Intensity fluctuations in a driven Dicke model

Physwa 103A (1980) 213-225 © North-Holland Pubhshmg Co

INTENSITY FLUCTUATIONS IN A DRIVEN DICKE MODEL S S HASSAN* AND R K BULLOUGH

UMIST-Department o[ Mathematics, P 0 Box 88 Manchester M60 IQD, U K and RR PURIANDSV

LAWANDE

Theoretwal Reactor Physws Section, Bhabha Atomw Research Centre, Bombay-400085, lndm Received i I March 1980 Exact analytic results for the steady-state atomic correlation functions of arbitrary order n are given for a system of N superimposed two-level atoms (the Dlcke model) collectively interacting with an imposed C W laser field Photon statistical studies through the normahsed intensityintensity correlation function, g~2)(0), show that when both N and the driving field become large, g~2)(0)---~l 2 This compares with an earlier approximated calculation I) wMch allows an independent atomic decay mechanism giving rise to g(2)(0) = 2 Cooperative interactions thus reduce intensity fluctuations Photon anti-bunching occurs for finite N There is a second-order phase transition critical bifurcation point in a thermodynamic limit in which N ~ c¢, a crlhcal exponent is determined

1. Introduction The equations of motion for N two-level atoms occupying the same site and driven by a c-number c.w. resonant laser field of Rabi frequency O (the "driven Dicke model") can be put, in dipole and rotating-wave approximations, in the form (cf. e.g. 2), dot means d/dr) (S+) = 2v0(S+Sz) - 2LO(Sz), (s_)

= (s+)*,

(sb

= - 2~,0(s+s_)

(1) - io((s+)

- (s_)).

The operators S+, u t,)(t), etc. and 3'0 is half _ Sz are the collective operators Y.,=~S_. the Einstein A-coefficient. The time-dependent solutions of these equations, within the context of a semiclassical factorisation, namely ( A B ) = (A)(B) were reported recently by Drummond and Hassan2). The solutions depend on a threshold value of the driving field g~o = yoNI2. Below this threshold value the behaviour is both cooperative, i.e. the intensity oc N 2, and stable. Above the threshold, the system is unstable and * On sabbancal leave from Ain Shams Universay, Faculty of Scmnce, Applied Mathematics Department, Cairo, Egypt 213

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S S HASSAN et al

b e h a v e s hke a family of L o t k a - V o l t e r r a cycles of period " B ' ( ~ 2 -- ~'~2)-1/2, whilst the semiclasslcal result 2,3) for the fluorescence spectrum shows (sharp) sidebands additional to the usual triplet Stark spectrum. Q u a n t u m - m e c h a m c a l l y , these equations have been solved m at least two significant cases (a) The operator products were f a c t o n s e d for pairs of distinct a t o m s i S j much as m the semlclasslcal theory 2) but the self-correlation of atoms was retained so that for example ~4-5)

(S+S.) = -½(S+) + (1 - N-')(S+)(S:)

(2)

