Collective resonance fluorescence in the presence of a thermal reservoir

Physica A 158 (1989) 656-671 North-Holland, Amsterdam

COLLECTIVE RESONANCE FLUORESCENCE IN THE PRESENCE OF A THERMAL RESERVOIR LE H O N G LAN, A.S. S H U M O V S K Y and T R A N Q U A N G Laboratroy of Theoretical Physics, Joint Institute for Nuclear Research, P.O. Box 70, Dubna, USSR

Received 15 November 1988

The spectral and statistical properties of the collective resonance fluorescence in the presence of a thermal reservoir are discussed. The influence of the thermal field intensity on expansion of the spectrum linewidths and on the asymmetry of the spectrum is shown. The photon statistics of the spectrum components, cross-correlation between them and violation of the Cauchy-Schwarz (CS) inequality in the dependence of the thermal field intensity are investigated.

1. Introduction Resonance fluorescence of two-level atoms in the presence of a strong laser field has received a great deal of attention in recent years [1-10]. In the work by Mollow [1], a three-peaked fluorescence spectrum in the interaction of a single two-level atom with a single-mode coherent state driving field has been predicted. This spectrum, a "dynamical Stark" spectrum, was first observed in the work by Schuda et al. [2]. Recently, the spectral [7, 8, 10], statistical [9, 10] properties and non-classical effects such as squeezing, violation of the CauchySchwarz inequality [11-14] in the collective resonance fluorescence have been discussed. In this paper we consider the problem of collective resonance fluorescence of N two-level atoms driven by an intense coherent field in the presence of an additional incoherent field. This incoherent field is typically a Gaussian chaotic field and can be a thermal black-body field. In the case of an intense external field the stationary solution for ~he density operator of the atomic system is given. Expansion of the lincwidths and asymmetry of the spectrum caused by a thermal black-booy field are shown. The influence on the thermal field intensity of the photon statistics of the spectrum components, the crosscorrelation between them and on the violation of the Cauchy-Schwarz ineq,,ality is investigated. 0378-4371/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

Le Hong Lan et al. / Collective resonance fluorescence

657

2. Basic equations Let N two-level atoms be concentrated in a region small compared with the wavelength of all the relevant radiation modes (Dicke model). The atoms are driven by a single-mode coherent field of the frequency wL and coupled to a reservoir containing all modes of the radiation field. In treating the external field classically and using the Markov and rotating wave approximation for describing the coupling of the system with the thermal reservoir, one arrives at the following master equation for the reduced atomic density operator p [15]: Op Ot = _i[½8(J2 2 _ j, t ) + G ( j , 2 + J2,), t'l 1

- ~ y ( r / + 1)(Y2,Y~2p

-

J,2PJ21 + H.C.)

--lyt~ (J12J2 1 p - - J21PJl2 + H . C . ) - Lp 2

(1)

where 8 = o)21- tOL is the detuning of the laser frequency toL from the atomic resonance frequency to21(o2~ = to2 - to1; h - 1); G = - d 2 ~ E o is the resonant Rabi frequency; 3' is the radiative spontaneous transition rate from the excited level ]2) to the ground state; [1); fi= f f ( w 2 ~ ) = [ e x p ( t o 2 ~ / k T ) - 1] -~ is the mean photon number in the broad-band thermal field provided by the reservoir at the atomic frequency to_,~; J,i (i, j = l, 2) are the collective operators (angular momenta) describing the atomic system and having in the Schwinger representation [9] the following form: t

J # = C, G

(i, j = 1 , 2 ) ,

where the operators C i and Cj obey the boson commutation relations

[ C,, G] = a, and can be treated as anvihilation and creation operators for atoms populating the level ]i). Further, we restrict our consideration to a strong laser field or to a large detuning 8 so that the Rabi frequency g~ satisfies the following relation: ~ 2 = ( aI6

2

+ G 2 )z z >> NT, fiT.

