# Cooperative Spontaneous Emission from a Single-quantum Excited Three-atom System.

код для вставкиСкачать~~ ~ Annalen der Physik. 7. Folge, Band 38, Heft 2, 1981, S. 106-122 J. A. Barth, Leipzig Cooperative Spontaneous Emission from a Single-quantum Excited Three-atom System By TH.RICHTER Zentralinstitut fiir Optik und Spektmskopie der Akademie der Wissenschaften der DDR, BerlinAdlershof Dedicated to Prof.Dr. &tau Richter on the Occasion of the 70th Anniversary of hie Birthday Abstract. The total radiation rate of the cooperative spontaneous emission from an initially single-quantum exoited three-atom system is calculated using Lehmberg's master equation and taking into account dipole-dipole near field interaction. We find a strong influence of the precise configuration of the atomic dipoles and of the position of the initially excited atom on the temporal behaviour of the total radiation rate, especially if the atoms are separated by distances much smaller than the resonance wavelength. Kooperative spontane Emission eines einfach angeregten Drei-Atom-Systems Inhaltsubersicht. Es wird die Gesamtintensitiit der spontanen Strahlung eines kooperativ emittierenden, anfiinglich einfach angeregten Drei-Atom-System unter Beriicksichtigung der DipolD i p o l - N a ~ e I d - ~ ' e c h s e mit l ~ ~Hilfe g der Lehmbergschen Mastergleichung berechnet. Ihr zeitlicher Verlauf hgngt, insbesondere bei Atomabstiinden, die vie1 kleiner als die Resonanzwellenliinge sind, ganz wesentlich von der genauen Anordnung der atomaren Dipolmomente und von dem Ort des anfanglich angeregten Atoms ab. 1. Introduction Superfluorescence - the cooperative spontaneous emission from a large number N of initially excited atoms (or molecules) - has attracted some interest in the last years. Recent experiinental [l- 31 as well as theoretical studies [4- 81 investigate superfluorescenre from atoms randomly arranged in a pencil shaped volume V = SL of cross section S A2 (A resonance wavelength) and of such a length L that the diffraction angle equals the geometric angle fi1.L. Starting from a microscopic formulation of the interaction of the atoms with the electromagnetic field current theoretical treatments of supeduorescence assume that due to the shape of the volume containing the atoms the evolution of the superfluorescent pulses is governed by the interaction of the atoms with two plane waves with slowly varying amplitudes travelling in opposite directions along the pencil axis. Therefore all atoms contained in a thin transverse slice of the pencil shaped volunie with thickness d A feel the same electric field. To this end it is necessary that the density e of the atoms is on the one hand so large that there are many atoms in every such slices, but on the other hand so small, ~ ( f / L ?<n 1, ) ~that dipole-dipole nearfield interaction can be neglected. This is the case in most of the experiments. If however there are inany atoms in a cube of the resonance wavelength each atom would feel a < Cooperative Spontaneous Emission from a Three-atom System 107 different electric field which depends strongly on the actual positions of the sorrounding atoms due to the quick fall off of the near-field component of the atomic dipoles. In the Dicke point model of superradiance widely used in the literature, e.g. [9-131, this fact is not taken into account. I n this model the cooperative spontaneous emission of N atoms with mutual distances much smaller than the resonance wavelength is studied supposing that “all atoms see the same field”. At the very beginning of the emission process the atoms radiate incoherently in all directions and the description by two plane waves travelling in opposite directions along the pencil axis fails, of course. In fact, one has to start from atoms coupled to the continuum of all field modes (especially in all directions)to make evident during the initiation of superfluorescencethe formation of essentially two field modes travelling in opposite directions along the pencil axis. But this is of course a very difficult problem unsolved up to now. Therefore in order to understand cooperative radiation effects from a macroscopic system of atoms it seems to be useful to study as completely as possible the radiation from a few atoms. The spontaneous radiation from two atoms coupled