Fluctuation in superradiance

Plzysica 84A (1976) 48-67

0 North-Holland Publishing Co.

FLUCTUATION

IN SUPERRADIANCE M. SUZUKI

Department of’ Physics, University of Tokyo, Hotzgo, Butzk~~o-kzc,Tokyo; Japan

Received 24 November

1975

The fluctuation of the intensity of superradiant pulse is calculated by applying a general asymptotic evaluation method of quanta1 macrovariables to a superradiant master equation. It is shown that the fluctuation of superradiant intensity is proportional to the cube of the number of the relevant atoms. The time-dependence of the proportional coefficient has been evaluated explicitly. It is predicted that there occurs an enhancement of fluctuation of superradiant intensity at an intermediate time. The intensity and fluctuation of higher order are also calculated. General expressions of expansion coefficients of a generating function are also given rigorously.

1. Introduction of papers by the present author r - 5), Kubo’s extensivity Ansatz6 - s, has been proved under general conditions and has been extended to quanta1 macrovariables3) with applications to Bose systems and spin systems, in particular to superradiance. Our theory gives also an extension of van Kampen’s systemsize expansiong) to quanta1 macrovariables. The purpose of the present paper is to give a detailed derivation of the fluctuation of superradiance partly discussed in section 8 of ref. 3. In ref. 3, we have formulated general asymptotic evaluation methods of quanta1 macrovariables. It is convenient to summarize here the asymptotic evaluation method for equations of motion. That is, we start with the following quanta1 master equation on quanta1 macrovariables {Xi> : 1n a series

(1.1) where Q denotes the density-matrix, such as a commutation operation. adjoint operator P Tr Pr*Q

.Q the system size and r is a general operator We have also defined in ref. 3 the following

= Tr QrP

(1.2) 48

FLUCTUATION

for arbitrary

E

$

operators

t

IN SUPERRADIANCE

P and Q. Then, we obtain the following

49 equation

= (r*Q>,.

(1.3)

In terms of cumulants3~10), d 8
of motion

eq. (1.3) is rewritten = Fk ,,..., B, ({(X;l

in the form3)

.‘. X,Y$,J),

(1.4)

for Q = X:l ... X$‘, where Fk, . . . *,(( Wj}) may be a certain polynomial of Wj in most cases. As was shown in theorem 6 of ref. 3, the extensivity of e assures the extensive property of all cumulants (X:l ... XLn),,, (i.e., to be of order Sz). By the help of this extensive property of cumulants, it is possible to evaluate the orders of many terms appearing in Fk, . h, and to expand it in the form Fk, ...k, = 9” (Fk(?.k. + ,F’l’

+ .z’F’~’ + . ..>

(1.5)

for a large 9, where (F(j)) are all of order unity and nz is assumed to be a common fixed exponent depending on the structure of r. Making a scale transformation as Pt + t, we obtain the following most-dominant equation of motion for a large L2 d E
= F;;!.k,.

(1.6)

Thus, we get coupled equations of motion asymptotically, and consequently the solution of (1.6) gives the asymptotic behavior of fluctuations for a large D. This is our main procedure to be applied to superradiance. According to Agarwal”) and Bonifacio et uZ.~~~‘~),the quanta1 master equation for superradiance is given by

OS+1+ K-e, S+l);

I 1 =2g2 3t

,

(1.7)

where S, = S, + is,, g denotes the coupling constant between the atoms and radiation and 3cthe damping constant of field. Fluctuations of superradiant field described by Boson operators B and Bt are related12s13) to those of the corresponding “macroscopic” spin S as ((B+)” B”), = l,@

((S+)n (S-)“>,;

Thus, we study fluctuations of spin operators ing behavior of superradiance. .

/J = s/x. in order to understand

(1.8) the fluctuat-

50

M. SUZUKI

It is convenient

to make transformations

z = sr,t

and

of variables

as

si = S,/S,

(1.9)

where S is the magnitude of spin, and it is of order iV (the number of the relevant atoms). Namely, S plays a role of the largeness parameter 52 in our problem, and E = l/S. The transformed equation is rewritten as ^ F

2 = 3 {[S_)

The adjoint r*Q

operator

QS+] + [5-Q, s,])

P

of I’is

E

l-0.

(1.10)

given by (1.11)

[Q, s-1 -t b+, PI s->.

