**7. Connection between correlation functions of different nature and some suitable representations for them**

One can notice that simultaneous correlation functions of field amplitudes of (37) type arise in a natural way in the framework of the reduced description method. At the same time Glauber correlation functions of (19) type (including positive-frequency and negativefrequency parts of the electric field operator (11) in the interaction picture) seem to be observable quantities from the point of view of experimental possibilities. The most interesting effects of quantum optics can be described with non-simultaneous Glauber functions (Lyagushyn & Sokolovsky, 2010a; Lyagushyn et al., 2011). Nevertheless we can insist that there are no real contradictions between the approaches. Correlation functions (19) characterize properties of electromagnetic field described by the statistical operator . In the previous section we have been constructed a reduced description for electromagnetic field in emitter medium and in plasma medium. These theories lead not only to equations for the reduced description parameters but also to the expression for corresponding nonequilibrium statistical operators. For the field-emitters system a nonequilibrium statistical operator has the form

$$\rho\_l(\boldsymbol{\xi}, \boldsymbol{\varepsilon}) = \rho\_l(\mathbf{Z}(\boldsymbol{\xi})) \rho\_\mathbf{m}(\mathbf{X}(\boldsymbol{\varepsilon})) - \frac{i}{\hbar} \int\_{-\eta}^{0} d\boldsymbol{\tau} \Big[ \mathrm{d}\boldsymbol{\tau} \Big[ \rho\_l(\mathbf{Z}(\boldsymbol{\xi})) \rho\_\mathbf{m}(\mathbf{X}(\boldsymbol{\varepsilon})), \hat{E}\_\mathbf{n}^\ell(\mathbf{x}, \boldsymbol{\tau}) \hat{P}\_\mathbf{n}(\mathbf{x}, \boldsymbol{\tau}) \Big] + O(\lambda^2) \,. \tag{73}$$

where <sup>ˆ</sup> (,) *<sup>t</sup> E x <sup>n</sup>* , <sup>ˆ</sup> (,) *P x <sup>l</sup>* are operators <sup>ˆ</sup> ( ) *<sup>t</sup> E x <sup>n</sup>* , <sup>ˆ</sup> ( ) *P x <sup>n</sup>* in the interaction picture. Analogously, a nonequilibrium statistical operator for the field-plasma system is given by the formula

$$\rho(\xi) = \rho\_l(Z(\xi))w\_m - \frac{i}{c\hbar} \int\_{-\alpha}^{0} d\tau \Big[ \mathrm{d}\tau [\rho\_l(Z(\xi)) w\_{m'} \hat{A}\_n(\mathbf{x}, \tau) \hat{j}\_n(\mathbf{x}, \tau)] + O(\lambda^2) \tag{74}$$

where <sup>ˆ</sup> (,) *A x <sup>n</sup>* , <sup>ˆ</sup> (,) *nj <sup>x</sup>* are operators <sup>ˆ</sup> ( ) *A x <sup>n</sup>* , <sup>ˆ</sup> ( ) *nj <sup>x</sup>* in the interaction picture. According to general theory of the Peletminskii-Yatsenko model (Akhiezer & Peletminskii, 1981), the following relations for the field-emitters system