to the continuum of all field modes, and taking into account the dipole-dipolenear-field interaction, was investigated by many authors, e.g. LEEVIBERG [14], STEUDEL[15], S ~ ~ I R NetOal. V [16] and RIOETER [17]. Recently COFFEYand FBIEDBERG [18] studied the influence of the near-field interaction on the spontaneous decay of an initially fully excited system of three atoms separated by distances much smaller than the resonance wavelength. They could derive approximate solutions and found that the atomic configuration has a strong effect on the total radiation rate. Collective radiation of several charged harmonic oscillators interacting essentially through near-field dipole forces was also studied by Zmomcz [19] and LEWENSTEIN and R ~ Z E W S [20]. K I In the latter paper explicit results for the total radiation rate from four oscillators placed on the vertices of a tetrahedron are presented. Related problems concerning two and three atoms excited continuously by an incoherent pumping mechanism were discussed by STEUDEL and RICHTER[21] and STEUDEL [22]. In the present paper we investigate the cooperative spontaneous emission from three atoms one of which is excited initially. The influence of the dipole-dipole near-field interaction on the total radiation rate is discussed in some detail. As we did inforegoing papers we start in sect. 2 from the master equation due to LEHIKBEBQ [23] and describe in sect. 3 how to derive the total radiation rate. In sect. 4 we discuss the equations of motion and give in sect. 5 the general solutions. Some limiting cases of atomic configurations are discussed in sect. 6 while numerical results for general cases are presented in sect. 7. 2. Model and Method We consider three identical atoms a t the positions x1 = (0, $v+., 0) 9 x, = forming an isosceles triangle of base length 2% and lateral side length 1. This three-atom system is coupled to a quantized multimode electromagnetic field. The distances between the atoms should be such that only electromagnetic interactions are possible. Each atom A,, (p= 1, 2, 3) is approximated by a two-level system; the ground state Is,) and the excited state leJ being connected by an electric dipole transition. The polarisation vector e, of the transition dipole moments e, d of the atoms are assumed to be adjusted parallel t o each other, and especially perpendicular to the plane of the three atoms, although this condition is in no case a neces8’ TE.RICHTER 108 sary one and some more general cases can be treated without too much additional calculations. Initially only one atom should be in the excited state and the electroningnetic field in the vacuum state. This model is still susceptible t o a n analytic treatment but general enough t o allow the study of some near- field effects in detail, which disappear in the simple case of a n equilateral triangle configuration of the atoms with a coninion polarisation vector e, = e perpendicular t o the atomic plane. The time evolution of the expectation value of any atomic operatorQis described by the master equation [12, 21, 231 (units in which ti = 1 are used throughout) if we restrict ourselves t o such a size of the atomic system for which secular changes of the atomic states can be neglected during the time of light propagation from one atom t o the other ones. The subscripts p, v = 1, 2, 3 denumerate the atoms. a,’ and a, are the usual raising and lowering operators for the atom number y. w - w = kc and 2yrc = 2y represent the transition frequency and the Einstein ,4rlr coefficient of a n undisturbed single atom, where k = 2n/A and Iz being the resonance wavelength. w,,, and yr,, in case p =/= v describe the frequency shift and the change of the single atom decay rate, respectively, due t o the mutual interaction through the electromagnetic field : where we have introduced two new quantities .4, and p,,,,, which depend only on the distance x,,,, = Ixp,,( = Ix,, - x,1 between the atoms p and Y and on the spatial orientation of the corresponding dipole moments described by CJI,”’) and (Of”, py) measured for each pair of given p and v in a separate polar coordinate system of which the polar axis is taken parallel t o the vector xpU. According to eq. (2) Iq,,,,l increases like x;: as x,,, tends t o zero. In.the following, as already mentioned