= 5 {s+

In the succeeding sections, we evaluate the intensity fluctuation of it, with the use of the above scheme.

of superradiance

and the

2. The intensity of superradiance and fluctuations of spin operators As preparations for evaluating the fluctuation of superradiant pulse in the next section, we study here the temporal evolutions of spin operators. From eq. (1.3) with (1.11) for a transformed time t the expectation value of S,K at a time z is easily seen to satisfy the following equation of motion

s2

(S,)

Transforming

5

= (S,S,)

nt, = mxm, + E* ((S,S,),

(S->,

(2.1)

&(S_).

this into a cumulant

where WI, = (s,)

Similarly,

-

form, we obtain

- 3 (S_),),

etc. and we have used the relations and

= (S->


(2.2) that

= (SXSZ>, + C
(2.3)

we have

d m, = mgn, dt

+ 2

(2.4)

FLUCTUATION

IN SUPERRADIANCE

51

and $

mZ = -(s+s_)

=

ml

-

1 + E’ ((S,‘),

-

(SZ),

-

(2.5)

S).

all As discussed in section 1, all the cumulants are of order S, and consequently the second terms on the right-hand sides of (2.2), (2.4) and (2.5) can be neglected for a large 0. The solutions of such asymptotic equations are easily given by W,(T) = tanh (z, pi?,

- t),

(2.6)

= m,(O) (cash z,) {cash (t - z,,,)}-‘,

(2.7)

111?(r) = m,(O) (cash tm) {cash (Z - zm)J-‘.

(2.8)

and

The result (2.6) is well known. Note that the above solutions satisfy the condition that I?T:(x) + m;(z) + m:(z) = 1, if it is satisfied at z = 0. Next, we study fluctuations of spin operators, some of which will be used in the succeeding sections. It is convenient to define the following symmetrized fluctuation G+ijz ((SiSj), + C*C.)/(2S);

i

=x,yandz.

(2.9)

In order to evaluate, for example, o,,, we study first the equation of motion the spin correlation (S,“). From eq. (1.3) for a transformed time T, we obtain

S $

Noting



= 2 - 2s (S -I- 1) (S,)

- 3

for

+ S (S + 1) + (&>. (2.10)

that

CC>

(2.11a)

= C + (SX,

and =
+ 3 C (SZ),

eq. (2.10) can be transformed d -a,, dz

= {4%%,

(2.11b)

+
into the following

+ (1 - mZ>} + EZ (2 CC>,

cumulant

form

- 3 , + (S,),

+ S), (2.12)

52

M. SUZUKI

where we have used eq. (2.5). The second term on the right-hand side of (2.12) is of higher order and consequently it can be neglected for a small E limit. The solution of eq. (2.12) for this limit is given by 0

zs = coshe4

(z - z.,,,) [- sinh 2 (t - t,J

(2.13)

+ 3 (z - t,,,) + ~$1,

such as where o,“, denotes the value of G==at z = T,. For other spin fluctuations see appendix. c,, and or,,, Spin fluctuations of higher order such as (S,“) and (Sz”) will be necessary later for discussing the fluctuation of superradiant light pulse. Thus, we study them here briefly for convenience. As before, it is easily shown that (S,“) and {S,“) satisfy the following equations :

S$

(S,“)

= 3
+ (3~ -

1) (S,)

- c. (2.14)

and S d (s,“) dr

= 4 (s,“)

10 (Sz”) + (10 - 4~) (S,“)

-

c = S(S

+ (1 - 4C) (As,) + c; In deriving the above equations relations :

[S,“, s-1 = =

s- ((S, s_

{4;-

+ .. . (_])“-’

+ (6~ - 5) (2.15)

+ 1).

from (1.3) with (I .I l), we have used the following

I)‘, - (SJ”:

+ (;) SF

-

(‘I)xl_3

(;)sz+i-v],

(2.16)

and

[S,“, S,] = -((Sz - 1)” - (&>n> s, These relations

are easily obtained

[eCxSz, S-1 = S_ (ex -

1) e-+.

.

from the following

(2.17) identity’

‘) (2.18)

It will be easy to derive the cumulants bzzz and ~~~~~from the above equations (2.14) and (2.15) for a small F limit, where G

zz”’

V

_(t) = (S,“),,, S-l. n

(2.19)

FLUCTUATION

Now we discuss the intensity

IN SUPERRADIANCE

of superradiant

pulse. This is proportional

photon number (B+B),, which is again proportional we have the relation that

s+s_ = the intensity

s(s+

Z(t) = -s2niz = s2 (1 -

to Z(z) = (S+S_),

to the . Since

(2.20)

1) - S,” + S,

Z(t) is expressed

53

by

(T)

W&t)) + s (1 + nzz(r) - 0&)}.