This a p p r o x i m a t i o n thus admits independent atomic decay It leads to a theory of steady-state optical blstablhty m its usual (mean fielde-9)) f o r m which suggests that the approximation is soundly based m the physics H o w e v e r , it breaks a mathematical p r o p e r t y of the model that S 2, the square of the total angular m o m e n t u m , ~s a constant of the motion Atomic o b s e r v a b l e s - t h e emission and absorption spectrum, the photon s t a t i s t i c s exhibit a first order 'phase transition '~ 4). (b) The equations were solved exactly m the atomic coherent-state representation ~°) without the assumption (2) The a t o m s d e c a y collectively with ( $ 2 ) = constant: the atomic o b s e r v a b l e s show what appears to be a second-order phase transition associated with a ' c o o p e r a t i v e fluorescence' critical point ~-~3) We thus have the situation that the exact solution of the model provides a poorer description of the physics of optical b~stabihty It Is the purpose of this p a p e r to analyse the exact solution more completely therefore and to investigate the character of its phase transition. Calculations of the fluorescence spectrum14-t6), the absorption spectrumS7), and the mtensity-intensity correlation function '4 is) have been carried out in the exact case (b) only for small values of N (up to N = 5 in some of the calculations and never for larger N ) In order to study the exact solution further we calculate in this p a p e r the exact atomic correlation functions of arbitrary order n In the steady state using the exact analytic steady-state solution for the reduced atomic density operator derived recently by P u n and L a w a n d e 1°) The effect of the c o o p e r a t i v e fluorescence field on the intensity fluctuation is examined both for all fimte N and all values of [g{, the normalised Rabl frequency T h e s e results allow us to investigate a ' t h e r m o d y n a m i c hmW defined by N, ]gJ ~ o~ w~th JgJl(NI2) = constant--- 0. We can then identify an 'order p a r a m e t e r ' and a critical bifurcation point at 0 2= 1: the quantity 0 2 plays the role of t e m p e r a t u r e T in a t h e r m o d y n a m i c phase transition. In phase-transition theory 'critical e x p o n e n t s ' are defined" for the so called ' m e a n field' theory (compare7)) a scaling law provides the connection ~ + 2/3 + 3' = 2 b e t w e e n

INTENSITY FLUCTUATIONS IN A DRIVEN DICKE MODEL

215

three 'critical e x p o n e n t s ' a,/3, y with y = 1 a n d / 3 = / . A g a r w a l et al. 7) w o r k i n g on a single-mode t h e o r y identify a first o r d e r transition a n a l o g o u s to that resulting in the case (a) a b o v e But t h e y also d r a w a c o m p a r i s o n with a s e c o n d - o r d e r p h a s e transition in 'mean-field' t h e o r y finding /3 = ~ and y = l R e m a r k a b l y we also find the value o f the critical e x p o n e n t / 3 = ~ for the e x a c t solution case (b). This is not a mean-field t h e o r y * h o w e v e r and o u r e x p e c t a t i o n has b e e n that the critical e x p o n e n t s y and t~ w o u l d not take their mean-field values U n f o r t u n a t e l y we have so far b e e n unable to calculate these e x p o n e n t s 3' and a, and the q u e s t i o n is left o p e n b y the paper. T h e p a p e r is o r g a n i s e d as follows: the c o h e r e n t - s t a t e r e p r e s e n t a t i o n IS i n t r o d u c e d and the e x a c t s t e a d y state solution f o r the r e d u c e d d e n s i t y matrix calculated in section 2. T h e a t o m i c correlation f u n c t i o n s are calculated e x a c t l y in section 3, and the nature of the p h a s e transition associated with these e x a c t results is investigated in section 4 T h e c o n c l u s i o n s f r o m the w o r k are p r e s e n t e d in section 5.

2. Exact steady-state solutions F o r the driven point s y s t e m (the Dicke model) d e s c r i b e d b y eqs. (1), the e q u i v a l e n t m a s t e r e q u a t i o n f o r the r e d u c e d a t o m i c d e n s i t y o p e r a t o r ~A reads (e.g.2.1°)) d ~-~ OA = - i O [ S + + S_, ~A] + y0[2S_~AS+ - S+S-~A - ~AS+S-].

(3)

This m a s t e r e q u a t i o n c a n also be derived using o p e r a t o r r e a c t i o n field t h e o r y 19) w i t h o u t f o r m a l l y tracing o v e r the field states. T h e collective a t o m i c dipole o p e r a t o r s S± t o g e t h e r with the collective a t o m i c inversion o p e r a t o r Sz o b e y the c o m m u t a t i o n relations [S+, S_] = 2S~,

[Sz, S_+] = -+S_+.

Eq. (3) s h o w s that the D i c k e states 2°) f o r m a eigenstates It, m ) o f the and S~lr, m ) = mlr, m ).

