dec 2014 11:40:30

Page 13

phase-squeezed field. These recurrence behaviors originate

in the preservation of the amplitudes of off-diagonal density

matrices, ??24

10?c?, 12?b?, and 12?d??, and the stationary ?constant? behav-

ior of ?44

also observe the exciton recurrence motion between sites a

and c, similar to the coherent field case ?Fig. 10?e??. As pre-

dicted previously, the length of recurrence region for the co-

herent and amplitude-squeezed fields becomes larger than

that for the phase-squeezed field ?see Figs. 10?e?, 12?e?, and

12?g??. The slower decay of the amplitudes of off-diagonal

density matrices, ??24

?22

escent region II?–III?. Indeed, the decay region of ??24

and ??46

amplitude-squeezed fields than for the phase-squeezed field

?see Figs. 10?f?, 12?f?, and 12?h??. In summary, similar to the

dimer ?D? case, the coherent and amplitude-squeezed fields

are found to provide longer exciton recurrence region than

the phase-squeezed field.

agg?t??, which are larger than ?22

agg?t? ?see Figs.

agg?t? in the quiescent region. For pentamer ?P?, we

agg?t?? and ??46

agg?t?, leads to the recurrence motion in the qui-

agg?t??, which are larger than

agg?t? and ?66

agg?t??

agg?t?? is more extended for the coherent and

3. Exciton dynamics in the case of strong coupling

with a coherent field

The strong coupling between dimer model ?D? and a

coherent field is realized by using a small cavity volume, V

=5.0?107Å3, without changing other parameters in Sec.

III C 1. In this case, the aggregate-field coupling is about 12

times as large as that in Sec. III C 1 ?see Eq. ?7??. Because

the collapse-revival behavior occurs in earlier time region

than that in Sec. III C 1 ?see Eq. ?25??, we examine the time

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 133.1.24.40

On: Mon, 08 Dec 2014 11:40:30

evolution until 3500 optical cycles. Figure 13 shows the

variation of exciton population ?a?, population on sites ?b?,

and on-and off-diagonal density matrices ?c? for dimer model

?D?. In contrast to the case of weak coupling ?Fig. 9?, in

which the collapse-revival behavior is observed only for

states 1 and 3, the collapse-revival behaviors are observed

for all exciton states ?Fig. 13?a??. The populations of exciton

states 2 and 3 exhibit collapse-revival oscillations with simi-

lar phase though the averaged population of state 2 is smaller

than that of state 3. Also, the exciton population of state 4,

which is a two-exciton state, exhibits a collapse-revival be-

havior with an opposite phase to that for states 2 and 3. This

is caused by the transition between states 2 and 4, and be-

tween states 3 and 4. Namely, the population changes of state

4 originate in the populations of states 2 and 3. Therefore,

the relative phase between the Rabi oscillations of the popu-

lations of states 2?3? and 4 is opposite to each other. It is

noted that states 2 and 3 involve ?01? and ?10? as the main

configurations, while state 4 involves a two-exciton configu-

ration ?11?. Because the amplitude of population ?diagonal

density? of each state is large, the exciton recurrence origi-

nating in the off-diagonal density matrices is not observed.

On the other hand, the revival oscillations with mutually op-

posite phase, though this gives an apparent exciton recur-

rence motion, occur between sites a and b ?see Fig. 13?c??.

This is caused by the transition of exciton population be-

tween states 2?3? and 4, not by the off-diagonal density ma-

trices ?coherency? between states. Indeed, the off-diagonal

density matrix, ??23

nal density matrices ?population?, ?22

agg?t??, turned out to be smaller than diago-

agg?t?, ?33

agg?t?, and ?44

agg?t?

FIG. 11. Variation in exciton population ??a? for the amplitude-squeezed field ?r=0.5 and ?=0? and ?d? for the phase-squeezed field ?r=0.5 and

?=???, population on sites ??b? for the amplitude-squeezed field and ?e? for the phase-squeezed field?, and amplitudes of some on- and off-diagonal density

matrices ??c? for the amplitude-squeezed field and ?f? for the phase-squeezed field? for dimer ?D?. See Fig. 9 for further legends.

234707-12Nakano et al.J. Chem. Phys. 125, 234707 ?2006?

Page 14

?see Figs. 13?a? and 13?d??, in contrast to the weak coupling

case ?Fig. 9?d??. These results indicate that the exciton recur-

rence behavior originating in superposition states does not

appear in the case of strong aggregate-field coupling in

contrast to the weak coupling case.