(2.21)

The second term on the right-hand side of (2.21) is of order S and consequently it can be neglected for a large S. Thus, the intensity Z(t) for a large S is found to be (2.22)

Z(r) = S2 sech2 (z - z,)

with the use of (2.6). This is a wellknown result 11,12*13*15). That is, Z(z) is proportional to the square of the atom number N (or S), which is characteristic of superradiance.

3. Fluctuation

of superradiant pulse intensity

We study here the,fluctuation fined by

of superradiant

pulse intensity,

CT,= ((B+B - (B+B>)‘). This is related

to a quadratic

CT= (B+B+BB)

-

(B+B)’

o,, which is de-

(3.1) fluctuation

G defined by (3.2)

as 0,

=

(T -

S21p12(s+s_),

(3.3)

through the relations (1.8) and (1.9). Bonifacio et LIZ.‘~,‘~) have used such a normalized quadratic fluctuation as 6 = CJ(BtB>-2. As was shown in theorem 3 of ref. 3, (s+s_) is of order unity. Therefore, the second term of (3.3) is of order Si while (rl will be seen later to be of order S3 and consequently it can be neglected for a large S. Thus, the fluctuation of the intensity, 0, is essentially proportional to uI for a large S. Owing to the relation (1.8), (Tis expressed by a spin fluctuation

M. SUZUKI

54 of the form o!‘(pl” =

{(s$sl>

This is easily rewritten

- (s+s-)2:. in the following

(3.4) cumulant

form

“ilP14= 2 c ts2 - <&>Z+ 2 c) + {2S (S,), +
+ 4 c (AS,), + 3 f -

(S;“)‘

- 29)

+ {c - 4 (Sz”), + 4 c where we have used the following
10 (SJi

= , + 4 (X3>,

(S,),

+ 2S),

(3.5)

formula

(X),

+ 3 (X2>,” + 6 (X2),

(X)2

+ (X),“,

(3.6)

in addition to the expressions (2.1 la) and (2.11 b). As was shown in theorem 6 of ref. 3, all cumulants {(S,“)} are of order S. Consequently, the first term on the right hand side of (3.5) is of order S3 and it is dominant (“S3-law”), the remaining term being at most of order S. Thus, o, takes the form OI = 2/W=(t) (1 - m;(z)

+ 2m, (r) oz_(t>} S3/‘D14 + O(S2)

= ~/,LL[’S3 tanh (t,,, - z) sech” (T,,, - z) x (1 + (202 + z - z,) tanh (r, The normalized 8

=

z/77=

fluctuation (t)

{ 1

-

m;(S)

x (1 - Ilzf(r);-2

of the intensity, +

S-’

24(t)

- x)> + O(S’).

(3.7)

6, is given by

&z(z))

+ O(S-2),

(3.8)

where m,(t) and G==(S)are given by (2.6) and (2.13), respectively. The dominant term of (3.7) with m,(z) given by (2.6) vanishes at t = z,. Thus, near and beyond 2 = t,, we have to consider the remaining term of higher order as well as an expression of 117=(~), (2.5), more rigorous than (2.6). In this region, uI should be used instead of 0, because the latter may become negative. In order to perform a calculation of higher order in general, we have to solve the temporal evolution eqs. (2.14) and (2.15) together with more complicated hierarchy. This will be discussed in the succeeding section. Tt should be noted that the fluctuation of superradiant pulse is proportional to the cube of the number of the relevant atoms for T =I r, and that the fluctua-

FLUCTUATION

tion at z = z, is proportional

55

IN SUPERRADIANCE

to the square

of the atom number

(which will be

discussed in more detail in the succeeding section). These results agree qualitatively with those obtained by Haake et al. 11,12,13,15*16)in terms of the P-representation or the directed angular momentum representation. Qualitative behaviors of Z(z) and CT~(Z)are shown there occurs an enhancement

in fig. 1 and fig. 2, respectively. of fluctuation of superradiant

Fig. 1

It is predicted that pulse near z = +z,,

Fig. 2

Fig. 1. A typical time-dependence of the superradiant pulse intensity. Fig. 2. Enhancement of the fluctuation ul(t) of the superradiant intensity: we denote a set of parameters 7, and o,“, by (z,, u,“,); a = (l,O.l), b = (1, OS), c = (1, I), d = (1,2), e = (1.5, l), f = (1.5, 1.5), g = (1.5, 2) and h = (1.5, 2.5).

as was generally discussed by Kubo et al. 6*‘). These will be observed hopefully by experiments in future. From observations of the intensity Z(t), the parameter of the time scale and the value of t, can be determined. Two more parameters are involved in cr. One of these is 1pj4 and it determines the scale of (TVand the other is &, which is related to the initial value o,(O) as o,(O) = ~],LL]~S3 tanh e, sech4 tm (1 + (202 - tJ

tanh zm}.