(4)

o p e r a t o r S 2 is a c o n s t a n t o f the motion. H e n c e the g o o d set o f basis states. T h e s e are s i m u l t a n e o u s o p e r a t o r s S 2 and Sz satisfying S2[r, m ) = r(r + 1)It, m), T h e q u a n t u m n u m b e r r is the Dicke c o o p e r a t i o n

* This use of 'mean field' is the usage in phase transition theory and is of course not related to mean field theories of optical blstablllty s-9) The cnUcal exponents /3 = ½, 3' = I m mean field theory differ slgmficantly from the observed values /3 = 0 31 to 0 37 and 3' = 1 22 to I 38 Our value /3 = ½ for the exact solution case (b) also differs from this, but /3 depends also on the dimenslondeg / 3 = 0 1 2 5 ( d = 2 ) , / 3 = 0 3 1 to 0 3 7 ( d = 3 ) and /3 = l (d = 4) The case d = 4 1 s however totally equivalent to mean field Our result apparently ~s not, since there msno reason to think of the problem as one with an effective d = 4

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n u m b e r (0 <~ r <~ N/2) and m is half the difference b e t w e e n the numbers of excited and unexcited a t o m s ([ml<~r). The condition that all a t o m s are initially m the ground state rn = - N / 2 , m e a n s that r = N/2. Only two q u a n t u m n u m b e r s r and m are needed w h e n r = N/2. T h e r e f o r e we can label the states by m alone. Then it is c o n v e n i e n t to set p = m + N/2 and label the state IN/2, p - N/2) = IP). We then have (cf.21)) S2lp) =

s(s + DIp),

Szlp) = (s - P)lP), S+[p) = (p(2s - p + 1))'nip - 1), S-Ip) = ((p + 1)(2s _ p)),/2[p + 1),

(5)

where 0 ~< p ~< 2s, s = N/2 and IP = 0) is the ground state The " c o h e r e n t spin s t a t e s " introduced first b y Radcliff z~) are then defined through I/~) = (1 + [/xlz)-s e"S-10) +1//,12)-s

(

2s),

p=0 p V ( 2 s - p ) Q

, tz p )

(6)

T h e s e states f o r m an o v e r - c o m p l e t e set. By using th~s coherent-state representation Puri and L a w a n d e ~°) d e r w e d the equivalent F o k k e r - P l a n c k e q u a h o n of the m a s t e r equation (3), and then obtained an exact f o r m for t~A in the steady state (for which d~A/dt = 0). This steady-state solution is valid for any N and g = iO/y0, and is given by 2s

(PA)ss = Nf ~ (g*)2s-m(g)2S-"(s_)m(s+)"

(7)

m,n

with

~, (__2s-_p_+k),p, ig12,2,_,,

N ?~ = pzT,=o~__o(p _ k )!(2s _ p )!

(8)

T h e weak-field and strong-field limits for (#ALs are (i) (t~A)s~

(S_Y'(S+) 2" '" Igloo (2s*)2 ,

t (ii) (#A)~s I~1-~ (2S + 1)

( f = unit operator).

(9)

The b e h a v i o u r of the steady-state expectation values of the atomic variables and their fluctuations were given in detail in ref. 10: the results show that in the t h e r m o d y n a m i c limit Igl, N - - , ~ the atomic polarisation and population inversion h a v e a discontinuous derivative at 0 =-21gl/N = 1, and

INTENSITY FLUCTUATIONS IN A DRIVEN DICKE MODEL

217

the fluctuations are finite at this transition point Similar conclusions were reached numerically I~) and asymptotically u) by a different approach.

3. Atomic correlation functions

We now calculate the nth-order atomic correlation function m the steady state for arbitrary N and Igl. The generalisation of relations (5) gives

(S+)"Jp) =

[ (2s - p + m ' p ' i_(~n~s~-~!j

1'/~ IP - n ) ,

n ~
(10)

and its hermltian adjoint. Thus the nth-order correlation function Is G(,)(0) _-- (( S+).( S_) ~) = Tr(( S+)n( S_)"(f~A)ss)

= N/E2' [~- [ (__2_s;p_+k+ ni!p' llgl2,2,_k, ,=.

L(p - k - n)W(2s - p)V]

(11)

Of p a m c u l a r interest are the first-order and second-order correlation funcaons, where for the simple cases N = 1, 2 we have N = 1: Gin(0)= 1 +21gl z' G(2)(0) -- 0, G(2)(0) gt:)(0) ---iGm(0)12 = 0.