IV. CONCLUSION

We have investigated the exciton dynamics of atomic/

molecular aggregates in the presence of a classical electric

field and a quantized optical field. It is found that the sym-

metric aggregate models ?dimer ?D?, trimer ?T?, and pen-

tamer ?P?? composed of two-state monomers with mutually

orthogonal configurations provide near-degenerate exciton

states with a significant transition moment from the ground

state. A pair of near-degenerate states involves in- and out-

of-phase exciton wave function distributions on a pair of

different sites in the aggregate system, respectively. The

weak applied fields in resonance with one of these near-

degenerate states are shown to create a superposition state

composed of a dominant contribution of the resonant states

and slight contributions of other near-resonant states. For a

classical laser field, the exciton recurrence behavior between

the pairs of different sites appears after cutting off the field,

while for the continuous irradiation such behavior disappears

and alternatively the Rabi oscillation with a large amplitude

between the ground and the resonant states appears. In con-

trast, in the case of applying continuous quantized optical

fields, which lead to weak aggregate-field coupling, the ex-

citon recurrence behavior appears during the quiescent re-

gion, in which the population of the resonant exciton state

remains stationary ?constant?. This exciton recurrence mo-

tion is caused by satisfying two conditions in the quiescent

region that ?i? the amplitude of the resonant state is station-

ary ?constant?, and ?ii? the amplitudes of off-diagonal density

matrices between the near-degenerate states are larger than

the population of near-resonant states. The quantum statisti-

cal feature affects the length of such recurrence region: the

coherent and amplitude-squeezed fields cause the longer-

term exciton recurrence motion than the phase-squeezed

field. These variations originate in the dependence of col-

lapse and revival times on the initial average value and vari-

ance of photon number. In the case of strong coupling be-

tween the aggregate system and the quantized optical field,

the resonant and near-resonant states are significantly excited

FIG. 12. Variation of exciton populations on sites ??a? trimer ?T? and ?e? pentamer ?P? for the amplitude-squeezed field and ?c? trimer ?T? and ?g? pentamer

?P? for the phase-squeezed field?, and on- and off-diagonal density matrices in the exciton state basis ??b? trimer ?T? and ?f? pentamer ?P? for the amplitude-

squeezed field and ?d? trimer ?T? and ?h? pentamer ?P? for the phase-squeezed field?. See Figs. 10 and 11 for further legends.

234707-13Exciton recurrence motion in aggregate systemsJ. Chem. Phys. 125, 234707 ?2006?

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 133.1.24.40

On: Mon, 08 Dec 2014 11:40:30

Page 15

together with two-exciton states, and the exciton recurrence

motion does not appear during the quiescent region but ap-

pears during the revival oscillation. This recurrence motion

does not originate in the superposition between near-

degenerate states but in the relative population changes be-

tween the one- and two-exciton states.

From these results, we could propose a novel generation

scheme of exciton recurrence motion in aggregates interact-

ing with continuously applied quantized optical field. The

recurrence features are expected to be controlled by the ag-

gregate architectures, the quantum statistical properties of

the initial quantized optical field, and the magnitudes of

aggregate-field coupling. At the next stage, we investigate

the effects of various types of relaxations on the exciton

recurrence motions, which is important for the realization of

the measurement of these phenomena.

ACKNOWLEDGMENTS

This work was supported by Grant-in-Aid for Scientific

Research ?No. 18350007? from Japan Society for the Promo-

tion of Science ?JSPS? and was also supported by the Min-

istry of Education, Science, Sports and Culture, Grant-in-Aid

for Scientific Research on Priority Areas ?No. 18066010?.

1P. O. J. Scherer, in J-Aggregates, edited by T. Kobayashi ?World Scien-

tific, Singapore, 1996?, p. 95.

2E. Collini, C. Ferrante, and R. Bozio, J. Phys. Chem. B 109, 2 ?2005?.

3R. V. Markov, A. I. Plekhanov, V. V. Shelkovnikov, and J. Knoester,

Microelectron. Eng. 69, 528 ?2003?.

4F. C. Spano and S. Mukamel, Phys. Rev. A 40, 5783 ?1989?.

5T. Ogawa, E. Tokunaga, and T. Kobayashi, Chem. Phys. Lett. 410, 18

?2005?.

6K. D. Belfield, M. V. Bondar, F. E. Hernandez, O. V. Przhonska, and S.

Yao, Chem. Phys. 320, 118 ?2006?.

7F. C. Spano and J. Knoester, Adv. Magn. Opt. Reson. 18, 117 ?1994?.

8C. R. Stroud, J. H. Eberly, W. L. Lama, and L. Mandel, Phys. Rev. A 5,

1094 ?1972?.

9C. M. Bowden and C. C. Sung, Phys. Rev. A 19, 2392 ?1979?.

