super-radiance from a thin solid film of three-level atoms: local-field effects

1 June 2000

Ž .Optics Communications 180 2000 59–68www.elsevier.comrlocateroptcom

Super-radiance from a thin solid film of three-level atoms:local-field effects

V.A. Malyshev a,), I.V. Ryzhov b, E.D. Trifonov b, A.I. Zaitsev b

a National Research Center ‘VaÕiloÕ State Optical Institute’ BirzheÕaya Liniya 12, 199034 Saint-Petersburg, Russiab Herzen Pedagogical UniÕersity, Moika 48, 191186 Saint-Petersburg, Russia

Received 8 September 1999; received in revised form 22 December 1999; accepted 3 April 2000

Abstract

We analyse the collective spontaneous emission from a thin solid film comprised of three-level atoms with aL-configuration of operating transitions. The thickness of the system is assumed to be smaller than the emission wavelengthso that the local-field correction to the acting field appears to be of importance and affects significantly the three-levelsuper-radiance. Because of this correction, there exists a competition between the operating transitions, resulting, undersuitable conditions, in the complete suppression of one of them. q 2000 Published by Elsevier Science B.V. All rightsreserved.

PACS: 42.50.Fx; 42.50.Md

1. Introduction

Ž .Dicke developed the theory of super-radiance SRw xfor an inverted system of two-level atoms 1 . Fur-

ther studies generally dealt with the same modelw x2–9 . Recently, it was shown that in a system of

Žthree-level atoms having, for instance, a doublet inw x w x .the ground 10–14 or in the upper 15 state SR

without inversion, being analogous to the amplifica-w xtion without inversion 16–21 , can be observed.

) Corresponding author. Fax: q34 91 394 45 47; e-mail:[email protected]

The goal of this contribution is to show that somenew features of the three-level SR also appear if oneaccounts for the local-field correction. In particular,we analyse the influence of the local-field correctionon the SR from a thin-film system of initially in-verted three-level atoms assuming the film thicknessto be much smaller than the emission wavelength. Asis well known, for a similar system of two-levelatoms, the local-field correction results in a popula-

w xtion-dependent resonance frequency shift 22,23 andhas to be taken into account for adequate descriptionof the optical response from a dense saturable ab-

w xsorber 24–38 and especially from a thin-film sam-w x.ple 9,39–45 . We show that in the case of a thin

film comprised of three-level atoms, the above shiftresults in a competition between the operating transi-

0030-4018r00r$ - see front matter q 2000 Published by Elsevier Science B.V. All rights reserved.Ž .PII: S0030-4018 00 00678-7

( )V.A. MalysheÕ et al.rOptics Communications 180 2000 59–6860

tions, which makes possible under suitable condi-tions to suppress completely one of them. The effectis driven by the ratio of the lower level splitting to

Žthe magnitude of the local-field correction in fre-.quency units and, thus, can be governed by an

external field.The bulk of the paper is organized as follows. In

Section 2, the model we are going to exploit isformulated. Section 3 deals with a particular case ofdegenerated doublet in the ground state, allowing ananalytical treatment of the problem. Results of nu-merical simulations are presented in Section 4. InSection 5, we provide an interpretation of the pecu-liarities found numerically, basing our considerationon the linear stage of SR. Section 6 summarizes thepaper.

2. Description of the model

Let us consider the SR from a thin slab comprisedof three-level atoms with a doublet in the ground

Ž .state so-called L-configuration , choosing the dou-blet splitting v to be much smaller than the fre-21

quencies v and v of the optical transitions31 32

between the upper state 3 and those of the doublet, 1Ž .and 2 see Fig. 1 . We suppose that the slab thick-

ness is much smaller that the emission wavelength,so that one can neglect the spatial dependence of allthe functions characterizing the problem. Besides, all

Žthe vectors the SR electric field EEEEE and the transition.dipole moments d ,d and d are assumed to be12 13 23

parallel to each other as well as to the slab plane.Hence, all the quantities can be considered as scalars.For the sake of simplicity, we also assume that the