(12)

N = 2: G(')(0) = 4/gl2(1 + Igl2)/(3[gl 4 + 41gl 2 + 4), Ga)(0) = 41g14/(31gl4 + 41glz + 4),

g(Z)(o) -- (31gl' + 41g/2 + 4)/(4(1

+

Igl2) z)

(13)

The results (12) are the ones expected for single-atom resonance fluorescence2:); whilst the result (13) for the normalised intensity-intensity correlation function, g(Z)(0), is m excellent agreement with the results of Agarwal et al ~4). The weak-field and strong-field limits of these results are

(0 Weak field. lim g(")(0) -------IGO)(0)l G(")(0). = 1, I~l-.0

n = l, 2, 3 . . . . ( N I> 2).

(14)

This result characterises a coherent-state field. [In c o m m e n t note that the weak-field limit might be thought to describe some aspect of the superradiance described by the Dicke model without external field:°). H o w e v e r , the system is then supposed typically to be placed

218

S S HASSAN et al

initially In a pure D, c k e state [r, m), and the e m i s s m n Is a transient acc o m p a n y i n g m o t i o n to the state [r, - r ) T h e s t e a d y state c a n still be t h o u g h t of as one m w h i c h s u c c e s s w e e x c i t a t m n s b y the w e a k laser field are b a l a n c e d b y s u c c e s s w e t r a n s , e n t d e c a y s H o w e v e r our solution ,s an a p p r o p r m t e m , x t u r e of states [r, m ) , r ~> m >~ - r , r e p r e s e n t e d m the trace t a k e n m the e x p r e s s i o n (11) It is t h e r e f o r e an interesting but o p e n question w h e t h e r one c a n really d e d u c e a n y results f o r s u p e r r a d m n c e f r o m the results f o r the driven s y s t e m o b t a i n e d in this p a p e r W e m a k e o n e s u g g e s t m n in section 5, below.] (fi) Strong field lim Gin(0) = ~N(N + 2),

(15)

in a g r e e m e n t with ref. 15. This result s h o w s that the intensity o~ N 2 ( c o o p e r a t w e b e h a w o u r ) , u n h k e the case (a), which allows i n d e p e n d e n t a t o m i c decayL4), and has the intensity oc N ( n o n - c o o p e r a t i v e ) F u r t h e r m o r e , since p ' ( N - p + k + 2)'

g(2)(0) =

(N - p)'(p - k - 2)' Igl-2k

/ [p~=,~_~l ° p ' ( N - p + k + l )k' hmg(2)(O)=[~2 P ' ( N - p + 2 ) ' x

[i

N 2 (N p !(N - p +- k)k)vv Igl-2k = k=0 - p)'(p

Ig/-2'12, ]

p'(N-p)' p'(N ~

N

p'(__N-p_+ l)']2 (N-p)'(p1)']

Let

I,= ~ p'(N-p+2)! ~,2(q+2_~(_N_--q_)_ p~o(-N-Z-P~-(PU'~ ' - ~"=o( N - q - 2)'q" then I

~2

If = q=o ( N -

(N_+_32

q-2)Vq

1

f zq+2(1- r) N-q d z

vJ 0

=(N+3)'[r2(I-T)2dT= 4 (N+3)' (N - 2)'J 5'( N - 2)' 0

Simdarly,

pW(N-p+l)V 1 ( N + 2)! p=, (--'N----pSi~-- ] ) ' = 3' ( N - 1)!" Thus 6(N-1)(N+3)~I.2

hm g~2>(0)--- 3 fgf-*~

-~tN-7 ~

as

N~oo.