After performing the canonical (dressing) transformation CI=Q~cosq~+Q_,sin~, C2 = - Q t sin ~ + Q2 cos ~o,

-3

658

Le Hong Lan et al. / Collective resonance fluorescence

where t a n 2q~ = 2 G / 6 ,

one can split the Liouville operator appearing in eq. (1) into the slowly varying part and ~he terms oscillating at frequencies 2 0 and 40. Since we assume that the Rabi frequency O is sufficiently large according to the relation (2) the secular approximation [7] is justified and we retain only the siowly varying part of the Liouville operator and the master equation (1) reduces to Ot

=

- i O[R3,

pl +

B(2RaOR3 - R ~ -

+ X,(2R,2~R2,-

~R~)

R2,R,2 ~ - Rz,R,2)

+ Xz(2RE,pR,2- R,2R2, ~ - t~R,2RE,),

(4)

where t~ = Up U*, :vhere U is the unitary operator representing the canonical transformation (3) R~ = R 2 2 -

Rll

(5)

,

B - y(r~ + ½) sin2~o cos2q~,

(6)

X~ = ~,y(ff + 1) ces4q~ + ~yh sin4q~,

(7)

X~_= -~y(r~ + 1) sin4q~ + ~yr~ cos 4~,

(8)

R~j = Q [ Q j (i, / = 1,2) are the collective operators of the dressed atoms. t The operators Qi and Q/satisfy the boson commutation relation t

[Q,, Q/l = 6#.,

(9)

so that [R#, Ri,j, ] = Ri/,6,, i - R,,j6ij, .

(10)

The exact stationary solution of cq. (4) takes the form N

L = z - ' Z x", n 1= 0

where

)

(11)

Le Hong Lan et al. / Collective resonance fluorescence

X-

Xi (ff + 1)cos4~0 + ti sin4~o , X2 (ri + 1 )sin4q~ + ri cos4~o X N+I

Z =

M

X- 1

659

(12)

1

'

(13)

the state In1) is the eigenstate of the operators R~I and R~ + R22. The solution (14) allows one to calculate all the stationary expectation values of the atomic observables. Some of the results that will be needed for our further consideration are given in the Appendix. In the case of the exact resonance cosec0- ½ the solution (11) reduces to N

£=(N+ 1)-'E In,)(n,I.

(14)

nl=O

The solution (14) is independent of the mean photon number of the thermal field and has the same form for the case when the thermal reservoir is in the vacuum state (i.e. T = 0). Consequently in the exact resonance case all the one-time expectation values of the atomic observables are independent of ri.

3. Fluorescence spectrum

In this section we consider the effects that may arise in the collective steady-state fluorescence spectrum due to the thermal reservoir. Following the work of rcf. [6], the steady-state spectrum of the fluorescent light has been calculated as the Fourier transform of the atomic correlation function,

(15)

(Jz,(Z)J~z)s = lim ( Jz,(t + 0 J , z(t ) ) , t...-~ ~

where ( . - - ) ~ denotes an expectation value over the steady-state (11). By using the transformation (3) one finds ~l

"t

J2~(t) = sin q~ cos q~ R3(t ) + cos-q~ R2~(t ) - sin-q~ R~2(t ) . The equation of motion for ( R , ( I ) ) can be derived bv using tile masZcr equation (4) and have the following forms:

d dt (R3(t)) = - 2 ( X , + X 2 ) ( R 3 ( t ) ) - ( X 2 - X , ) ( R ~ ( t ) ) + ( X 2 - XI) (N 2 + 2 N ) ,

(17)

Le Hong Lan et al. / Collective resonance fluorescence

660

d

dt (R'2(t)> = - 2 i j 2 (R,2(t)) - ( 4 B + X, + X2)(Rt2(t)) - !(X 2 - X,)< {Rt2(t ) ~ R3(t)} > 2 d

d

(18) (19)

dt (R2'(t) ) = d't (Rt2(t))*'

where {R,j, R3} = RijR 3 + R 3 R q. Eqs. (17)-(19) are so far exact. They contain, however, terms with the products of operators which make them unsolvable in the general case. For the one-atom case one can use the operator relation R q Ri, j, -- Rij,~i, j

(i,j,i',j'=l,2),

(20)

then eqs. (17)-(19) reduce to the linear, exact solvable equations d

dt (R3(t)> = - 2 ( X , + X2) (R3(t)> + 2(X 2 - X , ) , d dt

(R,2(t)) = - 2 i a

d (R,,(t)> = d

dt

(R,2(t))-(4B

dt ( R'z(t) ) *

+ X, + X 2 ) ( R , z ( t ) ) ,

(21) (22) (23)

Eqs. (21)-(23) are in agreement with the previous work [6] by Hildred et al. concerning resonance fluorescence of one atom in the presence of the thermal reservoir. For the case of exact resonance cos2q~ = ½ we have X z = X~ and the terms with the products of operators vanish; then all eqs. (17)-(29) reduce to the exact solvable linear differential equations. For the off-resonance case, according to the works [8, 10], we use the decorrelation scheme R,}> =