$$\operatorname{Sp} \operatorname{\rho}(\xi, \varepsilon) \mathbf{c}\_{\alpha k} = \operatorname{\operatorname{Sp}} \operatorname{\rho}\_{\mathbf{i}}(\operatorname{Z}(\xi)) \mathbf{c}\_{\alpha k} \, \, \mathsf{y} \operatorname{\,} \rho(\xi, \varepsilon) \mathbf{c}\_{\alpha k}^{+} \mathbf{c}\_{\alpha' k'} = \operatorname{\operatorname{Sp}} \operatorname{\rho}\_{\mathbf{i}}(\operatorname{Z}(\xi)) \mathbf{c}\_{\alpha k}^{+} \mathbf{c}\_{\alpha' k'} \, \, \mathsf{y} \, \, \operatorname{\,} \mathsf{R} \mathbf{x} \, \operatorname{\,} \mathsf{R} \mathbf{x} \, \operatorname{\,} \mathsf{R} \mathbf{x} \, \operatorname{\,} \mathsf{R} \mathbf{x} \, \operatorname{\,} \mathsf{R} \mathbf{x} \, \operatorname{\,} \mathsf{R} \mathbf{x} \, \operatorname{\,} \mathsf{R} \mathbf{x} \, \operatorname{\,} \mathsf{R} \mathbf{x} \, \operatorname{\,} \mathsf{R} \mathbf{x} \, \operatorname{\,} \mathsf{R} \mathbf{x} \, \operatorname{\,} \mathsf{R} \mathbf{x} \, \operatorname{\,} \mathsf{R} \mathbf{x} \, \operatorname{\,} \mathsf{R} \mathbf{x} \, \operatorname{\,} \mathsf{R} \mathbf{x} \, \operatorname{\,} \mathsf{R} \mathbf{x} \, \operatorname{\,} \mathsf{R} \mathbf{x} \, \operatorname{\,} \mathsf{R} \mathbf{x} \, \operatorname{\,} \mathsf{R} \mathbf{x} \, \operatorname{\,} \mathsf{R} \mathbf{x} \, \operatorname{\,} \mathsf{R} \mathbf{x} \, \operatorname{\,} \mathsf{R} \mathbf{x} \, \operatorname{\,} \mathsf{R} \mathbf{x} \, \operatorname{\,} \mathsf{R} \mathbf{x} \, \operatorname{\,} \mathsf{R} \mathbf{x} \, \operatorname{\,$$

and for the field-plasma system

18 Quantum Optics and Laser Experiments

( <sup>ˆ</sup> <sup>ˆ</sup> ( ) *N dxn x a a* ). (66)

ˆ ˆ ( ) Sp ( )[ , ] *<sup>i</sup> M H*

(see (29)). As a result, evolution equations for the reduced

 

 

. (70)

( ) *k* and magnetic susceptibility

*k*

, (71)

; (72)

*c* 

*<sup>i</sup> G xt t w j xt j* 

Re , ( ) *<sup>k</sup>*

*G k <sup>k</sup>*

   . (67)

*k* . (69)

( ) *k* of the

 

define the right-hand sides of evolution equations for the reduced

*<sup>k</sup>* is a frequency of photon emission and absorption. These quantities are

 *k* , 2 () *<sup>k</sup>* 

, (68)

m ˆ ˆ ( ) <sup>m</sup> *a a <sup>a</sup> H NT*

> 

{1 2 ( )} *k k* 

description parameters take the form (Sokolovsky & Stupka, 2004)

, mf

Integral equation (65) is solvable in a perturbation theory in plasma-field interaction based

<sup>3</sup> ( ) ( )( ) () *t kk k k kk k k kk k kk gi g gn O*

<sup>3</sup> \* , ( ) ( ) () *t k k k k k kk k x i x ixO*

where *k* is photon spectrum in the plasma, *nk* is the Planck distribution with the plasma

The second equation in (68) is a form of the Maxwell equations (53) with similar to (54)

<sup>3</sup> (,) ( ) ( ) ( ) ( ) ( ) *n nn J x dx x x E x c dx x x Z x O*

This material equation takes into account spatial dispersion and Fourier transformed

*mn mn m n* , m m ˆ ˆ *mn* , Sp [ ( , ), (0)] *m n*

2 2 4

*a a a a n e m*

.

 

*w e*

( ) ( ) ( ( )) *<sup>t</sup> t ic tM t*

 

 , <sup>2</sup> 2 ˆ *H* ~ 

Functions ( ) *M*

 

1 ˆ *H* ~ 

plasma medium. Their values are given by relations

<sup>1</sup> ( , ) ( , )( ) <sup>2</sup> *Gk G k k k*

  ( ) *x* give conductivity

Im , ( ) *<sup>k</sup>*

*G k <sup>k</sup>*

*k*

,

is a transversal part of current-current Green function:

In fact, the obtained results are valid for *kc* where is Langmuir frequency.