Fig. 1. Diagram of the energy levels and allowed transitions.

transition dipole moments are real. The evolution ofsuch a system is governed by a set of equations for

Ž .the density matrix r a ,bs1,2,3ab

d EE X

31r syiv r y i r yrŽ .˙31 31 31 33 11

"

d EE X

32q i r , 1aŽ .21

"

d EE X

32r syiv r y i r yrŽ .˙32 32 32 33 22

"

d EE X

31q i r , 1bŽ .12

"

d EE X d EE X

31 32r syiv r y i r q i r , 1cŽ .˙21 21 21 23 31

" "

d EE X

31r s i r yr , 1dŽ . Ž .˙11 31 13

"

d EE X

32r s i r yr , 1eŽ . Ž .˙22 32 23

"

d EE X

31r syi r yrŽ .˙33 31 13

"

d EE X

32y i r yr , 1fŽ . Ž .32 23

"

where the dots denote time derivatives and EE X standsfor the acting field

4pXEE sEEq PP . 2Ž .

3

The latter differs from the emission field by theŽ .local-field correction 4pr3 PP, where PP s

Ž .N d r qd r qc.c. is the electric polariza-0 31 31 32 32

tion, N being the atom concentration. The emission0w xfield itself is given by the equation 9,44

2p L˙EEsy PP . 3Ž .

c

Here, L and c stand for the slab thickness and thespeed of light, respectively. Neither relaxation ofpopulation nor de-phasing of the electric polarizationare not taken into account since the SR is assumed to

Žbe the fastest process estimates of the corresponding.constants are presented in Section 6 . Transitions

between the states of the doublet are not considered.

( )V.A. MalysheÕ et al.rOptics Communications 180 2000 59–68 61

Ž .We seek a solution of the set of equations 1 inŽ .the form: EE s Eexp yi v t q c.c., r sc 31

Ž . Ž .R exp yiv t , r sR exp yiv t , where v s31 c 32 32 c cŽ . Žv qv r2; E is the complex slowly varying in31 32

.scale of 2prv amplitude of the field; R and Rc 31 32

are the amplitudes of the off-diagonal density matrixŽ .elements further named as optical coherences . Note

that with regard to the low-frequency coherence r21

the analogous assumption is not used.Ž .Passing from Eqs. 1 to equations for the ampli-

tudes, one gets

v21R syi R31 312

1q y i D m r yr ym rŽ .L 31 33 11 32 21ž /tR

= m R qm R , 4aŽ . Ž .31 31 32 32

v21R s i R32 322

1q y i D m r yr ym rŽ .L 32 33 22 31 12ž /tR

= m R qm R , 4bŽ . Ž .31 31 32 32

r syiv r˙21 21 21

12< <qm m q i D R31 32 L 31ž /tR

12< <q y i D RL 32ž /tR

22 2q y i D m ym R R , 4cŽ .Ž .L 31 32 31 23

tR

222 < <r s m R˙11 31 31

tR

1qm m R R qR RŽ .31 32 31 23 13 32

tR

qi D R R yR R , 4dŽ . Ž .L 31 23 13 32

222 < <r s m R˙22 32 32

tR

1qm m R R qR RŽ .31 32 31 23 13 32

tR

yi D R R yR R , 4eŽ . Ž .L 31 23 13 32

22 22 2< < < <r sy m R qm R˙33 31 31 32 32

tR

qm m R R qR R . 4fŽ . Ž .31 32 31 23 13 32

Here, the following notations are introduced: m31ds rd and m s d rd , where d s31 32 32

2 2 2d q d r2 ; D s 4p d N r3" ; t s(Ž .31 32 L 0 R2 Ž ."r2p k d N L k sv rc . The quantities D andc 0 c c L

ty1 stand for the magnitudes of the local-field cor-RŽ .rection and of the SR field in frequency units ,

respectively. Since D t s2r3k L, for a thin filmL R cŽ . y1k L<1 one has D 4t , i.e., the local-fieldc L R

magnitude exceeds that of the SR field. However,the latter cannot be neglected as compared to theformer because they play different roles in the emis-sion process: the local-field correction produces thedynamical resonance-frequency shift while the SR

Žfield is responsible for the radiative damping for.more details, see Section 3 .