(f6)

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We also find the same result g(2)(0)---) 1.2 for the limits taken in reverse order, namely (hm [gl ~ oo) (lim N ~ oo) Thus for small values of the driving field the normalized intensity-intensity correlation function and the result gt")(0)~ 1 indicates complete coherence. In contrast as (N, Igl)~oo, g(Z)(O)-~1.2, which means that the scattered ratiation becomes only partially coherent but not totally coherent This value of 1.2 agrees with that reported numerically (ga)(0)-~ 1.18) ~3) and with that obtained by Drummond23); but it seems to be in disagreement with that obtained by Agarwal et al. t8) who found gt2)(0)~ 1 65. We analyse this discrepancy in the appendix and show that the value 1.65 is not correct. The earlier calculation of Hassan and Walls 1) gives g(2)(0) ~ 2 as (N, Igl)~ o0 and this is a manifestation of totally Incoherent radiation But in this case the authors adopted the decorrelation scheme which allows independent atomic decay (the case (a)) as noted in section ! Thus one can conclude that the enhanced cooperative decay contained within the exact solution of the problem reduces the fluctuation m the scattered intensity, as one might expect. We now calculate the thermodynamic limit, N ~ o~, Igl ~ oo so that 21gl/N -~ 0 remains finite. It is convenient to rewrite the expression for G")(0) and G(2)(0) in (11) in the forms (N/2)2 = 02[1 -

F(N,

0)],

(17)

+2 0)], G'2)(0) 0 4 1 1 - ( 1 + 2 Nfq~f)F(N, ] (N/2) 4 3 where the asymptotic expression for 202N

F(N, O) ----or ~0,

as

large N is given by ~°)

X/l - 0 ~ e2NVr~ (1+ X/1 - - 0 2 ) 2 N + 2

N~oo,

V'~- 1 ~02 arcsin(l/0)'

F(N, O) for

(18)

0<1,

0>1.

(19)

(20)

Thus for 0 < 1 in the thermodynamic hmlt G°)(O)/(N]2) 2 = 02; whilst for 0 > 1 these quantities are given by 02[1 - F ( N , 0)] and 0411 - ( 1 + 2 / 3 0 2 ) F ( N , 0)] with F ( N , O) given by (20). Although F ( N , O) is continuous at 0 = 1 its derivative is not. Thus Ga)(O)/(N/2) 2 and G(2)(O)/(N[2) 4 have discontinuous derivatives at 0 = 1. This is charactenstic of a second-order phase transition, a result we look at next.

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It Is worth noting that by using the expansions of V ~ - 1 and arcsin (1/0) in powers of 0 -I one easily finds the value (6/5) for g(2)(0) as 0--> ~. Nevertheless agreement with (16) does not seem to be a necessary consequence of the result 6/5 for that limit and for that limit taken in reverse order. The exact behavior of G°)(0) and g(2)(0) as a function of 0 is displayed in Figs. 1 and 2 for some finite values of N and also in the thermodynamic limit N ~ ~. N o t e that a photon antibunching effect 24'2~) can be seen in Fig. 2 near 0 = 1 for N < oo. 4. T h e s e c o n d - o r d e r p h a s e t r a n s i t i o n

In the thermodynamic limit the inversion r3 takes the f o r m '°'11'~3)

r3 ----N-I(s~) = -½(1 -

f~2/f~2)l/2

={-~ 1-02)'/2'

(2])

0>1.0<1'

This suggests that r3 is the natural order parameter with 0 2= T/Tc the thermodynamic temperature. These ideas motivate the thermodynamic limit defined in section 3. M o r e o v e r further analysis shows that there is a second root for r3, namely [ + ½ ( 1 - 02) I/2,

r3=~

0

,

0 <1, 0>1,

(22)

20 N=I / (~) 1 5 G (O._~))

- -

N =2

(N/2) 21 0

N=3 N=10 N .-. ¢o

O5

0 1

2

8

3

4

Fig 1 Steady-state fluorescent intensity G(~)(O)I(N/2)2 plotted for differel~t numbers of atoms, N = 1, 2, 3, 10, ~ against the scaled parameter 0 = 21g]lN (Igl Is the normalised Rab~ frequency).

