# Single Photon Ignition of Two-photon Super-fluorescence through the Vacuum
of Electromagnetic Field

Nicolae A. Enaki
Academiei str.5, Institute of Applied Physics of Academy of sciences of
Moldova,
Chisinau, MD 2028, Republic of Moldova

(December 8, 2010 )

###### Abstract

The ignition of two-quantum collective emission of inverted sub-ensemble of
radiators due to mutual interaction of this sub-ensemble with other two dipole
active atomic subsystems in process of two-photon exchanges between the atoms
through the vacuum field is proposed. The three particle resonances between
two-photon and single quantum transitions of inverted radiators from the
ensemble are proposed for acceleration of collective decay rate of bi-photons,
obtained relatively dipole-forbidden transitions of excited atomic sub-
ensemble. This mutual interaction between three super-fluorescent processes in
subatomic ensembles take place relatively dipole-forbidden transitions in one
of radiator subsystem. The collective resonance emission and absorption of
two-quanta have nontraditional behavior, accompanied with acceleration and
inhibition of collective emission processes of photons.

##  1 Introduction

A great deal of attention is currently devoted to the problem of coherence
which appears not only between the quanta but between groups of quanta too.
The generation of non-classical coherent electromagnetic field in multi-photon
emission and the interaction of coherent radiation with matter (nuclei, atoms
and solids) have been subjects of a number of theoretical and experimental
studies in recent years [1]-[3]. Examples include the higher-order coherence
in multi-photon generation of light the two-photon micro-maser emission [3],
two-photon lasers the parametric down conversion, four-wave mixing and other
effects in optical diapason [2], and the possibility of coherent generation of
photons in $x$-ray and $gamma$\- ray spectral regions.

In this article it is proposed to investigate the cooperative two-photon
emission from inverted system of radiators stimulated by single photon super-
fluorescent pulses in two-quantum resonance with dipole forbidden atomic
transition. Since the two-photon cooperative phenomenon has the small two-
photon cooperative emission time [9], we propose to extend our attention to
the new tape of cooperative resonance interaction between three radiators in
which single photon transitions of two radiators which enter in two-photon
resonance with dipole forbidden transition of third atom. This cooperative
three particle interaction take place through the vacuum fluctuations of
electromagnetic field and can amplify or diminish the spontaneous emission
rates of the atoms. In order to obtain more powerful pulses of entangled
photons it is proposed the cooperative interaction between three atomic
subsystems in which one of them are inverted relatively dipole forbidden
transition $|2S>$ $-$ $|S>$ of Hydrogen like or Helium Like atoms [4]-[8].
Taking in to account the elementary acts of two photon interaction between
radiators we archived the improvement of two-photon emission rate of the
system of radiators in comparison with two-photon super-fluorescence [9]. In
this article it is examined the mutual influence of two single-photon super-
fluorescence processes and two-photon cooperative emission of the atomic
system relatively dipole forbidden transition. The phenomenon of new
cooperative emission takes in to account the three particle mutual interaction
with vacuum of electromagnetic field in which the product of vacuum
polarization of two atoms enter in to resonance with two-photon polarization
of dipole forbidden transition of Hydrogen-like or Helium-like radiator. It
has been shown that in the process of spontaneous radiation, the radiators
(nuclei, atoms) enter a regime of single and two-photon super-radiance and the
rate of photon pair (bi-photon) emission increases (or decreases) due to new
three particle cooperative phenomenon, which appear between single and two-
photon spontaneous emission subgroups of radiators. It has been demonstrated,
that for hydrogen-like and helium-like atoms [10] the dipole-forbidden
transitions can generate more powerful pulse of entangled photon pairs (bi-
photons) under the influence of single photon super-radiance.

It is important to note that for coherent radiation of such system, was
studied for the dimension of a radiating system smaller than the radiation
wavelength. It is, however, interesting to study this type of cooperative
emission between three radiator subsystems in extended system of radiators.
The possibilities of two-photon cooperative resonances between three radiators
replaced at distance larger than emission wavelength are studied too. I
emphasize here, that the problem of cooperation between two single photon
cooperative emission subsystems and one two-photon cooperative emission
subsystem is more complicated than the similar problem of single [11] or two-
photon [9] super-radiances in extended system. In Dicke’s super-radiance, the
exchange integral between $j$-th and $l$-th atoms is described by more simple
exchange integral proportional to $\sin[k_{0}r_{jl}]/(k_{0}r_{jl})$ while in
two-photon super-radiance by more complicated function
$\sin[(2k_{0}-k)r_{jl}]/[(2k_{0}-k)r_{jl}]\times\sin[kr_{jl}]/(kr_{jl})$,
where $r_{jl}$ and $k_{i}\ $are the distance between the radiators and wave
vector of emitted photons respectively. The three particle exchange integral
between $j$-th, $m$-th and $l$-th atoms was obtained in this paper taking in
to account the two-quantum exchanges between two radiators proposed in papers
[9], [12]. The more complicated exchange integrals between two radiators with
dipole active transition and one radiator with dipole forbidden transition is
given in Appendix of this paper.

##  2 Interaction Hamiltonian and Master Equation

Let us consider the interaction of three subsystems of radiators $R$, $S$, and
$D$ thorough vacuum of electromagnetic field. The first two groups, $R$ and
$S$, are prepared in excited state $|e_{r}\rangle\otimes|e_{s}\rangle$ and can
pass in to Decke super-radiance regime [11] relatively the dipole active
transitions $e_{r}\rightarrow g_{r}$ and $e_{s}\rightarrow g_{s}$ at
frequencies $\omega_{r}$ and $\omega_{s}$ (see figure 1). The $D$ atomic
subsystem is prepared in excited state $|e_{d}\rangle$ and relatively dipole
forbidden transition $e_{d}\rightarrow g_{d}$ and can pass in the ground state
$|g_{d}\rangle$ simultaneously generation two quanta [9]. Let us consider the
simple cooperative stimulation of two-photon emission of $D$ system stimulated
by $R$ and $S$ radiator subsystems.

Figure 1: The resonance between two-photon transitions of D atomic subsystem
and two dipole active atomic subsystems$R$ and $S$ . As an example is
represented three atoms $D$, $R$ and $S$ situated at relatively distances
$r_{ds}$, $r_{dr}$ and $r_{rs}$. One of condition of exchange energies between
the subsystems is the e resonance between two-photon and single photon
transitions $2\omega_{0}=\omega_{r}+\omega_{s}$ .

In this case it is established the resonance between the dipole active atomic
subgroups $R$, $S$ and dipole forbidden radiators of $D$ ensemble and the
cooperative stimulation of two-quantum collective transition is possible.
Indeed, considering that the conservation energy law is established between
these groups, $\hbar(\omega_{r}+\omega_{s})=2\hbar\omega_{0}$, one can
proposed the following Hamiltonian of interaction of radiators with
electromagnetic field

$\displaystyle H$ $\displaystyle=$ $\displaystyle H_{0}+\lambda H_{I};$
$\displaystyle H_{0}$ $\displaystyle=$
$\displaystyle\sum\limits_{k}\hbar\omega_{k}a_{k}^{\dagger}a_{k}+\sum\limits_{j=1}^{N_{r}}\hbar\omega_{r}R_{zj}+\sum\limits_{l=1}^{N_{s}}\hbar\omega_{s}S_{zl}+2\sum\limits_{m=1}^{N}\hbar\omega_{0}D_{zm};$
$\displaystyle\lambda H_{I}$ $\displaystyle=$
$\displaystyle-\sum\limits_{k}\sum\limits_{j=1}^{N_{a}}(\mathbf{d}_{r},\mathbf{g}_{k})\\{R_{j}^{+}a_{k}\exp[i(\mathbf{k},\mathbf{r}_{j})]+R_{j}^{-}a_{k}^{\dagger}\exp[-i(\mathbf{k},\mathbf{r}_{j})]\\}$
(1) $\displaystyle-$
$\displaystyle\sum\limits_{k}\sum\limits_{l=1}^{N_{b}}(\mathbf{d}_{s},\mathbf{g}_{k})\\{S_{l}^{+}a_{k}\exp[i(\mathbf{k},\mathbf{r}_{l})]+S_{l}^{-}a_{k}^{\dagger}\exp[-i(\mathbf{k},\mathbf{r}_{l})]\\}$
$\displaystyle-$
$\displaystyle\sum\limits_{k_{1},k_{2}}\sum\limits_{m=1}^{N_{b}}(\mathbf{n}_{eg},\mathbf{e}_{\lambda_{1}})(\mathbf{n}_{ei},\mathbf{e}_{\lambda_{2}})q(\omega_{1},\omega_{2})$
$\displaystyle\times$
$\displaystyle\\{D_{m}^{+}a_{k_{2}}a_{k_{1}}\exp[i(\mathbf{k}_{1}+\mathbf{k}_{2},\mathbf{r}_{m})]$
$\displaystyle+$ $\displaystyle
D_{m}^{-}a_{k_{1}}^{\dagger}a_{k_{2}}^{\dagger}\exp[-i(\mathbf{k}_{1}+\mathbf{k}_{2},\mathbf{r}_{l})]\\}.$

Here

$q(\omega_{1},\omega_{2})=\frac{d_{23}d_{31}g_{k_{2}}g_{k_{1}}}{2\hbar}\left\\{\frac{1}{\omega_{32}+\omega_{k_{1}}}+\frac{1}{\omega_{31}-\omega_{k_{2}}}\right\\},\
\ \
\mathbf{g}_{k}=\sqrt{\frac{2\pi\hbar\omega_{k}}{V}}\mathbf{\epsilon}_{{}_{\lambda}},$

$a_{k}$ and $a_{k}^{\dagger}$ are annihilation and creation operators of EMF
photons with wave vector $\mathbf{k,}$ polarization  $\mathbf{\
\epsilon}_{{}_{\lambda}}$ and frequency $\omega_{k}$; $\mathbf{d}_{r}$ and
$\mathbf{d}_{s}$ are dipole momentum transition between the ground and excited
states for $R$ and $S$ atomic subsystems; $d_{ei}$ and $d_{eg}$ are dipole
momentum transitions in the three level system of atomic group $D$. The
operators of $R$, $S$, and $D$ atomic subsystems satisfy the commutation
relations for $SU(2)$ algebra $[J^{+},J^{-}]=2J_{z}$; $[J_{z}\ ,J^{\pm}]=\pm
J^{\pm}$, where $J^{\pm}$ is equivalent with $R^{\pm}$, $S^{\pm}$ and
$D^{\pm}$. Invertin operator $J_{z}$ is consider similar to $R_{z},$ $S_{z}$
and $D_{z}$ respectively. The operators of electromagnetic field satisfy the
commutation relation $[a_{k},a_{k^{\prime}}^{\dagger}]=\delta_{k,k^{\prime}}\
;[a_{k}^{\dagger},a_{k^{\prime}}^{\dagger}]=0$, where $k=(\mathbf{k,}\lambda)$
is the wave vector and polarization of the photon. Taking in to account the
Hamiltonian 1, let us represent the solution of Haisenberg equation through
the sources and free part operators

$a_{k}(t)=a_{k}(0)\exp[-i\omega_{k}t]+a_{ks}(t),$ (2)

where the source part is

$\displaystyle a_{ks}(t)$ $\displaystyle=$
$\displaystyle\frac{i(\mathbf{d}_{a},\mathbf{g}_{k})}{\hbar}\sum\limits_{l=1}^{N_{a}}\exp[-i(\mathbf{k},\mathbf{r}_{l})\int\limits_{0}^{t}d\tau\exp[-i\omega_{k}\tau]R_{l}^{-}(t-\tau)$
$\displaystyle+i\frac{(\mathbf{d}_{b},\mathbf{g}_{k})}{\hbar}\sum\limits_{j=1}^{N_{b}}\exp[-i(\mathbf{k},\mathbf{r}_{j})\int\limits_{0}^{t}d\tau\exp[-i\omega_{k}\tau]S_{j}^{-}(t-\tau)$
$\displaystyle+2i\sum\limits_{n=1}^{N_{b}}\sum\limits_{k_{1}}\frac{(\mathbf{n}_{eg},\mathbf{e}_{\lambda_{1}})(\mathbf{n}_{ei},\mathbf{e}_{\lambda})q(\omega_{1},\omega)}{\hbar}\exp[-i(\mathbf{k}_{1}+\mathbf{k},\mathbf{r}_{n})$
$\displaystyle\times$
$\displaystyle\int\limits_{0}^{t}d\tau\exp[-i\omega_{k}\tau]D_{n}^{-}(t-\tau)a_{k_{1}}^{\dagger}(t-\tau);\
\ \ [a_{ks}^{\dagger}(t)=[a_{s}(t)]^{+}.$