The initial conditions to the system of equationsŽ . Ž . Ž . Ž .4a – 4f are chosen in the form: r 0 s1, r 033 11

Ž . Ž . Ž . Ž .sr 0 sr 0 s0, R 0 sR 0 sR . Setting22 21 31 32 0

a fixed not-fluctuating value for the initial electricpolarization corresponds to the trigger scheme of theSR initiation by an ultrashort external pulse of small

w xarea and with a duration T -t 9,46,47 .p R

Concluding this section, we would like to specifythe model we will be dealing with along this articlemore in detail. It should be related to a thin dielectriccrystalline film rather than to a dense gas system. Inthe latter case, the pressure broadening terms, notincluded in the present model, appear to be of greatimportance. Indeed, these terms, being on the onehand of relaxation nature and of the same order of

w xmagnitude as the local-field correction 27 on theother hand, will act as a destructive factor relative to


SR, thus altering sufficiently this regime of emissionand accompanying effects. On the contrary, in thecase of a dielectric crystalline sample, the line broad-ening usually has a different nature than the localfield. It frequently arises due to the crystal imperfec-tions as well as coupling to phonons. Under certainconditions, its magnitude may occur to be smallerthan that of the local field. An undoubted confirma-tion of this fact is the existence of the Frenkelexciton states in dielectric solids due to the inter-

Žatomic dipolar coupling or, in other words, due to. w x .the local field 48 , see Section 4 .

( )3. Degenerated doublet v s021

Let us consider first the simplest case of the zerodoublet splitting, allowing to carry out an analyticaldescription of the SR. It can be done by passing from

< : < :the doublet states 1 and 2 to the new ones'< : Ž < : < :. < : Ž < :q s m 1 qm 2 r 2 and y s m 131 32 32'< :.ym 2 r 2 . The convenience of this subset of31

< :states becomes clear from the fact that q is theŽonly superposition coupled to the upper state 3 the

< :dipole moment of corresponding transition 3 lˆ '< : ² < < : .q is not equal to zero: 3 d q s 2 d while the

ˆ< : < : Ž² < < : .transition 3 l y is forbidden 3 d y s0 ,and in this sense has no effect on the SR. We will

< :further refer to the state y as to the dark state.< : < : < :In the frame of the new basis 3 , q , y the

density matrix elements can be expressed as follows

1 2 2r s m r qm r q2m m R r ,Ž .qq 31 11 32 22 31 32 212

5aŽ .

1 2 2r s m r qm r y2m m R r ,Ž .yy 32 11 31 22 31 32 212

5bŽ .

1r s m m r yrŽ .qy 31 32 11 222

2 2qm r ym r , 5cŽ .32 21 31 12

1R s m R qm R , 5dŽ . Ž .3q 31 31 32 32'2

1R s m R ym R . 5eŽ . Ž .3y 32 31 31 32'2

Now, r and r stand for the populations ofqq yyŽ < :. Ž < :.the operating q and dark y states respec-

tively; r represents the low-frequency coherence;qyR and R are the optical coherences, respec-3q 3y

Ž < : < :. Ž < :tively, of the operating 3 l q and dark 3 l< :.y channels; the diagonal element r remains33

unchanged. The corresponding set of equations reads

v21 2R syi 1ym R qm m RŽ .3q 32 3q 31 32 3y2

1q2 y i D R r yr , 6aŽ . Ž .L 3q 33 qqž /tR

v 421 2< <r syi m m r yr q R ,Ž .˙qq 31 32 qy yq 3q2 tR

6bŽ .