(3.9)

4. Intensity and fluctuation of higher order When we study fluctuations in a system which is not so large as is described by a dominant term for S + 00, or when we study such a case in which a dominant term happens to vanish as at z = t, for superradiance, we have to investi-

M. SUZUKI

56

gate terms of higher order in l/S. General

formulations

for such higher order ex-

pansions have been given already in refs. 2 and 3. Anyway, for our quanta1 macrovariables whose temporal evolution is described by (1 .l), we can obtain systematically higher order terms by expanding the both sides of (1.4) in a power of F. Here, we discuss explicitly fluctuations of superradiant pulse intensity up to the order S’. From

(2.5) and (2.12), W,(Z) and CT=,(Z)satisfy the following

equations

respectively - d nzz(t) = rnz dt

1 + ~(cr;,

-

mz -

I>,

(4.1)

and d g=:,,(t) = {4nz,o,z + (1 - I?/“,>>+ E (20,,, .dz If we expand

(4.2)

m=(x), ozz(z) and ~zzz(t) as

m=(z) = ml”(t)

o&)

- ACT,,+ mz + 1).

+ em:)

= o:;‘(t)

(z) + .?m~)

+ my

(z) + ... ,

(4.3a)

(7) + . ..)

(4.3b)

I CT..;_(z)= cT:ff(T) -i- &a;:; (7) + ... ) then the expansion

coefficients

(4.3c)

WZ:“(~), m:“(t),

CT::’ etc., are easily

found

to be

given by ~j’)(t)

= C-~(Z)

mu”

=

i c2(s) (oaf’ l0

C-~(Z)

ic2(.s)

(a::‘(s)

- WICK’ -

-

PZ~~‘(.S)

-

1) ds + m”‘(O)),

(mI1)(s))‘)

(4.4a)

ds + m:“(O)/,

(4.4b)

1

and o(l)(r) ZZ

= c-“(t)

(ECU

(4nz~“a~P’ - 2mj0’m~”

f m,(‘I + 1) ds + a:;‘(O)

i

+ 2~7::; -

306:’

,

where ~R?~(T) and 061;“(~)are given by (2.6) and (2.13), respectively, c(7) = cash (7 - z,)/cosh

z,.

(4.4c) and (4.5)

The zeroth order term &T) of ozzz(r) is obtained by transforming eq. (2.14) into a cumulant form and by solving it, as was discussed in general in ref. 3 and in

FLUCTUATION

IN SUPERRADIANCE

57

section 1 of the present paper. That is, eq. (2.14) is transformed into the form d -LG

I

= 6mzczzz + 6~ (G - m,) + (& - 1) + E (3,0ZZZZ - 6oZZZ-t- 40,~ - m, - 1).

(4.6)

Thus, the zeroth order solution of this equation is given by &!(rj

= c-“(t)

s’P(S) (6019)(~1:) - m:‘)

+ (m~0’)2 -

l> ds f &‘&O) . (4.7)

0

I

Thus the intensity of superradiant pulse I(z) is given by I(t) = S2 (1 - m:(z)) = S2

+

S (1 + m,(z) - a,,(t)>

(0) - 2m~‘m~“) sech’ (Z - z,J + S (1 + rnp’ - (szz

+
0::’

-

[mL1j]2 -

2m~‘m~“)

+ O(E).

(4.8)

The fluctuation of higher order for the superradiant intensity will be obtained by substituting such expressions of m,, oi3, u,,, and ~~~~~as are given by (4.3), (4.4) and (4.7) into (3.5). To evaluate (TV_ explicitly, we have to transform (2.15) into a cumulant form with the use of the following relation: =
+ 5VL

+ 15(X%

(X4>, + 10(X3)C
, + 10(X2), (X),” + (X),‘.

Now we are interested in the case of z = t,, in which rn~‘(~,) = CT,,. m Then, we obtain oI(T,) = S’[P]~ [(2mj1’ (7) + 3 (cg’)’

(4.9) = 0

- 2) l/J]* + I].

and (TV:)

(4.10)

Thus, oI(z,) is of order S2 and the time dependence of the coefficient is given by (4.10) with (4.4) and (2.13). A systematic derivation of general higher order terms will be given in the succeeding section by using a generating function.