Taking in to account that $a_{k}(0)|0\rangle_{ph}=\langle
0|_{ph}a_{k}^{\dagger}(0)=0$ we can partially eliminate the EMF field
operators from the mean value of Hesenberg equation for arbitrary atomic
operator $O(t)$

$\displaystyle\frac{d}{dt}\langle
O(t)\rangle=-i\sum\limits_{k}\sum\limits_{j=1}^{N_{a}}\frac{(\mathbf{d}_{a},\mathbf{g}_{k})}{\hbar}\langle[R_{j}^{+}(t),O(t)]a_{ks}(t)\rangle\exp[i(\mathbf{k},\mathbf{r}_{j})]$
$\displaystyle-i\sum\limits_{k}\sum\limits_{l=1}^{N_{b}}\frac{(\mathbf{d}_{b},\mathbf{g}_{k})}{\hbar}\langle[S_{l}^{+}(t),O(t)]a_{ks}(t)\rangle\exp[i(\mathbf{k},\mathbf{r}_{l})]$
$\displaystyle-i\sum\limits_{k_{1},k}\sum\limits_{m=1}^{N_{b}}\frac{(\mathbf{n}_{eg},\mathbf{e}_{\lambda_{1}}(k_{1}))(\mathbf{n}_{ei},\mathbf{e}_{\lambda}(k))q(\omega_{k_{1}},\omega_{k})}{\hbar}$
$\displaystyle\times\langle[D_{m}^{+}(t),O(t)]a_{k_{1}}(t)a_{ks}(t)\rangle\exp[i(\mathbf{k}_{1}+\mathbf{k},\mathbf{r}_{m})]+H.C.(O^{+}\rightarrow
O).$ (3)

Here the mean values of Hesenberg operators are considered taking into account
the initial state of the system $|\Psi_{r}(0)\rangle\otimes|0\rangle_{ph},$.
where $|\Psi_{r}(0)\rangle$ is the state of radiator subsystem, and
$|0\rangle_{ph}$ is the vacuum state of EMF. We are interested in the total
elimination of operators of electromagnetic field from the expression (3) For
elimination of operators of electromagnetic field we formulate the lemma

###### Lemma 1

If Bose $a_{k}(t)$ and $a_{k}^{+}(t)$ operators lie between the two operators
of the atomic subsystem $A(t_{1})$ and $B(t_{2})$ ($A(t_{1})$ , $B(t_{2})$
don’t contain the operators $a_{k}$ and $a_{k}$ ) belonging to other times,
the elimination of the free part of these operators yields the following
expression for the correlation:

$\displaystyle\left\langle A(t_{1})a_{k}(t)B(t_{2})\right\rangle$
$\displaystyle=$ $\displaystyle\left\langle
A(t_{1})a_{ks}(t)B(t_{2})\right\rangle$ $\displaystyle-$ $\displaystyle
e^{-i\omega_{3}(t-t_{2})}\left\langle
A(t_{1})[a_{ks}(t_{2}),B(t_{2})]\right\rangle,$ $\displaystyle\left\langle
A(t_{1})a_{k}^{+}(t)B(t_{2})\right\rangle$ $\displaystyle=$
$\displaystyle\left\langle A(t_{1})a_{ks}^{+}(t)(t)B(t_{2})\right\rangle$ (4)
$\displaystyle-$ $\displaystyle
e^{i\omega_{3}(t-t_{1})}\left\langle[A(t_{1}),a_{ks}^{+}(t_{1})]B(t_{2})\right\rangle.$

Proof. The commutations in (4) play the highest role in the two-photon
spontaneous emission and only such commutations bring the main contribution to
the two-photon process. The problem is reduced to the elimination of vacuum
part lies between the operators $A(t_{1})$ and $B(t_{2})$

$<A(t_{1})a_{k}(t)B(t_{2})>=<A(t_{1})(a_{k}^{v}(t)+a_{ks}(t))B(t_{2})>$ (5)

Since $a_{k}(t)=a_{k}^{v}(t)+a_{ks}(t),\;$we will represent the vacuum part
$a_{k}^{v}(t)=a_{k}(0)\exp[-i\omega_{k}t]$ through the vacuum-operator at time
$t_{1\;\ }$and taking in to account the identity (2) we can represent the
vacuum part in the following form
$a_{k}^{v}(t)=a_{k}^{v}(t_{2})e^{-i\omega_{k}(t-t_{2})}=\\{a_{k}(t_{2})-a_{ks}(t_{2})\\}e^{-i\omega_{k}(t-t_{2})}.$
After substitution of $\ a_{k}^{v}(t)\;$into the correlation it is obtain

$\displaystyle\langle A(t_{1})a_{k}(t)B(t_{2})\rangle$ $\displaystyle=$
$\displaystyle\langle A(t_{1})a_{ks}(t)B(t_{2})\rangle$ $\displaystyle+$
$\displaystyle e^{-i\omega_{k}(t-t_{2})}\langle
A(t_{1})\\{a_{k}(t_{2})-a_{ks}(t_{2})\\}B(t_{2})\rangle,$

We observe that $a_{k}(t_{2})\;$commutes with the operator $B(t_{2})$.
Consequently taking into account that

$a_{k}(t_{2})\ B(t_{2})|0>=\ B(t_{2})a_{ks}(t_{2})|0>,\;$it is easily obtain
that

$\left\langle A(t_{1})\\{a_{k}(t_{2})-a_{ks}(t_{2})\\}\
B(t_{2})\right\rangle=-\left\langle
B(t_{2})[a_{ks}(t_{2}),B(t_{2})]\right\rangle$.

This relation proofs the Lemma.

This lemma (4) can be used in the last term of generalized equation (3) for
correlation functions $\
\langle[D_{j}^{+}(t),O(t)]a_{k}(t)S_{n}^{-}(t-\tau)\rangle$ , $\
\langle[D_{j}^{+}(t),O(t)]a_{k}(t)R_{l}^{-}(t-\tau)\rangle\ $ and
$\langle[R_{j}^{+}(t),O(t)]D_{n}^{-}(t-\tau)a_{k_{1}}^{\dagger}(t-\tau)\rangle,$
$\langle[S_{l}^{+}(t),O(t)]D_{n}^{-}(t-\tau)a_{k_{1}}^{\dagger}(t-\tau)\rangle$.
Indeed taking in to account the lemma (4) the above correlation functions can
be represented through atomic operators

$\displaystyle\langle[D_{j}^{+}(t),O(t)]a_{k}(t)S_{n}^{-}(t-\tau)\rangle$
$\displaystyle=$
$\displaystyle\left\langle[D_{j}^{+}(t),O(t)]a_{ks}(t)S_{n}^{-}(t-\tau)\right\rangle$
$\displaystyle-$ $\displaystyle
e^{-i\omega_{k}\tau}\left\langle[D_{j}^{+}(t),O(t)][a_{ks}(t-\tau),S_{n}^{-}(t-\tau)]\right\rangle,$
(6)
$\displaystyle\langle[R_{j}^{+}(t),O(t)]a_{k}^{\dagger}(t-\tau)D_{n}^{-}(t-\tau)\rangle$
$\displaystyle=$
$\displaystyle\langle[R_{j}^{+}(t),O(t)]a_{ks}^{\dagger}(t-\tau)D_{n}^{-}(t-\tau)\rangle$
$\displaystyle-$ $\displaystyle
e^{-i\omega_{k}\tau}\langle[[R_{j}^{+}(t),O(t)],a_{ks}^{\dagger}(t)]D_{n}^{-}(t-\tau)\rangle.$
(7)

The interaction between the atomic subsystems can be found in the third order
of interaction constants with the subsystems $S$, $R$ and $D$ respectively
$(\mathbf{d}_{r},\mathbf{g}_{k})(\mathbf{d}_{s},\mathbf{g}_{k})q(\omega_{1},\omega_{2})$
According with this condition the smooth correlation functions is obtained
only for the following terms of expressions (6) and (7) : $\left\langle
R_{l}^{-}(t-\tau^{\prime})[D_{j}^{+}(t),O(t)]S_{n}^{-}(t-\tau)\right\rangle$
and $\langle
R_{l}^{+}(t-\tau^{\prime})[R_{j}^{+}(t),O(t)]D_{n}^{-}(t-\tau)\rangle$. The
contribution of other terms of the expressions (6) and (7) give the
contribution more hair order on the decomposition on the small parameter
$\lambda$ of the interaction Hamiltonian (1).

The lemma (4) is non-applicable for correlation functions in which it is meet
simultaneously the creation and annihilations Boson operators belonging to
different time intervals:
$\langle[D_{j}^{+}(t),O(t)]a_{k}(t)D_{n}^{-}(t-\tau)a_{k_{1}}^{\dagger}(t-\tau)\rangle$
and its hermit conjugate part $\langle
a_{k_{1}}(t-\tau)D_{n}^{+}(t-\tau)a_{k}^{\dagger}(t)[O(t),D_{j}^{-}(t)]\rangle$.
In order to eliminate the vacuum part of operators $a_{k}(t)$ and
$a_{k}^{\dagger}(t^{\prime})$ let us formulate the following rule.

###### Lemma 2

If the operators $A(t_{1})$ contains the creation operators of EMF and
$B(t_{2})$ contains the annihilation operators of EMF the elimination of
vacuum part of annihilation $a_{k}(t)$ or creation $a_{k}^{{\dagger}}(t)$
operators situated between these operators $A(t_{1})$ and $B(t_{2})$ takes
place according with Lemma 1. In opposite case, when operator $A(t_{1})$ can
be represented through the product of atomic operator $\mathcal{A}(t_{1})$ and
annihilation field operators
$A(t_{1})$=$\mathcal{A}(t_{1})a_{k_{1}}(t_{1})a_{k_{2}}(t_{1})...a_{k_{n}}(t_{1})$
. the operator $B(t_{2})$ is represented through the product of creation field
operators and atomic operator $\mathcal{B}(t_{2})$ so that
B(t2)=$\mathcal{B}(t_{2})a_{k_{1}}^{{\dagger}}(t_{2})a_{k_{2}}^{{\dagger}}(t_{2})...a_{k_{m}}^{{\dagger}}(t_{2})$
the elimination of vacuum part of the operators $a_{k}(t)$ and
$a_{k}^{{\dagger}}(t)$ can be represented in the following form

$\displaystyle\left\langle A(t_{1})a_{k}(t)B(t_{2})\right\rangle=\left\langle
A(t_{1})a_{ks}(t)B(t_{2})\right\rangle$
$\displaystyle-\exp[-i\omega_{k}(t-t_{2})]\\{\left\langle
A(t_{1})[a_{ks}(t_{2}),B(t_{2})]\right\rangle$
$\displaystyle-\delta_{k,k_{1}}\left\langle
A(t_{1})\mathcal{B}(t_{2})a_{k_{2}}^{{\dagger}}(t_{2})...a_{k_{m}}^{{\dagger}}(t_{2})\right\rangle$
$\displaystyle-...-\delta_{k,k_{m}}\left\langle
A(t_{1})\mathcal{B}(t_{2})a_{k_{1}}^{{\dagger}}(t_{2})a_{k_{2}}^{{\dagger}}(t_{2})...a_{k_{m-1}}^{{\dagger}}(t_{2})\right\rangle\\},$
(8) $\displaystyle\left\langle
A(t_{1})a_{k}^{{\dagger}}(t)B(t_{2})\right\rangle=\left\langle
A(t_{1})a_{ks}^{{\dagger}}(t)B(t_{2})\right\rangle$
$\displaystyle-\exp[i\omega_{k}(t-t_{2})]\\{\left\langle[A(t_{1})a_{ks}^{{\dagger}}(t_{1})]B(t_{2})\right\rangle$
$\displaystyle-\delta_{k,k_{1}}\left\langle\mathcal{A}(t_{1})a_{k_{2}}(t_{1})...a_{k_{n}}(t_{1})B(t_{2})\right\rangle$
$\displaystyle-...-\delta_{k,k_{n}}\left\langle\mathcal{A}(t_{1})a_{k_{1}}(t_{1})a_{k_{2}}(t_{1})...a_{k_{n-1}}(t_{1})B(t_{2})\right\rangle\\}.$
(9)