description parameters

on estimations <sup>1</sup>

temperature,

given by formulas

material equation

( ) *x* , 

functions

where *G k*(, )

$$\text{Sp}\,\rho(\xi)\mathbf{c}\_{ak} = \text{Sp}\_{\text{f}}\,\rho\_{\text{i}}(\text{Z}(\xi))\mathbf{c}\_{ak}\,\,\text{,}\qquad \text{Sp}\,\rho(\xi)\mathbf{c}\_{ak}^{+}\mathbf{c}\_{a'k'} = \text{Sp}\_{\text{f}}\,\rho\_{\text{i}}(\text{Z}(\xi))\mathbf{c}\_{ak}^{+}\mathbf{c}\_{a'k}\,\,\text{,}\tag{76}$$

are valid. Average of products of three and more Bose operators should be calculated with taking into account the second term in expressions (73), (74) and using the Wick–Bloch–de Dominicis theorem. It is convenient to perform the calculation of correlation functions (23) for the field-plasma system through using formulas (11), (74). For the field-emitters system the following formula

$$
\hat{E}\_n^{(+)} (\mathbf{x}, t) = i \int d\mathbf{x}' \{ D\_+ (\mathbf{x} - \mathbf{x}', t) \hat{Z}\_n (\mathbf{x}') + \frac{1}{c} \dot{D}\_+ (\mathbf{x} - \mathbf{x}', t) \hat{E}\_n^{\dagger} (\mathbf{x}') \} \tag{77}
$$

Description of Field States with Correlation Functions and Measurements in Quantum Optics 21

ˆ ˆ <sup>ˆ</sup> ( ) { ( ), ( )} <sup>8</sup> *<sup>n</sup> nlm l m <sup>c</sup> q x ExB x* 

In the developed above theory average values of binary in the field quantities can be calculated exactly. For the field-plasma model the following result can be obtained in terms

> ( ) (,) ( , )f ( ) *iqx n kq kq n n k kq k*

*q xn k q e ki x V V x*

1 2 12 1 2 <sup>1</sup> (,) ( )( ) { } <sup>2</sup> *<sup>n</sup> nl ms ml ns l ks km l km ks k k kk k e e k e e*

For a weakly nonuniform states of the field formula (82) can be simplified and gives (at

3 <sup>3</sup> ( ) f () (2 ) *<sup>k</sup> <sup>n</sup> k k*

*d k q x x*

Formula (83) should be put in the basis of the theory of radiation transfer. The simplest consideration is based on the approximate expression (85). Radiation transfer can be

Therefore, an equation of radiation transfer can be based on the kinetic equation for the Wigner distribution function of the field. According to definition (43) and equation (68), for weakly nonuniform states in the absence of the average field this kinetic equation is written

f f <sup>1</sup> <sup>f</sup> 2 (f ) <sup>4</sup>

The radiation transfer equation follows from the definition (86) and kinetic equation (87)

*k k k k*

 

<sup>2</sup> (,) (,) (,) 2 { (,) } { }

*I nx I nx I nx c n I n x I a nn b*

*n n nl nl n k x kk xx*

> 

*a nn b*

 , *<sup>k</sup> k n <sup>c</sup>*

3 3 2 (,) f () (2 ) *<sup>k</sup> k n <sup>c</sup> I nx x*

*t k k k k kk*

*t l l m lm*

 

*k*

*k k*

*l m k n <sup>c</sup>*

 *c*

 

. (82)

, (84)

11 22 11 22

. (85)

(| | 1 *<sup>n</sup>* ) (86)

2 2

  

 

 

  *a*

; (89)

 

1 2 \* \*

*n*

'

*l l m*

*l m lm*

 

*x x x*

. (87)

 

, (88)

*k*

.

(83)

  2 2

*c c*

, ,

of the one-particle density matrix and Wigner distribution function

 ( , ) ( /2, /2) *n n kq k q k q* 

described with specific intensity of radiation in the form

*l*

*c n*

, <sup>2</sup>

*k*

*k*

*<sup>l</sup> k n <sup>c</sup>*

 

1 2

 

*V* ) a classic expression

where

as follows

where the notations

 /2, /2

 

can be useful. Here *D xt* ( ,) is a standard function widely used in electromagnetic theory (Akhiezer A. & Berestetsky V., 1969) and defined by expression

$$D\_{+}(\mathbf{x},t) \equiv \frac{1}{2} \int \frac{d^3k}{(2\pi)^3} \frac{1}{k} e^{i(kx - \alpha\_k t)} \,. \tag{78}$$