above, we restrict ourselves to the cage of dipole moments parallel to each other and perpendicular to the atomic plane so that according t o eqs. (2), (3) p12 = p13 and qI2 = q13’ plrutends t o 1 as the distance between the atoms p and Y tends t o zero. Our objective is t o calculate the normally ordered one-time correlation function of the electromagnetic field a t a point r = r? in the wave zone, which is proportional t o the radiation intensity I(?,t ) (Or, r2c 2nw I(?,t ) = -<E(-)(r, t ) E ( + ) ( r ,t ) ) . Here we have introduced the factor (r2c/2no)so that I(?, t ) dl2;dt is the probability to find one photon inside the solid angle element dQ; around the direction i. in the time intervall dt a t time t in the far field zone of the radiation emitted by the atomic system. The field operator E(r, t ) in the far zone r % Ixc - x, 1, iland for t > r/c is given by the well Cooperative Spontaneous Emission from a Three-atom System 109 known expression [12, 231 E ( r , t ) = E ( + ) ( r ,t ) + E(-)(r,t ) (5) with [e E ( + ) ( r ,t ) = E&+)(r,t ) - k2 d - (e?)31 r 2 u,(t - r/c)exp(--ikx,t), (6) ,=I where E $ ( r , t ) denotes the positive frequency part of the vacuum field and the second terin in eq. (6) describes the retarded field of the atomic point dipoles. Because the field is initially in the vacuum state, the vacuum part does not contribute to the expectation value of the normally ordered correlation operator in eq. (4) and we get for I(?, t ) the expression, I(;, t ) = 2yu(?) 2 (u,'(t) a&)) exp (-ikz,,?), P,V where from now on t means the time of emission of the observed light intensity and u(?)= (3/8n) [l - (e;),] denotes the radiation characteristic of a single atomic dipole. On summing over all solid angles dQ; eq. (7) yields the total radiation rate I ( t ) given in photons per second : = 27 c p,,(a,+ ( t ) a,(t)> (8) 3 CP where we have used the relation p,, = Ju(;)exp (--ikx,,?) dQ;. (9) 3. Calculation of the Radiation Rate The problem is simplest if in zeroth order the symmetric and antisymmetric combinations of the single atomic states are used. To facilitate comparison with previous work [lo-121 we use the following well known set of angular-momentum states or so called DICKE states 1 r, m ; s) as basic states of the atomic system : a ) all atoms in the ground state (10) I3/2, -3/2 ; 1) = 10) = ISi) IS,) 193), b) the fully symmetric single excited state 1 13/29 -1/2; 1) = 11) = [ISl> IS,> - 1%) v3 f 191) 1%) 193) 4-1%) IS,) 193)Ij (11) c ) the symmetric single excited state with respect to exchanging atoms A, and A, 1 11/22 -1/2; 1) = 11') = [ISJ 192) 1%) f (91) lea) 193) - 2 lei> IS,> ISs>I. V6 (12) d) the antisymmetric single excited state with respect to exchanging atoms A, and A, p/2, -1/2; 2) = 1 11") = -[Is VF Now we define the operators Prms,rlmlrl = Ir, m ; s) (r', m ' ; s'I which have the property <P-,r~a.) =erm,r~6#, 1) 192) 1%) - 191) Ie,> 193)l. (13) TH.RICHTER 110 where erM,r#,,,n'sn are the matrix elements of the reduced atomic density operator in the representation chosen. The expectation values <a,' (t) a,(t)) can be expressed by ernur,r~,,,~8~ in the following manner (for abbreviation we put eZ,-L,,;. - L ; ~= ell etc.) 3 2 2 ' 2 <utu,,) = {n,, ) gives the probability to find atom A,in the excited state. Inserting (16)(21) into (8) and putting p12= p13= p and q12 = qls = q the expression for the total ra- diation rate reads as follows Therefore we have to determine the density matrixelementsgll, eltlt,el-l,,and ella+el.l. 4. Equations of Motion Fortunately the master equation (1)couples only expectation values of operators with the same value of A = m - m' [all. The expression shows that for calculations of radiation rates only the expectation values of operators with A = 0 are needed. Now the master equation (1)yields for the expectation values of the operators Pru,r.ms. two sets Cooperative Spontrtneous Emission from a Three-atom System 111 of four coupled differential equations and two single equations. The set of equations we are interested in prove to be a at 0 0 X eqs. (23) show explicitely that in contrast t o the well known two-atom case already for a three-atom system it is in general no longer possible t o iptroduce orthogonalized basic states of the atomic system of which the temporal developments are independent from