Proof. Taking in to account the lemma (4), we can represent the third
correlation $\left\langle A(t_{1})a_{k}(t)B(t_{2})\right\rangle$ of expression
(8) in the following form

$\displaystyle\left\langle A(t_{1})a_{k}(t)B(t_{2})\right\rangle$
$\displaystyle=$ $\displaystyle\left\langle
A(t_{1})a_{ks}(t)B(t_{2})\right\rangle$ (10) $\displaystyle+$
$\displaystyle\exp[-i\omega_{k}(t-t_{2})]\left\langle
A(t_{1})(a_{k}(t_{2})-a_{ks}(t_{2}))B(t_{2})]\right\rangle$ $\displaystyle-$
$\displaystyle\exp[-i\omega_{k}(t-t_{2})]\\{\left\langle
A(t_{1})[a_{ks}(t_{2}),B(t_{2})]\right\rangle.$

According with explicit expression of operator,
$B(t_{2})=\mathcal{B}(t_{2})a_{k_{1}}^{{\dagger}}(t_{2})a_{k_{2}}^{{\dagger}}(t_{2})...a_{k_{m}}^{{\dagger}}(t_{2})$,
let us introduced it in the third term of right hand site of expression (10).
Following the commutation roles of boson operators of electromagnetic field,
the operator $a_{k}(t_{2})$ can be permuted in the right hand site of the
correlation. Taking in to consideration that
$(a_{ks}(t_{2})+a_{kv}(t_{2}))\left|0\right\rangle=a_{ks}(t_{2})\left|0\right\rangle,$
this term becomes

$\displaystyle\left\langle
A(t_{1})a_{k}(t_{2})\mathcal{B}(t_{2})a_{k_{1}}^{{\dagger}}(t_{2})a_{k_{2}}^{{\dagger}}(t_{2})...a_{k_{m}}^{{\dagger}}(t_{2})\right\rangle=$
$\displaystyle\delta_{k,k_{1}}\left\langle
A(t_{1})\mathcal{B}(t_{2})a_{k_{2}}^{{\dagger}}(t_{2})...a_{k_{m}}^{{\dagger}}(t_{2})\right\rangle$
$\displaystyle+...+\delta_{k,k_{m}}\left\langle
A(t_{1})\mathcal{B}(t_{2})a_{k_{1}}^{{\dagger}}(t_{2})a_{k_{2}}^{{\dagger}}(t_{2})...a_{k_{m-1}}^{{\dagger}}(t_{2})\right\rangle$
$\displaystyle+\left\langle
A(t_{1})\mathcal{B}(t_{2})a_{k_{1}}^{{\dagger}}(t_{2})a_{k_{2}}^{{\dagger}}(t_{2})...a_{k_{m}}^{{\dagger}}(t_{2})a_{ks}(t_{2})\right\rangle.$
(11)

Introducing this relation in (11) it is not difficult to observe that the new
expression for correlation $\left\langle
A(t_{1})a_{k}(t)B(t_{2})\right\rangle$ coincides with (8). The similar
procedure of permutation of vacuum part of creation operator
$a_{k}^{{\dagger}}(t)$ demonstrates the identity (9) of Lemma 2.

According with lemma (8) we obtain the following expression for correlation
function

$\displaystyle\langle[D_{j}^{+}(t),O(t)]a_{k}(t)D_{n}^{-}(t-\tau)a_{k_{1}}^{\dagger}(t-\tau)\rangle=\langle[D_{j}^{+}(t),O(t)]a_{ks}(t)D_{n}^{-}(t-\tau)a_{k_{1}}^{\dagger}(t-\tau)\rangle$
$\displaystyle-\exp[-i\omega_{k}\tau]\\{\langle[D_{j}^{+}(t),O(t)][a_{ks}(t-\tau),D_{n}^{-}(t-\tau)a_{k_{1}}^{\dagger}(t-\tau)]\rangle-\delta_{k,k_{1}}\langle[D_{j}^{+}(t),O(t)]D_{n}^{-}(t-\tau)\rangle\\}.$
(12)

The next step of elimination of operator $a_{k_{1}}^{\dagger}(t-\tau)$ from
this expression must be do taking in-to account the lemma (4). When the first
and second order interaction constants have the same small magnitude
$\lambda\sim(\mathbf{d}_{s},\mathbf{g}_{k})\eqsim
q(\omega_{k_{1}},\omega_{k})$, in Born approximation we take in to account
only the last term of expression (12). the interference contribution of which
is proportional to $\lambda^{3}$. As follows from the representation (2) and
(3) the procedure of elimination mast continue. Indeed introducing again this
equation in the right hand cite of equation (14) we obtain the following
master equation for arbitrary operator $O(t)$ in thread approximation on the
interaction constant $\lambda$

$\displaystyle\frac{d\langle O(t)\rangle}{dt}$ $\displaystyle=$
$\displaystyle\sum\limits_{k}\sum\limits_{l,j=1}^{N_{r}}\frac{(\mathbf{d}_{a},\mathbf{g}_{k})^{2}}{\hbar^{2}}\int\limits_{0}^{t}d\tau\exp[-i\omega_{k}\tau+i(\mathbf{k,r}_{j}-\mathbf{r}_{l})]\langle[R_{j}^{+}(t),O(t)]R_{l}^{-}(t-\tau)\rangle$
(13) $\displaystyle+$
$\displaystyle\sum\limits_{k}\sum\limits_{l,l=1}^{N_{s}}\frac{(\mathbf{d}_{b},\mathbf{g}_{k})^{2}}{\hbar^{2}}\int\limits_{0}^{t}d\tau\exp[-i\omega_{k}\tau+i(\mathbf{k,r}_{j}-\mathbf{r}_{l})]\langle[S_{j}^{+}(t),O(t)]S_{l}^{-}(t-\tau)\rangle$
$\displaystyle+$
$\displaystyle\sum\limits_{k_{1},k_{2}}\sum\limits_{l,j=1}^{N}\frac{(\mathbf{n}_{eg},\mathbf{e}_{\lambda_{1}})^{2}(\mathbf{n}_{ei},\mathbf{e}_{\lambda_{2}})^{2}q^{2}(\omega_{1},\omega_{2})}{\hbar^{2}}\int\limits_{0}^{t}d\tau\langle[D_{j}^{+}(t),O(t)]D_{l}^{-}(t-\tau)\rangle$
$\displaystyle\times$
$\displaystyle\exp[-i(2\omega_{0}-\omega_{k_{1}}-\omega_{k_{2}})\tau]\exp[i(\mathbf{k}_{1}+\mathbf{k}_{2},\mathbf{r}_{j}-\mathbf{r}_{l})]$
$\displaystyle+$ $\displaystyle
i\sum\limits_{k,k^{\prime}}\sum\limits_{n=1}^{N}\sum\limits_{j=1}\sum\limits_{l=1}\frac{(\mathbf{n}_{eg},\mathbf{e}_{\lambda}(\mathbf{k}))(\mathbf{n}_{ei},\mathbf{e}_{\lambda}(\mathbf{k}^{\prime}))q(\omega_{k},\omega_{k^{\prime}})}{\hbar^{3}}\int\limits_{0}^{t}d\tau\int\limits_{0}^{t}d\tau^{\prime}\exp[i(\mathbf{k,r}_{n}-\mathbf{r}_{l})-i\omega_{k}\tau]$
$\displaystyle\times$
$\displaystyle\exp[i(\mathbf{k}^{\prime}\mathbf{,r}_{n}-\mathbf{r}_{l})-i\omega_{k^{\prime}}\tau^{\prime}][(\mathbf{d}_{s},\mathbf{g}_{k^{\prime}})(\mathbf{d}_{r},\mathbf{g}_{k})\langle[D_{n}^{+}(t),O(t)]R_{j}^{-}(t-\tau)S_{l}^{-}(t-\tau^{\prime})\rangle$
$\displaystyle+$
$\displaystyle(\mathbf{d}_{r},\mathbf{g}_{k^{\prime}})(\mathbf{d}_{s},\mathbf{g}_{k})\langle[D_{n}^{+}(t),O(t)]S_{j}^{-}(t-\tau)R_{l}^{-}(t-\tau^{\prime})\rangle]$
$\displaystyle-$ $\displaystyle
i\sum\limits_{k,k^{\prime}}\sum\limits_{m=1}^{N}\sum\limits_{j=1}\sum\limits_{l=1}\frac{(\mathbf{n}_{eg},\mathbf{e}_{\lambda}(\mathbf{k}))(\mathbf{n}_{ei},\mathbf{e}_{\lambda}(\mathbf{k}^{\prime}))q(\omega_{k},\omega_{k^{\prime}})}{\hbar^{3}}\int\limits_{0}^{t}d\tau^{\prime}\int\limits_{0}^{t}d\tau\exp[i\omega_{k}\tau-i(\mathbf{k,r}_{m}-\mathbf{r}_{l})]$
$\displaystyle\times$
$\displaystyle\exp[-i\omega_{k}\tau^{\prime}-i\omega_{k^{\prime}}\tau^{\prime}+i(\mathbf{k}^{\prime}\mathbf{,r}_{j}-\mathbf{r}_{m})][(\mathbf{d}_{s},\mathbf{g}_{k})(\mathbf{d}_{r},\mathbf{g}_{k^{\prime}})\langle
S_{l}^{+}(t-\tau)[R_{j}^{+}(t),O(t)]D_{m}^{-}(t-\tau^{\prime})\rangle$
$\displaystyle+$
$\displaystyle(\mathbf{d}_{s},\mathbf{g}_{k})(\mathbf{d}_{r},\mathbf{g}_{k^{\prime}})\langle
R_{l}^{+}(t-\tau)[S_{j}^{+}(t),O(t)]D_{m}^{-}(t-\tau^{\prime})\rangle]+H.c.(O^{+}\rightarrow
O).$

The traditional Born-Marcov approximation in the right hand site of equation (
13) give us the divergent functions. In order to understood this we
approximate the right hand site of equation (2) with following expression

$\displaystyle a_{ks}(t)$ $\displaystyle=$
$\displaystyle\frac{(\mathbf{d}_{a},\mathbf{g}_{k})}{\hbar}\sum\limits_{l=1}^{N_{a}}R_{l}^{-}(t)\exp[-i(\mathbf{k},\mathbf{r}_{l})\zeta^{\ast}(\omega_{k}-\omega_{a})$
(14)
$\displaystyle+\frac{(\mathbf{d}_{b},\mathbf{g}_{k})}{\hbar}\sum\limits_{j=1}^{N_{b}}S_{j}^{-}(t)\exp[-i(\mathbf{k},\mathbf{r}_{j})\zeta^{\ast}(\omega_{k}-\omega_{b})$
$\displaystyle+2\sum\limits_{n=1}^{N_{b}}\sum\limits_{k_{1}}\frac{(\mathbf{n}_{eg},\mathbf{e}_{\lambda_{1}})(\mathbf{n}_{ei},\mathbf{e}_{\lambda})q(\omega_{1},\omega)}{\hbar}$
$\displaystyle\times
D_{n}^{-}(t)a_{k_{1}}^{\dagger}(t)\exp[-i(\mathbf{k}_{1}+\mathbf{k},\mathbf{r}_{n})\zeta^{\ast}(\omega_{k}+\omega_{k_{1}}-2\omega_{0}),$

in the Born-Marcovian approximation [15] , [13]. The small parameter in this
approximation is the ratio of retardation time to cooperative spontaneous
emission times of the subsystem, $\tau/\tau_{i}<<1$. Here
$i\zeta(x)=iP/x+\pi\delta(x)$ is the Heitler function [13],[14]. represents
k-summation in analogy with Cauchy principal value [15]. Introducing the
operators (14) in equation (3) and eliminating the boson operators of EMF, it
is obtain the following equation for operator $O(t)$ in Born-Marcov
approximation