10V. Malyshev and P. Moreno, Phys. Rev. A 53, 416 ?1996?.

11M. Nakano and K. Yamaguchi, Chem. Phys. Lett. 290, 216 ?1998?.

12M. Nakano and K. Yamaguchi, J. Phys. Chem. A 102, 6807 ?1998?.

13M. Nakano, S. Yamada, H. Nagao, and K. Yamaguchi, Int. J. Quantum

Chem. 71, 295 ?1999?.

14M. Nakano, M. Takahata, H. Fujita, S. Kiribayashi, and K. Yamaguchi,

Chem. Phys. Lett. 323, 249 ?2000?.

15M. Takahata, M. Nakano, H. Fujita, and K. Yamaguchi, Chem. Phys.

Lett. 363, 422 ?2002?.

16K. Yamaguchi, M. Nakano, H. Nagao et al., Bull. Korean Chem. Soc. 24,

864 ?2003?.

17M. Takahata, M. Shoji, S. Yamanaka, M. Nakano, and K. Yamaguchi,

Polyhedron 24, 2563 ?2005?.

18M. Takahata, M. Shoji, H. Nitta, R. Takeda, S. Yamanaka, M. Okumura,

M. Nakano, and K. Yamaguchi, Int. J. Quantum Chem. 105, 615 ?2005?.

19M. Nakano and K. Yamaguchi, Chem. Phys. 295, 328 ?1998?.

20M. Nakano and K. Yamaguchi, Chem. Phys. Lett. 324, 289 ?2000?.

21M. Nakano and K. Yamaguchi, J. Chem. Phys. 116, 10069 ?2002?.

22M. Nakano and K. Yamaguchi, J. Chem. Phys. 117, 9671 ?2002?.

23M. Takahata, M. Nakano, and K. Yamaguchi, J. Theor. Comput. Chem.

2, 459 ?2003?.

24H. Nitta, M. Shoji, M. Takahata, M. Nakano, D. Yamaki, and K. Yamagu-

chi, J. Photochem. Photobiol., A 178, 264 ?2006?.

25Y. R. Kim, P. Share, M. Pereira, M. Sarisky, and R. M. Hochstrasser, J.

Chem. Phys. 91, 7557 ?1989?.

26F. Zhu, C. Gralli, and R. M. Hochstrasser, J. Chem. Phys. 98, 1042

?1993?.

27I. Yamazaki, S. Akimoto, N. Aratani, and A. Osuka, Bull. Chem. Soc.

Jpn. 77, 1959 ?2004?.

28I. Yamazaki, N. Aratani, S. Akimoto, T. Yamazaki, and A. Osuka, J. Am.

Chem. Soc. 125, 7192 ?2003?.

29I. Yamazaki, S. Akimoto, T. Yamazaki, S.-I. Sato, and Y. Sakata, J. Phys.

Chem. A 106, 2122 ?2002?.

30E. T. Jaynes and F. W. Cummings, Proc. IEEE 51, 100 ?1963?.

31L. Allen and J. H. Eberly, Optical Resonance and Two-level Atoms

?Wiley, New York, 1975?.

32P. L. Knight and P. W. Milonni, Phys. Rep. 66, 21 ?1980?.

33P. W. Milonni and S. Singh, Adv. At., Mol., Opt. Phys. 28, 75 ?1990?.

34D. Stoler, Phys. Rev. D 1, 3217 ?1970?; 4, 1925 ?1971?.

35H. P. Yuen, Phys. Rev. A 13, 2226 ?1976?.

36D. W. Walls, Nature ?London? 306, 141 ?1983?.

37M. Nakano and K. Yamaguchi, J. Phys. Chem. A 103, 6036 ?1999?.

38M. Nakano and K. Yamaguchi, J. Chem. Phys. 112, 2769 ?2000?.

39M. Nakano and K. Yamaguchi, Phys. Rev. A 64, 033415 ?2001?.

40S. M. Barnett, Methods in Theoretical Quantum Optics, Oxford Series on

Optical and Imaging Sciences Vol. 15 ?Clarendon Press, Oxford, 2003?.

41G. J. Milburn, Opt. Acta 31, 671 ?1984?.

FIG. 13. Variation of exciton population ?diagonal density matrix? ?a?, popu-

lation on sites ?b?, and on- and off-diagonal density matrices ?c? for dimer

?D? in the case of strong aggregate-coherent field coupling using a small

cavity volume V=5.0?107Å3.

234707-14 Nakano et al.J. Chem. Phys. 125, 234707 ?2006?

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 133.1.24.40

On: Mon, 08 Dec 2014 11:40:30