A~ ! \t O,..--r

By using the density matrix (11 ) one can show that in the case of large N the decorrelation (24) yields a small error (of the order of N -'/2) in the calculation of the steady-state fluorescent spectrum. With the approximation (24), eqs. (17)-(19) have simple exponential solutions. Applying relation (16), solutions of eqs. (17)-(19) and using the quantum regression theorem [16], one obtains the following expressions for the correla-

Le Hong Lan et al. / Collective resonance fluorescence

661

tion function ( 15): s = sin4q~ (R,2R2,)s e x p ( - F~ r - 2iar) + COS4q~ (R21R12}s exp(-F~ r + 2iOr)

+ sin2~ cos2~ (( R2)~ - I~)exp(- For) + sin2¢ cosZ~ I~, (25) where F~ = 4 B + X, + X: + (X 2 - X , ) ( R 3 ) ~ = y(r~ + ~ )(1 + 2 cos2q~ sinZq~) + ~ (sinZ,w - c0s2~) ( R3 ), F0 = 2(Xt + X2) + (X 2 -

Xl)

(26)

s

(R3)

= 2y(rT + ~ )(cos4~o + sin4q0 + ~y(sin2~o - cos2q~)(R3) ~ 6 = ½(sin2~ - COS2~)(N2 + 2 N ) (

R3)JF

(27) (28)

o .

T h e expressions for the weighting factors of the particular exponents (R~zR2,)~, ( R z , R , 2 ) . ,, (R-'3}s are given in the appendix. The steady-state spectrum of the fluorescence is proportional to the Fourier transform of the correlation function (25) and takes the form

Re{f 0

v,

4 = c o s

_

+ sinaq~ ( R l eRe, )~ .... - ......

-._ll

+ 2 n ) 2 + F~

v, (to - w,. + 2 n ) 2 + V~ \

r

~

(~o - cg. )- + I ':;

i29)

+ ~ sin-q~ cos-,¢ 1 . 6 ( 0 - % ) .

As for the case when the thermal reservoir is in the vacuum state [8]. the steady-state spectrum (29) contains three spectral lines centered at frequencies w = % ; %. _+ 2g2. In the off-resonance case cos-,# #- { the central line contains the elastic component with the intensity I.. being proporth-nal to N" (which "3

662

Le Hong Lan et al. / Collective resonance fluorescence

vanishes for the exact resonance case) and the Lorentzian shaped component with the iinewidth F~j and intensity sin"~0 cos"~o((R]).~- Ic). The two sidebands are Lorentzians of the linewidth F~ centered at frequencies to = (DE - - 2 1 2 and (D = (DE + 212, and having the intensities which are proportional to s i n ~ (R t2REt )~ and c o s ~ (R21R t2 )~ respectively. As is seen from the relations (26), (27), in the general case the linewidths of the three inelastic components are expanded due to the presence of the thermal field. The expansion of the linewidths F0, F t are sufficiently large for the case of ri ~> 1 and exact resonance when the collective part [½y(sin2~0cos2~0)(R3)~ ] of the iinewidths vanishes. For the off-resonance case and N ~> 1 from relation for (R3) ~ given in the appendix one finds !y(sin2q ~ 2 - c0s2¢)( R3)~ ~ ½Y N ]sio",,, - cos: l

if cos2~o # ~. '

(30)