$\displaystyle\frac{d}{dt}\langle O(t)\rangle$ $\displaystyle=$
$\displaystyle\sum\limits_{k}\sum\limits_{l,j=1}^{N_{r}}\frac{(\mathbf{d}_{a},\mathbf{g}_{k})^{2}}{\hbar^{2}}\langle[R_{j}^{+}(t),O(t)]R_{l}^{-}(t)\rangle$
$\displaystyle\times$
$\displaystyle\exp[i(\mathbf{k,r}_{j}-\mathbf{r}_{l})]i\zeta^{\ast}(\omega_{r}-\omega_{k})+\sum\limits_{k}\sum\limits_{l,j=1}^{N_{s}}\frac{(\mathbf{d}_{a},\mathbf{g}_{k})(\mathbf{d}_{b},\mathbf{g}_{k})}{\hbar^{2}}$
$\displaystyle\times$
$\displaystyle\langle[S_{j}^{+}(t),O(t)]S_{l}^{-}(t)\rangle\exp[i(\mathbf{k,r}_{j}-\mathbf{r}_{l})]i\zeta^{\ast}(\omega_{s}-\omega_{k})$
$\displaystyle+$
$\displaystyle\sum\limits_{k,k^{\prime}}\sum\limits_{l,j=1}^{N}\frac{(\mathbf{n}_{eg},\mathbf{e}_{\lambda})^{2}(\mathbf{n}_{ei},\mathbf{e}_{\lambda^{\prime}})^{2}q^{2}(\omega_{k},\omega_{k^{\prime}})}{\hbar^{2}}$
$\displaystyle\times$
$\displaystyle\langle[D_{j}^{+}(t),O(t)]D_{l}^{-}(t)\rangle\exp[i(\mathbf{k-k}^{\prime}\mathbf{,r}_{j}-\mathbf{r}_{l})]i\zeta^{\ast}(\omega_{a}-\omega_{k}-\omega_{k^{\prime}})$
$\displaystyle+$ $\displaystyle
2i\sum\limits_{k,k^{\prime}}\sum\limits_{n=1}^{N}\sum\limits_{l=1}^{N_{s}}\sum\limits_{l=1}^{N_{r}}\frac{(\mathbf{d}_{a},\mathbf{g}_{k^{\prime}})(\mathbf{d}_{b},\mathbf{g}_{k})(\mathbf{n}_{eg},\mathbf{e}_{\lambda})(\mathbf{n}_{ei},\mathbf{e}_{\lambda^{\prime}})q(\omega_{k},\omega_{k^{\prime}})}{\hbar^{3}}$
$\displaystyle\times$
$\displaystyle\langle[D_{n}^{+}(t),O(t)]R_{j}^{-}(t)S_{l}^{-}(t)\rangle$
$\displaystyle\times$
$\displaystyle\exp[i(\mathbf{k,r}_{n}-\mathbf{r}_{l})+i(\mathbf{k}^{\prime}\mathbf{,r}_{n}-\mathbf{r}_{l})]i\zeta^{\ast}(\omega_{r}-\omega_{k})i\zeta^{\ast}(\omega_{s}-\omega_{k^{\prime}})$
$\displaystyle-$ $\displaystyle
i\sum\limits_{k,k^{\prime}}\sum\limits_{m=1}^{N}\sum\limits_{l=1}\sum\limits_{j=1}\frac{(\mathbf{d}_{a},\mathbf{g}_{k_{1}})(\mathbf{d}_{b},\mathbf{g}_{k_{2}})(\mathbf{n}_{eg},\mathbf{e}_{\lambda_{1}})(\mathbf{n}_{ei},\mathbf{e}_{\lambda_{2}})q(\omega_{1},\omega_{2})}{\hbar^{3}}$
$\displaystyle\times$ $\displaystyle[\langle
S_{l}^{+}(t)[R_{j}^{+}(t),O(t)]D_{m}^{-}(t)\rangle
i\zeta^{\ast}(\omega_{r}-\omega_{k})i\zeta(\omega_{s}-\omega_{k^{\prime}})$
$\displaystyle+$ $\displaystyle\langle
R_{l}^{+}(t)[S_{j}^{+}(t),O(t)]D_{m}^{-}(t)\rangle
i\zeta^{\ast}(\omega_{s}-\omega_{k})i\zeta(\omega_{r}-\omega_{k^{\prime}})]$
$\displaystyle\times$
$\displaystyle\exp[i(\mathbf{k,r}_{j})+i(\mathbf{k}^{\prime}\mathbf{,r}_{l})-i(\mathbf{k+k}^{\prime},\mathbf{r}_{m})]+H.C.(O^{+}\rightarrow
O).$ (15)

In the right hand part of the equation (15) the third order terms contain the
resonances between the single photon radiators $A$ ,$B$ and two-photon
radiator $D$ described by the correlation functions $\langle
S_{l}^{+}(t)[R_{j}^{+}(t),O(t)]D_{m}^{-}(t)\rangle$,
$\langle[D_{n}^{+}(t),O(t)]R_{j}^{-}(t)S_{l}^{-}(t)\rangle$ and $\langle
R_{l}^{+}(t)[S_{j}^{+}(t),O(t)]D_{m}^{-}(t)\rangle$. As it is observed from
equation (15), these terms contain the product of the functions
$[P/(\omega_{b}-\omega_{k})][P/(\omega_{b}-\omega_{k^{\prime}})]$ which
describe the principal value in the integration procedure on the variables $k$
and $k^{\prime}$. It is not difficult to observe that these integrals become
divergent expressions In order to avoid these divergence in Appendix1 it is
proposed the integration procedure which takes in to account the retardation
between the radiators in the representation of right hand site of equation
(13). According with Appendix1 the right hand site of master equation (13)
takes the following non-divergent form

$\displaystyle\frac{d}{dt}\langle O(t)\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2\tau_{r}}\sum\limits_{l,j=1}^{N_{r}}\chi_{r}(j,l)\langle[R_{j}^{+}(t),O(t)]R_{l}^{-}(t)\rangle+\frac{1}{2\tau_{s}}\sum\limits_{l,l=0}^{N_{s}}\chi_{s}(j,l)\langle[S_{j}^{+}(t),O(t)]S_{l}^{-}(t)\rangle$
(16) $\displaystyle+$
$\displaystyle\frac{1}{2\tau_{b}}\sum\limits_{l,j=1}^{N}\chi_{d}(j,l)\langle[D_{j}^{+}(t),O(t)]D_{l}^{-}(t)\rangle$
$\displaystyle+$
$\displaystyle\frac{i}{2\tau_{bsr}}\sum\limits_{m=1}^{N}\sum\limits_{l=1}^{N_{r}}\sum\limits_{l=0}^{N_{s}}U(j,l,m)\langle[D_{m}^{+}(t),O(t)]R_{j}^{-}(t)S_{l}^{-}(t)\rangle$
$\displaystyle-$
$\displaystyle\frac{i}{4\tau_{sbr}}\sum\limits_{m=1}^{N}\sum\limits_{j=1}^{N_{r}}\sum\limits_{l=0}^{N_{s}}V(j,l.m)[\langle
S_{l}^{+}(t)[R_{j}^{+}(t),O(t)]D_{m}^{-}(t)\rangle+\langle
R_{j}^{+}(t)[S_{l}^{+}(t),O(t)]D_{m}^{-}(t)\rangle]$ $\displaystyle+$
$\displaystyle H.C.(O^{+}\rightarrow O).$

Here the spontaneous emission $\tau_{i}$ and exchange integral between
radiators $j$ and $l\ $, $\chi_{i}(j,l)$ are defined in expressions (28),
(32), (34) and (37) of Appendix. This equation can be used for description of
interaction between the dipole forbidden and dipole active systems of
radiators.

##  3 Kinetic equations for correlation functions

In order to found the correlation process between dipole forbidden transitions
of $D$ subsystem and dipole active transitions in $S$ and $R$ subsystems of
radiators let us found the equations for arbitrary atomic correlation
functions of subsystems of radiators. According with the generalized equation
(16) it is obtain the following chain of equation for atomic correlations

$\displaystyle\frac{d}{dt}\langle R_{zj}(t)\rangle$ $\displaystyle=$
$\displaystyle-\frac{1}{2\tau_{r}}\sum\limits_{l,=1}^{N_{a}}[\chi_{r}(j,l)\langle
R_{j}^{+}(t)R_{l}^{-}(t)\rangle+\chi_{r}^{\ast}(j,l)\langle
R_{l}^{+}(t)R_{j}^{-}(t)\rangle]$ (17)
$\displaystyle+\frac{i}{4\tau_{sbr}}\sum\limits_{m=1}^{N}\sum\limits_{l=0}^{N_{s}}[V(j,l.m)\langle
S_{l}^{+}(t)R_{j}^{+}(t)D_{m}^{-}(t)\rangle-V^{\ast}(j,l.m)\langle
D_{m}^{+}(t)R_{j}^{-}(t)S_{l}^{-}(t)\rangle],$
$\displaystyle\frac{d}{dt}\langle S_{zl}(t)\rangle$ $\displaystyle=$
$\displaystyle-\frac{1}{2\tau_{s}}\sum\limits_{p=0}^{N_{s}}\chi_{s}(l,p)[\langle
S_{l}^{+}(t)S_{p}^{-}(t)\rangle+\chi_{s}^{\ast}(l,p)\langle
S_{p}^{+}(t)S_{l}^{-}(t)\rangle]$ (18) $\displaystyle+$
$\displaystyle\frac{i}{4\tau_{sbr}}\sum\limits_{m=1}^{N}\sum\limits_{l=0}^{N_{s}}[V(j,l.m)\langle
S_{l}^{+}(t)R_{j}^{+}(t)D_{m}^{-}(t)\rangle-V^{\ast}(j,l.m)\langle
D_{m}^{+}(t)R_{j}^{-}(t)S_{l}^{-}(t)\rangle],$
$\displaystyle\frac{d}{dt}\langle D_{zn}(t)\rangle$ $\displaystyle=$
$\displaystyle-\frac{1}{2\tau_{b}}\sum\limits_{l=1}^{N}[I_{{}^{b}}(j,l)\langle
D_{n}^{+}(t)D_{l}^{-}(t)\rangle+I_{{}^{b}}^{\ast}(j,l)\langle
D_{l}^{+}(t)D_{n}^{-}(t)\rangle$ (19) $\displaystyle-$
$\displaystyle\frac{i}{2\tau_{srb}}\sum\limits_{j=1}^{N_{r}}\sum\limits_{l=0}^{N_{s}}[U(j,l,n)\langle
D_{n}^{+}(t)R_{j}^{-}(t)S_{l}^{-}(t)\rangle-U^{\ast}(j,l,n)\langle
S_{l}^{+}(t)R_{j}^{+}(t)D_{n}^{-}(t)\rangle,$

$\displaystyle\frac{d}{dt}\langle R_{j}^{+}(t)R_{l}^{-}(t)\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{\tau_{r}}\sum\limits_{n=1}^{N_{r}}[\chi_{r}(l,n)\langle
R_{j}^{+}(t)R_{zl}(t)R_{n}^{-}(t)\rangle+\chi_{r}(n,j)\langle
R_{n}^{+}(t)R_{zj}(t)R_{l}^{-}(t)\rangle]$ (20) $\displaystyle-$
$\displaystyle\frac{1}{2\tau_{sbr}}\sum\limits_{m=1}^{N}\sum\limits_{k=0}^{N_{s}}[V(j,k,m)\langle
S_{k}^{+}(t)R_{j}^{+}(t)R_{zl}(t)D_{m}^{-}(t)\rangle$ $\displaystyle+$
$\displaystyle V^{\ast}(j,k.m)\langle
D_{m}^{+}(t)R_{zl}(t)R_{j}^{-}(t)S_{k}^{-}(t)\rangle];$

