**6. Reduced description of electromagnetic field in medium. Role of field correlations**

Here we discuss kinetics of electromagnetic field in a medium. This theory must connect dynamics of the field with dynamics of the medium. The problem can be solved only on the basis of the reduced description of a system. One has to choose a set of microscopic quantities in such way that their average values describe the system completely. Therefore, the Bogolyubov reduced description method (Akhiezer & Peletminskii, 1981) can be a basis for the general consideration of the problem. In this approach its starting point is a quantum Liouville equation for the statistical operator ( )*t* of a system including electromagnetic field and a medium

$$
\hat{\mathcal{O}}\_t \varphi(t) = -\frac{\dot{\mathcal{I}}}{\hbar} [\hat{H}, \varphi(t)] \,, \qquad \hat{H} = \hat{H}\_\mathbf{f} + \hat{H}\_\mathbf{m} + \hat{H}\_\mathbf{mf} \,. \tag{30}
$$

The method is based on the functional hypothesis describing a structure of the operator ( )*t* at large times (Bogolyubov, 1946)

$$
\rho(\mathbf{t}) \xrightarrow[t \gg \tau\_o]{} \rho(\tilde{\mathbf{s}}(\mathbf{t}, \rho\_0), \eta(\mathbf{t}, \rho\_0)) \equiv \rho^{(+)}(\mathbf{t}) \qquad \left(\rho\_0 \equiv \rho(\mathbf{t} = 0) \text{ )}\right) \tag{31}
$$

where reduced description parameters of the field 0 (, ) *t* and matter 0 (, ) *<sup>a</sup> t* are defined in a natural way

$$
\xi\_{\mu}(t,\rho\_0) = \text{Sp}\,\rho^{(+)}(t)\hat{\xi}\_{\mu} \, \, \, \qquad \eta\_a(t,\rho\_0) = \text{Sp}\,\rho^{(+)}(t)\hat{\eta}\_a \, \, \, \, \tag{32}
$$

( <sup>0</sup> is a characteristic time determined by an initial state of the system 0 and a used set of reduced description parameters). The set of parameters 0 (, ) *t* , 0 (, ) *<sup>a</sup> t* is determined by the possibilities and traditions of experiments as well as by theoretical considerations (for simplicity we will drop 0 in the parameters). The development of the problem investigation has resulted in finding the main approximation for the statistical operator (,) , so called a quasiequilibrium statistical operator ( ( ), ( )) *<sup>q</sup> Z X* (though it describes states which are far from the equilibrium) defined by the relations

$$
\rho\_q(Z, Z\_m) = \rho\_\mathbf{i}(Z)\rho\_\mathbf{m}(Z\_m);\tag{33}
$$

$$\rho\_{\mathbf{i}}(\mathbf{Z}) = \exp\{\Phi(\mathbf{Z}) - \sum\_{\mu} \mathbf{Z}\_{\mu} \hat{\tilde{\xi}}\_{\mu}\} \quad , \quad \text{Sp}\_{\mathbf{i}} \,\rho\_{\mathbf{i}}(\mathbf{Z}) = 1 \quad , \quad \text{Sp}\_{\mathbf{i}} \,\rho\_{\mathbf{i}}(\mathbf{Z}(\xi)) \hat{\tilde{\xi}}\_{\mu} = \tilde{\xi}\_{\mu} \; ; \tag{34}$$

$$\rho\_{\mathbf{m}}(X) = \exp\{\Omega(X) - \sum\_{a} X\_{a}\widehat{\eta}\_{a}\} \ , \quad \text{Sp}\_{\mathbf{m}}\rho\_{\mathbf{m}}(X) = 1 \ , \quad \text{Sp}\_{\mathbf{m}}\rho\_{\mathbf{m}}(X(\eta))\widehat{\eta}\_{a} = \eta\_{a} \ . \tag{35}$$