5. General expressions of higher order terms and a generating function

When we study systematically fluctuations of higher order, {S:), and try to find rigorously compact expressions of them, it is convenient to introduce the following generating function’) of a macrovariable S,: f(x, 7) =
(5.1)

M. SUZUKI

58

It is easily shown from (1.3), (1 .Il) and (2.18) with its complex f(x, z) satisfies the following partial differential equation

Sif(r_

t) = (emX -

1)

S(S C

The fluctuation

(S,“),

=

(S:),

is calculatedl)

[

$logf(x, i

z)

1.

+ 1) + Zix

conjugate,

- E f(x, t). ,= )

from this generating

function

that

(5.2)

through

(5.3)

x=0

Essentially the same equation as (5.2) has been obtained by Agarwal’r), but he has not studied it at all. Here we solve eq. (5.2) by expanding the logarithm of .f(x, r) as f(x, t) = exp (Sy, (x, z) + YI + &y2 + ...I. As was discussed generally in ref. 2, the expansion the following differential equations

%o=

-

at

(eex-

l){l - (zr),

(5.4) coefficients

{Y,,} of (5.4) satisfy

(5.5)

(5.6)

and in general .

(5.7

where Q (x, T) = 2 (eeX -

I) %J ax

(5.8)

and R, for n 3 2 is given by

R, = (ewx - I)

-_

d2yn-1

___

a.9

(5.9)

Equation (5.7) for IZ 3 1 can be solved by the Lagrange method, if yO, y, , . . . , and yn_r are known. On the other hand, eq. (5.5) is easily shown to have a com-

FLUCTUATION

plete solution yo(x,z)

59

of the form = +Ja

+ 1 log

where a and b are arbitrary

Thus, a general teristic equation

constants

(a -I l))/(e-”

L’(X) = {(emx -

dr --=-=-9 1

IN SUPERRADIANCE

-

and

1))“.

(5.11)

term Y),,(x, t) can be obtained of (5.7) is given by

dx

dy,

Q (x, 7)

R,

interatively.

In fact, the charac-

(5.12)

where Q (x,

t)

=

T2

for the complete two integrals :

(e-x

-

solution

1) (1 -

a (e-’

-

I>>” E Q (xi a),

(5.10). The characteristic

(5.13)

eq. (5.12) has the following

(5.14)

t - S(x) = a, ; and y

= n

s

with certain

JR, (x’, a,, + SW)) dx’ Q (x’, 4

constants

by

+ b,,

(5.15)

a, and b,. Thus, the general

solution

for y,, (x, t) is given

X R, (x’, S(x’) - S(x) + t)

Yn 6% t) =

Q b’, a)

s

dx’ + fn (z - W),

(5.16)

where f,(x) is an arbitrary function. Then, the solution of the Cauchy problem for y,, (x, r) is also easily obtained from (5.16) by the standard method. In particular, the solution for y0 (x, t) with an initial condition y0 (x, 0) = v(x) is easily obtained using the Hamilton-Jacobi method’, ‘) as YO

(x, 7) = 945) + (e-r - 1) (1 - [#(O]‘) - Ja +

‘t

+ 1 log I U (u(x), 4/U (v(t), a)1

log Iu W), O)/u ME), O)l,

(5.17)

60

M. SUZUKI

where U(v, a) = {z’ - (a + l)*>/{u + (a f lowing nonlinear equation

1)) and E is the solution

of the fol-

(5.18) with a = (e-” -

1) (1 -

[v’(6)12).

For the purpose of evaluating fluctuations {(S:),) explicitly by the help of (5.3), it is more convenient to expand yn (x, z) in a power series of x as follows:

Then, the fluctuation

(XL

(5.19)

on, m(T)*

yl, (.% T) = ,rzi, 5

(S,“), is expressed

in terms of {~~,,,,(z)l as

& = l/S.

= s 2 &Q&n (t);

(5.20)

k=O

First we find a general expression of ~~,~(t) for v. (x, T) from (5.5). It is easily found that {co, n(r)j satisfy the following equations b,,,(t)

= &?,I -

1,

~a,&)

= (2~~0,1)~0*,

and + &(~O,l,

for

. . ..GO.n-!)