42< <r sy R , 6cŽ .˙33 3q

tR

v21 2R syi 1ym R qm m RŽ .3y 31 3y 31 32 3q2

1q2 i D y R r , 6dŽ .L 3q qyž /tR

v21 2 2r syi m ym r˙ Ž .qy 32 31 qy2

qm m r yrŽ .31 32 qq yy

1q2 q i D R R , 6eŽ .L q3 3yž /tR

v21r s i m m r yr . 6fŽ . Ž .˙yy 31 32 qy yq2

< : < :At v s0, the operating channel 3 l q , Eqs.21Ž . Ž . Ž .6a , 6b and 6c , evolve independently of the dark

< : < : Ž . Ž . Ž .one 3 l y , Eqs. 6d , 6e and 6f . r re-yymains unchanged during the emission process, Eq.Ž .6f . Thus, the problem can be reduced to only twoequations

1R s4 y i D R Z , 7aŽ .3q L 3qž /tR

42˙ < <Zsy R . 7bŽ .3q

tR


Ž .Here, the notation Zs r yr r2 is introduced.33 qqŽ .From Eq. 7a , it is clearly seen the role of the

Ž .local-field correction term D Z and of the emis-LŽ y1 .sion field term t Z, mentioned in Section 2: theR

former results in the resonance frequency shift whilethe latter stands for the radiative damping, both,however, being dependent on the population differ-ence Z.

Ž .The set of equations 7 has an analytical solution

ty td1Zsy tanh , 8aŽ .2 ž /t r2R

ty td1 ifR s e sech , 8bŽ .3q 2 ž /t r2R

tfsy4D Z t dtŽ .HL

0

ty t td dsD t lncosh y lncosh , 8cŽ .L R

t r2 t r2R R

being analogous to that for a two-level system ofw xsize smaller than the emission wavelength 9,49–51

with the substitutions of t by t r2 as well as DR R LŽ . Ž . < Ž . <y1by 2 D . In Eqs. 8 , t s t r2 ln R 0 is theL d R 3q

delay time of the SR pulse; f describes the SRŽphase modulation ffy2 Dt at t- t and ffd

Ž . .2 D ty2 t at t) t . The populations r andL d d 33Ž .r can be found using Eq. 8a and the formulasqq

Ž .Ž . Ž .Ž .r s 1r2 1qZ and r s 1r2 1yZ .33 qq

4. Results of numerical simulations

In order to gain insight into the peculiarities of thethree-level SR in a more general case, we carried outnumerical simulations of the SR emission, assumingthe dipole moments of the optical transitions 3l1and 3l2 to be equal to each other: m sm s1.31 32

At ts0, all the atoms were assumed to be excited inŽ . Ž . Ž .their upper level: r 0 s 1, r 0 s r 0 s33 11 22

Ž .r 0 s0. The initial value of the electric polariza-21Ž . Ž . y8tion was set R 0 sR 0 s10 .31 32

In Fig. 2, we depicted the numerical data of theSR behaviour versus the doublet splitting v at a21

< < < <Fig. 2. Kinetics of the SR field modulus ´ s d E t r" as wellRŽ .as of the level populations r , dotted line; r , solid line11 22

versus the doublet splitting v at a fixed magnitude of the21

local-field correction D s5. Numbers denote the values of v .L 21

All quantities are given in units of ty1.R

Žfixed magnitude of the local-field correction in thisy1 .particular case, D s5, hereafter in units of t .L R

From this figure, it is clearly seen a competitionbetween the transitions 3l1 and 3l2 depending

Žon the ratio v rD . Indeed, at v F1 also in21 L 21y1 .units of t , both transitions are strongly coupled toR

each other by the emitting field and evolve syn-chronously, which results in approximately equalfinal populations of the doublet states. Because ofthe interference between the transitions, this regimeof the SR is characterized by the delay time t andd

the pulse duration t which is two times shorter thanp

those for the standard two-level SR, in full accor-Ž .dance with Eqs. 8 .


