J. Mod. Phys., 2010, 1, 163-170
doi:10.4236/jmp.2010.13024 Published Online August 2010 (http://www.scirp.org/journal/jmp)
Copyright © 2010 SciRes. JMP
Quantum Statistical Properties of Resonant Radiation
Scattered on Excited Systems
Boris A. Veklenko
Joint Institute for High Temperatu r e of Russian Academy of Science, Moscow, Russia
E-mail: VeklenkoBA@yandex.ru
Received April 15, 2010; revised May 21, 2010; accepted June 10, 2010
Abstract
The scattering of resonant radiation on an excited atom is considered. It is shown that the scattering cross
section calculated with the help of quantum theory of radiation is five times larger than the one calculated
using semi-classical theory. The quantum theory predicts, in general, the change in internal quantum statisti-
cal properties of light due to the scattering processes on excited atoms.
Keywords: Quantum Theory, Semi-Classical Theory, Resonant Radiation
1. Introduction
The quantum excited systems possess remarkable prop-
erties. They manifest themselves most prominently in
lasers and masers, which were created in the middle of
the last century. The theory of these devices was elabo-
rated by W. Lamb [1] on the base of a semi classical
theory of radiation which deals with classical electro-
magnetic field. Later the quantum theory was proposed
[2]. It is possible to state omitting the fluctuations prop-
erties that both the semi classical and the quantum theo-
ries result practically in the same results for quantum
means values. Such a fact resulted in overestimation of
the applicability of the semi-classical theory. In 1966
year, Ch. Koester predicted the effect of light enhance-
ment [3] by selective reflection of resonant radiation
from excited media. All efforts of quantitative ex plaining
this effect on the base of semi-classical theory of radia-
tion discussed in monograph [4] were unsuccessful [5,6].
It was shown later that quantum field theory should be
used instead [7], but the mathe matically problems on th is
way occurred very difficult [8]. The consequences of
such a theory manifest themselves on a macroscopic
level. The correct description of stimulated radiation
plays an especial role when the resonant reflection of
light from excited media is considered. Nevertheless,
there are recent works [9] which make use the semi-
classical theory and Fresnel’s formulae to describe the
reflection of light from enhanced media.
Much attention has been paid recently to the effect of
the enhanced transmission of light through the metallic
films [10,11]. There is no agreement between theory and
experiment. It is believed that the enhancement of radia-
tion may be explained through the interaction of light
with induced standing surface—plasmon waves. Thus we
deal with effects of stimulate radiation, which means that
one should use the quantum field theory.
Examples shown above made us revise the theory of
resonant radiation scattering on excited systems. The
conventional perturbation technique is not adequate to
describe the resonant scattering and it is necessary to
sum up (Dyson summation) the infinitely long subsets of
Feynman’s ladder diagrams. It was V. Weissk opff and E.
Wigner who constructed such a theory for the first time
by considering the interaction of resonant radiation with
atomic systems [12]. Such a summation of Feynman’s
diagram proved to be useful for the shape of spectrum
line of resonant radiation and effects of resonant light
scattering on non excited systems. The difficulties
emerge in the theory of combined resonant scattering
processes when one of the photons after stimulation emi-
ssion of excited atom undergoes of elastic scattering on
the ground state of the same atom. Such combined scat-
tering is non-analytic in charge. The summation of the
Feynman’s diagrams like this one is not performed up to
now [8]. We propose indirect way to estimate this sum.
Present work demonstrates insufficiency Weisskopff-
Wigner’s method and Dyson’s method of summation
Feynman’s ladder diagrams for the calculations the
cross-sections of light scattering on resonant excited
systems and failure of semi-classical theory of radia tion.
Let the resonant radiation scatters on some system the
initial state of which in interaction representation is de-
scribed by wave function 0
. The total wave function
of electromagnetic field and scattering system is denoted
as
. The expansion of such function over a base of
scattering system wave function i
is
B. A. VEKLENKO
Copyright © 2010 SciRes. JMP
164