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221

-m |

1

0

12-| ' / 1

N ---~oo

N=,0 1

~

N=4

~

. . . N= 3

08 ~N=2

06

0

1

2

3

4

0 Fig 2 Steady-state normahsed intensity-intensity correlation function g~2~(0) plotted for N = the scaled parameter 19 (Inset g~2~(0) as calculated m ref I (the case (a) of secUon 1) for N = 40)

2, 3,4, 10,ooagainst

so that 0 = 1 is a bifurcation point. The second root is remarkable since for 0 = 0 (no field) r3 = +1 is not a steady state due to the effects of spontaneous emission. We conclude from this that the whole branch r3 = +½(1- 02) I/2 is unstable for 0 < 0 < 1. Notice however that r3 = +½ is usually held to be a non-steady state because vacuum (or matter) fluctuations drive the system out of the state. The present analysis presents r3 = +½ as a steady (i.e. equilibrium) state which is unstable to fluctuations. If r3 is the order parameter (analogous to the magnetisation M in a magnetic phase transition) one sees from (22) that the critical e x p o n e n t / 3 (see section 1) is one half. From this one could go on to define a free energy in powers of 02 and deduce a simple jump in the specific heat and a second-order phase transition (compare~)). This is Landau t h e o r y ' ) , 'mean field' (see section 1), and is known to be in disagreement with experiments. Plainly the present theory

222

S S HASSAN et al

leading to (22), which is an exact theory, Js not ' m e a n field' in this sense even though/3 = ½ tn pressing the analogy with a magnetic phase transition we have identified M with r3, T/Tc with 02 and might expect to identify an analogue of the magnetic field H, define a susceptibility X = (OM[OH)r, and find the critical e x p o n e n t y in X ~-(1 - 0 2 ) - L So far we have not been able to see how to do this. A candidate for H is the 'effective field' X (the usual X in bistability theory, compareL4"6"7) this is essentially the Maxwell D - v e c t o r m the theory) In the t h e r m o d y n a m i c limit X - - 0 - ( 2 [ N ) ( S v ) (with N ~ ) , Sy = (1/2i) ( S + S_) Certainly X = 0 for (22) when 0 < 1, since (Sy)/(NI2)= 0, 0 < 1 ; = 0(1 - F(N, 0)), 0 > 1. But the second result means X is a function of 0, 0 > 1 and certainly (0, X ) do not constitute a pair of independent p a r a m e t e r s as T,H do. A second free p a r a m e t e r for finite N is the effective " c o o p e r a t i o n n u m b e r " for the present problem y 0 N - c o m p a r e the identification of cooperation n u m b e r C with T m ref 7. This, or its reciprocal, m a y be a candidate for an analogue of H in the present problem. H o w e v e r , in order to find 0 = 1 as a bifurcation point, we are obhged in this problem to go to the t h e r m o d y n a m i c hmit w~th ~/yoN=-O. This step ehminates one of two otherwise free p a r a m e t e r s g2 and yoN of the problem. Thus it may not be possible to extend (22) to f o r m a part of the usual description of a second-order phase transition And in particular it m a y not be possible to define a critical e x p o n e n t y for the problem.

5. Summary and conclusions T h e steady-state correlation functions ((S+)"(S_)")-= G(")(0) have been calculated exactly for the driven Dicke model in which N two-level atoms o c c u p y a single site and are driven by an imposed C.W laser field. The limiting value of g(2)(0) --- G(2)(O)/(G(I)(O))2 as N and g ----ig2y0-1, essentially the Rabi f r e q u e n c y scaled to the A-coefficient, both tend to infinity is 6/5. This ~s in contrast to a previous a p p r o x i m a t e resuW) obtained by decorrelating expectation values of o p e r a t o r products retaining h o w e v e r self-interactions which describe single atom d e c a y p r o c e s s e s as explained in section l (the case (a) described this). H e r e g(2)(0)-02, the chaotic value. Thus the exact solution of the driven Dicke model shows that this model IS more cooperative than the a p p r o x i m a t e d model in the strong-field limit. The exact model nevertheless undergoes a second-order type phase transition in a t h e r m o d y n a m i c limit in which [gJ~o¢ and N ~ o o , but Ig[/(N/2)=0 = finite. The transition is at 0 = 1 and the ' o r d e r - p a r a m e t e r ' r3 = N-~(sz) bifurcates there: for 0 > 1 the s y s t e m is fully disordered (r3 = 0) and the order changes as r3 = -½(1 - 02) 112 for 0 < 1.

