The evolution of a completely excited system of N two-level atoms, distributed over a large region and interacting with all modes of radiation field, is studied. The distinction between r-conserving (RC) and r-nonconserving (RNC) processes is emphasized. Considering the number of photons emitted as the discrete independent variable, the evolution is described by a partial difference equation. Numerical solution of this equation shows the transition from RNC dominance at the beginning to RC dominance later. This is also a transition from incoherent to coherent emission of radiation

Lee, C. T.

English

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https://ntrs.nasa.gov/search.jsp?R=19750016367 2020-03-22T21:15:59+00:00Z
Difference Equation for Superradiance*
C. T. Lee
Department of Physics and Mathematics, Alabama A5M University
Normal, Alabama 35762
We study the evolution of a completely excited system of N two-level is
atoms, distributed over a large region, interacting with all modes of
radiation field. We pay special attention to the distinction between r-,
ii
conserving (RC) and r-nonconserving (RNC) processes. Considering the i,
number of photons emitted as the discrete independent variable, the evo-
lution is described by a partial difference equation. Numerical solution
of this equation shows the transition from RNC dominance at the beginning
to RC dominance later. This is also a transition from incoherent to cohe-
rent emiss_ •-, of radiation.
(NASA-CR-142875) DIFFERENCE EQUATION FOR
	 N75-24439
SUPERRADIANCE (Alabama A 6 M Univ., Normal.)
10 p HC $3.25
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Unclas
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l_
	
I	 ^l^_	 ^
In 1954 R. H. Dickel
 discussed the spontaneous emission of radiation
fron in excited system of N identical two-level atoms. By considering the
entire collection of atoms as a single quantum mechanical system, he found
that under certain conditions the individual atoms cooperate to emit radia-
tion at a rate proportional to N2
 which is much greater than their incohe-
rent emission rate. This phenomenon is called super;adiance. The quantum
number that plays the central role in the description of such system is the
so-called cooperation number r which can be any integer between 0 and N/2,
assuming N to be an even number.
In the vast literature on superradiance2 , most of the investigation
have been limited to cases in which the emiting atoms either are confined
to-a region smaller than the wavelength of emitted radiation or are able to
couple with only one radiation mode. In these situations r is conserved;
which permits important mathematical simplifications. Current attention in
this field is being focused on the more difficult, but more realistic,
problem of N two-level atoms, distributed over a space of dimension greater
than the radiation wavelength, interacting with all the modes of radiation
field. 3
 In this situation r is no longer conserved. 4
 However, the effects
of r-nonconserving processes have not received sufficient attention in the
literature.
Recently, we  have developed a simple and systematic diagrammatic
technique for calculating the matrix elements of collective operators; which
gives the transition probabilities of both r-conserving (RC) and r-noncon-
serving (RNC) processes as follows:
Tk (r,m) _ (r+m)(r-m+l) r(k-kl) ,
	 (1)
1k C (r,m) = (r+m)2r+m-1) [I - r(k--k1)) _,	 (2)
!_	 1	 1	 1 	 1 	 1 	 1 	 1
3
where k is the wavevector of the photon emitted and k l is a specific mode
which may be the wavevector of the exciting pulse or the end -fire mode; m
is 1/2 of the difference of the numbers of atoms in the excited state and
the ground state. r (k kl) is independent of r as long as r is very close
to N/2; but it depends on the geometry of the sample. For a circular
cylinder of length R and radius a, it is known to be
sin2 [(k k 
1)
	
1
	
z 
R/2)	 4J2[(k--k 
1 ) 
a)
f(k-kl)	 + +
	
^ —+ +	
0	
,	 (3)
	[(k kl) Zi/2 ) 2	[(k-k1)pa)2
where(k-kl) E 'p are the transversal and axial components of (k4k 1) and J 
is the Bessel function of order 1. Rehler and Eberly6 have carried out
+
the numerical calculation of r(k-k+ l) which shows a very sharp peak along
the direction of kl . This implies that a photon emitted through RC process
will be predominantly along the direction of kl ; while RNC process will
give radiation along all other random directions.
When (1) and (2) are integrated over all possible directions, we obtain
	