Calculation of the simplest correlation function (1,1) 1 1 (,) *G yy nl* can be done according to (75), (76) exactly. For example, for the field-plasma system one has

$$\mathbf{G}^{(1,1)}\_{\rm mm}(\mathbf{x},t;\mathbf{x}',\mathbf{t}') = \sum\_{\mathbf{k}\alpha,\mathbf{k}'\alpha'} \frac{2\pi\hbar c}{V} (\mathbf{k}\mathbf{k}')^{\mathbf{1}\dagger 2} \varepsilon\_{\alpha\mathbf{k}\mathbf{m}}^{\*} \varepsilon\_{\alpha'\mathbf{k}'\mathbf{n}} e^{i(\alpha\_{\mathbf{k}}t - \mathbf{k}\mathbf{x})} e^{-i(\alpha\_{\mathbf{k}}t' - \mathbf{k}'\mathbf{x}')} n\_{\mathbf{k}\mathbf{k}'}^{\alpha\alpha'} \tag{79}$$

An exact expression for this correlation function of the field-emitters system is given by the formula

$$\begin{aligned} G\_{mn}^{(1,1)}(\mathbf{x},t;\mathbf{x}',t') &= \int d\mathbf{x}\_1 d\mathbf{x}'\_1 \left\{ D\_+^\dagger(\mathbf{x}-\mathbf{x}\_1,t) D\_+(\mathbf{x}'-\mathbf{x}'\_1,t') \langle Z\_m^{\mathbf{x}\_1} Z\_n^{\mathbf{x}\_1'} \rangle + \\ &+ \frac{1}{c^2} D\_+^\dagger(\mathbf{x}-\mathbf{x}\_1,t) \bar{D}\_+(\mathbf{x}'-\mathbf{x}'\_1,t') \langle E\_m^{\mathbf{x}\_1'} E\_n^{\mathbf{x}\_1'} \rangle + \\ \\ &+ [D\_+^\dagger(\mathbf{x}-\mathbf{x}\_1,t) \bar{D}\_+(\mathbf{x}'-\mathbf{x}'\_1,t') + \frac{1}{c} \bar{D}\_+^\dagger(\mathbf{x}-\mathbf{x}\_1,t) D\_+(\mathbf{x}'-\mathbf{x}'\_1,t')] \times \\ \\ &\times [\langle E\_m^{\mathbf{x}\_1'} Z\_n^{\mathbf{x}\_1'} \rangle + 2i\pi\hbar c \langle \delta\_{mn} \Delta\_1 - \frac{\bar{\sigma}^2}{\mathfrak{D}\mathbf{x}\_{1m}\mathfrak{D}\mathbf{x}\_{1n}} \rangle \delta(\mathbf{x}\_1 - \mathbf{x}'\_1)] \Big]. \end{aligned}$$

Correlation function (2,2) 12 12 (,) *G yy yy nl* can be calculated only approximately. For example, for the field-plasma system the formula

$$\mathbf{G}\_{mn}^{(1,1)}(y\_1, y\_2; y\_1', y\_2') = \mathbf{G}\_{m\_1 n\_1}^{(1,1)}(y\_1, y\_1') \mathbf{G}\_{m\_2 n\_2}^{(1,1)}(y\_2, y\_2') + \mathbf{G}\_{m\_1 n\_2}^{(1,1)}(y\_1, y\_2') \mathbf{G}\_{m\_2 n\_1}^{(1,1)}(y\_2, y\_1') + \mathbf{O}(\boldsymbol{\lambda}^1) \,. \tag{81}$$

is obtained.

So, the method of the reduced description of nonequilibrium states allows calculating Glauber correlation functions in important models. It gives possibility to analyze correlation properties of electromagnetic field interacting with emitters and plasma in the considered examples. Such analysis can be performed in terms of average electromagnetic field and binary correlations of the field.

Quantum theory of radiation transfer is an important part of quantum optics (Perina, 1984). The problem is: to choose parameters that describe radiation transfer in a medium and obtain a closed set of equations for such parameters. This problem can be solved in the reduced description method.