$\displaystyle\frac{d}{dt}\langle S_{j}^{+}(t)S_{l}^{-}(t)\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{\tau_{r}}\sum\limits_{n=1}^{N_{s}}[\chi_{r}(l,n)\langle
S_{j}^{+}(t)S_{zl}(t)S_{n}^{-}(t)\rangle+\chi_{r}(n,j)\langle
S_{n}^{+}(t)S_{zj}(t)S_{l}^{-}(t)\rangle]$ (21) $\displaystyle-$
$\displaystyle\frac{i}{2\tau_{sbr}}\sum\limits_{m=1}^{N}\sum\limits_{k=1}^{N_{r}}[V(k,l.m)\langle
R_{k}^{+}(t)S_{j}^{+}(t)S_{zl}(t)D_{m}^{-}(t)\rangle$ $\displaystyle-$
$\displaystyle V^{\ast}(k,j.m)\langle
D_{m}^{+}(t)S_{zj}(t)S_{l}^{-}(t)R_{k}^{-}(t)\rangle];$
$\displaystyle\frac{d}{dt}\langle D_{n}^{+}(t)D_{l}^{-}(t)\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{\tau_{b}}\sum\limits_{m=1}^{N}[I_{{}^{b}}(l,m)\langle
D_{n}^{+}(t)D_{zl}(t)D_{m}^{-}(t)\rangle+I_{{}^{b}}^{\ast}(n,m)\langle
D_{m}^{+}(t)D_{zn}(t)D_{l}^{-}(t)\rangle]$ (22) $\displaystyle+$
$\displaystyle\frac{i}{\tau_{bsr}}\sum\limits_{j=1}^{N_{r}}\sum\limits_{k=0}^{N_{s}}[U(j,k,m)\langle
D_{n}^{+}(t)D_{zl}(t)R_{j}^{-}(t)S_{k}^{-}(t)\rangle$ $\displaystyle+$
$\displaystyle U^{\ast}(j,k,m)\langle
S_{k}^{+}(t)R_{j}^{+}(t)D_{zn}(t)D_{l}^{+}(t)\rangle].$

$\displaystyle\frac{d}{dt}i\langle
D_{m}^{+}(t)S_{l}^{-}(t)R_{k}^{-}(t)\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\tau_{bsr}}\sum\limits_{j=1}^{N_{r}}\sum\limits_{n=1}^{N_{s}}[U^{\ast}(j,n,m)\langle
S_{n}^{+}(t)R_{j}^{+}(t)D_{zm}(t)S_{l}^{-}(t)R_{k}^{-}(t)\rangle$ (23)
$\displaystyle+$
$\displaystyle\frac{1}{2\tau_{sbr}}\sum\limits_{n=1}^{N}\sum\limits_{j=1}[V(j,l.n)\langle
S_{zl}(t)R_{j}^{+}(t)R_{k}^{-}(t)D_{m}^{+}(t)D_{n}^{-}(t)\rangle$
$\displaystyle+$ $\displaystyle V(k,j.n)\langle
D_{m}^{+}(t)D_{n}^{-}(t)R_{zk}(t)S_{j}^{+}(t)S_{l}^{-}(t)\rangle]$
$\displaystyle+$ $\displaystyle
i\sum\limits_{j=1}\bigl{[}\frac{1}{\tau_{s}}\chi_{r}(j,l)\langle
D_{m}^{+}(t)S_{zl}(t)S_{j}^{-}(t)R_{k}^{-}(t)\rangle$
$\displaystyle+\frac{1}{\tau_{r}}\chi_{r}(j,k)\langle
D_{m}^{+}(t)R_{zk}(t)S_{l}^{-}(t)R_{j}^{-}(t)\rangle$
$\displaystyle+\frac{1}{\tau_{b}}I_{{}^{b}}^{\ast}(m,l)\langle
D_{j}^{+}(t)D_{zm}(t)S_{j}^{-}(t)R_{k}^{-}(t)\rangle\bigr{]}$

Let us consider the interaction three different atoms in interaction through
vacuum EMF. Introducing the exited numbers for the atomic subsystems $\langle
N_{\alpha}\rangle=\langle J_{z\alpha}(t)\rangle+0.5$ (here $J\leftrightarrow
S,\ R,\ D$ $\ \alpha=s,\ r,\ d$ ) and correlation function between the atoms
$\langle F\rangle=i[\langle D^{+}(t)S^{-}(t)R^{-}(t)\rangle-\langle
S^{+}(t)R^{+}(t)D^{-}(t)\rangle]$ we can obtain the closed system of equations
from the chain of equations (17-23). Indeed considering that the distance
between the radiators is smaller then radiation wavelength: $\Re$
$\\{U(j,k,m)\\}=\Re\\{V(j,k,m)\\}=1$ , we obtain the following closed system
of equation

$\displaystyle\frac{d}{dt}\langle N_{s}(t)\rangle$ $\displaystyle=$
$\displaystyle-\frac{\langle
N_{s}\rangle}{\tau_{s}}-\frac{1}{2\tau_{sbr}}\langle F\rangle;$
$\displaystyle\frac{d}{dt}\langle N_{r}(t)\rangle$ $\displaystyle=$
$\displaystyle-\frac{\langle
N_{r}\rangle}{\tau_{r}}-\frac{1}{2\tau_{sbr}}\langle F\rangle;$
$\displaystyle\frac{d}{dt}\langle N_{d}(t)\rangle$ $\displaystyle=$
$\displaystyle-\frac{\langle
N_{d}\rangle}{\tau_{d}}-\frac{1}{\tau_{sbr}}\langle F\rangle;$
$\displaystyle\frac{d}{dt}\langle F(t)\rangle$ $\displaystyle=$
$\displaystyle-\bigl{[}\frac{1}{2\tau_{s}}+\frac{1}{2\tau_{r}}+\frac{1}{2\tau_{d}}\bigr{]}\langle
F(t)\rangle$ $\displaystyle+$ $\displaystyle\frac{1}{\tau_{bsr}}[6\langle
N_{s}N_{r}N_{d}\rangle-2\langle N_{s}N_{r}\rangle-\langle
N_{s}N_{d}\rangle-\langle N_{r}N_{d}\rangle],$
$\displaystyle\frac{d}{dt}\langle N_{s}N_{r}N_{d}\rangle$ $\displaystyle=$
$\displaystyle-[\frac{1}{\tau_{r}}+\frac{1}{\tau_{b}}+\frac{1}{\tau_{s}}]\langle
N_{s}N_{r}N_{d}\rangle,$ $\displaystyle\frac{d}{dt}\langle N_{s}N_{r}\rangle$
$\displaystyle=$ $\displaystyle-[\frac{1}{\tau_{r}}+\frac{1}{\tau_{s}}]\langle
N_{s}N_{r}\rangle,$ $\displaystyle\frac{d}{dt}\langle N_{s}N_{d}\rangle$
$\displaystyle=$ $\displaystyle-[\frac{1}{\tau_{r}}+\frac{1}{\tau_{b}}]\langle
N_{s}N_{b}\rangle,$ $\displaystyle\frac{d}{dt}\langle N_{s}N_{d}\rangle$
$\displaystyle=$ $\displaystyle-[\frac{1}{\tau_{r}}+\frac{1}{\tau_{b}}]\langle
N_{s}N_{b}\rangle,$

in which the new correlation functions between the atomic excitation is
introduced $\langle\hat{N}_{s}\hat{N}_{r}\hat{N}_{d}\rangle$,
$\langle\hat{N}_{s}\hat{N}_{r}\rangle$,
$\langle\hat{N}_{s}\hat{N}_{d}\rangle$, and
$\langle\hat{N}_{r}\hat{N}_{d}\rangle$. This system of equation is exactly
solvable. The solution is

$\displaystyle\langle N_{s}N_{r}N_{d}\rangle$ $\displaystyle=$
$\displaystyle\exp(-At);\ \ \ \langle N_{s}N_{r}\rangle=\exp(-Bt),\langle
N_{s}N_{d}\rangle=\exp(-Ct);\ \ \ \langle N_{d}N_{r}\rangle=\exp(-Dt)$
$\displaystyle\langle N_{i}N_{j}\rangle$ $\displaystyle=$
$\displaystyle\exp\bigl{[}-\bigl{(}\frac{1}{\tau_{i}}+\frac{1}{\tau_{j}}\bigr{)}t\bigr{]},\
\ \ i,j\rightarrow s,\ r,\ b;$ $\displaystyle\langle F(t)\rangle$
$\displaystyle=$
$\displaystyle\frac{\exp(-At/2)}{\tau_{bsr}}\bigl{[}6\frac{1-\exp[-At/2]}{A}-4\frac{1-\exp[-(B-\tau_{b}^{-1})t/2]}{(B-\tau_{b}^{-1})}$
$\displaystyle-\frac{2(1-\exp[-(C-\tau_{r}^{-1})t/2])}{(C-\tau_{r}^{-1})}-\frac{2(1-\exp[-(C-\tau_{s}^{-1})t/2])}{(C-\tau_{s}^{-1})}\bigr{]};$
$\displaystyle\langle N_{i}\rangle$ $\displaystyle=$
$\displaystyle\exp[-t/\tau_{i}]-\frac{1}{\tau_{bsr}}\int\limits_{0}^{t}\exp[-(t-t^{\prime})/\tau_{i}]\langle
F(t^{\prime})\rangle,\ \ \ \ \ i\equiv s,\ r,\ d,$

where the collective rates are defined $A=1/\tau_{r}+1/\tau_{b}+1/\tau_{s}$;
$B=1/\tau_{r}+1/\tau_{s}$; $C=1/\tau_{b}+1/\tau_{s}$;
$D=1/\tau_{b}+1/\tau_{s}$. The solution of this system of equation is plotted
in figure 2. It is observed the influence of single photon transition on the
two quanta transitions. This influence drastically depends on the cooperative
rate $1/\tau_{bsr}$.

Figure 2: The decay law of inversion, $D_{z}(t)$, of dipole forbidden radiator
stimulated by single photon decay processes of two radiators for following
parameters of the system: relative decay rates of S- and R- atoms
$\tau_{b}/\tau_{s}=\tau_{b}/\tau_{r}=5$.

As follows from figure 3 the cooperative exchanges between the radiator
accelerate the two-photon decay processes so that the the derivation
$-\frac{dD_{z}}{dt}$ achieved the maximal value decreasing after that till
zero value.

Figure 3: The two-photon decay rate $dD_{z}(t)/dt$ stimulated by single photon
processes for same parameter of the system as in figure 2.

Neglecting the quantum fluctuations of inversion operators $\langle
R_{z_{j}}\rangle$, $\langle S_{zl}\rangle,$ $\langle D_{zn}\rangle$, dipole-
dipole correlations between the same radiators $\langle
R_{l}^{+}(t)R_{j}^{-}(t)\rangle$, $\langle S_{l}^{+}(t)S_{p}^{-}(t)\rangle$
and $\langle D_{n}^{+}(t)D_{l}^{-}(t)\rangle$ and between subsystems $\langle
S_{l}^{+}(t)R_{j}^{+}(t)D_{m}^{-}(t)\rangle$, $\langle
D_{m}^{+}(t)R_{j}^{-}(t)S_{l}^{-}(t)\rangle$ we can de-correlated the chain of
equations (17-23) in order to obtain the closed system of equations