TRC(r,m) a (r+m)(r-m+l)N
	 ,	 (4)
T 
RNC (r,m) a [(r+m)(r+m-1)/2r](1 - u)
	 ,	 (5)
where u has the sums meaning as given by Ref. 6 and can have values much
smaller than 1.
Suppose we start with a completely excited system %Ath r - m - N/2.
Then at the beginning photons will be emitted predominantly through RNC
processes and, therefore, along random directions. For each photon emitted
i}
4
through RNC process, both r and m decrease by 1; which does not improve the
chance of RC processes. On the other hand, for each photon emitted through
RC process, r remains the same while m decreases by 1; which will improve
the chance of RC processes. Hence, the more photons emitted along kl
direction, the higher the probability of additional photons emitted along
this direction. The RC and RNC processes will be about equally important
when (r-m)u is of the order of 1.
In this article, we are most interested in studying the transition
from RNC dominance to RC dominance. To study the evolution of a system of N
two-level atoms from a completely excited state, we will replace the time
as the continuos independent variable by the number of photons emitted as
the discrete independent variable. Then our "dynamic equation" will be a
difference equation.
Let P(r,m) be the probability that our system has cooperation number
r after N/2-m photons have been emitted. Then (4) and (5) gives the follow-
ing partial difference equation for P(r,m)
P(r,m)	 2r(r-m)a	 P(r,m+l) +	
(rfn±I	 P(r+l,m+l)
2r(r-m)a + (rl	 (2r+2) (1	 + (r+m+l)
(6)
with the initial condition P(N/2,N/2) - 1 and the obvious restrictions:
0 S r S N/2 and -r S m S r. We have defined a- u/(1 -u) which is essentially
the same as p whek P is a'^zch less than 1. In obtaining (6), we have ignored
the possibility that a photon may be absorbed by the atomic system and the
possibility that r may increase. The latter is justified in Ref. 5.
It is convenient to define two new variables
s	 =_	 N/2 - r. , t =_	 r - m (7)
and let Q(s,t) a P(r,m).	 Assuming that both s and t are much less than N,
5(6) becomes
Q(s,t) - [to/(i+to)]q(s,t-1) + 11 + (t+l)a]-1Q(s-1,t) 	 (8)
with initial condition Q(0,0) - 1.
By direct observation, the solution to (8) can be seen to be
t+l
8 	t	 1	 tQ(s,t) -	 II (1+io) it II (l+ia) 1 t! a	 ,	 (9)
s i-1 	 it	 J
where 61 - 0, 1, 2, •-•; and the summation is over all possible choices of
Si such that
t+l
61 - s	 (10)
i-1
The result of t':is summation can be expressed as a contour integral as follows:
t+1
Z	 n (1 + io)-St
s i-1
f
z-s-lt+12ai 	 n [l + (l+ia) -lz + (l+ia) -2z2 + ••• ] dz . (11)
 i-1
A good approximation for Z can be obtained by integrating through a saddle
point on the real axis, the so-called steepest descent method , as follows:
 s t+1 (1 + ia) 1 t+l ( 1 + ia)A	 8
Z = (27r) -'F A	 II	 , (12)
i=1 (1 - A + ia)J i=1 (1 - A + io) 2 ]
where z = A is the location of the saddle point to be determined by the
equation
t+1
A	 (1 - A + io)
-1
	s+1	 (13)
i=1
rn
6
u
	Substituting (12) into (9) and using Stirling ' s formula for tl, we obtain 	 (;
	
rt+l	 t+l ^- 1/2
Q(s,t) = A s[1+(t+l ) o]I 11 (1-A+ia)-11	 l+ia A 1-
	
i-1	 J i-1 (1-A+io)2 J
x	 t tt+1/2 at	 (14)'
Replacing s and t by the original variables r and m, we have a good
approximation for P (r,m). We want to know what cooperation number has the
highest probability after a certain number of photons have been emitted;
i.e., we want to locate the maximum of P(r,m) with respect to r for fixed m.
We will defined
x e (N/2 - m) o	 ,	 y =_ (N/2 - r ) o	 ,	 (15)
consider x and y as continuos variables, and replace all summations encoun-
tered by integrations.- Then, from (13) and (14), we have
	
A ln[l + (x-y)/(1-A)] - y	 (16)
1nP(y,x) - ln(l+x-y) - (y/a+1 /2)lnA + (1/2)ln[(x-y)/a]
+ [ (1-A+x-y ) ln(1-A+x-y) + (1-A)ln (1-A) + (x-y ) ln(x-y)]/a
(( 
rY1 lnSll + _!. -	 A	 1 + 1 ^	 (17)2 (o L A 1^-A 1-A+x-yJ	 A
Letting alnP (y,x)/ay - 0 and using (16), we obtain
1/(l+x-y) + 1/[2(x-y)] + (1/a)ln[A (x-y)/(1-A+x-y)]
+ (1+x-y)[A2 (x-y) - (1-A) 2 (l+x-y)y 1/2[A2 (x-y) + (1-A) (1-A+x-y)12
0	 (18)
7Simultaneous numerical solution of (16) and (18) gives y as a function
of x which, in turn, gives the most probable cooperation number as a function
of the number of photons emitted. The results for five different values of
U are presented in Fig. 1. A slop of l of these curves indicates pure RNC
processes and a slop of 0 indicates pure RC processes.
The author is grateful to Mr. T. L. Kuo for his help with the numerical
calculation.
8REFERENCES
*Work supported by NASA under Grant No. NSG-8011.
1R. H. Dicke, Phys. Rev. 93, 99 (1954).
2See the review article in "Quantum Optics" by G. S. Agarwal, Springer
Tracts in Modern Physics, Vol. 70, Springer-Verlag, 1974.
3See, for example, P. Schwendimann, Z. Physik 265, 267 (1973) and
references therein.
4D. Dialetis, Phys. Rev. A2, 599 (1970).
SC. T. Lee, to be published in Phys. Rev. A.
6N. E. Rehler and J. H. Eberly, Phys. Rev. A3, 1735 (1971).
7See, for example, "Principle of Quantum Mechanics" by W. V. Houston
and G. C. Phillips, North-Holland Publishing Co., Amsterdam, 1973, P. 329.
i9
FIGURE CAPTIONS
Fig. 1. Drop in cooperation number (N/2-r) as a function of the number of
photons emitted (N/2-m) with a as a parameter.
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Difference equation for superradiance

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