each other. This qualitative new effect compared with the two-atom case is mainly due to the different frequency shifts caused by the dipole-dipole near-field interaction. Only if the three atoms form an equilateral triangle and have a common polarisation vector perpendicular t o the atomic plane, 80 that according to eqs. (2, 3) p = pZ3 and q = qB3,the excited states (1)and 1')' are dynamically decoupled not only from the state Il"> but also from each other, as can be seen immediately from eqs. (23). TH.R ~ m m 112 If we now reduce the size of the atomic equilateral triangle configuration similar to itself we arrive (but only in this case) in the limit I A at the DICKE point model for a three-atom system, with the superradiant state )1>and the subradiant states Il') and 11") being independent from each other. But we want to emphasize once more that in the general configuration case the temporal evolution of the atomic system is muchmore complicated due to the different frequency shifts, which cause a dynamical coupling of the superradiant and subradiant states especially in the limit of atomic distances much smaller than the resonance wavelength (contrary to the statement in many papers in which the influence of the frequency shift on the decay laws are overlooked even in case of small atomic distances compared with 1, e.g. [lo, 11, 131). < 5. General Solution We have solved the Eqs. (23), (24) by Laplace transform methods. Fortunately the secular equation for the system (23) proves to be a biquadratic one: [S + Y(2 f Putting P23)I4 + Y2(8P2 f !7%3 - 8P2 - P;3) [s + + Y(2 + Pz3)I2 - Y4(8w+ 2)23q23) = O. 2 = [S Y ( 2 P23)I2 Eq. ( 2 5 ) reduce to a quadratic equation with the roots (25) (26 ) 1 21.2 = - pr2[(8n24-&3) - @P2 f P%1 Since q 2are real, z1 > 0 and x2 < 0, we find for the biquadratic equation ( 2 5 ) two pure real roots Cooperative Spontaneous Emission from a Three-atom System 113 and where we have introduced the quantities = 2 - r(q31/2 1 3 = -y('q f (34) 423)' Finaly eq. (24) can be solved immediately and we obtain el#el#t(t)= exp [-2y(l - pas) t ] &,,lpn. (35) Henceforth we consider the special case that the common polarisation vector e of the atomic dipole moments is perpendicular to the plane of the three atoms and either atom A, on the vertex or atom A, on the base is excited initially. According to eqs. (16)-(21) these initial conditions expressed in terms of eP1 etc. which appear in the general solutions are given by: = (1/3), ,&el, = (2/3), elet1t. = 0, ey1, = ell = -(2/3 if atom A, is excited initially, i.e. n,(O) = 1, and e& = (1/3), e h = (1/6), e!& = (W), 1/2) (36) - = e y e 1 = (1/3 1 2 ) (37) if atom A, is excited initially, i.e. n,(O) = 1. 6. Discussion of Some Limiting Cases First of all we want to discuss the solutions in some limiting cases of atomic configurations. . TIT.RICHTER 114 6.1. Case Ze 128, Equilateral Triangle Configuration I n this case the atom8 are lying on the corners of a n equilateral triangle, so that eqs. (2), (3) give p = pz3 and q = qZ3.Inserting this into the equations of inotion (23) we get: el1 -2y(1 2 p ) ell7 (38) = -2y(l - P)elT> (39) (ell. e1*1)* = --I4 P) (ell. e l d 3+de11* - @ 1 d , (-10) (ell. - e d = --33iy~(e~ e l 4 - ~ ( 2 P) (ell. - e d , (41) which are decoupled as much as possible from each other and which lead t o simple exponential decay laws for the single excited DICREstates Il>and 11') just as for 11") (see eq. (24)). We emphasize once more that only for a n equilateral triangle confignration (and more generally for all those atomic configurations and directions of dipole moments for which plz = p13= p2, and qlz = ql, = qS) such simple decay laws exist, a fact which is in general overlooked in the literature, e.g. [lo-121. For all other atomic configurations especially in the small sample limit xpv = (x, - x,I < 1 the various frequency shifts strongly influence the decay laws of the DICKEstates 11) and Il'), as can be seen from the general solutions (30), (31). Assuming that initially only atom A, is excited and using A=1=3yp, P=E=3yq, B = C = O (42) the general solution reduces t o + + + + + + + + + e114) @l,l(t)= -(4/3 exp [-r('3 P)tl cos (3yqt) (45) in accordance with the results following directly from the above eqs. (38)-(41). The probability of finding atom A,. in the excitted state