$\displaystyle for\ \ \ J\leftrightarrow S,\ R,\ D;\ \langle
J_{j}^{+}(t)J_{zl}(t)J_{m}^{-}(t)\rangle$ $\displaystyle=$
$\displaystyle\langle J_{zl}(t)\rangle\langle J_{j}^{+}(t)J_{m}^{-}(t)\rangle\
\ \ j\neq l\neq m;$ $\displaystyle\langle
S_{k}^{+}(t)R_{j}^{+}(t)R_{zl}(t)D_{m}^{-}(t)\rangle$ $\displaystyle=$
$\displaystyle\langle R_{zl}(t)\rangle\langle
S_{k}^{+}(t)R_{j}^{+}(t)D_{m}^{-}(t)\rangle_{l\neq j}-\delta_{l,j}\langle
S_{k}^{+}(t)R_{j}^{+}(t)D_{m}^{-}(t)\rangle;$ $\displaystyle\langle
D_{n}^{+}(t)D_{zl}(t)R_{j}^{-}(t)S_{k}^{-}(t)\rangle$ $\displaystyle=$
$\displaystyle\langle D_{zl}(t)\rangle\langle
D_{n}^{+}(t)R_{j}^{-}(t)S_{k}^{-}(t)\rangle_{l\neq n}-\delta_{l,n}\langle
D_{n}^{+}(t)R_{j}^{-}(t)S_{k}^{-}(t)\rangle;$ $\displaystyle\langle
S_{l}^{+}(t)R_{j}^{+}(t)D_{zn}(t)R_{p}^{-}(t)S_{k}^{-}(t)\rangle$
$\displaystyle=$ $\displaystyle\langle D_{zn}(t)\rangle\langle
R_{j}^{+}(t)R_{p}^{-}(t)\rangle\langle S_{l}^{+}(t)S_{k}^{-}(t)\rangle_{l\neq
k;j\neq p}$ $\displaystyle+\delta_{j,p}\delta_{l,k}\langle
D_{zn}(t)(R_{zj}(t)+0.5)(S_{zl}(t)+0.5)\rangle;$ $\displaystyle\langle
D_{m}^{+}(t)D_{n}^{-}(t)S_{zl}(t)R_{j}^{+}(t)R_{k}^{-}(t)\rangle$
$\displaystyle=$ $\displaystyle\langle S_{zl}(t)\rangle\langle
D_{m}^{+}(t)D_{n}^{-}(t)\rangle\langle R_{j}^{+}(t)R_{k}^{-}(t)\rangle_{m\neq
n;j\neq k}$
$\displaystyle+\delta_{m,n}\delta_{j,k}\langle(D_{zn}(t)+0.5)S_{zl}(t)(R_{zj}(t)+0.5)\rangle;$
$\displaystyle\langle
D_{m}^{+}(t)D_{n}^{-}(t)R_{zj}(t)S_{l}^{+}(t)S_{k}^{-}(t)\rangle$
$\displaystyle=$ $\displaystyle\langle R_{zj}(t)\rangle\langle
D_{m}^{+}(t)D_{n}^{-}(t)\rangle\langle S_{l}^{+}(t)S_{k}^{-}(t)\rangle_{m\neq
n;l\neq k}$ (24)
$\displaystyle\delta_{m,n}\delta_{l,k}\langle(D_{zn}(t)+0.5)R_{zj}(t)(S_{zl}(t)+0.5)\rangle.$

Taking in to account the de-correlation (24), we obtain the following closed
system of equations

$\displaystyle\frac{d}{dt}R_{z}(t)$ $\displaystyle=$
$\displaystyle-\frac{1}{\tau_{r}}\\{N_{r}(N_{r}+2)/4-R_{z}^{2}+R_{z}\\}-\frac{1}{2\tau_{sbr}}F,$
$\displaystyle\frac{d}{dt}S_{z}(t)$ $\displaystyle=$
$\displaystyle-\frac{1}{\tau_{r}}\\{N_{s}(N_{s}+2)/4-S_{z}^{2}+S_{z}\\}-\frac{1}{2\tau_{sbr}}F,$
$\displaystyle\frac{d}{dt}D_{z}(t)$ $\displaystyle=$
$\displaystyle-\frac{1}{\tau_{r}}\\{N(N+2)/4-D_{z}^{2}+D_{z}\\}-\frac{1}{\tau_{sbr}}F;$
$\displaystyle\frac{d}{dt}F$ $\displaystyle=$
$\displaystyle\bigl{[}\frac{1}{\tau_{s}}[S_{z}(t)-1]+\frac{1}{\tau_{r}}[R_{z}(t)-1]+\frac{1}{\tau_{d}}[D_{z}(t)-1]\bigr{]}F$
$\displaystyle+$
$\displaystyle\frac{1}{\tau_{bsr}}\bigl{[}2D_{z}\\{N_{s}^{2}/4-S_{z}^{2}\\}\\{N_{r}^{2}/4-R_{z}^{2}\\}$
$\displaystyle+$ $\displaystyle
R_{z}\\{N_{s}^{2}/4-S_{z}^{2}\\}\\{N^{2}/4-D_{z}^{2}\\}+S_{z}\\{N_{r}^{2}/4-R_{z}^{2}\\}\\{N^{2}/4-D_{z}^{2}\\}$
$\displaystyle+$ $\displaystyle(4N_{s}N_{r}N\exp[-A\ast t]-N_{s}\ast
N_{r}\ast\exp[-Bt]-0,5N_{s}N\exp[-Ct]$ $\displaystyle-$ $\displaystyle
0.5N_{r}N\exp[-Dt]\bigr{]}.$

From this system of equations follows the oscillatory behavior of the decay
rate of the inversion $D_{z}$. Taking in to account the following relative
expressions of the decay rates we obtain the numerical simulation of the
inversion $D_{z}$ and its derivative (see figure 4 and figure 5). As follows
from this system of equation the increasing of decay rate of two-photon
spontaneous emission is possible under the influence of single photon
cooperative emission of two atomic subsystems. In figures 4 is plotted the
time dependence of the inversion $<D_{z}(t)>$ of dipole forbidden radiators as
function of the relative coupled parameter between the radiators
$\tau_{b}/\tau_{srb}$.

Figure 4: The time dependence of inversion $D_{z}(t)$ for dipole forbidden
radiator subsystem for following values of the parameter of the system:nuber
of atoms in the subsystems $S,R$ and $D$ are, $N_{s}=N_{r}=N=50$ respectively;
the relative decay times of the subsystems are
$\tau_{b}/\tau_{s}=\tau_{b}/\tau_{r}=6$ ; the coupling parameter of these
three system is changed between $0$ and $1$. The oscillatory behavior of decay
rate is observed. Figure 5: The cooperative decay rate of inversion
$\frac{dD_{z}(t)}{dt}$ of dipole forbidden radiator subsystem for same values
of the parameters of the system as in figure 4. The increasing of decay rate
of bi-photons is observed.

The same dependence is represented in figure 5 for the intensity of two-photon
emission proportional to $-d<D_{z}(t)>/dt$. As follows from these plots it is
observed the mutual influences between single and two-photon super-radiance
processes of three particle interaction. This effect plays an important role
in the collective decay process of the systems of radiators with the dimension
smaller than wavelength.

##  4 Conclusion

In this paper the effective interaction between three radiator subsystems in
two-photon resonance is found using the method of elimination of operators of
vacuum field. The new cooperative interaction between dipole-forbidden atomic
subsystem and two-dipole active subsystems of radiators was proposed. The
master equation 16, which describes the energy dissipation from the system due
to mutual interaction between the radiators through the vacuum of
electromagnetic field, was obtained. Using the chain of equation17-23, which
describes the cooperative interaction between three radiator subsystems, it is
obtained the closed system of equations for three radiators. Neglecting the
quantum fluctuation of the inversion, the de-correlation method of the this
chain of equation is proposed 24 in order to describe numerically the behavior
of mutual influences of single and two-photon super-radiance processes. As a
consequence of effective interaction between three radiators through two-
photon resonance processes of inverted systems increase substantially in
process cooperative decay of the system The three particle exchange integral
has been established and the influence of this effect on the behavior
cooperative decay of the atomic subsystems was estimated (see figure 4 and
figure 5). Similar experimental situation can bi realized in exited atomic
(for example transitions in Cs atoms [17]) or nuclei (for example
${}^{193m}Ir$, ${}^{195m}Pt$ and ${}^{103m}Rh$ nuclei[16]) subsystems in
resonance interaction through vacuum field.

##  5  Appendix: Exchange integrals

In order to estimate all exchange integrals in equation (13) let us firstly
found the well known exchange integral between two radiators in single photon
interaction with vacuum of electromagnetic field. In the first terms of
equation (13) the retardation can be found integration firstly on the $k$
vector respectively

$\displaystyle V_{jl}^{i}$ $\displaystyle=$
$\displaystyle\frac{d_{\alpha}^{2}}{(2\pi)^{2}\hbar
c^{3}}\sum\limits_{l,j=1}^{N_{a}}\int\limits_{0}^{\infty}\omega_{k}^{3}d\omega_{k}\int
d\Omega_{k}\int\limits_{0}^{t}d\tau\exp[i(\omega_{i}-\omega_{k})\tau]$
$\displaystyle\times(1-(\mathbf{e}_{k},n_{d}))\langle[\tilde{J}_{j}^{+}(t),O(t)]\
\tilde{J}_{l}^{-}(t-\tau)\rangle\exp[i\omega_{k}r_{jl}\cos\theta],$
$\displaystyle i$ $\displaystyle\equiv$ $\displaystyle a,\ \ \ ,b.$ (25)

Here the frequency $\omega_{i}$ corresponds to $\mathit{A}$ and $\mathit{B}$
atomic systems; $i=r,s$; for $i=r$ operators $\tilde{J}_{j}^{+}$,$\ \
\tilde{J}_{l}^{-}$ corresponds to $\tilde{R}_{j}^{+}$ , $\tilde{R}_{l}^{-}$
and for $i=s$ these operators corresponds to $\tilde{S}_{j}^{+}$,
$\tilde{S}_{l}^{-}$ Passing to new variable $\upsilon=\omega_{k}-\omega_{i}$
and considering that the smooth function $\omega_{k}$ under integral can be
approximation with $\omega_{i}^{3}$, we obtain the following approximate
expression of thirst order exchange integrals

$\displaystyle\omega_{i}^{3}\int\limits_{-\omega_{k}}^{\infty}dv\exp[iv(\tau-
r_{jl}\cos\theta/c)]$ $\displaystyle\eqsim$
$\displaystyle\omega_{i}^{3}\int\limits_{-\infty}^{\infty}dv\exp[iv(\tau-
r_{jl}\cos\theta/c)]$ (26) $\displaystyle=$ $\displaystyle
2\pi\omega_{i}^{3}\delta(\tau-r_{jl}\cos\theta/c).$

$\omega_{i}$ is the emission frequency relatively the dipole active
transitions of the $R$ and $S$ atomic subsystems. In this approximation I
obtain the following integral on angle $\theta$ and retardation $\tau$

$\displaystyle V_{jl}^{i}$ $\displaystyle=$
$\displaystyle\frac{\omega_{i}^{3}d_{i}^{2}}{2\hbar
c^{3}}\int\limits_{0}^{\pi}d\theta\int\limits_{0}^{t}d\tau\sin\theta\delta(\tau-
r_{jl}\cos\theta/c)D_{i}(\theta)\langle[\tilde{J}_{j}^{+}(t),O(t)]\
\tilde{J}_{l}^{-}(t)\rangle$ (27) $\displaystyle=$
$\displaystyle\frac{\omega_{i}^{3}d_{i}^{2}}{2\hbar
c^{3}}\int\limits_{0}^{\pi}d\theta\sin\theta\Theta(\cos\theta)D_{jl}\bigl{[}\frac{\partial}{\partial\omega_{i}}\bigr{]}\exp[i\omega_{i}r_{jl}\cos\theta]\langle[\tilde{J}_{j}^{+}(t),O(t)]\
\tilde{J}_{l}^{-}(t)\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2\tau_{i}}\chi(j,l)\langle[\tilde{J}_{j}^{+}(t),O(t)]\
\tilde{J}_{l}^{-}(t)\rangle.$

Here $\tau_{i}$ and $\chi(j,l)$ are the spontaneous emission time and exchange
integral between the single photon radiators respectively [13]

$\tau_{i}=\frac{3\hbar c^{3}}{4d_{i}\omega_{i}^{3}},\ \ \
\chi(j,l)=D_{jl}\bigl{[}\frac{\partial}{\partial\omega_{i}}\bigr{]}\frac{3}{4}\frac{\exp[i\omega_{i}r_{jl}/c]-1}{i\omega_{i}r_{jl}/c},$
(28)

the expressions in equation (27) and (28) are defined below

$\displaystyle D_{i}(\theta)$ $\displaystyle=$ $\displaystyle
1+\cos^{2}\xi_{i}-\cos^{2}\theta(3\cos^{2}\xi_{i}-1),$ $\displaystyle
D_{jl}\bigl{[}\frac{\partial}{\partial\omega_{i}}\bigr{]}$ $\displaystyle=$
$\displaystyle\bigl{[}1+\cos^{2}(\xi_{i})+(3\cos^{2}\xi_{i}-1)\frac{c^{2}}{r_{jl}^{2}}\frac{\partial^{2}}{\partial\omega_{i}^{2}}]^{2}\bigr{]}$
(29)

where $\cos\xi_{i}$ is the scalar product between the unitary vectors along
the direction of dipole momentum of the $j$ (or $l$) atom
$\mathbf{n}_{d_{i}}=\mathbf{d}_{i}/d_{i}$ and the direction of the distance
between the $j$ and $l$ atoms $\mathbf{n}_{jl}=\mathbf{r}_{jl}/r_{jl}$.