Thus, for the case of N ~> 1 and COS2~ 5~ / the collective part of the spectrum linewidths is independent of ri and dominates over the other one-atom parts of the linewidths. Consequently, in this case the spectrum linewidths are approximately independent of the thermal field intensity ri. In contrast with the spectrum line widths, the intensities of the spectrum components are independent of ti for the exact resonance case. In this case, the elastic component vanishes and intensities of the three inelastic components ") i have the super-radiant behaviour (--- N2). For the off-resonance case: cos-q~ ~ _~ the intcnsities of the spectrum components strongly depend on the thermal field intensity. The dependence of the intensities of the spectrum components on the thermal reservoir is shown in figs. 1, 2 where the relative intensities of the sidebands (fig. la, b), i.e. the quantities l_t/N = c o s 4 t p ( g E 1 R 1 2 ) / N (solid curves) and l+l/N = cos4~o(R21Rt2)/N (dashed curves) and of the central line containing the elastic and inelastic components (fig. 2a, b), i.e. the quantities I,/N 2= sin2q~ cos2~o(R~)JN 2 a r e plotted as functions of the parameters cosec0 for various values of N and ri. As is shown in fig. la, b, in the case of ri = 0 (the curves 1) the intensities of the two sidebands are equal and spectrum is syminetric. In the case of d # 0 (the curves 2) the intensities of two sidebands are quite different for the off-resonance case cosec0 # ½thus spectrum asymmetry is substantial. Analogous conclusions for the resonance fluorescence frem a single atom in the presence of thermal reservoir has been made in the work by Hiidred et ai. [6]. Using the relations for (R2~Rt2)~, (RI2R21), and (R~)~ given in the appendix, one can show that for a large number of atoms N >> 1 and the off-resonance case the intensities of the spectrum components are strongly dependent on the thermal field intensity ff and the fluorescence spectrum is quite as,,mmetric while the spectrum linewidtbs are independent of d.

Le Hong Lan et al. / Collective resonance fluorescence

I31 /N

l

N=2S

t

1.0

\\

0.8 2

e,63

/ ./"

".._i

/ !

0.6

/ ! I I I

0.~ I

2/ /

/

0.2

/

//

1

/ /

0

a

-0.2

0.~,

I-+I/N

20

0.6

0.8

COS 2 (,p

t~~ il N --50

1.6

1.2 £ f

0.8

f f

O.&

///

.-y 0

0.2

Y 04

0.6

0.8 cos 2

b Fig. 1. T h e quantities I , ~/N ( d a s h e d curves) a n d ! ~,''N (solid curves) as functk~ns of cos-,= for N = 25 (a) and N = 50 (b). T h e curves I and 2 c o r r e s p o n d to p~ = 0, 1.0, respectively.

664

Le

Hong

/

Lnn

et al.

I

Collective

resonnnce

fluorescence

I, /N N=2S

0.2

0.16

, 0.12

ROE

0.01

0

0.2

0.6

0.4

0.8

0.16

0.12

0.08

[,,

0.04

0

0.2

“,

,

0.4

,

0.6

,

,

0.8

,

\i COS2yl

Fig. 2. The quantity ,lN’ as function of cob’cp for IV = 15 (a) snd N = SO (b). correspond to Ii = 0. L.2. 1.0, respectively.

The curves 1 to 3

Le H, ng Lan et al. / Collective resonance fluorescence

665

4. Statistical pcoperties In the recent works [9, 10, 13], the statistical properties of the collective resonance fluorescence in the case when the thermal reservoir is in vacuum state have been investigated. The anticorrelation between sidebands and central spectrum components [9] and the non-classical correlation (violation of the Cauchy-Schwarz inequality) between two sidebands [13] have been predicted. In this section, we discuss the influence of the thermal reservoir on the statistical properties of the fluorescence field. We consider the independence of the photon statistics of the spectrum component and non-classical con qations between them on the thermal field intensity ri. It is easy to see from the previous section and relation (16) that the operators --sin2to R12, sintocos,p R 3 and cosEq~R21 can be considered as operator-sources of the spectrum components of the fluorescence field at frequencies toE - - 2 ~ ' ~ , toE and toE "]- 2 ~ ' ~ , and for simplicity these operators will _~, S O be denoted by S ÷ + and S~ , respectively. As Loudon [17], we define the degree of second-order coherence between the spectrum components S~ and Sj in the form +

+

a ! , _= ( s, s j s, -'.' (s, s , ) , ( s , sj), +

(i, j = 0 , - + 1).

+

(31)

Since the operator Si does not commute with the operator S, in the general case, we have G i,j (2) ~k G (2) j,i

(i~j).

The correlation functions GI.2i) (i = 0 +- 1) describe the photon statistics of the spectrum components s and the correlation function~ _,.~G!27describe the cross-correlation between the spectrum components Si and Sj. By using the steady-state density matrix (11) and commutation relation (9), (10) one finds the quantities G(2)i.i (i = 0, -+ 1) in the form ¢.-g (2) v0.0

=

(

4

G (2) G (2' 1.1 - -I.-I

R2

/ ((3) =

)2 s

"

(R,.R .R.,R. .1! ) s / ( ( R , .. R .-1) s )2 .- 1~ .

