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15 July 1997

OPTICS

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ELSEVIER Optics Communications 140 (1997) 83-88

Effect of exciton-exciton annihilation on optical bistability of a linear molecular aggregate

V.A. Malyshev ‘, H. Glaeske, K.-H. Feller *

Fuchhochschule Jena, Fachbereich Medi:intechnik/ Phyikalische Technik, Tutzendpromenade lb, O-07745 Jew, Gemum~

Received 2 January 1997; accepted 13 March 1997

Abstract

A theoretical study of the optical bistable response of a linear molecular aggregate modelled as a linear chain of molecules is carried out, making use of the one-molecule density matrix approach in which the intermolecular dipole-dipole interaction is included explicitly. The effect of exciton-exciton annihilation on the bistable behaviour is analyzed. One point

of view seems to be that the annihilation channel of relaxation can destroy the bistability. However, we found that the bistable behaviour may be observable even when the annihilation constant greatly exceeds the intermolecular dipole-dipole

interaction being responsible for the aggregate bistability. 0 1997 Elsevier Science B.V.

PAC.% 42.65.P~; 36.4O.V~

Keywords: Molecular aggregates; One-molecule density matrix; Bistability; Exciton-exciton annihilation

1. Introduction

Linear and non-linear optical properties of linear molecular aggregates have been a subject of great interest

up to date. The collective (excitonic) character of aggre-

gate eigenfunctions is responsible for some extraordinary features as radiative rate enhancement [1] and anomalous

high values of non-linear susceptibilities [2,3] (see also the review [4] and references therein).

These features make linear molecular aggregates promising candidates for applications in all-optical switch-

ing. One possible realization is in a waveguide-based

arrangement. On the other hand, linear and non-linear optical losses are high in the on-resonance regime, where

* Corresponding author. E-mail: ckf@rz.uni-jena.de.

’ Permanent address: AR-Russian Research Center “Vavitov

State Optical Institute”. Birzhevaya Liniya 12, 199034 Saint- Petersburg. Russia.

these extraordinary features are most pronounced. There- fore, it seems to make sense to look for switching devices of another geometrical form, based on bistability as light

passes through a thin sample.

Recently [5], the possibility of observing bistable be- haviour in the optical response of aggregates was pre-

dicted. As was shown, bistability originates from the dy- namical resonance frequency shift dependent on the level of population. Interesting intensity-dependent spectral fea-

tures due to population changes in molecular aggregates have been observed and explained for several years, such as the first experimental observation of the intensity-de-

pendent changes in absorption around the collective ab- sorption band of J-aggregates (the so-called J-band) by Gadonas et al. 161. Theoretically, the observed blue shift was later explained in terms of the fermion character of one-dimensional excitons [7,8]. In turn, also the dynamical frequency shift mentioned above results from the fact that one-dimensional Frenkel excitons are weakly interacting fermions (see, for example, Ref. [4]). The main feature of the effect consists of a sudden switching of the population level with increasing amplitude of the external field. Thus.

OON-4018/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved.

PIf SOO30-401 X(97)00 148-X

84 V.A. Malysher et al. /Optics Communications 140 (1997183-88

one may observe a consequent change in transmittivity of ized by including the exciton-exciton annihilation can be a system made up of linear aggregates. written as

The main goal of the present communication is to

improve the model used in Ref. [5] taking into considera- tion exciton-exciton annihilation, which plays an impor- tant role as a process competing with saturation [9- 131. As

linear aggregates are systems with collective (lD-ex-

citonic) excitations, a multi-particle approach has to be used to obtain the correct energy spectrum (see, e.g., Ref.

[ 141). However, in practice, treating dynamical multy-exci- ton-effects in the exact picture is rather complicated. Nev-

ertheless, it is possible, as an approximation, to use the one-molecule (two-level) density matrix formalism, in

which the field acting on each molecule consists of the

external field plus the field produced by the rest of the molecules in the aggregate. Thereby taking into account

the intermolecular retarded interaction exactly, collective

radiative damping is included, as well as a dynamical shift of the resonance frequency with increasing level of excita- tion. This shift, the limits of which are equal to the limits

of the exact excitonic spectrum, is responsible for the bistability effect, as above mentioned. Such an approach

was previously used to treat Dicke super-radiance [ 15,161

and to describe optical response of a collection of two-level atoms disposed in a volume with linear dimensions smaller than the emission wavelength [17]. It was justified in Ref.