2. The two-photon exchange integral between dipole forbidden transition of the radiators of $D$ subsystem is described by third term in the right hand site of equation (13)

$\displaystyle V_{jl}^{b}$ $\displaystyle=$
$\displaystyle\frac{V^{2}}{(2\pi)^{6}}\int\limits_{0}^{2\pi}d\varphi_{1}\int\limits_{0}^{\pi}d\theta_{1}\sin\theta_{1}\int\limits_{0}^{\infty}k_{1}^{2}dk_{1}\int\limits_{0}^{2\pi}d\varphi_{2}\int\limits_{0}^{\pi}d\theta_{2}\sin\theta_{2}\int\limits_{0}^{\infty}k_{2}^{2}dk_{2}\frac{(\mathbf{n}_{eg},\mathbf{e}_{\lambda_{1}})^{2}(\mathbf{n}_{ei},\mathbf{e}_{\lambda_{2}})^{2}q^{2}(\omega_{1},\omega_{2})}{\hbar^{2}}$
(30) $\displaystyle\times$
$\displaystyle\int\limits_{0}^{t}d\tau\exp[-i(2\omega_{0}-\omega_{1}-\omega_{2})\tau+i(\mathbf{k}_{1}+\mathbf{k}_{2},\mathbf{r}_{j}-\mathbf{r}_{l})]\left\langle[\tilde{D}_{j}(t),O(t)]\tilde{D}_{l}(t-\tau)\right\rangle.$

The exchange integral in Born-Marcov approximation for two-photon emission was
obtained in paper [9]. Here we will estimate the exchange integral of
expression (30) integration firstly on the wave vectors $\mathbf{k}_{1}$ and
$\mathbf{k}_{2}.$ Indeed considering that the amplitude
$q^{2}(\omega_{1},\omega_{2})$ is the smooth function of the variables $k_{1}$
and $k_{2}$ in comparison with rapid oscillation functions
$\exp[i(\mathbf{k}_{1}+\mathbf{k}_{2},\mathbf{r}_{j}-\mathbf{r}_{l})]$ and
$\exp[-i(2\omega_{0}-\omega_{k_{1}}-\omega_{k_{2}})\tau]$, we can approximate
the amplitude $q^{2}(\omega_{1},\omega_{2})$ according with the resonance
frequencies of the atomic system $q^{2}(\omega_{1},\omega_{2})\eqsim
q^{2}(\omega_{0},\omega_{0})$. It is not difficult to observe that the maximal
value of the smooth function under the integral is obtained for frequencies
$\omega_{k_{2}}=\omega_{k_{1}}\eqsim\omega_{0}$. In order to integrate this
expression on the variables $k_{1}$ and $k_{2}$ we change the variables
$x_{i}=\omega_{k_{i}}-\omega_{i}$ as in expression (26), and consider that
$\tilde{D}_{j}^{+}(t)=D_{j}^{+}(t)\exp[-2i\omega_{0}t]$ is smooth operator. In
these approximations it is obtain the following expression for $V_{jl}^{b}$

$\displaystyle V_{jl}^{b}$ $\displaystyle=$
$\displaystyle\frac{V^{2}q^{2}(\omega_{0},\omega_{0})}{(2)^{4}\pi^{2}}\int\limits_{0}^{t}d\tau\int\limits_{0}^{\pi}d\theta_{1}\sin\theta_{1}\int\limits_{0}^{\pi}d\theta_{2}\sin\theta_{2}D_{0}(\theta_{1})D_{0}(\theta_{2})\exp[i\omega_{0}r_{jl}(\cos\theta_{1}+\cos\theta_{2})/c]$
(31) $\displaystyle\delta(\tau-r_{jl}\cos\theta_{1}/c)\delta(\tau-
r_{jl}\cos\theta_{2}/c)\left\langle[\tilde{D}_{j}(t),O(t)]\tilde{D}_{l}(t-\tau)\right\rangle$
$\displaystyle\eqsim$
$\displaystyle\frac{V^{2}q^{2}(\omega_{0},\omega_{0})}{(2)^{4}\pi^{2}}D_{jl}^{2}\left(\frac{\partial}{\partial\omega_{0}}\right)\frac{\exp[2i\omega_{0}r_{jl}]-1}{2i\omega_{0}r_{jl}}\left\langle[\tilde{D}_{j}(t),O(t)]\tilde{D}_{l}(t-\tau)\right\rangle,$

where expressions $D_{0}(\theta_{1})$ and
$D_{jl}(\partial/(\partial\omega_{0})$ are defined by the expressions (29).
Integrating the right hand site of the equation (31) on the solid angle and
retardation, we obtain the following approximative expression

$V_{jl}^{b}=\frac{1}{2\tau_{b}}\chi_{b}(j,l)\left\langle[\tilde{D}_{j}^{+}(t),O(t)]\tilde{D}_{l}^{-}(t)\right\rangle,$

where

$\displaystyle\frac{1}{2\tau_{b}}$ $\displaystyle=$
$\displaystyle\frac{2^{2}}{3^{2}}\frac{\omega_{0}^{7}d_{23}^{2}d_{31}^{2}}{2^{2}\pi\hbar^{2}c^{6}}\\{3/2\\}\left\\{\frac{1}{\omega_{32}+\omega_{0}}+\frac{1}{\omega_{31}-\omega_{0}}\right\\}^{2},$
$\displaystyle\chi_{b}(j,l)$ $\displaystyle=$
$\displaystyle\frac{3^{2}}{4}\frac{\pi
c}{4\omega_{0}r_{jl}}D^{2}\bigl{[}\frac{\partial}{\partial\omega_{0}}\bigr{]}\frac{\exp[2i\omega_{0}r_{jl}/c]-1}{i\omega_{0}r_{jl}/c}$
(32)

This exchange integral diverges, when the distance between the radiators
$r_{jl}$ is less than the wavelength $\lambda_{0}=2\pi c/\omega_{0}$. In order
to take in to account the value of the exchange integral for the small
parameter, $r_{jl}/\lambda_{0}<<1,$ let us integrate the expression (30)
taking in to account the method proposed in papers [9] and [12]. In this case
we obtain the following expression for $ReV_{jl}^{b}$

$\displaystyle F(j,l)$ $\displaystyle=$ $\displaystyle
ReV_{jl}^{b}=\frac{d_{31}^{2}d_{32}^{2}}{4\pi\hbar^{2}c^{6}}\int\limits_{0}^{2\omega_{0}}d\omega_{k}\omega_{k}^{3}(\omega_{21}-\omega_{k})^{3}$
$\displaystyle\times$
$\displaystyle\chi_{jl}(\omega_{k})\chi_{jl}(\omega_{21}-\omega_{k})\left\\{\frac{1}{\omega_{31}-\omega_{k_{1}}}+\frac{1}{\omega_{32}+\omega_{k_{1}}}\right\\}^{2},$

where

$\chi_{jl}(\omega)=(1-\cos^{2}{\xi})\frac{\sin{\frac{\omega
r_{jl}}{c}}}{\frac{\omega
r_{jl}}{c}}+(1-3\cos^{2}{\xi})\left\\{\frac{\cos{\frac{\omega
r_{jl}}{c}}}{(\frac{\omega r_{jl}}{c})^{2}}-\frac{\sin{\frac{\omega
r_{jl}}{c}}}{(\frac{\omega r_{jl}}{c})^{3}}\right\\}.$

3\. In the right hand part of this equation (13) the third order terms
contains the resonances between dipole active radiators $\mathit{A}$
,$\mathit{B}$ and dipole forbidden radiators $\mathit{D}$, described by the
correlation functions $\langle
S_{l}^{+}(t)[R_{j}^{+}(t),O(t)]D_{m}^{-}(t)\rangle$,
$\langle[D_{n}^{+}(t),O(t)]R_{j}^{-}(t)S_{l}^{-}(t)\rangle$ and $\langle
R_{l}^{+}(t)[S_{j}^{+}(t),O(t)]D_{m}^{-}(t)\rangle$. Let us introduced the
similar approximation in the chronological interaction between the atomic
subsystems $\mathit{A}$, $\mathit{B}$, and $\mathit{D}$. The exchange integral
between three atoms is represented in the similar form as in the expression
(30)

$\displaystyle V_{jln;as-d}^{c}$ $\displaystyle=$ $\displaystyle
i\frac{V}{(2\pi)^{3}}\frac{V}{(2\pi)^{3}}\frac{2\pi\hbar
d_{s}d_{r}}{Vc^{6}\hbar^{3}}\frac{(2\pi)^{3}}{V}(2\pi)^{2}\left(\frac{1}{2}\right)^{2}\int\limits_{0}^{\infty}\omega_{1}^{2}d\omega_{1}\int\limits_{0}^{\infty}\omega_{2}^{2}d\omega_{2}\sqrt{\omega_{1}\omega_{2}}\chi(\omega_{1},\omega_{2})$
(33) $\displaystyle\times$
$\displaystyle\int\limits_{-1}^{1}dx_{1}\int\limits_{-1}^{1}dx_{2}\int\limits_{0}^{t}d\tau_{1}\int\limits_{0}^{t}d\tau_{2}\exp[-i(\omega_{1}-\omega_{r})\tau_{1}-i(\omega_{2}-\omega_{s})\tau_{2}]$
$\displaystyle\times$ $\displaystyle
D_{nl}\bigl{[}\frac{\partial}{\partial\omega_{s}}\bigr{]}D_{nj}\bigl{[}\frac{\partial}{\partial\omega_{r}}\bigr{]}\exp[i\omega_{1}r_{nj}x_{1}/c+i\omega_{2}r_{nl}x_{2}/c]$
$\displaystyle\langle[\tilde{D}_{n}^{+}(t),O(t)]\tilde{R}_{j}^{-}(t-\tau_{1})\tilde{S}_{l}^{-}(t-\tau_{2})\rangle;$

After the substitution of variables $\omega_{1}-\omega_{r}=\tilde{\omega}_{1}$
and $\omega_{2}-\omega_{s}=\tilde{\omega}_{2}$, we can approximate the smooth
amplitude
$\omega_{1}^{2}\omega_{2}^{2}\sqrt{\omega_{1}\omega_{2}}\chi(\omega_{1},\omega_{2})$
with expression
$(\omega_{r})^{2}\omega_{s}^{2}\sqrt{\omega_{r}\omega_{s}}\chi(\omega_{r},\omega_{s})$.
Integrals on the new variable $\tilde{\omega}_{1}$ and $\tilde{\omega}_{1}$
give the following aspect of expression (33)

$\displaystyle V_{jln;as-d}^{c}$ $\displaystyle=$ $\displaystyle 2\pi
i\left(\frac{1}{2}\right)^{2}\frac{d_{s}d_{r}}{c^{6}\hbar^{2}}\omega_{s}^{2}(\omega_{r})^{2}\sqrt{\omega_{r}\omega_{s}}\chi(\omega_{s},\omega_{r})\int\limits_{0}^{1}dx_{1}\int\limits_{0}^{1}dx_{2}$
$\displaystyle\times$ $\displaystyle
D_{nl}\bigl{[}\frac{\partial}{\partial\omega_{s}}\bigr{]}D_{nj}\bigl{[}\frac{\partial}{\partial\omega_{r}}\bigr{]}\exp[i\omega_{r}r_{nj}x_{1}/c+i\omega_{s}r_{nl}x_{2}/c]$
$\displaystyle\times$
$\displaystyle\langle[\tilde{D}_{n}^{+}(t),O(t)]\tilde{R}_{j}^{-}(t-r_{nj}x_{1}/c)\tilde{S}_{l}^{-}(t-r_{nl}x_{2}/c)\rangle$

from which follows that $x_{1},$and $x_{2}>0$ . In the Born approximation the
expression for $V_{jln;as-d}^{c}$