In the theory of radiation transfer (Chandrasekhar, 1950) energy fluxes in medium and polarization of the radiation are problems of interest. Operator of energy flux is given by the formula

$$
\hat{q}\_n(\mathbf{x}) = \frac{c}{8\pi} \varepsilon\_{nlm} \{ \hat{E}\_l(\mathbf{x}), \hat{B}\_m(\mathbf{x}) \}\ . \tag{82}
$$

In the developed above theory average values of binary in the field quantities can be calculated exactly. For the field-plasma model the following result can be obtained in terms of the one-particle density matrix and Wigner distribution function

$$\eta\_n(\mathbf{x}) = \frac{\hbar c^2}{V} \sum\_{\mathbf{k}\eta,\alpha\alpha'} \eta\_{\mathbf{k}-\mathbf{q}/2,k+q/2}^{\alpha\alpha'} \langle k,\mathbf{q} \rangle e^{i\mathbf{q}\mathbf{x}} = \frac{\hbar c^2}{V} \sum\_{\mathbf{k},\alpha\alpha'} \phi\_n^{\alpha\alpha'}(\mathbf{k},-\mathbf{i}\frac{\partial}{\partial \mathbf{x}}) \mathbf{f}\_{\mathbf{k}}^{\alpha\alpha'}(\mathbf{x}) \tag{83}$$

where

20 Quantum Optics and Laser Experiments

can be useful. Here *D xt* ( ,) is a standard function widely used in electromagnetic theory

3

(1,1) 1 2 \* ( )( )

*<sup>c</sup> G xtx t kk e e e e n*

<sup>2</sup> ( ,; , ) ( ) *k k i t kx i t k x mn km k n kk*

An exact expression for this correlation function of the field-emitters system is given by the

 1 1 (1,1) \* 11 1 <sup>1</sup> ( ,; , ) ( ,) ( , ) *x x G x t x t dx dx D x x t D x x t Z Z mn m n*

> 2 1 1 <sup>1</sup> ( ,) ( , ) *x x D x x tD x x t E E m n <sup>c</sup>*

\* \*

*m n mn*

 

1 1

Correlation function (2,2)

binary correlations of the field.

reduced description method.

is obtained.

formula

for the field-plasma system the formula

1 1 \*

11 11 <sup>1</sup> [ ( , ) ( , ) ( , ) ( , )] *D x x tD x x t D x x tD x x t <sup>c</sup>*

[ 2( ) ( )] *x x*

*EZ i c x x*

2 1 1 1 1 1

*x x*

1 1 2 2 1 2 2 1 (1,1) (1,1) (1,1) (1,1) (1,1) 1 <sup>1212</sup> 11 22 12 21 *mn* (,;,) (,) (,) (,) (,) () *G yy yy G yyG y y G yyG y y O m n m n m n m n*

So, the method of the reduced description of nonequilibrium states allows calculating Glauber correlation functions in important models. It gives possibility to analyze correlation properties of electromagnetic field interacting with emitters and plasma in the considered examples. Such analysis can be performed in terms of average electromagnetic field and

Quantum theory of radiation transfer is an important part of quantum optics (Perina, 1984). The problem is: to choose parameters that describe radiation transfer in a medium and obtain a closed set of equations for such parameters. This problem can be solved in the

In the theory of radiation transfer (Chandrasekhar, 1950) energy fluxes in medium and polarization of the radiation are problems of interest. Operator of energy flux is given by the

*m n*

.

12 12 (,) *G yy yy nl* can be calculated only approximately. For example,

 

1 1 ( ,) <sup>2</sup> (2 ) *<sup>k</sup> d k i kx t D xt <sup>e</sup>*

3

*k*

( )

(79)

(80)

. (78)

1 1 (,) *G yy nl* can be done according to (75),

. (81)

 

(Akhiezer A. & Berestetsky V., 1969) and defined by expression

(76) exactly. For example, for the field-plasma system one has

,

*V*

*k k*

 

Calculation of the simplest correlation function (1,1)

formula

$$
\phi\_n^{\alpha\alpha^\*} (k, q) \equiv \phi\_n^{\alpha\alpha^\*} \left( k - q \mid \text{ $\heartsuit$  k + q  $/ \text{$ \heartsuit $}$ } \right), \tag{84}
$$

$$\log \sigma\_n^{a\_1 a\_2} (k\_1, k\_2) = \frac{1}{2} (\delta\_{nl} \delta\_{ms} - \delta\_{ml} \delta\_{ns}) (k\_1 k\_2)^{1/2} \{ \tilde{k}\_{1l} \underline{e}\_{a\_1 k\_1 s}^\* e\_{a\_2 k\_2 m} + \tilde{k}\_{2l} \underline{e}\_{a\_1 k\_1 m}^\* e\_{a\_2 k\_2 s} \} \dots$$