[5], for the case of large aggregates (L B A, when the

collective radiation constant, ya, exceeds the energy inter- vals of the lD-exciton spectrum, and therefore the quanti-

zation may be neglected), and working only in the low-en- ergy region of the ID-exciton spectrum. We extended the

system of equations used in Ref. [5] by including the exciton-exciton annihilation in a phenomenological way and nevertheless found that the bistable behaviour may exist at very high values of the annihilation constant which

exceed any interaction scale of the problem. In view of the

complexity of the problem under consideration, we leave aside here an explicit treatment of localization effects on

the aggregate bistability reserving them for further studies.

b$“,) = i- “,“” [ pi;’ - pi:‘] - 2(Y&‘[ pi:- ‘) + pi;+ “1

+ r* P:;‘, (lb)

pi:) = - iw,, pii’ + i- y [ P\.’ - &‘]

- a&‘[ pi;- ‘) + pi;+ “1, (‘cl

p&i’ = - rpjy + a!&‘[ pi;- ‘) + pg+ “1 , (Id)

where p is the matrix element of the dipole operator of the transition 1 + 2, for any molecule (we assume all dipoles are parallel and form an angle 0 with the chain axis); w2, is the resonance frequency of a single molecule; E, is the electric field acting on the kth molecule, and includes the

external field EF’ and the field induced by the remaining molecules in the location of the kth molecule,

CN ,( ~ kJ= , E,,. Regarding the external field, we will assume it to be a plane wave (w,, k,) travelling perpendicular to

the chain axis and polarized along p. The incident fre- quency is on-resonant with the molecule transition 1 -+ 2.

The field emitted by the Ith molecule is considered as a classical field produced by a classical dipole with momen-

tum equal to the mean value of dipole operator d,(t) = F[ pjy(r> + p$(t)] (we do not show here the explicit

expression for E,, addressing the reader, for instance, to Ref. [5]).

The paper is organized as follows. In Section 2 we formulate the model and describe equations of the one-

molecule density matrix approach adapted to the exciton- exciton annihilation. Section 3 deals with the mean-field approximation, allowing us to analyse the problem in question in an analytical way. Finally, Section 4 concludes the paper.

The parameter (Y phenomenologically describes the contribution of the exciton-exciton annihilation. With re- spect to this process, we assume first, that two excitations

disappear when they are nearest-neighbours, and second,

that the annihilation rate is proportional to the sum of the nearest-neighbour population, p$ ‘) + pi;’ I). Corre- sponding terms in the equations driving pii) and R, differ from each other by a factor of two. Through the third

level, the excitation comes back to the ground and excited

states, with rates T, and r, (r= r, + r,). Thus, we include in our scheme both the ‘ ‘Auger”-kind [9,11,12] (described by r2) and the fusion model [IO] (associated with r,) of excitonic annihilation ‘. The magnitudes of r

z 2. Theoretical background

Here, we will consider the aggregate as an ensemble of N identical three-level molecules equally spaced along a chain (the third level, being non-resonant to the external field, serving as an intermediate state, through which annihilation occurs). The system of equations for the den- sity-matrix elements (pT:‘> of the kth molecule general-

As well-known, the Huang-Rhys factor of the J-band is rather

small what signifies that the exciton-phonon interaction is quite

suppressed, for the collective excitonic states. Thus, the transfer of

two-exciton energy to a suitable molecular level either of an own

molecule of the aggregate or of one of the aggregate environment,

strongly coupled to the vibrations seems to be a more probable

process of the electronic energy degradation. Due to this coupling,

the molecular energy then relaxes to the vibrational degree of

freedom.