According to the common idea, electromagnetic field in medium is usually described by average values of electric ( ,) *E xt <sup>n</sup>* and magnetic ( ,) *B xt <sup>n</sup>* fields. So, it seems possible to choose operators ˆ in (32) as <sup>ˆ</sup> ( ) *<sup>n</sup> <sup>x</sup>* : 1 ˆ ˆ () () *<sup>t</sup> n n x Ex* , 2 ˆ ˆ () () *n n x Bx* . However, in this case the statistical operator f ( ) *Z* does not exist (its exponent contains only linear in Bose amplitudes form and f ( ) *Z* is non-normalized). Therefore, one has to use a wider set of parameters ˆ in conformity with the observation made in (Peletminkii et al., 1975). At least, exponent in (34) should contain quadratic terms. So the simplest quasiequilibrium statistical operator of the field can be written as

$$\rho\_{\mathbf{i}}(\mathbf{Z}) = \exp\{\Phi(\mathbf{Z}) - \sum\_{\alpha k, \alpha' k'} Z\_{kk'}^{\alpha \alpha'} c\_{\alpha k}^+ c\_{\alpha' k'} - \{\sum\_{\alpha k, \alpha' k'} \tilde{Z}\_{kk'}^{\alpha \alpha'} c\_{\alpha k}^+ c\_{\alpha' k'}^+ + \sum\_{\alpha k} Z\_k^\alpha c\_{\alpha k}^+ + h.c.\}\tag{36}$$

Kinetics of the field based on this statistical operator describes states with zero average fields at 0 *Zk* . Quadratic terms in (36) correspond to binary fluctuation of the field *x x m nt* (or two binary correlations ( ) *x x m nt* ) as additional reduced description parameters

$$\text{Sp } \rho^{(+)} (\mathbf{t}) \frac{1}{2} \{\hat{\xi}\_{am} (\mathbf{x}), \hat{\xi}\_{\alpha' n} (\mathbf{x'})\} = \langle \xi\_{am}^{\mathbf{x}} \xi\_{\alpha' n}^{\mathbf{x'}} \rangle\_{\mathbf{t}}, \qquad \langle \xi\_{am}^{\mathbf{x}} \xi\_{\alpha' n}^{\mathbf{x'}} \rangle\_{\mathbf{t}} = \langle \xi\_{am}^{\mathbf{x}} \xi\_{\alpha' n}^{\mathbf{x'}} \rangle\_{\mathbf{t}} - \xi\_{am} \langle \mathbf{x}, \mathbf{t} \rangle \xi\_{\alpha' n} \langle \mathbf{x'}, \mathbf{t} \rangle \tag{37}$$

In other words, the quasiequilibrium statistical operator (34) corresponds to field description by average values of operators

$$
\hat{\xi}\_{\mu} \colon \quad \hat{\xi}\_{\alpha n}(\mathbf{x}) \; , \quad \frac{1}{2} \{ \hat{\xi}\_{\alpha m}(\mathbf{x}) , \hat{\xi}\_{\alpha' n}(\mathbf{x'}) \} \; . \tag{38}
$$

The theory can be significantly simplified in the Peletminskii-Yatsenko model (Akhiezer & Peletminskii, 1981) in which

$$\frac{1}{\hbar}[\hat{H}\_{\text{f}}, \hat{\xi}\_{\mu}] = \sum\_{\mu'} c\_{\mu\mu'} \hat{\xi}\_{\mu'} \ . \tag{39} \\ \qquad \qquad \frac{1}{\hbar}[\hat{H}\_{\text{m}}, \hat{\eta}\_{a}] = \sum\_{\text{a}\text{i}'} c\_{\text{a}\text{i}'} \hat{\eta}\_{a'} \tag{39}$$

where *c* , *aa <sup>c</sup>* are some coefficients. Operators of electromagnetic field <sup>ˆ</sup> ( ) *<sup>t</sup> E x <sup>n</sup>* , <sup>ˆ</sup> ( ) *B x <sup>n</sup>* and operator ˆ( ) *x* satisfy these conditions

$$[\hat{H}\_{\mathbf{f}'}\hat{E}\_{n}^{t}(\mathbf{x})] = -ic\hbar \operatorname{rot}\_{n}\hat{B}(\mathbf{x}) \, \, \, \qquad [\hat{H}\_{\mathbf{f}'}\hat{B}\_{n}(\mathbf{x})] = ic\hbar \operatorname{rot}\_{n}\hat{E}^{t}(\mathbf{x}) \, \, \, \, \tag{40}$$

$$[\hat{H}\_{\mathfrak{m}'}\hat{\varepsilon}(\mathfrak{x})] = 0\_{\mathfrak{m}'}$$

therefore, relations (39) are valid for all field operators in (38).