(32~ (33)

" , ( R4)~,, (RI2R12R21R2~}.,, (Rl,R21)~are given in the where the values (R3) appendix. For the one-atom case, by using the ope,ator relation (20) one can show that

Le Hong Lan et al.

666

G(2) = (92) 1.1

I

0

-1,-l

=

!i

thus the photon statistics of the central component remains Poissonian; the sidebands have sub-Poissonian statistics and photon statistics of all three Mollow’s spectrum components are independent of the thermal field intensity n’. Contrary to the one-atom case the photon statistics is strongly dependent on the thermal field intensity rZ; as is shown in figs. 3,4 where the ccrrelation functions G gi, G(2)rc-I.21 are plotted as functions of the parameter cos2q for fixed iV= 50 and for , .various values of K It is clear from figs. 3,4 that except for the point of exact resonance cos2q = $, the thermal fixed intensity 6 plays an improtant role in the photon statistics for Mollow’s triplet of the collective resonance fluorescence. By an analogous approach one finds the cross-correlation functions Gi.‘,’ (i # j) for the single-atom case in the following form: G(2) = G(9) = G(2) o,_1 0.1 I .o G”’ 1,-l

G”,‘,

=

G’;.,,= 1 ,

(36)

=x+1>1,

(37)

=l+l/X>l.

Thus, for the one-atom case the central component

S,, remains uncorrelated

‘(2) 40

1.8

1.6

1 0

Fig. 3. The

quantity

ti = 0, 0.2.

1 .O,

0.2

G:,f,: as function

rcspccti~ylv.

0.4

0.6

0.8 cos2y

of cos’~ for the cast’ N = SO. The curvt’s 1 to 3 correspond

to

Le Hong Lan et al. / Collective resonance fluorescence

2.0 t u_.+l t-.(2l ' -+1

667

N= 50 1

1

1.8

1.6

1./,

1.2

I

0

i

0.2

,.i

I

0.t,

t

.

I

0.6

l

I

i

=_

0.8 c0s 2 ce

Fig. 4. The quaa~2ty G ~" _-~,_-3 as function of cos2q for the case N = 50. The curves 1 to 3 correspond to fi = 0, 0.2, 1.0, respectively.

with the sidebands S. z in spite of the presence of the thermal field while the cross-correlation between the sidebands is directly dependent on the thermal field intensity ft. For the collective case N > i, by using the steady-state density matrix (11) 2 I one can show that except for the point of exact resonance cos ,~= :. the ~2~ ( i ¢ j ) is strongly dependent on the thermal cross-correlation function C, _i./ field intensity. Further, we fix our attention on the correlation between the sidebands and discuss the influence of the thermal field on the non-classical e f f e c t - t h e violation of the Cauchy-Schwarz (CS) inequality [17-22] that has been predicted for the collective resonance fluorescence in the work [13]. We mention that the violation of the Cauchy-Schwarz inequality holds for the correlation between the spectra~ components S, and S, if the following condition [13] is satisfied:

K,., = ( e l . ,

•

,.,)/(el,


139t

The factor K,.j describes the degree of violation of the Cauchy-Schwarz inequality. One shows by the relations (34)-(38) that Ki. J = 0 {i -¢ j: i,] = 0 , +_ 1) for the single-atom case. This means that the CS inequality is violated for any two

668

Le Hong Lan et al. / Collective resonance fluorescence

K-1 ,* I

3.5 N=SO

3.0

2.5 2.0

1.S 1.0

O.S I

I

I

0

0.2

O.t,

i

0.6

0.8

c os z ~e

a

K4,-t

3.S N :S0

3.0

2\ 2.S 2.0 1.S 1.0 ,,4

0

0.2

O.t,

0.6

0,8

cos 2 q~-

b Fig. 5. The quantities K t.,, (a) and K~._ ~ (b) as functions of cos2,p for N = 50. The curves 1 to 3 correspond jq = 0, {).2, 1.0, respectively.