at time t is ac*cordingt o eq. (16) I/% + 3 e x P [--Y(Z + Y)tI('0s (+?I4 -l and shows a sinusoidal iriodulation with frequency 3yq superposed on a composed exponential decay. Similarly the probability to find the atonis A, or A, in the excited state is given by 2 - -exp 9 [-y(B + p) t] cos (3ypt). (47) On the other hand the probability of finding any one of the three atonis in the excited state decreases monotonically with t iiiie (without sinousoidal modulation) 1 2p) t ] 2 exp[--2y(l - p ) t ] } . (?zl(t) it,(t) it&) = 3 {exp [-+(I + + + + (48) Therefore a part of the excitation energy is oscillating back and forth from Atom A, t o the other two while the whole atomic excitation decreases monotonically with time. 115 Coopera;tiveSpontaneous Emission from a Three-atom Systsm Moreover the above result shows the effect of repressing the radiation (in the current literature called “radiation trapping”) at small distances 1 A, for which p approaches the value 1. Then the trapped part of the energy (-(2/3) exp [ - 2 y ( l - p ) t ] ) amounts t o two third of the energy stored initially in the atomic system, instead of one half in the case of two atoms. The same effect of course appears in the total radiation rate < I ( t ) = 2Y 1 [T(1 + 2P) exp [-%(I + 22-4 tl 2 + -j(1 - P ) exp [-2Y(l (49) - PVI) * 6.2. Case 1 4 298, 1 Here according to eqs. ( 2 ) , ( 3 )p and q vanish, so that atom A, ceases to interact with the other two ones. From eqs. (28), (29) follow immediately = YP23, E (50) Depending on which specific atom is excited initially we obtain for the probability of finding atom A, excited and for the total radiation rate the results a) atom A, excited initially = Yq23’ <nlW = exp ( - 2 ~ 4 , I ( t ) = 2y exp ( - W , b) atom A, excited initially (n,(t)) = 0, I ( t ) = 2y exp ( - 2 y t ) [cosh (2yp,t) (51) - p , sinh (2yp, t ) ] . (52) These expressions are of course in accordance with the well known behaviour of the spontaneous emission from a) a single undisturbed atom or b) from a single-quantum excited two-atom system. 6.3. Case 138 Q 1Q A, Very Small, Very Acute Triangle Configuration This time the three atoms form a very acute isosceles triangle with side lengths much smaller than the resonance wavelength. Due to t h i s configuration we have according to eqs. (2), ( 3 ) p w p,, w 1 and qa3 & q & 1 . (53) Now we give approximate solutions valid to zeroth order in the parameter q.z;‘ 4 1. From eqs. (2H), (29) we get approximately and E = ~ q , ~ . Furthermore the quantities (33), (34)reduce in this approximation to A =y (54) - A = 37, B = 0, C = - (1/3 1 / 2 ) yqs, F = ( 1 / 3 )yq,. (55) Depending on which specific atom is excited initially we obtain for the probability of finding atom A, excited and for the total radiation rate to zeroth order in the parameter qG1: a) atom A, excited initially ( n l ( t ) ) M exp (-Byt), b) atom A, excited initially I ( t ) M 2y exp (-Byt), (56) I ( t ) M 2y exp (-4yt). (57) Thus we obtain the same results as in the second limiting case G.2., considering the vertex atom so far away from the other two atoms, that it is dynamically decoupled from these (nl(t)) M 0, TH.RICHTER 116 latter two (see eq. (52) in which in the present case we have to put pZa 1).The physical reason for this behaviour of three atoms arranged in mutual distances much smaller than the resonance wavelength is the frequency shift due to the dipole-dipolenear-field interaction, depending on the relative atomic positions and the directions of the atomic dipole moments. The two atoms on the base of this acute isosceles triangle are strongly coupled to each other, causing such a big frequency splitting that the third more distant atom comes out of resonance and therefore hardly interacts with the two base atoms. In this way the three-atom system can decompose itself into two approximately dynamically independent subsystems of one and two atoms, respectively. The same effect was found [22] in connection with the total radiation rate of a continuously incoherentby STEUDEL ly pumped three-atom system by numerical computations. 7. Discussion of General Cases and Numerical Results Next we discuss in some detail the excitation probability of a specific atom and the total radiation rate from a single-quantum excited three-atom system for different atomic configurations and for various positions of the atom excited initially. 