II > 2,

(5.21)

where * s,z=n!C

n-2

(-I),’

~o,n!+l~o,n-m

m=z

c,_,

+

II! c

m!

m=l

m!(n

-

m -

(5.22)

l)! ’

co= 1 - bo, c, = -b,

b, = f:

and

(p > 1)

q!(p

4=0

It will be instructive

to present

so,3

=

6~0,1~i,3

+

ri a,4

=

~~0,1~0.4

+

Cd,,

(1

-d,d

(5.23)

~0,4+1~O*P-_q+l*

- q)!

here the first few terms explicitly:

-

1)

+

fXd,,

+24~,,,~,,,

-

~0.100,2),

-

12(00*,00,3

+

&),

etc.

(5.24)

FLUCTUATION

IN SUPERRADIANCE

These results agree with those obtained is expressed

61

in section 4. In fact, the solution

of (5.21)

by

for n 3 1, which yield (4.4a) and (4.4b) for IZ = 1 and IZ = 2, respectively. Similarly, given by

the temporal

evolution

equations

of {~,,~(r)}

&..,W = 2moo,,o,,, + R,,,,,

for a general

yn are

(5.26)

where m-1 m! (-l)“-*

R n. nr = (- l),’ 6,,1 + c

4=0

_;g

2 ;;“6;;,“!

6,. , being the Kronecker

The solution

(r

(m - q)! b”,, +;.g

frn_,,q+l i

-Qn--l.g+z

q! m;,;;cJ;“;,;P,

- %, (1 - &,I) i (5.27)

delta and

of (5.26) is

n,d4 = c-Y4

(JkY” R,,,(s) ds + ~,,..O)-

(5.29)

In particular, G, ,2 and @o,3 yield (4.4~) and (4.7), respectively. Thus, we have obtained a formal solution of (5.2) in a power series of F = l/S. This is a typical example of exactly soluble systems in the above sense. It is easily seen from the above solution that all y” (x, t) for n z 1 go to zero as t goes to infinity, except the case of relaxation from the complete inversion. This is consistent with the fact that the stationary solution of (5.2) is give by (5.30) although region.

the superradiant

master

eq. (1.7) may be applicable

only to a transient

M. SUZUKI

62

6. On the exact formal solution of the generating function It will be of interest Planck

equation

c:Y -=

_2zE_Ez(1_Z)~;

at

with

to find an exact formal

(5.2). It is easily transformed

solution

of the non-linear

into the following

Fokker-

equation

F=s-‘l,

(6.1)

dZ

’ Y (z, z) = e”sf(.x, z)

z=

and

1 -ee”.

(6.2)

This differential eq. (6.1) is quite similar to the Fokker-Planck equation on the quasi-probability distribution function derived by Narducci et a/.‘*), as it should be. Thus, eq. (6.1) can be solved rigorously using the method of eigenfunction expansions, under the boundary condition Y (0, z) = 1, in which our problem requires a caution different from that of Narducci et al.“). The Laplace transform yields the following eigenvalue equation z(l

az!P

-z)-

+zszE-Iyr=O az

822

(6.3) ”

’

with the time-dependence exp (-&AZ). Since eq. (6.3) is Gauss’ differential tion, the eigenvalue ii of it is easily determined to be 1, = (n + 1) (2s - n) ;

O
equa-

(6.4)

from the boundary conditions that Y(O) = I for 1 = 0 and Y(0) = 0 for /1 =I=0, and that Y’(z) is a polynomial at most of order 2S, as is easily seen from the definition (5.1). Clearly, Y(z) = 1 (or constant) is an eigenfunction of (6.3) with 3, = 0, which yields the stationary solution (5.30). As the eigenvalues (6.4) are doubly degenerate and eigenfunctions zG, (I -2S, 2, z) 3 F (-n, n + 1 - 2S, z; z) (where G, (a, y, z) is Jacobi’s polynomial) are linearly dependent on those for 0 < YE< [S - 51, we have to consider also solutions of (6.1) in the form y(z,

r)

Substituting z(l

=

e-ci.nT . z

{F,(z) + tGn Cd}.

(6.5)

(6.5) into (6.1), we obtain

- z)L

d2F

+ (2 - (a + 1) z> z

+ n (a + n) F,, = -C,SG,,

(6.6)

+ (2 - (a + 1) z> 2

+ n (LX + n) G, = 0,

(6.7)

dz2 z(1

d2G - z)L dz2

FLUCTUATION

with OL= 1 - 2S, and with an appropriate consistently.