Le Hong Lan et al. / Collective resonance fluorescence

669

spectrum components from Mollow's triplet as in the case when the thermal field intensity ri is equal to zero. For the collective case (N ~>2), as for the case of ri = 0 [13], the violation of the CS inequality is presented only for two sidebands of the fluorescent spectrum while the factors K t _ 1 and K_~.~, describing the degree of the violation of the CS inequality between the sidebands S±~, is dependent on the thermal field intensity ri. One can write the factors K~_~ and K_~.~ in the following form:

(R2,R2,R,2R,2)s (R,2R,2R2,R2,)~ K, _, = (( R2,R,2R2,R,2)~)2 ,

(40)

( R2,R2tR,2R,2)s ( R,2R,2R2,R2, )s

K_~,~ =

((R12R2~R~ER2)),)2

,

(41)

expectation values (R21R21R12R12)s, (R21R12R21R~2)s, (R~2R12R2~R21)s and (RI2R2~R~2R2~)s can be found in the appendix. The behaviour of the factors K~.~ and K~_~ as functions against the parameter cos2~o for N = 50 and various values of r~ is plotted in figs. 5a, b, respectively. As is seen from fig. 5, the thermal reservoir reduces the factors K_i. t and K i _ ~ describing the degree of violation of the CS inequality.

where

the

5. Conclusions We have considered the problem of collective resonance fluorescence of the small system of N two-level atoms driven by an intense coherent driving field in the presence of an additional incoherent field. In the case of an intense external field the stationary solution for the density operator of the atomic system is given. We have shown the expansion of linewidths and asymmetry of the collective resonance spectrum caused by the thermal field. It has been shown that except for the point of exact resonance cos2~0 = ½the thermal field intensity r~ plays an important role in determining the photon statistics of the spectrum component, cross-correlation between them. The thermal reservoir reduces the degree of the violation of the CS inequality for the correlations between the sidebands.

Appendix In this Appendix we give the explicit expressions for the steady-state averages of the atomic operators that car, be calculated with the use of the

670

Le Hong Lan et al. / Collective resonance fluorescence

density matrix (11)

(R,,),~ = Z - ' [ N X u+2 - ( N + 1)X N+~ + X ] / ( X (R2~)~ = Z - ~ [ N 2 X N+3 - ( 2 N z + 2 N -

1) 2 ,

(A.1)

1 ) X N+2 + ( U + 1)2X N+~ - X 2 - X]

/ ( X - 1) 3 ,

(A.2)

= Z-I[N3X~t+4(3N a + a N 2 - aN + 1)X 'v+3

(g~}s

+ (3N 2 + 6 N - 4 ) X N+2 - ( N a + 3 N 2 + 3N + 1)X 'v+~ + X 3 + 4X 2 + X]/(X-

1) 4 ,

(A.3)

(g4~}s = Z-~[N4XN+S-(4N4 + 4 N 3 - 6N 2 + 6 N 2 + 6 N + ( 6 N 4 + 12N 3 - 6 N -(4N

1)X N+4

2 - 12N + 1)X/v+3

4 + 12N 3 + 6N 2- 12N-

+ (N 4 + 4N 3 + 6N 2 +

l l ) X N+2

4N + 1)X 'v' ~ - X 4 -

llX 3 -

11X 2 - X]

/ ( X - 1)s,

(A.4)

(Ra},~=N-E(R~I)~,

(A.5)

(gZ3)~ = 4 ( R ~ I ) ~ - 4 N ( R ~ ) ~ + N2,

(A.6)

( g l E R 2 1 ) s = -(R211}s + (N + 1 ) ( g l l } s ,

(A.7)

(g2~g~2}~ = - { R ~ } s + ( N - 1)(R~)~ + N ,

(A.8)

(g~}~ = 16(R~!} - 32N(R~l}~ + 24NE(R2 }s 8Na(R~l}~ + N 4 '

(g.9)

(g12R12R2~g2~}s = (g4~)~- 2(N + 2 ) ( g ~ } ~ + (N 2 + 5N + 5) {g~l}, - ( N 2 + aN + 2 ) ( R ~ ) , , { g21g.,1Rl.,Rl2} -

-

s

=

( g ~ ) - (2N - 4)( R 3 ), + ( N2 - 7N + 5)( R;l

),,

+(3NZ-7N+Z)(R,.),

(A.11)

s

(R2,R,2R2,R,2) = (R~,) - ( 2 N - 2 ) ( R s

(A.10)

,s

11

+2N2-ZU,

3l l ) s + ( N2 - 4 N + 1)(R;,)~

+ (2N 2 - 2 U ) ( R , i ) , + U 2 ,

(A.12)

Le Hong Lan et ai. I Collective resonance fluorescence

(R,2R,,R,2R,,) _ . ,, =

(U411 } ,, -- 2 ( N + 1 ) ( R 3, , } . ~ + ( N + I

)2 ( R ~ ,,} . ,

671

(A.13)

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