0000 00
0ii
i
fff f
 
  
.
The term containing 0
is written separately. The
scalar product due to orthogonality of scatter’s wave
functions
0000 0ff


is equal to zero. Assume that the incident light is in
quantum coherent state [13] and its quantum mean elec-
trical strength is not equal to zero ˆ(,)0t
rEin all
space points r at arbitrary instant of timet. We are
interesting in quantum mean value of operator ˆ
E of
the reflected light
0000 0000
() ()
ˆˆ
(,) (,)
ˆˆ
(,) (,)
cn
tt
ftf ftf



 
 
 

rr
rr
EE
EE
EE
(1)
We state that the first term of the right hand side of
Equation (1) describes the so-called coherent scattering
channel with medium returning to the initial quantum
state after scattering (e.g. elastic scattering). The second
term of the right hand side of Equation (1) describes the
non-coherent scattering processes with the medium
changing initial quantum state (Compton scattering, Ra-
man scattering and induced radiation of light). The latter
is very important. We stress once again that the coherent
Heisenber-Kramers scattering and induced radiation of
light are described by different scattering channels. It
means that if the scattering media consisted only of the
non-excited atoms the first term of Equation (1) would
describe the coherent Heisenberg-Kramers scattering
while the second one would describe the diffusion scat-
tering. If the excited atoms are present in the medium
then due to the induced radiation processes it is impossi-
ble to avoid the presence of the non-coherent channel
even if only the selective scattering is under our investi-
gation. The total measured electrical strength ˆ(,)t
rE,
that is the left hand part of Equation (1), may be evalu-
ated separately using the semi-classical theory of radia-
tion if one neglects the fluctuation optical processes and
their influence on ˆ(,)t
rE. The region of validity of
the semi-classical theory of radiation is very large but it
does not mean that ˆ(,)t
rE describes the bilinear
field characteristics.
Let us consider the energy characteristics of electro-
magnetic field described by normal operator product

2
ˆˆ
N
E. Such value should be estimated from below
using the following procedure. One takes into account
that

ˆˆˆ
(,) 2
i ickti ickt
ck
tie ee
V

 



kr kr
kk k
k
r
E,
where ˆ
k and ˆ
k are the annihilation and creation
photon operators in states describing by wave vector k
and polarization index
. These operators obey the
conventional commutation relations
ˆˆ
;





kkkk
Consider electromagnetic field as a transverse
one(1,2)
,
k
e denotes the unite linear polarization
vectors, V is the quantization volume. Since the op-
erators ˆ
k and ˆ
k are mutual conjugate than

()()
ˆˆ ˆˆ0
i ickkt
ekek e



 

 
 


kkr
kkkkkk
kk
Now
()()
ˆˆ i ickkt
kk eee

 




 

kkr
kkkk
kk
()()
ˆˆiickkt
kk eee

 




 

kkr
kk kk
kk
If the electromagnetic field possesses the characteristic
frequency 0
and characteristic wave length 0
and
we are interesting in time and space values much larger
then 0
1/
and 0
the following in equality occurs
()()
ˆˆ i ickkt
kk eee

 




 

kkr
kkkk
kk
()()
ˆˆ iickkt
kk eee

 




 


kkr
kkk k
kk
()()
ˆˆ iickkt
kk eee

 




 

kkr
kkk k
kk
()()
ˆˆ iickkt
kk eee

 



  
 

kkr
kkk k
kk
Now it is non difficult to see that

2
ˆˆ
(,)Nt
rE
() ()
ˆˆ iicktt
ckk eee
V

 




 

kkr
kkkk
kk
() ()
2
ˆˆ
ˆ(,)
iicktt
ckk eee
V
t

 


 
 
kkr
kk kk
k
r
E
(2)
Thus ˆ
E proposes the opportunity to estimate
B. A. VEKLENKO
Copyright © 2010 SciRes. JMP
165

2
ˆˆ
N
Efrom below. The validity of obtained ine-
quality does not depend on particular quantum state on
which the averaging is performed and does nothing to do
with perturbation theory. But if such ineq uality is app lied
to each term of right hand site of
22
00 00
2
00 00
ˆˆ ˆˆ
() ()
ˆˆ
()
NfNf
fN f




 
EE
E
(3)
We find that
2
22
00000000
ˆˆ
()
ˆˆ
N
fff f

 

E
EE
The last formula can be rewritten in as
2
2
2
00000000
ˆˆ
()
ˆˆˆ
N
ff ff

 
 