V.A. Malwhec et al./Optics Communications 140 f19Y7) X3-88 X5

vary in several papers from 0.1 ps- ’ [IO] to 5 ps- ’ [I 1,121.

Making use in Eqs. (1) of the standard rotating wave

approximation, we pass to the set of differential equations for the amplitude of the off-diagonal matrix element, R,, and for the populations of the kth molecule, p{:’ and pi!) [Eq. (Id) remains unchanged],

5 A,,(R,R; -RJR,) /fzk)= I

+f i y,,(R,R; +R:R,)-+R; -R,) /(#kJ= I

+ apg[ p:;- ‘) + pb$’ “1 + r, P::‘, (?a)

&;I=$ ; A,,(R,R; -R;R,) /(#k)=l

-; f y,,(RiR;+R:R,)+i;(R;-R,) K#!,)= I

li,= -iAR,+i f (A,, - iy,,)R,[ ~$5’ - &‘I /(#k)= I

where 3 = o?, - w0 is the resonance detuning of the external field frequency w,, and the molecular transition

frequency wz, ; R = pcexr/fz is the Rabi frequency of the

external field. Matrices A,, and -yIk are given by formulae

151 2

4, = ;

cos( k,all- “1) + k a sin( k,all- kl)

I1 - kl’ 0 L /I - kj’ 1

X(1 -3cos20)-(k,n)-

(3a)

P2 Y,m = 3

k sin( k,all - kl)

fin 0

u cos( koall- kl) I/-kl’ - I1 - kl” 1

X(1 -3cos%) - (k”(l)- 2 sin( kOd - kl) sin28

II- kl i

(3b)

Here, a is the distance between neighbouring molecules and k, = oa/c_ The matrix A,, - iy,, may be identified with the matrix of the intermolecular retarded interaction. The imaginary part, ylk, results from the molecular interac-

tion through the transverse field (see, for instance, Ref.

[16]) and describes the collective radiative damping. The

real part, A,,, gives rise to phase modulation in the optical

response of the aggregate and is responsible for the bistable behaviour. To recall the meaning of these parameters, let us consider the particular case of an aggregate with a

length which is small compared with the wavelength of the incident field (L = Nu << Al. In this limit, Y,~ equals half

of the radiative constant of an isolated molecule, whereas A,, is just the near-zone dipole-dipole interaction [ 161. In

the homogeneous case (see below), A,, reduces to a

dynamical (depending on the level of excitation) frequency shift. We do not include in our scheme any Z--processes,

assuming the annihilation-induced dephasing to be the

dominant one. In the absence of any relaxation terms ( (Y = 0, I-, = ~1,

it is straightforward to get from Eqs. (I ) N integrals of

motion I R,j’ + [ pii’ - pi:‘]’ = const which is the conser-

vation law of the Bloch vector length of the k th molecule in the on-resonance channel. It should be pointed out that,

precisely according to this law, one can interpret the radiative damping associated with the matrix y,,. to be of a collective origin (like Dicke super-radiance).

3. Mean-field approximation

In this communication, we restrict our study to the

simplest model in which all functions in Eqs. (2) are assumed to be independent of the site number. Such an

approximation has already been used in Ref. [5] to shed light on the origin of the aggregate bistability (this model is exact in the case of an infinite chain). It simplifies

drastically the analysis and is very useful in obtaining a qualitative picture of the process. The system of equations (2) then transforms into

A, = $IR/‘- i$ R* -R) + 2ap;, + r, prj. (3a)

&= -+‘+i;(R- -R)-4q&+lyTp3,.

(3b)

R={-i[A-A~(pl,-p,,)l+y,(p,,-p,,)}R

- ifi( pl? -~,,)-2aRp~~. (4c>

&3 = - I$,, + 2 cup,‘, . (Id)

where

N/?

A,=2 c A,,, (5a) I-k=1

N/’

YR = - 7 c -Y/A’ (5b) /k=I

86 V.A. MalwheL> et al. /Optics Communications 140 (1997) 83-88

In the limit k,a < 1 -K k, L the summations can be evalu-

ated explicitly and give

A,= 3Yo

74(3)(1 - 3cos28), 2( ,%,a)

37rYo -sin20.