For a weakly nonuniform states of the field formula (82) can be simplified and gives (at *V* ) a classic expression

$$\eta\_n(\mathbf{x}) = \sum\_{\alpha} \int \frac{d^3k}{(2\pi)^3} \alpha\_k \hbar \frac{\partial \alpha\_k}{\partial k\_n} \mathbf{f}\_k^{\alpha \alpha}(\mathbf{x}) \,. \tag{85}$$

Formula (83) should be put in the basis of the theory of radiation transfer. The simplest consideration is based on the approximate expression (85). Radiation transfer can be described with specific intensity of radiation in the form

$$\left.I\right|\_{\alpha}^{\alpha\alpha'}\left(n,\infty\right)\equiv\frac{\alpha^3\hbar}{\left(2\pi\right)^3c^2}\left.\mathbf{f}\_k^{\alpha\alpha'}\left(\mathbf{x}\right)\right|\_{k=n\frac{\alpha}{c}}\qquad\left(\left|\left.n\right|\equiv 1\right)\tag{86}$$

Therefore, an equation of radiation transfer can be based on the kinetic equation for the Wigner distribution function of the field. According to definition (43) and equation (68), for weakly nonuniform states in the absence of the average field this kinetic equation is written as follows

$$
\hat{\sigma}\_t \mathbf{f}\_k^{\alpha a'} = -\frac{\partial \mathfrak{Q}\_k}{\partial \mathbf{k}\_n} \frac{\partial \mathbf{f}\_k^{\alpha a'}}{\partial \mathbf{x}\_n} - 2\nu\_k (\mathbf{f}\_k^{\alpha a'} - \mathfrak{n}\_k \delta\_{kk'} \delta\_{\alpha a'}) + \frac{1}{4} \frac{\partial^2 \nu\_k}{\partial \mathbf{k}\_n \partial \mathbf{k}\_l} \frac{\partial^2 \mathbf{f}\_k^{\alpha a'}}{\partial \mathbf{x}\_n \partial \mathbf{x}\_l}. \tag{87}
$$

The radiation transfer equation follows from the definition (86) and kinetic equation (87)

$$\left[\hat{\sigma}\_{l}\mathbf{I}\_{oo}^{\alpha\alpha'}\left(\mathbf{n},\mathbf{x}\right) = -\sigma\_{oo}n\_{l}\frac{\hat{\sigma}\mathbf{I}\_{oo}^{\alpha\alpha'}\left(\mathbf{n},\mathbf{x}\right)}{\hat{\sigma}\mathbf{x}\_{l}} - 2\nu\_{oo}\{\mathbf{I}\_{oo}^{\alpha\alpha'}\left(\mathbf{n},\mathbf{x}\right) - \mathbf{I}\_{o}\delta\_{\alpha\alpha'}\} + \left\{a\_{oo}n\_{l}n\_{m} + b\_{oo}\delta\_{lm}\right\}\frac{\hat{\sigma}^{2}\mathbf{I}\_{oo}^{\alpha\alpha'}\left(\mathbf{n},\mathbf{x}\right)}{\hat{\sigma}\mathbf{x}\_{l}\hat{\sigma}\mathbf{x}\_{m}}\right\},\tag{88}$$

where the notations

$$\left. \frac{\partial \tilde{\alpha}\_{k}}{\partial k\_{l}} \right|\_{k = \underbrace{\alpha \nu}\_{c} n} \equiv c\_{\alpha \nu} n\_{l'} \qquad \left. \frac{\partial^{2} \nu\_{k}}{\partial k\_{l} \partial k\_{m}} \right|\_{k = \frac{\alpha \nu}{c} n} \equiv a\_{\alpha \nu} n\_{l} n\_{m} + b\_{\alpha \nu} \delta\_{lm'} \qquad \nu\_{k} \Big|\_{k = \frac{\alpha \nu}{c} n} \equiv \nu\_{\alpha \nu} \tag{89}$$

Description of Field States with Correlation Functions and Measurements in Quantum Optics 23

picture in terms of *P*-function is very attractive from the point of view of analysis of field

Kinetic theory of electromagnetic field in media has choosing a set of parameters describing nonequilibrium states of the field as a starting point with necessity. The minimal set of such parameters includes binary correlations of field amplitudes. The corresponding mathematical apparatus uses different structures of averages: one-particle density matrices, Wigner distribution functions, and conventional simultaneous correlation functions of field operators. All approaches can be connected with each other due to the possibility of expressing the main correlation parameters in various forms. The reduced description method elucidates the construction of kinetic equations in electrodynamics of continuous media (field-plasma, field-emitters systems) and radiation transfer theory. Electromagnetic field properties are discussed in quantum optics in terms of Glauber correlation functions measured in experiments. Theoretical calculation of such functions requires information about the statistical operator of the system under investigation. In the framework of the reduced description method we have succeeded in obtaining the statistical operator of the

field in the form that is convenient for calculations in a number of interesting cases.