E
EEE
(4)
That stresses the importance of the coherent scattering
channel when the scattered light is not classical and.
2
2
ˆˆ ˆ
() N

EE.
Inequality (4) allows to estimate 2
ˆˆ
() N
Ein the
semi-classical approximation. The value ˆ
EE
can be calculated using the conventional semi-classical
theory operating with non quantum electromagnetic field.
The calculation ()
00 00
ˆ
c
ff


EE can be per-
formed using only the coherent scattering channel. Even
in extensive media such procedure may be performed
with the help of wave functions [14]. Thus one can avoid
of matrix density formalism specific for non coherent
scattering chan nel .
2. Principal Equations
Let the electromagnetic field scatters on an atom situated
at a point with radius-vector Rand for the sake of sim-
plicity possesses only one orbital electron with coordi-
nate r. Let the atom possesses only two energy levels.
Zeeman`s sublevels with different magnetic numbers are
possible. Let the frequency of incident radiation
is
in a quasi resonance 00

 with th e atom
transition frequency 0
. Let Schroedinger equation for
atom and radiation is as follows
ˆˆ ˆ
aph
iHHH
t




,
where
2
ˆ
ˆˆˆ
()() ()
2
ap
H
Ud
m





r
rrRrr
ˆ
ˆˆˆ
ˆ
()() ()
e
H
pA d
mc


 r
rrrr
are the Hamiltonian of the non-interacting atoms and an
interaction Hamiltonian in Schroedinger representation.
Than
ˆ
ˆ()( )
j
j
j
b


rrR, ˆ
ˆ()( )
j
j
j
b



rrR,
ˆ
pi

rr
.
The following communitation relations are assumed
ˆˆ
;
j
jjj
bb



for the electron creation operator ˆ
j
b and annihilation
operator ˆ
j
b in the state described by wave function
j
.
The particular form of communication relations in our
case of one electron in the atom does not play any role.
By ()U
rRwe denote the potential energy of atom
electron. The Einstein summation rule is assumed over
all repeating indices
throughout the paper. The Ham-
iltonian of free electromagnetic field and vector-potential
operator are as follows
ˆˆˆ
ph
Hck
kk
k

ˆˆˆ
() 2
ii
c
Aeee
kV

 



kr kr
kk k
k
r
In order to realize the calculation project mentioned in
introduction we switch to the interaction representation
with the help of unitary operator
1
ˆˆˆ
() exp()
aph
UtHH t
i




.
In this picture
00
ˆ
() ,tStt
,

01
ˆˆˆ
,exp ()
t
Stt THtdt
i

 


(5)
ˆ
ˆˆˆ
ˆ
()()() ()
e
H
txpAxxd
mc


 rr,
,
x
tr,
ˆ
ˆ()() j
it
j
j
j
x
eb


rR
ˆ
ˆ()() j
it
j
j
j
x
eb



rR .
where
0
ˆ,Stt is the scattering operator,
j
is the
atomic energy in state
j
, ˆ
T is the time-ordering op-
erator and
B. A. VEKLENKO
Copyright © 2010 SciRes. JMP
166
() ()
ˆˆ ˆ
()()()
ˆˆ
22
i ikcti ikct
AxAx Ax
cc
ee ee
kV kV


 






kr kr
kk kk
kk

(6)
3. Coherent Scattering Channel
Suppose that the initial state of the field was described
by 00
(,)
k and was in quantum coherent state [13]
2
00
1
02
0
ˆ
()
0
!
n
n
fe n

k.
The amplitude of initial radiation was

00 00
00 0000
00
00 0000
ˆˆ
() ()
,
iikctiikct
AxfAxf
eae ae

 


 
kr kr
kk k
00 2
c
akV
k
.
We are interested in the radiation amplitude after scat-
tering in second order of perturbation technique. The
problem of Feynman’s diagrams summation will be dis-
cussed below. In Equation (5) it is sufficient to consider
the sum
(1) (2)(3)
ˆˆˆˆ
1SSSS  ,
where
(1) 1
ˆˆ()SHtdt
i


2
(2) 2
ˆ
ˆˆ()
2( )
T
SHtdt
i 
,

3
(3)
3
ˆ
ˆˆ()
3!()
T
SHtdt
i 
.
If the photon scatters in the coherent channel then the
atom rests in initial state. So in second order of perturba-
tion technique we are interested in the cons truction
() (2)
00
ˆˆ
() ()..
c
x
AxS cc