At v 4D , the transition 3l2 is essentially21 L

suppressed, so that the upper level of the doublet is,in fact, depopulated after the SR pulse has beenemitted. All the population is finally transferred tothe lowest level 1. Notice that such a regime ofemission takes place even at large values of v ,21

exceeding the spectrum of the SR pulse. At firstglance, it seems that under such a condition, thetransitions 3l1 and 3l2 might be decoupled onefrom the other. However, to really decouple themone should push up v to higher magnitudes.21

It is worth to mention the fact that a low magni-tude of r at t™` does not mean that the state 222

was never populated before. As follows from Fig. 2,r behaves in time non-monotonically, increasing22

initially and then dropping to a low level whichdepends on the ratio v rD .21 L

Fig. 3. Kinetics of the level populations versus the local-fieldcorrection at a fixed magnitude of the doublet splitting v s3.21

Numbers denote the values of D . All quantities are given in unitsL

of ty1.R

The changes in kinetics of the lower level popula-tions upon raising D , at a fixed value of the doubletL

splitting v , are drawn in Fig. 3. One observes here21

that the lowest level also gets advantageous condi-tions for its population as the local-field correctiongoes up, similar to what takes place upon rising thedoublet splitting v at a fixed magnitude of the21

local-field correction D .L

5. Discussion

In this section, we provide an interpretation of thepeculiarities of the three-level SR, found in thenumerical simulations. It is useful to look at thelinear stage of the emission process. Hence, we startthe discussion with the linear analysis of SR.

5.1. Linear stage

Ž .Linearization of the set of equations 4 assumesthe fact that during some period of time, the popula-

Ž .tions r is1,2,3 and the low-frequency coher-i i

ence r maintain their initial values, while the only21

evolving quantities are the optical coherences R31

and R . Under the conditions adopted in the numer-32Ž Ž . Ž .ical simulations m sm s1, r 0 s1, r 031 32 33 11

Ž . Ž . Ž . Ž . .s r 0 s r 0 s 0, R 0 s R 0 s R one22 21 31 32 0

arrives at

1 v21R s y i D q R31 L 31ž /t 2R

1q y i D R , 9aŽ .L 32ž /tR

1 v21R s y i D y R32 L 32ž /t 2R

1q y i D R , 9bŽ .L 31ž /tR

The solution of these coupled equations has the form

R v0 21R s v q exp yiv tŽ .31 1 1ž /v yv 2Ž .1 2

v21y v q exp yiv t , 10aŽ . Ž .2 2ž /2


R v0 21R s v y exp yiv tŽ .32 1 1ž /v yv 2Ž .1 2

v21y v y exp yiv t , 10bŽ . Ž .2 2ž /2

where v are the roots of the corresponding secular1,2

equation

iv s D q1,2 Lž /tR

2 2i v21" D q q . 11Ž .) L ž /ž /t 2R

A simple analysis of the solution found showsthat, first, the imaginary part of v differs from that1

of v , and second, both of them are positive, inde-2

pendently of the values of D and v . Hence, Eqs.L 21Ž . Ž .10a and 10b yield the divergent solutions, andwhat is more important, with different increments.The real parts of v and v also differ one from1 2

another. This may result in a modulation of the SRw Ž .xsignal with a period R 2pr v yv .1 2

( )5.2. Large doublet splitting v 4D21 L

It seems reasonable to assume that at v 4D ,21 L

transitions 3l1 and 3l2 are coupled to eachother only through the upper level as a source ofpopulation so that there is no reason for the prefer-ence in development of any of them. Thus, for thelevel populations one can use the solution

ty td1 1r s y tanh , 12aŽ .33 2 2 ž /tR

ty td1 1r sr s q tanh . 12bŽ .11 22 4 4 ž /tR

y1'Ž < <.Here, t st ln 2 R is the SR delay time.d R 0Ž .Eqs. 12 describe the synchronous population of

levels 1 and 2 as the time increases up to the value1r2. As it nevertheless follows from Fig. 2, in the

Žpresence of the local-field correction D s5 for theL.case at hand , even a fairly large value of the doublet

splitting v s30 is not sufficient to arrive at equal21

populations of the doublet levels.