YR= 8k,a

In this form the approximation describes adequately linear

chains of lengths L > h. Here, Y,, is the spontaneous emission rate of a single molecule and r(3) = 1.202 is the

third-order Riemann 5 function. We will assume that A,

is negative (the case of J-aggregates, see Ref. [4]). Further-

more, as follows from Eqs. (6), 1 A, I> yR [ 161 except for a narrow range of values of 13 where 1 - 3cos28 is very

close to zero. In order to check whether the bistability effect exists in

the presence of excitonic annihilation, one should perform

a stationary analysis of (4) substituting zero in Eqs. (41 for the time derivatives. Then an algebraic equation of the

fifth order appears, for the population of the second level

(7)

where A’ = A - 1 ALI is the detuning of resonance renor- malized by the dipole-dipole interaction.

As was shown in Ref. [5], in the absence of the exciton-exciton annihilation bistability may occur even at

a low level of excitation owing to the inequality I A, I z+

yR. In our opinion, this is the most interesting limit from an utilitarian point of view. Thus, first of all, we are going to analyse the effect of the exciton-exciton annihilation precisely in that case. If one neglects in Eq. (7) all terms

proportional to LY as well as taking the necessary limit p2? -=z 1, one then obtains

This is exactly the same as Eq. (19) from Ref. [5]. It has the three-valued solutions (and, consequently, demon- strates a bistable behaviourl if A’ < - fiYR, which means that, in order to observe the effect, one should excite the system slightly above the resonance. Comparing Eq. (71 with (8) one can conclude, without any calculations, that under the conditions (Y 2 ya, r there is no effect of the exciton-exciton annihilation on the aggregate bistability. We may expect this provided either one or both of the conditions: (Y >> YR and (Y B r apply.

From recent experimental data [11,12], it usually fol- lows that the inequality cy > r> Ya is satisfied. Neverthe- less, first we will omit in Eq. (7) all terms proportional to cu/T assuming (cu/T )pZ2 K 1. In addition, one can also neglect p22 ( pzz < 1 in the case of interest) in the

right-hand side of Eq. (7). Finally, Eq. (7) is transformed

into

(91 \ YR)

\ ,

Further, we will assume that A’ < - fi yR, i.e., as regards the detuning of resonance we are above the threshold of bistability which would be valid without annihilation and are going to examine the precise effect of excitonic anni-

hilation, described by (Y, on the bistable behaviour. Since one has again an algebraic equation of the third

order (91, the conditions of bistability can be derived from examining zeros of the derivative d022/dp22. The exis-

tence of two zeros of dn2/dpZ2 is equivalent to the case of Eq. (9) having a three-valued solution. This takes place

if

4(;+&)ZL3[l+(;)Z][l+(;)2~> (10)

where the inequality gives the threshold for bistability. From this, one can find a critical value of the annihilation

constant ff

a, ~(A’/Y,) + J”;[(Ar/~R>2 + 11 Lq= 3(d’/yR)2 - 1

(11)

If (Y > (Y,, the inequality (10) is never satisfied and conse-

quently, p22 monotonically depends on R. In the opposite

case, (Y < (Y,, a three-valued solution of Eq. (91 appears, which results in bistable behaviour.

It should be mentioned that (Y,, as a function of A’, has

an upper limit at 1 A’[/ yR z+ 1. Taking large values of ]&I/y,, one thenobtainsfrom(ll1: max{a!,]=]A,_]/&. An experimental value ’ of cr is about 10 ps- ‘. On the other hand, IA,_ = 650 cm-’ = 20 ps- ’ is accepted as a

typical value. Thus, under real conditions, we have the magnitude of the annihilation constant slightly less than max(Ly,} = /ALI/ &, i.e., the case of bistable behaviour.