 A (7)
We change the time-ordering product of the atom op-
erators by the normal ordering one. For the scattering
operator (2)
ˆ
S we get
12
12
12
(2)
2
1112 2212
ˆ
ˆˆˆ
ˆˆ
ˆˆ
()() (, )()()
Ar
S
T
e
x
pAxGxxpAxxdxdx
mc i





 rr
where ˆA
Tis the time-ordering operator acting only on
the electromagnetic field operators and
12
ˆˆ
ˆˆ
()( )(,)
r
TNxx iGxx

,
12
12
()
12
(, )
()()2
r
Ett j
jj r
j
Gxx
EdE
eG



 

rRr R

1
0
j
rj
E
GEi

 

(8)
If the atom undergoes the action of external random
fields the finite width of its energy levels can be taken
into account by replacing the term 0iby /2
j
i
with the same sign because it is governed by the causal-
ity principle. The same result follows from summing up
(Dyson summation) the ladder Feynman`s diagrams for
excited atoms due to their interaction with electromag-
netic vacuum. For the same reason formula (8) can be
written as 1
2
j
r
j
j
E
G
Ei


 
.
without specifying the value
j
. We take into account
that,
12
1212
12
1212
ˆˆ
ˆ() ()
ˆˆ
ˆ
(, )()()
A
TAx Ax
iDxxNAx Ax

 
(9)
where 12 12
(, )Dxx

is not the operator function. The
first term in (9) does not play any role in electromagnetic
field scattering process. Finitely
12
12
21 12
2
(2) 112
() ()() ()
211 2212
1
ˆˆˆˆ
()(, )
ˆˆˆˆ ˆ
()()()() ()
r
e
SxpGxxp
imc
A
xAx AxAxxdxdx

 
 





rr
(10)
The right hand side terms of this equality are respon-
sible on scattering processes of electromagnetic field by
both the non excited atom and excited one.
3.1. Scattering on Non-Excited Atom
Substituting (6) and (8) into Equation (10) and taken the
limit t, we find
12 1012
00 11
022011 00
2
211
02
2
0 0
10 ()
(2) 00 00
00
ˆˆ
ˆˆ
2
i
jj
jj jjj
jjj
kc kc
ce
Sfppbbe eef
iVk mckc i
 







 
Rk k
k
kk
k
B. A. VEKLENKO
Copyright © 2010 SciRes. JMP
167
Through 0
j one denote here the quantum number of
initial state of atom. In dipole approximation
020ˆ
() ()
jj j
ppd


ρ
ρρρ.
The limit t is not necessary but it makes the
calculations simpler. According to (7) we need to calcu-
late the construction
00
()()(2)
00 00
ˆˆ
()() ..
c
x
fAxSf cc


A
Let us use the following equalities connecting any
smooth function ()
f
k and limits V, rR

11
11 11
11
()
101
()
i
ee kckcefk

krR
kk
k


1
11
1
1
() 10 11
3
0
00
2
()
(2)
sin
()
2
i
Vekckcfkd
rr
k
Vkfk nn
 






krR k
rR
rR
where
n
rR
rR
Here we take into account only the term describing
diverge wave. The neglected term turns into zero by infi-
nitely small interval of integration over 0
k that is sup-
posed. Finitely

1
100
0122
00 02 2000
22
2
()
00
11
() ..
4
2
ikik ct
i
cjj jjjj
nn
e
x
ae ppeecc
mc ck i
  





 
rR
kR
kk
rR 
A (11)
3.2. Scattering on Excited Atom The second term in Equation (10) after the same type of
transformation sho w n in part 3. 1 yi elds

2
200
012 1
00 02 2000
22
2
()
00
11
() ..
4
2
ikik ct
i
cjj jjjj
nn
e
x
ae ppeecc
mc ck i
  




 
rR
kR
kk
rR 
A (12)
If one takes into account the width of atom’s energy
level in state described by 0
j
than it is necessary to
replace 202
j
jj
 
 in Equation (12). The va-
lidity of Equations (11) and (12) are restricted by domain
/1
r
 where r
is the radiation width of excited
state of atom.
4. Non Coherent Scattering Channel
The second order perturbation technique gets
00
()(1) (1)
00 0
ˆˆ ˆ
() ()
nxfSAxSf