INTENSITY FLUCTUATIONSIN A DRIVEN DICKE MODEL

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The exact solution of the system therefore shows no optical bistability, whilst the approximate solution has a first-order phase transition and the usual cusp-catastrophe-type optical bistability. Thus the exact solution shows that the driven Dicke model is not so good a model of the observed physics of optical bistability as is the approximated model. Our conclusion from this is that the mathematical requirement of the model namely that S 2 = constant of the motion, if preserved, does not allow single atom damping processes to take their usual course. Although radlatwe damping is contained within the model as the approximated theory shows by isolating these processes as self interactions, the condition S 2 = constant seems to inhibit this damping. We conclude that an essential feature of optical bistability is a clearly defined single-atom damping process either simply radiatwe or enhanced by other relaxation processes. We also note that the Dicke model is a point system model and that optical bistability has been observed only in systems large compared with a wavelength. In an extended system S 2 is no longer a good quantum number even for the simple two-level atom systems considered in this paper. Spatial effects in the real extended system are in any case known to modify results for cooperative emission (cf.26'27)) and in optically bistable systems2S). Other interactions, collisional effects and dipole-dipole interactions break the constancy of S 2. The indication of a'ntibunching which can be seen in the fig. 2 suggests that it might be useful to make intensity-intensity correlation measurements in a super-radiance experiment. The idea here is that m the cooperative region 0 < 1 the cooperative relaxahon is a transient phenomenon (on a time scale much less than yo ~) m which excitation due to the driving field ~s balanced m the steady state by this cooperative decay. We have noted earlier that it is an open question whether this point of view is relevant to the transient phenomenon of super-radiance itself however We add finally that the time development of the spectrum and correlation functions for the system approximated as in (a) (section I) are also under investigation. We believe that a complete analysis based on the exact solution (as m (b) (section 1)) and on the approximated solution (as in (a) (section 1)) will provide results instructive for the understanding of more realistic models of optical bistability.

Acknowledgement One of us (SSH) acknowledges useful correspondence with both D.F. Walls and P. Drummond.

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Appendix The apparent disagreement with the work of Agarwal et al. ~s) Is due to an algebraic error in the expression for the time-dependent normalised secondorder correlation function (eq. (11) oflS)), 3"(2)(~.) in their notation. If one proceeds correctly using the quantum regression theorem and eq. (10) oP s) one does arrive at the correct result (in the limit O >> 3"0) 3"~2,(~.)= 1 + 1 [ 1 3

N ( N3+ 2)] e-3"°" + 3 [1

N ( N + 2)

N ( N3 + 2)] e_5~o~cos(412r )

e -3v°~cos(20~'),

which gives 3"(2)(0) = 1.2

18 = 6 (N - 1)(N + 3) 5 N ( N + 2) 5 N ( N + 2)

for any N, in agreement with our result (16)

Note added in proof After the submission of this manuscript, we have found an exact quantummechanical steady-state solution for the master equation (3) with the inclusion of detuning effects, i.e. when the laser field is detuned to the atoms3°). The detuning parameter d ) - 280/3,0N; 80, the frequency mismatch between the atoms and the field, adds an extra parameter which we choose to Identify with the magnetic field H : H = ~b2X sgn r3. In this way31), we may identify the exponents a, y: the susceptibility Xo = (c9r3/002)0 ~ ( 1 - 02)-v and the specific heat C, = (~r3/002)~ ~ (1 - 02)-°. We find within the semiclassicai decorrelation and d~-->0 that a = ½, 3, = 1.5 and hence a + 2/3 + 3' = 3 > 2 There is no obviously necessary ground for the particular choice of the objective analogy we have drawn, so the question must remain open for further study. The least we have shown is that the exact solution of the driven Dicke model is not a critical mean field type result (cf. ref. 31).