7.1. Decay of a Single Atom Coupled to a Two-atom System At first we illustrate the influence of two neighboring atoms in the ground state on the decay law of a third excited atom. Fig. 1shows the temporal dependence of the occupation probability (nl(t))of the excited state of the vertex atom A, in an isosceles tri- 0' I 0.5 *Yt Fig. 1. Occupation probability <nl(t)) of the excited state of the vertex atom A, in an isosceles triangle configuration as function of time for different base lengths Is, a t fixed lateral side lengths ,Z = ,Z = 0.12. Initially the vertex atom A, is excited 0 and the two atoms A, and A, on the baee are in the ground state 0.I,, = 0 corresponds also to a single-atom decay (see 6.3.). 117 Cooperative Spontaneous Emission from a Three-atom System angle configuration for different base lengths I , a t fixed lateral side length I, = 113 = 0.U. Initially the atom A, is excited and the two atoms on the base are in the ground state. In this and the following figures and o indicate an atom initially in the excited state and ground state, respectively. For an equilateral triangle configuration we find a pronounced sinusoidal modulation superposed on the composed exponential decay (see eq. (46)). With decreasing base length lm the modulation frequency increases, while the modulation amplitude diminishes. For base lengths very short compared with the resonance wavelength the decay law approaches that one of a single undisturbed atom, as discussed in 6.3. 7.2. Total Radiation Rate Now we consider the total radiation rate in the most interesting case in which the atomic distances are all much smaller than the resonance wavelength. Then practically only the superradiant state 11) is emitting radiation, as can be confirmed by inserting p M pZ3M 1 into the expression (22) for the total radiation rate. For an equilateral triangle configuration this leads to the effect of "radiation trapping" since the energy initially stored in the subradiant states 11') and 11") can not leave the atomic system. In a more general configuration however the dipole-dipole near-field interaction induces an exchange of excitation energy between the superradiant state 11) and the subradiant state 11').In this way the atomic system can emit the energy initially stored in the subradiant state 11') also. 27 0' 0.5 *It Fig. 2. Normalized total radiation rate aa a function of time for an isosceles triangle configuration with fixed base length I,, = 0.11 and for different lateral side lengths I,* = I,, = al. Initially the vertex atom A , is excited 0 and the atoms A , and A , on the base are in the ground state 0. TH.R I ~ E B 118 Fig. 2 shows the temporal dependence of the normalized total radiation rate for an isosceles triangle configuration with fixed base length 1, = 0 . U and for different lateral side lengths I,, = Zl3 = &Im. Initially the vertex atom A , is excited and the atoms A , end A , on the base are in the ground state. For an equilateral triangle configuration the radiation rate decreases monotonically with time according to eq. (49). If we now reduce the lateral side length then in the total radiation rate a sinusoidal modulation appears which is most striking if the atoms are placed on a straight line. A remarkable fact is that in some cases the total radiation rate which always starts with a downward slope soon afterwards increases with time above its initial value, then decreases again and so on in an oscillatory manner. This can be explained most easily'in terms of the occupation probabilities of the states 11) and 11'). Due to the dipole-dipole near-field interaction the excitation energy is oscillating back and forth between the superradiant state 11) and the subradiant state 11'). Since practically only the superradiant state is emitting energy the oscillations of its occupation probability can be seen directly in the total radiation rate. Now in Fig. 2 atom A, is excited initially so that the states (1)and 11') are occupied with probabilities 113 and 213, respectively according to (36). Therefore soon after the beginning of the emission process there obviously can exist such a strong flux of excitation energy from the subradiant state 11') to the superradiant state 11) that ell and therefore the total radiation rate also increases. Later on the excitation energy is flowing back from state 11) to state 11') and the radiation rate decreases, and so on. A similar and RZ+ZEWSEI [201 investigating the collective radiaeffect was found by LEWENSTEIN tion from four harmonic oscillators. 