Equation

(6.7) is Jacobi’s

G,(z) = G, (LX,2; z). It is found is given by

F,(z) = SZS-1,n(2; z> where we have introduced

63

IN SUPERRADIANCE

constant

differential

C,, to be determined equation

after some considerations

and

c,s

the following

SM,.(y;4 = lim {GM-~(ol,y;z)

“associated” -

that a solution

= 2s - 211 -

self-

with the solution

1)

of (6.6)

(6.8)

Jacobi’s polynomial

G,(a,y;z))(~~

+

a+-M

M)-‘.

(6.9)

This “associated” Jacobi’s polynomial S,,, (y; z) is of order (A4 - n) in z. Therefore, the solutions Szs_ I,n (2, z) for 0 < n < [S - +] are linearly independent of (G, (1 - 2S, 2; z)}. Thus, the solution of (6.1) is expressed in the form

p(z,

z) = 1 + z c e-EnnT ja,,G,, (1 - 2S, 2; z) PI= 0 + b,(S,,_,,,(2;z) + t&(1

- 2S, 2; z)}.

(6.10)

by the initial condition Here, the coefficients {a,,} and {b,} will be determined ul(z, 0) = exsf(x, 0) in an appropriate method, for instance, using modified orthogonality relations such as _+

(1 - X)“-YG,(O(,y;X)G,(ol,y;x)dx

for Rey A,

> Oand =

Re(a

(_l)‘/+“-’

m,(t) =

n!F(oc

m,(t) is expressed

--E

(6.11)

+ II + m) < 0, where

x {T(ol + 12)F(l The moment

= A,,&,,

+ 2n)T(y)F(l

- 01 - 2n)

+ y - 01 - n)>-1.

(6.12)

by

C (a, + b,t) evEAn’ -

1,

(6.13)

n=o

and the variance

g=,,(t) is

0=,,(t) = E C e-“n’ n=o

{(n (12 + 1) + 2nS - 1) (a, + b,~)

+ b, (S - n - If) - (an + b,z)2 exp (-c&t)}.

(6.14)

M. SUZUKI

64 Explicit evaluations

of these expressions

simple initial condition as f(x) elsewhere. Agarwal’s expression

are rather complicated

even for such a

= exp (xS), and they will be discussed in detail may be more convenient for a small S.

7. Summary and discussions We have applied our general formulation presented in ref. 3 to fluctuations of superradiant light pulse and have shown that the transient fluctuation of superradiant pulse is proportional to the cube of the number of the relevant atoms and that there occurs an enhancement of fluctuation at an intermediate time. We have also discussed the importance of a higher order term at the special time z = r,, at which the intensity of superradiant pulse, Z(t), takes the maximum value. The above effects will be observed hopefully by experiments. General expansion coefficients of a generating function for the macrovariable S, have been obtained rigorously. It will be useful to remark here a relation between the conceptual microscopic proof of the extensivity in ref. 1 and the present treatment based on the generalized extensivity with the scale transformation of time, eq. (1.9). That is, our previous proof of ref. 1 can be straightforwardly extended to more general hamiltonians which are not necessarily of order J2 as in superradiance, with the use of an appropriate transformation of time Qmt -+ t. Our results on fluctuations, (3.7) and (4.10) may be obtained also with the use of the method of the directed angular momentum representation15‘*). Our treatment will be extended to a non-markoffian quanta1 master equation”).

Acknowledgements The author would like to thank Professor R.Kubo for his encouraging discussions and also thank Professor F.Haake, Dr. Shibata, Mr. Takahashi and Mr. K.Sogd for their useful discussions. This work was partly performed at the International Center of Theoretical Physics, Trieste, Italy. This study is partially financed by the Scientific Research Fund of the Ministry of Education and also by the Mitsubishi Foundation.

FLUCTUATION

65

IN SUPERRADIANCE

Appendix

Spin j7uctuation.s Oi j

Spin fluctuations {Uij> are obtained in the same way as for czz in section That is, it is easily shown that they satisfy the following equations:

S&= 2
2.

(S:) + +(S,>, J

S &

= 2 - 2i (SJ,)

S2

(S,S,>

S$

*

+ (S,2)

- (S,‘)

= 2 (L&!&y -
+ *(S,),

- 4 {S-S,

+ 2S,S_),

= 2 - (S,) S(S + I) -
S -$



= 2 (S,S,S,>

From these equations, d Gxx = (2m,o,, dT

- i ((S,‘)

we obtain

+ (S:)

the following

-k (S,S_))

equations

- (S-S,).

of cumulant

(A.l)

forms:

+ 2mxoxz + m;‘)

+ E2 (2
+ , - CS,“>= + 2i ,

+ p),

d %v = (2mznyv + 2myb,, + m:) dz + c2 (2 ,

+ CS,‘>C - = - 2i
+ +
d vxz = (3m,o,, + mgz-_ - m,m,) dt + c2 W%S,2), d

d,

- 2 J

gYz = (3m,o,, + m,o,, - mymz)

+ Jz2{(C&C>, - :
+ c.c. + aJ,

fJ’xY

= (2m,~,, +

2

+ CC. + &,

+ mxcY, + m,4

((,

-

4

(S,SY>,)

+

cc>.