In usual kinetic theory nonequilibrium states of quantum system are described by oneparticle density matrix ( ) *n t kk* 

$$m\_{kk'}^{aa'}(t) \equiv \text{Sp } \rho^{(+)}(t) c\_{ak}^+ c\_{a'k'}\,. \tag{41}$$

States, for which parameters

12 Quantum Optics and Laser Experiments

<sup>ˆ</sup> ( ) *nj <sup>x</sup>* is electric current, ˆ ( ) *n x <sup>a</sup>* is density operator of the *<sup>a</sup>* th component of the system.

**6. Reduced description of electromagnetic field in medium. Role of field** 

Here we discuss kinetics of electromagnetic field in a medium. This theory must connect dynamics of the field with dynamics of the medium. The problem can be solved only on the basis of the reduced description of a system. One has to choose a set of microscopic quantities in such way that their average values describe the system completely. Therefore, the Bogolyubov reduced description method (Akhiezer & Peletminskii, 1981) can be a basis for the general consideration of the problem. In this approach its starting point is a quantum

The method is based on the functional hypothesis describing a structure of the operator

( <sup>0</sup>

( )

 

 

, ( )

the possibilities and traditions of experiments as well as by theoretical considerations (for

investigation has resulted in finding the main approximation for the statistical operator

*q m* (, ) () ( ) *ZZ Z Z* 

 , f f Sp ( ) 1 

 , Sp ( ) 1 m m

According to the common idea, electromagnetic field in medium is usually described by average values of electric ( ,) *E xt <sup>n</sup>* and magnetic ( ,) *B xt <sup>n</sup>* fields. So, it seems possible to

( )*t* of a system including electromagnetic

 and matter 0 (, ) *<sup>a</sup> t* 

 , 0 (, ) *<sup>a</sup> t* 

f mm ; (33)

 *a a* . (35)

; (34)

 

( 0) *t* ) (31)

(32)

are defined

0 and a used set of

(though it describes

is determined by

, f m mf *HH H H* ˆˆ ˆ ˆ . (30)

 

<sup>0</sup> ( , ) Sp ( ) ˆ *a a*

0 in the parameters). The development of the problem

 *<sup>Z</sup>* , f f <sup>ˆ</sup> Sp ( ( )) *Z*

*X* , m m Sp ( ( )) ˆ

*X*

*<sup>q</sup> Z X* 

 

 *t t* 

> 

*Hm* is the Hamilton operator of plasma particles with account of Coulomb interaction,

Here ˆ

**correlations** 

field and a medium

in a natural way

simplicity we will drop

( <sup>0</sup> 

> (,)

Liouville equation for the statistical operator

( )*t* at large times (Bogolyubov, 1946)

*<sup>i</sup>* 

<sup>ˆ</sup> ( ) [ , ( )] *<sup>t</sup>*

 

where reduced description parameters of the field 0 (, ) *t*

 *t Ht*

0 0 ( ) ( ( , ), ( , )) ( ) *<sup>o</sup> <sup>t</sup> t tt t*

( ) <sup>0</sup> <sup>ˆ</sup> ( , ) Sp ( ) *t t*

is a characteristic time determined by an initial state of the system

, so called a quasiequilibrium statistical operator ( ( ), ( ))

 

reduced description parameters). The set of parameters 0 (, ) *t*

states which are far from the equilibrium) defined by the relations

*a*

 

<sup>f</sup> <sup>ˆ</sup> ( ) exp{ ( ) } *Z ZZ*

m( ) exp{ ( ) } ˆ *a a*

*X XX*

$$\tilde{m}\_{kk'}^{aa'}(t) \equiv \text{Sp} \,\rho^{(+)}(t) \mathfrak{c}\_{ak} \mathfrak{c}\_{a'k'} \,\,\, \text{or} \qquad \mathfrak{x}\_{ak}(t) \equiv \text{Sp} \,\rho^{(+)}(t) \mathfrak{c}\_{ak} \,\, \text{and} \tag{42}$$

Description of Field States with Correlation Functions and Measurements in Quantum Optics 15

correlations of dipole orientations are absent and their distribution is isotropic one.

and phenomenologically accounts for non-resonant interaction between the field and