0' I 0.5 I 1 Fig. 3. Normalized total radiation rate aa a function of time for an isosceles triangle configuration with fixed base length,Z = 0.11 and for different lateral side lengths Z, = Z, = al. Initially atom A , on the base is excited 0 , while the atoms A , and A , are in the ground state 0 . Cooperative Spontaneous Emission from D Three-atom System 119 Next we consider the case that only the base atom A, is excited initially so that the occupation probabilities ofthe states Il), 11') and 11') are equal to 1/3,1/6 and 112, respectively. The temporal dependence of the total radiation rate is shown in Fig. 3 for the same atomic configurations as in Fig. 2. This time in any case the total radiation rate a t the beginning of the emission process is lower than that of a free atom. Especially for an obtuse triangle configuration (a< 1)at this atomic distances the superradiant state [ 1) is again practically the only one emitting radiation in an appreciable amount. Since this time a t the beginning there is a net transfer of excitation energy from the superradiant state [I)to the subradiant state 11')the radiation rate at first decreases faster than in the case of an equilateral triangle configuration, as can be seen from Fig. 3. Later on the temporal behaviour of the total radiation rate is similar to that of the former cases already discussed, showing a sinuaoidal modulation with an amplitude decreasing in the course of time. I n the limit of very long lateral side lengths the total radiation rate approaches that one of a singly excited two-atom system as it must be. Finally we consider the total radiation rate of an isosceles triangle configuration with constant lateral side length lI, = lI3 = 0.1L for different base lengths 123. Fig. 4 and Fig. 5 show this quantity for the cases that atom A, or atom A, is excited initially. Again for an equilateral triangle configuration the total radiation rate decreases monotonically with time. Decreassing the base length I,, so that the triangle becomes more Fig. 4. Normalized total radiation rate as a function of time for an isosceles triangle configuration with fixed lateral side lengths Z12 = Zls = 0.11 and for different base lengths lzp Initially the vertex atom A, is excited and the atoms A , and A, on the base are in the ground state 0.ZZ3 = 0 corresponds also to a single-atom decay (see 6.3.). TH.R I ~ E R 120 and more acute a sinusoidal modulation of the decay occurs with increasing frequency but decreasing amplitude. For very small base lengths, l,, 1,the total radiation rate approaches that of a single undisturbed atom or of a two-atom system respectively, see Fig. 4 and Fig. 5. The reason is again the dipole-dipole near-field interaction which causea a dynamically decoupling of the vertex atom A , from the two base atoms A , and A,, due t o their increasing frequency splitting as discussed above. < 1 Iltl I(01 0.5 rL 0.3 0.5 *?df Fig. 5. Normalized total radiation rate as a function of time for an isosceles triangle configuration with fixed lateral side lengths I,, = lls = 0.11 and for different base lengths lSs. Initially atom A , on the base is excited 0 , while the atoms A, and A , are in the ground state 0 . I,, = 0 corresponds also to a two-atom decay (see 6.3.). 8. Summary The temporal evolution of the atomic states and of the total spontaneous radiation rate of a system of three cooperatively emitting atoms one of which is excited initially are found to depend significantly on the precise configurations of the atomic dipoles, especially in the small sample limit, in which all mutual atomic distances are much smaller than the resonance wavelength. Purthermore, the position of the initially excited atom influences strongly the temporal behaviour of the total radiation rate. Only in the special case in which the atoms are forming a n equilateral triangle and their common direction of polarisation is perpendicular t o the plane of the t h e e atoms (and more generally for all configurations in which p12 = pl, = pB