The work was partially supported by the State Foundation for Fundamental Research of

Akhiezer, A. & Peletminskii, S. (1981). *Methods of Statistical Physics*, Pergamon Press, New

Akhiezer, A. & Berstetsky, V. (1969). Q*uantum Electrodynamics*, Nauka, Moscow, USSR (in

Banfi, G. & Bonifacio, R. (1975). Superfluorescence and Cooperative Frequency Shift. *Physical* 

Bogolyubov, N. (1946). *Problems of Dynamical Theory in Statistical Physics*, Gostekhizdat,

Bogolyubov, N. (Jr.) & Shumovsky, A. (1987). *Superradiance,* JINR publication P17-87-176,

Bogolyubov, N. (Jr.), Kozierowski M., Trang, Q. & Shumovsky A. (1988). New Effects in

De Groot S. & Suttorp L. (1972). *Fundamentals of Electrodynamics*, North Holland Publishing

Dicke, R. (1954). Coherence in Spontaneous Radiation Processes. *Physical Review,* Vol.93,

Glauber, R. (1963). The Quantum Theory of Optical Coherence. *Physical Review,* Vol.130,

Glauber R. (1965). Optical Coherence and Photon Statistics (Les Houches, 1964), In: *Quantum* 

*Optics and Electronics,* C. DeWitt, A. Blandin, C. Cohen-Tannoudji (Eds.), pp. 63-107,

Issue 4, (August 1988), pp. 831-863, ISSN 0367-2026 (in Russian)

Quantum Electrodynamics. *Physics of Elementary Particles and Atomic Nuclei,* Vol.19,

*Review,* Vol.A12, (November 1975), pp. 2068-2082

Moscow-Leningrad, USSR (in Russian)

Chandrasekhar S. (1950). *Radiative Transfer*, Oxford, UK

Company, Amsterdam, Netherlands

No.1, (January 1954), pp. 99-110

Issue 6, (June 1963), pp. 2529-2539

Gordon and Breach, New York, USA

Dubna, USSR (in Russian)

properties under consideration in quantum optics.

**8. Conclusions** 

**9. Acknowledgement** 

**10. References** 

Ukraine (project No. 25.2/102).

York, USA

Russian)

$$I\_{\alpha} \equiv \frac{\alpha^3 \hbar}{(2\pi)^3 c^2} \frac{1}{e^{\hbar \alpha \dagger T} - 1}$$

are introduced. Usually this equation is written for stationary states and given without correction with the last term. So, the reduced description method provides an approach in which it is possible to justify the radiation transfer theory.

In quantum optics functional methods are widely used. Starting point of such methods is a definition of a generating functional (3) for average values calculated with considered statistical operator . This functional gives possibility of calculating all necessary average values

$$\mathbf{S}\mathbf{p}\boldsymbol{\rho}\mathbf{c}\_{\boldsymbol{\alpha}\_{1}\mathbf{k}\_{1}}^{\*}\ldots\mathbf{c}\_{\boldsymbol{\alpha}\_{s}\mathbf{k}\_{s}}^{\*}\mathbf{c}\_{\boldsymbol{\alpha}\_{1}'\mathbf{k}\_{1}'}\ldots\mathbf{c}\_{\boldsymbol{\alpha}\_{s}'\mathbf{k}\_{s}'} = (-1)^{s'} \frac{\partial^{s+s'}\mathbf{F}(\mathbf{u},\mathbf{u}^{\*})}{\partial\mathbf{u}\_{\mathbf{u}\_{1}\mathbf{k}\_{1}}\ldots\partial\mathbf{u}\_{\mathbf{u}\_{s}\mathbf{k}\_{s}}\partial\mathbf{u}^{\*}\_{\mathbf{u}\_{1}'\mathbf{k}\_{1}'}\ldots\partial\mathbf{u}^{\*}\_{\mathbf{u}\_{s}'\mathbf{k}\_{s}'}\Big|\_{\mathbf{u},\mathbf{u}^{\*}\mathbf{u}}.\tag{90}$$

