**8. Conclusions**

22 Quantum Optics and Laser Experiments

(2 ) <sup>1</sup> *<sup>T</sup> <sup>I</sup> c e*

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

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

. This functional gives possibility of calculating all necessary average values

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)

11 1 123

are introduced. The following evolution equation for \* *Fuu t* ( , ,) can be easily obtained

\* \* \* 11 1 \* <sup>1</sup> <sup>1</sup> <sup>1</sup> ( , ,) ( )( , ,) *<sup>t</sup> i Fuu t u u Fuu t*

 

\* \* \* <sup>3</sup> 1 2 \* \* <sup>123</sup> 3 3 1 2 1 2 (12,3) ( ) ( )( ) . . ( , , ) *<sup>u</sup> u u cc F uu t u u uu u u* 

Instead of the generating functional the Glauber-Sudarshan distribution (Glauber, 1969;

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

*u u*

(94)

*F u u d zP z z e*

s+s \* + + <sup>s</sup> α k α k α k α k \* \*

F(u,u ) Sp c ...c c ...c = (-1)

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

*H cc* 

1 123

analogously to (Akhiezer & Peletminskii, 1981) from the Liouville equation

2

*uz uz*

 

1

\* 11 ss 11 s s

u ... u u ... u

<sup>α</sup> <sup>k</sup> <sup>α</sup> <sup>k</sup> <sup>α</sup> <sup>k</sup> <sup>α</sup> <sup>k</sup> u u, =0

\* \* \* F(u,u ) = F(-u,-u ) . (91)

 *kkk* ,

\* <sup>2</sup> \* (, ) (, ) *kk kk*

*k*

{ (12,3) . .} *c c c hc* (92)

 *c* , 11 22 33 (12,3) ( , ; ) 

. (90)

*i i i k* 

\* \* ( )

*uz uz*

 

 (95)

(93)

 

3 3 2

which it is possible to justify the radiation transfer theory.

11 ss 11 s s

shows that the functional obeys the property

, *i i i k c c*

 , *<sup>i</sup> i k* 

\* \* ( ) \* 2\* <sup>1</sup> (, ) (, ) *kk kk*

*k*

,

operator

where notations

*i i i k c c* 

Klauder & Sudarshan, 1968)

*P z z d uF u u e*

 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.