To understand this peculiarity, let us analyse theŽ .linear stage of the SR. At v 4D , Eq. 11 is21 L

reduced to

v i 2 D21 Lv sD " q 1" . 13Ž .1,2 L ž /2 t vR 21

Ž .Accordingly, Eqs. 10 takes the formv21

R sR exp yi D q t31 0 Lž /2

2 D tL=exp 1q , 14aŽ .ž /v t21 R

v21R sR exp yi D y t32 0 Lž /2

2 D tL=exp 1y . 14bŽ .ž /v t21 R

Notice that the increment of R is slightly larger31

than that of R , i.e., R develops faster than R .32 31 32

In turn, the lower level of the doublet will be finallypopulated to a higher value than the upper one. Theresulting populations depend on the ratio D rv .L 21

< < < <Comparison of R with R at the moment of31 32

appearance of the SR pulse, t , allows us to estimated

the relative participation ratio of the transitions in theemission process. In doing so, we will use the fact

Ž .that Eqs. 14 provides a fairly good fit to theŽSR kinetics almost up to the SR maximum see

. < Ž . Ž . <Fig. 4 . Thus, the estimate R t rR t s32 d 31 dw Ž .xexp y4D t r v t appears to work well. FromL d 21 R

< Ž . < < Ž . <here, one gets R t < R t provided32 d 31 d

4D rv )t r4 t . The latter inequality representsL 21 R d

Ž .Fig. 4. Kinetics of the optical coherences R solid lines and31Ž . ŽR dashed lines as well as their linear approximations diver-32

.gent solutions for D s5. Numbers denote the values of v . AllL 21

quantities are given in units of ty1.R


the condition for discrimination of the transition3™2 as compared to that 3™1. For the SR delaytime f15t , the inequality found will read: D rvR L 21

)1r60, which is in good correspondence with theŽ .numerical data see Figs. 2 and 4 .

( )5.3. Large local-field correction D 4vL 21

Let us now turn to the opposite case of a largelocal field magnitude, assuming the inequality D 4L

v to be fulfilled. The solution of the secular Eq.21Ž .11 is then simplified to

iv f2 D q , 15aŽ .1 Lž /tR

v 2 i21v sy D y . 15bŽ .2 L2 y2 ž /t8 D qtŽ . RL R

y1 < < < < Ž .Recall that D 4t so that v 4 v . Eqs. 10L R 1 2

then reads

v 121R sR 1q exp 2 y i D t , 16aŽ .31 0 Lž / ž /4D tL R

v 121R sR 1y exp 2 y i D t . 16bŽ .32 0 Lž / ž /4D tL R

From here, it follows that R fR in the zeroth31 32

order approximation with respect to v rD <1. It21 L

seems that there is no preference in the developmentof the transition 3l1 relative to the other one3l2. This, however, evidently contradicts the nu-merical data presented in Fig. 3: there exists a suffi-cient difference in behaviour of the transitions 3l1and 3l2. We believe that the key points for under-

< <standing this discrepancy are, first, that R is still31< <larger than R , meaning a little advantage in popu-32

lation of the lower doublet level, and second, thatdue to the local-field, both levels of the doubletundergo a dynamical shift proportional to r yr33 i iŽ .is1,2 . As a result, they will cross each otherduring the SR emission. Indeed, at ts0, the doubletlevels are shifted due to the local-field effect by the

w Ž . Ž .xsame value equal to y2 D r 0 yr 0 sy2 DL 33 i i Lw Ž .xsee Eqs. 16 . As the SR pulse develops, the dy-namical shift of the lower doublet level will raise