References 1) 2) 3) 4)

SS P D IR HJ

Hassan and D F Walls, J Phys A l l (1978) L87 Drummond and S S Hassan, Phys Rev A (m press) Semtzky, Phys Rev L e n 40 (1978) 1334, Phys Rev A6 (1975) 1171 Carmlchael and D F Walls, J Phys BI0 (1977) L685

INTENSITY FLUCTUATIONS IN A DRIVEN DICKE MODEL

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5) C M Bowden and C C Sung, Phys Rev A19 (1979) 2392 6) R Bonlfaclo and L Lugtato, Opt Comm 19 (1976) 172, Phys Rev A18 (1978) 1129 7) G S Agarwal, L M Narduccl, D H Feng and R Gilmore, in Coherence and Quantum Opt.cs IV L Mandel and E Wolf, eds (Plenum, N Y , 1978) PP 281-292. also m Phys Rev Ai8, (1978) 620 8) S S Hassan, P D Drummond and D F Walls, Opt Comm 27 (1978) 480 9) The first successful observation of optical blstabdlty was reported by H M Gibbs, S L McCall and T N C Venkatesan, Phys Rev Lett 36 (1976) 1135 10) R R PUrl and S V Lawande, Phys Lett 72A (1979) 200, Physlca 101A (1980) 599 I 1) P D Drummond and H J Carmlchael, Opt Comm 27 (1978) 160 12) L M Narduccl, D H Feng, R Gdmore and G S Agarwal, Phys Rev AI8 (1978) 1571 13) D F Walls, P D Drummond, S S Hassan and H J Carmlchael, Prog of Theoret Phys Suppl 64 (1978) 307 14) G S Agarwal, A C Brown, L M Narduccl and G Vetn, Phys Rev AI5 (1977) 1613, G S Agarwal. D H Feng L M Narduccl, R Gdmore and R A Tuft, Phys Rev A20 (1979) 2040 15) A S Amm and J G Cordes. Phys Rev AI8 (1978) 1298 16) H J Carmlchael, Phys Rev Lett 43(1979)1106 17) A Suguna and G S Agarwal, Phys Rev A20 (1979) 2022 18) G S Agarwal, L M Narducc.. D H Feng and R Gdmore, Phys Rev Lett 42 1260 (1979), see also errata m Phys Rev Lett 43 (1979) 238 19) R Saunders, R K Bullough and F Ahmed, J Phys A8 (1975) 759, R Saunders, Ph D thes,s (1973) U Manchester S S Hassan, Ph D thesis (1976) U Manchester 20) R H Dlcke, Phys Rev 93 (1954) 99 21) J M Radchff, J Phys A4 (1971) 313 22) B R Mollow, Phys Rev 188 (1969) 88 S S Hassan and R K Bullough, J Phys B8 (1975) L147, and references quoted m 2) 23) P D Drummond, preprlnt and private commumcatlon 24) H J Carmlchael and D F Walls, J Phys B9 (1976) 1199 25) H J Klmble, M Dagenals and L Mandel, Phys Rev Lett 39 (1977)691, M Dagenals and L Mandel. Phys Rev AI8 (1978) 2217 26) See relevant articles m Cooperative Effects m Mater and Radiation Howgate and H R Bobl, eds (Plenum, N Y . 1977) 27) R Saunders, S S Hassan and R K Bullough, J Phys A9 (1976) 1725 J McGiIhvray and M Feld, Phys Rev A14 (1976)1169 28) E Abraham, R K Bullough and S S Hassan, Opt Comm 29 (1979) 109 E Abraham, S S Hassan and R K Bullough, Opt Comn 33 (1980) 93 29) L D Landau and E M Llfshltz, Statistical Physics, Chap 14 (Pergamon, Oxford, 1958) [30] R R Purl, S V Lawande and S S Hassan, Opt Comm (m the press) [31] S S Hassan and R K Bullough, invited paper presented at Int Conf and Workshop on Optical Blstablhty, 3-5 June (1980), Ashewlle, N C , USA (Plenum, New York), to be pubhshed