‘Thevalueof a=lOps-’ can be extracted from the experi-

mental data both of the annihilation constant y = 3 X IO-’ cm’/s [9,12] and of a typical size of the exciton coherence length of

about 20. We do not include the excitonic effects due to the third

level as well as 3-1 and 3-2 coherence, assuming a fast dephas-

ing within the third electronic state.

V.A. MalysheL: et al. /Optics Communications 140 (1997) 83-88 87

r- L1

1.62 1.64 1.66 I.68 1.1

Fig. 1. Results of numerical calculations of Eq. (9) demonstrating

the disappearance of the three-valued solution with increasing

exciton annihilation constant Q. Curves correspond to values of cy

of 41 to 47 from top to bottom, assuming r/r2 = 2. (A’ = - 10, \A,_/ = 100.) Bistability disappears at (Y = 45 in full correspon-

dence with Eq. (I I). (All numbers in units of ~a_)

In Fig. 1, we have drawn a few examples of the numerical solution of Eq. (91, displaying the disappearance

of the aggregate bistability with increasing LY. As one can see, the excitonic annihilation mostly affects the upper

branch of the three-valued field dependence of the popula- tion, thus diminishing the hysteresis loop.

Fig. 2 presents the dependence of the critical annihila- tion constant (Y, on the initial detuning of resonance il’. The bistability effect exists where cy < cr, (at a fixed A’); the shaded part in Fig. 2 shows this region.

Finally, in Fig. 3 we have plotted a family of solutions

of the original Eq. (7), in order to demonstrate the effect of

introducing a finite value of r (in fact, the case consid- ered earlier corresponds to r= 11. From this figure, it follows that introducing r affects the bistable behaviour

in such a way that the critical value of the annihilation constant increases, providing better conditions for display-

ing bistability.

4. Concluding remarks

We have examined the effect of exciton-exciton anni-

hilation on bistability of linear molecular aggregates origi- nating from the population-dependent resonance frequency shift. We have done this by making use of the system of

coupled equations for density matrices of individual molecules in which the intermolecular dipole-dipole inter-

action has been included explicitly. From our study based

on the mean-field approximation, the conclusion can be drawn that this influence is restrained from some critical

value of the exciton annihilation constant a which, in turn. depends on the initial detuning of resonance and has

an upper limit 1 A, I/ fi (1 A, 1 being the magnitude of the dipole-dipole interaction) at large values of detuning of

resonance (in the scale of the radiative damping constant). Radiationless relaxation may act in such a way that it gives

rise to an increase of this value. By this, we have not only increased the number of relevant parameters, on which the bistable effect depends, but also made our theory relevant for practical applications, as exciton-exciton annihilation

is an important feature present in real situations and limit- ing the occurrence of bistability (see Fig. 21. The question

-12 -10 -8 -6 -4 -2 Ah*

Fig. 2. Dependence of the critical value of annihilation constant (cy,) on the initial detuning of resonance ( A’). The shaded part corresponds

to the existence of aggregate bistability [a < a,( A’)].

88 V.A. Malyshev et al./ Optics Communications 140 (1997183-M

Fig. 3. Results of numerical solution of original Eq. (7) showing

that including a finite relaxation constant r the critical value of

the annihilation constant increases providing more suitable condi-

tions for bistability to be present. (d’ = - 10, IALl = 100. All

numbers in units of ~a. T/T? = 2.) (a) r = 3, (Y = 60; 90: 120:

150 in order of decreasing area of bistability, (Y, = 120. tb)

r = 0.2, a! = 300; 600; 900; 1200 in order of decreasing area of

bistability, LY, = 900.

remains open as to whether our results can be extended to

the general case where spatial inhomogeneity plays a role. This could be proved by solving the original set of equa-

tions in which spatial effects are naturally embedded.

Nevertheless, it seems to be reasonable to assume that excitonic annihilation will act to equalize the spatial inho- mogeneity. We intend to address this question in a forth-

coming paper.

Acknowledgements

V.M. thanks the Deutsche Forschungsgemeinschaft for a grant. The authors are indebted to Dr. J. Vickers for a

careful reading of the manuscript.

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