$V_{rs-d}(m,j,l)=\frac{i}{4\tau_{bsr}}U(j,l,m)\langle[\tilde{D}_{n}^{+}(t),O(t)]\tilde{R}_{j}^{-}(t)\tilde{S}_{l}^{-}(t)\rangle$

where

$\displaystyle\frac{1}{\tau_{bsr}}$ $\displaystyle=$
$\displaystyle\left(\frac{2}{3}\right)^{2}\frac{d_{s}d_{r}d_{23}d_{31}\omega_{s}^{3}(\omega_{r})^{3}}{4\pi
c^{6}\hbar^{2}}\left\\{\frac{1}{\omega_{32}+\omega_{s}}+\frac{1}{\omega_{31}-\omega_{r}}\right\\},$
$\displaystyle U(j,l,m)$ $\displaystyle=$
$\displaystyle-\left(\frac{3}{2}\right)^{2}D_{nl}\bigl{[}\frac{\partial}{\partial\omega_{s}}\bigr{]}D_{nj}\bigl{[}\frac{\partial}{\partial\omega_{r}}\bigr{]}\frac{c^{2}[\exp
i\omega_{r}r_{nj}/c]-1][\exp[i\omega_{s}r_{nl}/c]-1]}{\omega_{r}\omega_{s}r_{nj}r_{nl}}.$
(34)

4. Let now found the retardation in the last correlation function term of equation (13).Taking in to account the retardation in the rapid oscillation part of atomic operators $J^{\pm}(t-\tau)=\tilde{J}^{\pm}(t-\tau)\exp[\pm i\omega(t-\tau)]$ where $J^{\pm}$, $\omega$ and $\tau$ are the atomic operators, transition frequencies and delay time for atomic subsystems $\mathit{A}$, $\mathit{B}$ and $\mathit{D}$ respectively,

$\displaystyle V_{;ab-d}(t)$ $\displaystyle=$ $\displaystyle
i\sum\limits_{k_{1}k_{2}}\sum\limits_{m=1}^{N}\sum\limits_{l=1}^{N_{a}}\sum\limits_{j=0}^{N_{b}}\frac{(\mathbf{d}_{r},\mathbf{g}_{k_{1}})(\mathbf{d}_{s},\mathbf{g}_{k_{2}})(\mathbf{n}_{eg},\mathbf{e}_{\lambda_{1}})(\mathbf{n}_{ei},\mathbf{e}_{\lambda_{2}})q(\omega_{k_{1}},\omega_{k_{2}})}{\hbar^{3}}$
(35) $\displaystyle\times$
$\displaystyle\int\limits_{0}^{t}d\tau_{1}\exp[i(2\omega_{0}-\omega_{k_{1}}-\omega_{k_{2}})\tau_{1}]\int\limits_{0}^{t}d\tau_{2}\exp[-i(\omega_{s}-\omega_{k_{1}})\tau_{2}]$
$\displaystyle\times$
$\displaystyle\exp[-i(\mathbf{k}_{1},\mathbf{r}_{j}-\mathbf{r}_{m})+i(\mathbf{k}_{2},\mathbf{r}_{l}-\mathbf{r}_{m})]\langle\tilde{S}_{l}^{+}(t-\tau_{2})[\tilde{R}_{j}^{+}(t),O(t)]\tilde{D}_{m}^{-}(t-\tau_{1})\rangle.$

Passing from the summation to integration in expression (35) we obtain
following expression for correlation between the $j$, $l$ and $m$ atoms

$\displaystyle V_{rs-d}(m,j,l)$ $\displaystyle=$ $\displaystyle
i\frac{V}{(2\pi)^{3}}\frac{V}{(2\pi)^{3}}\frac{2\pi\hbar
d_{s}d_{r}}{Vc^{6}\hbar^{3}}\frac{(2\pi)^{3}}{V}(2\pi)^{2}\left(\frac{1}{2}\right)^{2}\int\limits_{0}^{\infty}\omega_{1}^{2}d\omega_{1}\int\limits_{0}^{\infty}\omega_{2}^{2}d\omega_{2}$
(36)
$\displaystyle\times\sqrt{\omega_{1}\omega_{2}}\chi(\omega_{1},\omega_{2})\int\limits_{-1}^{1}dx_{1}\int\limits_{-1}^{1}dx_{2}\int\limits_{0}^{t}d\tau_{1}\exp[i(2\omega_{0}-\omega_{1}-\omega_{2})\tau_{1}]$
$\displaystyle\times\int\limits_{0}^{t}d\tau_{2}\exp[-i(\omega_{s}-\omega_{2})\tau_{2}]$
$\displaystyle\times
D_{jl}\bigl{[}\frac{\partial}{\partial\omega_{2}}\bigr{]}D_{jm}\bigl{[}\frac{\partial}{\partial\omega_{1}}\bigr{]}\exp[-i\omega_{2}r_{ml}x_{2}/c-i\omega_{1}r_{jm}x_{1}/c)]$
$\displaystyle\times\langle\tilde{S}_{l}^{+}(t-\tau_{2})[\tilde{R}_{j}^{+}(t),O(t)]\tilde{D}_{m}^{-}(t-\tau_{1}).$

Introducing the new variables $u_{1}=\omega_{1}-\omega_{r}$;
$u_{2}=\omega_{2}-\omega_{s}$ in (36), and approximating the smooth amplitude
$\omega_{1}^{2}\omega_{2}^{2}\sqrt{\omega_{1}\omega_{2}}\chi(\omega_{1},\omega_{2})$
with expression
$\omega_{s}^{2}\omega_{r}^{2}\sqrt{\omega_{s}\omega_{r}}\chi(\omega_{s},\omega_{r})$
we get using the delta functions (26)

$\displaystyle V_{jlm;as-d}(j,l)$ $\displaystyle=$ $\displaystyle 2\pi
i\left(\frac{1}{2}\right)^{2}\frac{d_{s}d_{r}}{c^{6}\hbar^{2}}\omega_{s}^{2}\omega_{r}^{2}\sqrt{\omega_{s}\omega_{r}}\chi(\omega_{s},\omega_{r})\int\limits_{-1}^{1}dx_{1}\int\limits_{-1}^{1}dx_{2}\int\limits_{0}^{t}d\tau_{1}\int\limits_{0}^{t}d\tau_{2}$
$\displaystyle\times$
$\displaystyle\delta(\tau_{2}-\tau_{1}-r_{jl}x_{2}/c)\delta(\tau_{1}-r_{jm}x_{1}/c)$
$\displaystyle\times
D_{jl}\bigl{[}\frac{\partial}{\partial\omega_{s}}\bigr{]}D_{jm}\bigl{[}\frac{\partial}{\partial\omega_{r}}\bigr{]}\exp[i\omega_{s}r_{ml}x_{2}/c]\exp[-i\omega_{r}r_{jm}x_{1}/c]$
$\displaystyle\times\langle\tilde{S}_{l}^{+}(t-\tau_{2})[\tilde{R}_{j}^{+}(t),O(t)]\tilde{D}_{m}^{-}(t-\tau_{1})\rangle,$

the value of which can be estimated observing from the arguments of
$\delta$-functions that $x_{1}>0$ and $\ r_{jm}x_{1}+r_{jl}x_{2}>0$. In this
case, neglecting the retardation $\tau_{1}$ and $\tau_{2}$ in the smooth
correlation function we obtain

$V_{jlm;as-d}(j,l)=\frac{i}{4\tau_{sbr}}V(j,l.m)\langle\tilde{S}_{l}^{+}(t)[\tilde{R}_{j}^{+}(t),O(t)]\tilde{D}_{m}^{-}(t)\rangle,$

where cooperative rate is

$\frac{1}{\tau_{sbr}}=\left(\frac{2}{3}\right)^{2}\frac{d_{s}d_{r}d_{23}d_{31}\omega_{s}^{3}\omega_{r}^{3}}{4\pi
c^{6}\hbar^{2}}\left\\{\frac{1}{\omega_{32}+\omega_{s}}+\frac{1}{\omega_{31}-\omega_{r}}\right\\}.$

the integral $V(j,l.m)$ on the direction of the emitted photons is

$\displaystyle V(j,l.m)$ $\displaystyle=$
$\displaystyle\left(\frac{3}{2}\right)^{2}D_{ml}\bigl{[}\frac{\partial}{\partial\omega_{s}}\bigr{]}D_{jm}\bigl{[}\frac{\partial}{\partial\omega_{r}}\bigr{]}$
$\displaystyle\times$
$\displaystyle\bigl{[}\frac{c^{2}\\{2\exp[i(\omega_{s}r_{ml}-\omega_{r}r_{jm})/c]-2\exp[2i\omega_{s}r_{ml}]-\exp[-2i\omega_{r}r_{jm}]+1]\\}}{2\omega_{s}\omega_{r}r_{lm}r_{ml}}\theta(r_{ml}-r_{mj})$
$\displaystyle+.$
$\displaystyle\bigl{(}\frac{c^{2}(\omega_{s}+\omega_{r})[\exp(-i\omega_{r}r_{mj}/c)-1]\exp(i\omega_{s}r_{ml}/c)-\exp[-i\omega_{s}r_{ml}/c-i\omega_{r}r_{mj}/c]}{(\omega_{s}+\omega_{r})\omega_{s}\omega_{r}r_{mj}r_{ml}}$
$\displaystyle+$
$\displaystyle\frac{c^{2}[\omega_{s}\exp[-i(\omega_{s}+\omega_{r})r_{ml}/c]+\omega_{r}]}{(\omega_{s}+\omega_{r})\omega_{r}\omega_{s}r_{jm}r_{ml}}\bigr{)}\bigr{]}\theta(r_{mj}-r_{jl}).$
(37)

If we will consider that $x_{1}>0,$ and $x_{2}>0$ the expression for (37)
takes more simple form

$\displaystyle V(j,l.m)$ $\displaystyle\eqsim$
$\displaystyle\left(\frac{3}{2}\right)^{2}D_{ml}\bigl{[}\frac{\partial}{\partial\omega_{s}}\bigr{]}D_{jm}\bigl{[}\frac{\partial}{\partial\omega_{r}}\bigr{]}$
$\displaystyle\frac{c^{2}\\{\exp[i\omega_{s}r_{ml}/c]-1][\exp[-i\omega_{r}r_{jm}/c]-1]}{\omega_{s}\omega_{r}r_{lm}r_{ml}}$

The expression for exchange integrals described in points $1.-4.$ are used in
the master equation (16)

## References

  * [1] B. Nikolaus, D. Zhang, and P. Toschek, Phys. Rev. Lett. 47, 171(1981).
  * [2] D.J. Gauthier, Q. Wu, S.E. Morin, and T.W. Mossberg, Phys.Rev. Lett. 68, 464 (1992);
  * [3] M.Bruno, J. M. Raimond, S. Haroche, Phys. Rev. A 35, 3888 (1987); M. Bruno, J. M. Raimond, P. Goy, Phys. Rev. Lett. 59,1898 (1987).
  * [4] G.Breit and E.Teller, Astrophys. J . 91, 215 (1940);
  * [5] M. Goeppert-Mayer, Ann Phys. (Leipzig) 9, 273 (1931).
  * [6] L. Spitzer and J.L.Greenstain, Astrophys. J . 144, 215 (1940).
  * [7] J. Shapiro and G. Breit, Phys Rev. 113, 179 (1959).
  * [8] S.G.Alexander and P. M.Eszaros, Astrophysics J. 327, 554 (1991); 565 (1991).
  * [9] N.A.Enaki, Zh. Eksp. Teor.Fiz.94, 135 (1988); N.A.Enaki, Zh. Eksp. Teor.Fiz., 98, 783 (1990); N. Enaki, M. Macovei, Phys. Rev. A 56, 3274 (1997).R. Marrus and R. Schmieder, Phys. Rev. A 5, 1160 (1972).
  * [10] R. Marrus and R. Schmieder, Phys. Rev. A 5, 1160 (1972).
  * [11] R. H. Dicke, Phys. Rev. 93, 99 (1954).
  * [12] N. Enaki, M. Macovei, Phys. Rev. A 56, 3274 (1997).
  * [13] Gross M., Haroche.S. Phys.Repp. 93, 301-396 (1982).
  * [14] Andreev A.V., Emelyanov V.I., and Ilinskii Yu.A. Cooperative Effects in Optics (IOP Publishing, Bristol, 1992).
  * [15] W. Heitler, The Quantum Theory of Radiation 3-ed ed., London 1954, Sec. 8]
  * [16] Y. Cheng, B. Xia 2007 arXiv: 0706.0960v2.
  * [17] X. Lu, J. H. Brownell, and S. R. Hartmann, Laser Physics, 5, 522 (1995).