Hence, the generating functional gives complete description of a system and evolution equation for this functional is equivalent to the quantum Liouville equation. Definition (3) shows that the functional obeys the property

$$\left(\mathbf{F}(\mathbf{u}, \mathbf{u}^\*)\right)^\* = \mathbf{F}(\mathbf{-u}, \mathbf{-u}^\*) \cdot \tag{91}$$

Let us suppose that effective photon interaction in a system has the form

$$\hat{H} = \sum\_{1} \varepsilon\_{1} c\_{1}^{+} c\_{1} + \sum\_{123} \{\Phi(12, 3)c\_{1}^{+} c\_{2}^{+} c\_{3} + h.c.\} \tag{92}$$

where notations

$$\mathbf{c}\_{i} = \mathbf{c}\_{a;k\_{i}} \, \prime \qquad \mathbf{c}\_{i}^{+} = \mathbf{c}\_{a;k\_{i}}^{+} \, \prime \qquad \mathbf{c}\_{i} = a \mathbf{e}\_{k\_{i}} \, \prime \qquad \Phi(12.3) = \Phi(a\_{1}k\_{1}, a\_{2}k\_{2}; a\_{3}k\_{3}) \, \prime \qquad \sum\_{i} = \sum\_{a\_{i}k\_{i}} \tag{93}$$

are introduced. The following evolution equation for \* *Fuu t* ( , ,) can be easily obtained analogously to (Akhiezer & Peletminskii, 1981) from the Liouville equation

$$i\hbar \hat{\boldsymbol{\alpha}}\_{1} \boldsymbol{F}(\boldsymbol{\mu}, \boldsymbol{u}^{\*}, t) = \sum\_{1} \varepsilon\_{1} (\boldsymbol{u}\_{1} \frac{\partial}{\partial \boldsymbol{u}\_{1}} - \boldsymbol{u}\_{1}^{\*} \frac{\partial}{\partial \boldsymbol{u}\_{1}^{\*}}) \boldsymbol{F}(\boldsymbol{\mu}, \boldsymbol{u}^{\*}, t) + \tag{94}$$

$$+ \sum\_{123} \left\{ \Phi(12, 3) \Big( (\boldsymbol{u}\_{3} - \frac{\partial}{\partial \boldsymbol{u}\_{3}^{\*}}) \frac{\partial^{2}}{\partial \boldsymbol{u}\_{1} \partial \boldsymbol{u}\_{2}} + (\boldsymbol{u}\_{1}^{\*} - \frac{\partial}{\partial \boldsymbol{u}\_{1}}) (\boldsymbol{u}\_{2}^{\*} - \frac{\partial}{\partial \boldsymbol{u}\_{2}}) \frac{\partial}{\partial \boldsymbol{u}\_{3}^{\*}} \Big) + c.c. \right\} \boldsymbol{F}(\boldsymbol{u}, \boldsymbol{u}^{\*}, t)$$

Instead of the generating functional the Glauber-Sudarshan distribution (Glauber, 1969; Klauder & Sudarshan, 1968)

$$P(z, z^\*) = \frac{1}{\pi} \left[ d^2 u F(u, u^\*) e^{\frac{\sum \left( u\_{ak}^\* z\_{ak} - u\_{ak} z\_{ak}^\* \right)}{\epsilon}} \right], \quad F(u, u^\*) = \int d^2 z P(z, z^\*) e^{\frac{\sum \left( u\_{ak} z\_{ak}^\* - u\_{ak}^\* z\_{ak} \right)}{\epsilon}} \tag{95}$$

is widely used. Formula (95) shows that this distribution is the Fourier transformed generating functional. Note that an evolution equation for the Glauber-Sudarshan distribution can be easily obtained by substituting the second formula in (95) into equation (94). Such evolution equations can be a starting point for constructing the reduced description of a system (Peletminskii, S. & Yatsenko A., 1970). Obtaining the field evolution picture in terms of *P*-function is very attractive from the point of view of analysis of field properties under consideration in quantum optics.