A.
Or by taken into account only the scattering processes
we have in explicit form
12
12
12
200
() ()
() ()
00 1200
2
1ˆˆ ˆ
ˆˆˆ ˆ
ˆˆ
() ..
()
ne
x
fpAdxApAdxfcc
mc
i



 





 
rr
A
Follo win g the procedures described in part 3 .1 we have

2
202 00
012 1
00 02 2000
2
2
() 0
() ..
2
jj ikik ct
i
njj jj
j
nn
ie
x
ae ppkceecc
mc
  







 
rR
kR
kk
rR
A (13)
If we take into account the finite width of atom energy level than in formula (13) it is necessary to change
02
02 02
0
00
11
222
jj
jj jj
kc ikc ikc i

 



 







B. A. VEKLENKO
Copyright © 2010 SciRes. JMP
168
Let us find now the total amplitude of electromagnetic field scattered by excited atom

2
200
012 1
00 02 2000
2
2
() ()
00
11
()() ()..
42
ikik ct
i
cn jj jj
j
nn
e
x
xxaeppee cc
mc ck i
 
 




 
rR
kR
kk
rR 
AA A
(14)
5. Semi-Classical Theory of Radiation
The set equations for field operators ()
x
and ()
A
x
in Heisenberg representation is the following
2
()
ˆˆ
( )()()
2
x
it
pe
UAxpx
mmc





r
r
rR
,
1
()()
A
xjx
c
 , ˆˆ
() 2
e
jxp p
m

 
 






rr .
This set equations is equivalent to the one mentioned
at part 3. Now
1
1
1
011 11
ˆ
()()(,)()( )
r
e
A
xAxxxxp xdx
mc





r.(15)
Here 0()
A
x
is given by the formula (6) and

11
1
1
0
0
111
11
1
1
(,)();()()
()
4
r
x
xAxAxtt
i
nn
cct t
 






rr
rr
,1
rr .
(16)
We are interested in the second order perturbation ex-
pansion. This mean that the ()
x
operator has be
evaluated in the first order of perturbation technique
11
1
01111
ˆ
()()(,)()()
r
e
x
xGxxpAxxdx
mc
 


r. (17)
Substituting (16) and (17) into (15) we find
12
12
12
2
011 122212
ˆ
ˆˆ
ˆˆ
()()(,)()(,)()( )..
rr
e
A
xAxxxxpGxxpAx xdxdxHc
mc

 


 

 


rr (18)
For the mean values, the same result can be obtained
either by averaging (18) with subsequent breaking up the correlators, or using the semi-classical theory. After re-
alizing in (18) the substitution

20202 0202
22 2
00 00
00
0
22020
() ()()iikct iikct
AxxAxeaeae
 



kr kr
kk
k
A
for the scattered fi eld we ha ve
1
100
012 2
00 02 2000
202
2
0
11
() ..
40
ikik ct
ijj jj
jjj
nn
e
x
ae ppeecc
mcck i
  




 
 rR
kR
kk
rR
A (19)
The upper sign describes the scattering of electroma-
gnetic field on the non-excited atom while the low one
describes the scattering on excited atom. One should take
into account the width of atomic energy level by replac-
ing in dominator 0i with 02
/2 ()/2
jj
ii

 .
By comparing Equation (19) with Equations (11) and
(14) we find that in the approximation we used both the
quantum theory and the semi-classical theory result in
the same expressions for the scattered amplitude ()
A.
Namely the necessary coinciding in such results leads to
the equality of constants
in Formulas (19), (11) and
(14).
6. Bilinear Field Charasteristics
In this part we are interesting in the following construc-
tion shown in introduction
ˆˆ ˆ
()()Nx x

EE .
In order to calculate this value in forth order of per-
turbation expansion it should use the Formula (3). But it
is not worth to do it. The strait calculation shows that for
resonant field scattering 0
()
the construction
(1) (3)
00
ˆˆˆˆ
ˆ() ()SNAxAxS

,
B. A. VEKLENKO
Copyright © 2010 SciRes. JMP
169
which appears in such approximation at non-coherent
channel results in negative value. This fact evidently
contradicts with the positive definition of expression
2
00 00
ˆˆ
() 0fN f