Žfaster than that for the upper one just because of< < < <.R ) R , resulting finally in their crossing. It31 32

occurs at some instant of time t , which can bec

estimated as

2 D r t yr t sv . 17Ž . Ž . Ž .L 11 c 22 c 21

To carry out the calculations analytically, assumethat the levels cross each other during the linearstage of the SR. For evaluating r yr , we can22 11

then use the evident solution for R and R , Eqs.31 32Ž . Ž . Ž .16 . Substituting them into Eqs. 4d and 4e , weobtain for r yr :11 22

v 4 t212< <r yr s R exp . 18Ž .11 22 0 ž /2 D tL R

Ž .Respectively, Eq. 17 gives us

t 1Rt s ln . 19Ž .c < <2 R0

The time t , found in such a manner, is shorter thanc

the SR delay time t , which for the case we aredŽ .discussing is given by the expression t s t r2 =d R

y1'Ž < <.ln 2 R . This is in good correspondence with0

the numerical data presented in Fig. 3, a simpleanalysis of which shows that the dynamic crossing of

Ž Ž ..levels as determined by Eq. 17 , occurs slightlyŽbefore the SR pulse arrives at its maximum at ts td

.when r s0.5 . Afterwards, it appears a redistribu-33

tion of populations in favour of the lowest level.

6. Summary and concluding remarks

The super-radiance from a thin slab of initiallyinverted three-level atoms with a L-configuration ofoperating transitions is studied theoretically. It isfound that interplay between the local-field correc-tion and the doublet splitting affects significantly theemission process. Our finding are the following:

Ž .i In the case of a degenerated doublet, thethree-level problem is equivalent to that for a twolevel system with a re-normalized SR time scale. Therole of the local-field correction is also similar tothat for the two-level problem, manifesting itself in aphase modulation of the SR signal.

Ž . Žii As the doublet splitting raises at a fixed.magnitude of the local-field correction , a competi-

tion between the operating channel occurs due to thelocal-field effect, resulting in a nearly complete sup-pression of the transition to the upper doublet level.As a result, all the population is transferred during


the emission process from the excited level to thelowest one.

Ž .iii Raising the magnitude of the local-field cor-Ž .rection at a fixed value of the doublet splitting also

results in a sufficient suppression of the transition tothe upper doublet level.

In conclusion, we comment on some real systemswhich could serve as possible candidates for theexperimental observation of the predicted regimes ofemission. In this sense, thin films of some aromaticcompounds such as naphtalene, antracene etc seemto be suitable for this task. At low temperatures, theoptical excitations in these crystals are Frenkel exci-tons, meaning that the intermolecular dipole-dipole

Žinteraction or, in other words, the local-field correc-.tion dominates over the relaxation. An estimate for

the electric polarization de-phasing rate can be doneon the basis of the width of the exciton absorption

w xline. Due to the exchange narrowing effect 52 , itsŽ y1typical value appears to be rather low ;100 cm

y1 .or 3 ps in frequency units . On the other hand, thedensity of the optically active molecules is extremely

Ž 21 y3.large N ;10 cm , resulting in a very high0

magnitude of the super-radiant damping constanty1 Ž .2 2t s 2p d N Lr"l. In particular, for a film ofR 0

thickness Ls0.1l and under the assumption ofls5P10y5 cm and a dipole-allowed transition, thisconstant appears to be of the order of 100 psy1, i.e.,

Žit sufficiently exceeds the de-phasing constant 3y1 .ps . Thus, one has here the conditions needed for

the super-radiant regime of emission and the pre-dicted effects to occur. Summarizing, we believe thatour model is applicable for treating an isolatedFrenkel exciton transition with a doublet structure inthe ground state.

Acknowledgements

This work was supported by the Russian Ministryof Education.

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super-radiance from a thin solid film of three-level atoms: local-field effects

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