(A.3

66

M. SUZUKI

Here we have used the following

cumulant

expressions:

For more details, see refs. 3 and 10. The second terms on the reight hand sides of (A.2) are again of higher order and consequently they can be neglected for a small F limit. The asymptotic equations thus obtained for spin fluctuations agree with those obtained by Takahashi and Shibata14) using a coherent representation. Since such asymptotic equations are simple linear equations, they can be solved immediately. That is, we obtain

xc>

m s u,,(s) - m,(s) m,(s))

c3(s)ds + a0)

7 i

-

m,m,)

c3(s) ds + C,=(O),\ ,

+

myax&

c*(s)ds -I- o,,(O) I ,

+

aYY

i(2m,a,,

m:)

J

c2(s)ds + ox,,(O) , I

+ mz)c2(s) ds + a,,,,(O)I

with c(r) E cash (z - z,)/cosh fluctuations, see ref. 14.

(A.4)

1’

0

r,.

For more

explicit

evaluations

of these spin

Note added in proof Recently the present author has formulated a general method to evaluate asymptotically the intensity and fluctuation near the instability point [Prog. Theor. Phys. 56 (1976) No. 21 on the basis of the general scaling theory of transient phenomena [Prog. Theor. Phys. 56(1976) No. 1 and J. Stat. Phys., submitted]. This in the fluctuation-enhanced (or scaling) gives the result that oI CC{i(t)/i(O)}*

FLUCTUATION

IN SUPERRADIANCE

67

region. Thus, another peak of the fluctuation appears outside the region shown in fig. 2, correspondingly to the shape of the intensity shown in fig. 1. The relaxation and fluctuation from the complete inversion can be also discussed with the use of the scaling theory of transient phenomena, which will be reported in the near future.

References I) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18)

19)

M.Suzuki, Prog. Theor. Phys. 53 (1975) 1657. M.Suzuki, Prog. Theor. Phys. 55 (1976) 383. M. Suzuki, Prog. Theor. Phys. 55 (1976) 1064. M.Suzuki, .I. Stat. Phys. 14 (1976) 129. M. Suzuki, Lecture Notes in Physics 39. Intern. Symp. on Mathematical Problems in Theoretical Physics, H.Araki, ed. (Springer, Berlin, 1975). R.Kubo, in Synergetics, H.Haken, ed. (Teubner, Stuttgart, 1973). R.Kubo, K.Matsuo and K.Kitahara, J. Stat. Phys. 9 (1973) 51. R. Kubo, Lecture Notes in Physics 39 (see ref. 5). N.G.van Kampen, Can. J. Phys. 39 (1961) 551. R.Kubo, J. Phys. Sot. Japan 12 (1962) 1100. G. S. Agarwal, Phys. Rev. A2 (1970) 2038 and Springer Tracts in Modern Physics 70 (1974) 1. R.Bonifacio, P.Schwendimann and F.Haake, Phys. Rev. A4 (1971) 302 and 854. F.Haake, Springer Tracts in Modern Physics 66 (1973) 98. Y. Takahashi and F.Shibata, J. Stat. Phys. 14 (1976) 67. R. J.Glauber and F.Haake, in Cooperative Effects, H.Haken, ed. (North-Holland, Amsterdam, 1974) and preprint. F.Haake and R. J.Glauber, Phys. Rev. A5 (1972) 1457. F. T. Arrecchi, E. Courtens, R. Gilmore and H. Thomas, Phys. Rev. A9 (1974) 829. L.M.Narducci, C.A.Coulter and C.M.Bowden, Phys. Rev. A9 (1974) 829. L. M. Narducci, C. M. Bowden, V.Bluemel and G. P. Carrazana, Phys. Rev. All (1975) 280 and 973. L.M.Narducci and V.Bluemel, Phys. Rev. All (1975) 1354. R.Bonifacio, in Cooperative Effects, H.Haken, ed. (North-Holland, Amsterdam, 1974).