 E
Such contradiction was found before in Reference [25]
where different model has been considered. In order to
reconstruct the positive definition of the non-coherent
channel using perturbation set it is necessary to average
the product ˆˆ
()()
x
x

EE over the wave function

(1)(2)(3)0
ˆˆˆ
1SSS 
. But doing this we find the
terms proportional to the sixth order of charge. It means
that such reconstruction may be achieved only by using
higher order terms of perturbation technique. Thus one
can not restrict oneself here by the terms of lover order
of perturbation technique. So the conventional perturba-
tion theory for ˆˆ ˆ
() ()Nxx

EE is problematic.
For these reasons we estimate the contribution of
non-coherent processes using inequality (2)
2
00 00
ˆˆ ˆ
fN f


EE
2
00 00
ˆ
ff

 E (20)
Then we use the same method to estimate the contri-
bution of coherent channel. Thus according to the quan-
tum theory using Equation (12), Equation (13) and Equa-
tion (20) one gets for the scattering by excited atom in
two level approximation the following formula;


2
2112
00 0002 20
2
2
22
22
22
00 00
11
ˆˆ ˆ21
4
44
jj jj
qu j
nn
e
Naepp
mc ck ck
 




 

 
 


kk
rR  
EE
While according to the semi classical theory one gets


2
2112
00 0002 20
2
2
2
2
2
00
11
ˆˆ ˆ24
4
jj jj
scl j
nn
e
Naepp
mc ck
 



 
kk
rR 
EE
The ratio of results of these two calculation methods
for the resonant scattering frequency 0
is equal
to
ˆˆ ˆ
5
ˆˆ ˆ
qu
scl
N
N


EE
EE
.
The same value characterizes the ratio of scattering
cross sections /
qu scl
. This result does not depend on
. We note that for the scattering of electromagnetic
field on non- excited atoms this ratio is equal to one. The
dependence of ratio /
qu scl
for scattering on excited
atom as a function of scattering frequency by 0
is
shown in the Figure 1.
7. Conclusions
The evaluations the scattered field amplitude of reso-
nance scattering electromagnetic field on an excited atom
can be performed equally well using both the Heisenberg
representation and Schroedinger one. In our approxima-
tion the both calculations lead to the same results. The
same results follow also from the semi-classical theory
of radiation, which deals with classical electromagnetic
field. In general, the perturbation technique is not suffi-
cient to describe the resonance scattering process and we
need to sum up the ladder Feynman diagrams. Such pro-
cedure is not difficu lt to be performed using an y of theo-
ries mentioned above.
In the other case we deal with calculation of the quan-
tum mean values of bilinear products of the field opera-
tors ˆˆ
EE . Here it is more convenient to deal with Sch-
0 0.511.52
1
2
3
4
5
Figure 1. The typical dependence of ratio /
qu scl
for
scattering of electromagnetic field on excited atom as a
function of scattering frequency 00
/ck
.
00
/ck
/
qu scl
B. A. VEKLENKO
Copyright © 2010 SciRes. JMP
170
rödinger representation or with interaction representation,
which give additional opportunities to sum up the Feyn-
man diagrams. The letter representations allow us to
present the scattering process with the help of two com-
ponents: coherent (elastic) and non-coherent. Such com-
ponents could be evaluated independently. The analysis
of non-coherent channel shows that the Dyson’s summa-
tion of ladder Fynman’s diagrams by scattering of reso-
nant electromagnetic field on excited atoms is not suffi-
cient. Other summation methods are very unwieldy [8].
In present work we propose the simple method of esti-
mation from below the results of the non-coherent scat-
tering channel. As a result we find that the semi-classical
theory of radiation essentially underestimates the cross
section of resonance scattering. The quantum theory in
its turn shows the violation of equality ˆˆ ˆ
N

EE
ˆˆ
EE
in scattered radiation even if such equality
took place in the incident electromagnetic field. So the
quantum theory results in a change of quantum statistical
structure of electromagnetic field due to scattering. This
can not be obtained with the help of semi-classical theory
of radiation. This change of internal quantum field stru-
cture by its scattering on excited atom manifests itself on
macroscopic level. Namely such effect makes impossible
using here the semi-classical theory of ra diation.
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