and q12 = ql, = q2a) it is possible t o introduce orthogonalized basic states of the atomic system, SO called DICKE Cooperative Spontaneous Emission from a Three-atom System 121 states which are dynamically independent from each other and therefore decay according t o simple exponential laws. If we reduce this configuration similar t o itself down to an arbitrary small size we get back (but only in the case of this configuration) the results which are well known from the DICKE point model.of a three-atom system. This model which is widely used in the literature considers N atoms enclosed in a volume of diniensions much smaller than the resonance wavelength supposing that “all atoms see the same field”. But this assumption neglects from the very beginning the spatial variation of the electric field over distances small compared t o the resonance wavelength and which is due to the near-field component of the atomic dipole fields. Our results show explicitely that only for some special atomic configurations the frequency shifts due t o the dipoledipole near-field interaction are canceled out from the equations of motion and in these cases the atomic system evolves according t o the DICEEpoint model. Another remarkable result is the possible decomposition of a three-atom system in the small sample limit into two subsystems consisting of one atom and two atoms, respectively, evolving dynamically independent from each other. This occurs if two atoms are so near t o each other that their frequency splitting becomes so large that the third more distant atom comes completely out of resonance and therefore ceases t o interact with the other two atoms. Similar effects are to be expected in a N-atom system. Finally we point out the fact that in deriving the equations of motion (23), (24) we actually used only the conditions p12 = p.13and qls = q13’Therefore we can describe all atomic configurations satisfying these conditions including such cases in which the fixed dipole moments are even askew t o each other (see eqs. (2), (3)), which only lead t o a somewhat more complicated relationship between the atomic dipoles and the radiation pattern due t o the different directions of the atomic polarisations. The cooperative spontaneous emission from a n initially fully excited three atom system, and especially two-photon correlations in this radiation, will be investigated in a following paper. The author would like t o thank Dr. H. STEUDEL for interesting discussions. References [l] Q. H. F. VREHEN, H. M. J. HIKSPOORS, and H. M. GIBBS,Phys. Rev. Lett. 88, 764 (1977). [2] H. M. GIBBS,Q. H. F. VREHEN,and H. M. J. HIKSPOORS, Phys. Rev. Lett. 39, 547 (1977). [3] C. M. BOWDEN, D. W. HOWOATE, and M. R. ROBL(ed.) Cooperative effects in Matter and Radiation, Plenum, New York 1977. [4] J. C. MACGILLIVRAY and M. S. FELD, Phys. Rev. A 14, 1169 (1976). [5] R. GLAUBER, and F. HAAKE, Phys. Lett. A 68, 29 (1978). [6] M. F. M. SCWURMANS, P. POLDER, and Q. H. F. VREHEN,Phys. Rev. A 19, 1192 (1979). [7] F. HAAKE,H. b a , G. SCHRODER, J. HAUS,and R. GUUBER,Phys. Rev. -4 20, 2047 (1979). [8] J. A. HERMANN and R. K. BULLOUOH, Opt. Commun. 31, 219 (1979). [9] R. H. DICRE,Phys. Rev. 93, 99 (1954). [lo] M. DILLARDand H. R. ROBL,Phys. Rev. 184, 312 (1969). [11] D. DIALETIS,Phys. Rev. A 2, 599 (1970). Springer Tracts Mod. Phys. 70, 1 (1974). [12] G. S. AOARWAL, [la] R. PRAKASH and N. CE~ANDRA, Phys. Rev. A 21,1297 (1980). [l4] R. H. LEIWBERO, Phys. Rev. A 2, 889 (1970). [15] H. STEUDEL, Ann. Physik Leipz. 27, 57 (1971). J. W. SOKOLOV, and E. D. TRIFONOV, Zh. Eksp. Teor. Fiz. 68, 2105 (1972). [16] D. F. SMIRNOV, [17] TE.RICHTER,Ann. Physik Leipz. 38, 266 (1979). [18] B. COFFEYand F. FRIEDBERO, Phys. Rev. A 17, 1033 (1078). [19] ~AKOWICZ, Phys. Rev. A 17, 343 (1978). 9 Ann. Physik. 7. Folgc, Rd. 38 122 and K. RZ@EWSKI,J. Phys. Lond. A 13,1297 (1980). [20] M. LEWENSTEIN [Zl] H. STEUDEL and TH.RICHTE~, AM. Physik Leipz. 36, 122 (1978). [a21 € STEUDEL, I. J. Phys. Lond. B 12, 3309 (1979). [23] R. H.LE-BERG, Phys. Rev. A2, 883 (1970). Bei der R,edaktion eingegangen am 29. August 1980. Anschr. d. Verf. : Dr. TH.RICHTER Zentralinstitut fur Optik und Spektroskopie der Akademie der Wisenschaftens der DDR DDR-1199 Berlin-Adlemhof Rudower Chaussee 6 TH.RIUETER

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