The obtained integral equation is solved in perturbation theory in emitter-field interaction

to use the Wick–-Bloch–-de Dominicis theorem. However, one needs this theorem for calculating contributions of the third and higher orders of the perturbation theory to the

f f <sup>ˆ</sup> Sp ( ( )) ( ) ( )

<sup>1</sup> ˆ ˆ ˆ ˆ Sp ( ( )) ( ) ( ) ( ) ( ) ( ) [ ( ), ( )] <sup>2</sup> *x x Z xx m n mn m n m n xx x x*

Moreover, according to the general theory of the Peletminskii-Yatsenko model (Akhiezer & Peletminskii, 1981) the same formulas are valid for calculations with the statistical operator

<sup>ˆ</sup> Sp ( , ) ( ) ( ) *n n x x*

<sup>1</sup> ˆ ˆ ˆ ˆ Sp ( , ) ( ) ( ) ( ) ( ) ( ) [ ( ), ( )] <sup>2</sup>

Averages with a quasiequilibrium statistical operator of the medium are calculated by the method developed for spin systems (Lyagushyn et al., 2005). It gives, for example, an expression for energy density of emitter medium via its temperature *T x*( ) and density *n x*( )

Integral equation (46) solution gives evolution equations for all parameters of the reduced

*<sup>n</sup>* , ( , ) rot ( , ) *t n B xt c Ext <sup>n</sup>* (53)

 

(54)

, ( , ) rot ( , ) *Z xt Bxt n n*

 

 

 *Z xx* 

> 

 

> 

*m n mn m n m n x x xx x x*

 

> 

 

.

1 () ( ) *<sup>a</sup> a N nx x x* 

 

 

> 

 

.

). Important convenience is provided by the structure of f

0

*d w* () 1 

. Averages that are linear and bilinear in the field can be

(1

 ( ( )) *Z* 

*n n* , (50)

 

, (51)

 

 

). (52)

 

 

 

 

    ) (49)

allowing

is defined by formulas

 

> 

 

 

( ) ( )th 2 2() *x nx*

(

( , ) rot ( , ) 4 ( , ( ), ( )) *t n E xt c Bxt J x t t <sup>n</sup>*

<sup>2</sup> ( ,) ( ,) ( ) *<sup>t</sup> E xt E xt O n n*

 

 

() ()( )

 

(,) 

calculated on the basis of relations:

f f

 

 

*w c* 

2 2 0

 

 

> 

 

*x x*

*T x*

description. Average electric and magnetic fields satisfy the Maxwell equations

 

where average current density in terms of the total electric field is given by the relation

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

 

 

,

Function *w* ( ) 

emitters.

mf ˆ *H* ~ (1 

(,) :

statistical operator

are not equal to zero, are considered as states with a broken symmetry. Therefore, ( ) *n t kk* is called an anomalous one-particle density matrix. However, average electromagnetic fields are expressed through ( ) *<sup>k</sup> x t* . Instead of density matrices Wigner distribution functions are widely used (de Groot, S. & Suttorp L., 1972)

$$\mathbf{f}\_{k}^{\alpha\alpha'} (\mathbf{x}, t) \equiv \operatorname{Sp} \rho^{(+)} (t) \hat{\mathbf{f}}\_{k}^{\alpha\alpha'} (\mathbf{x}) \,, \qquad \tilde{\mathbf{f}}\_{k}^{\alpha\alpha'} (\mathbf{x}, t) \equiv \operatorname{Sp} \rho^{(+)} (t) \hat{\mathbf{f}}\_{k}^{\alpha\alpha'} (\mathbf{x}) \, \tag{43}$$

where

$$\hat{\mathbf{f}}\_{\mathbf{k}}^{\alpha\alpha'}\{\mathbf{x}\} = \sum\_{q} \mathbf{c}\_{\alpha,k-q/2}^{+} \mathbf{c}\_{\alpha',k+q/2} \mathbf{e}^{iqx}, \qquad \qquad \hat{\mathbf{f}}\_{\mathbf{k}}^{\alpha\alpha'}\{\mathbf{x}\} = \sum\_{q} \mathbf{c}\_{\alpha,k+q/2} \mathbf{c}\_{\alpha',-k+q/2} \mathbf{e}^{iqx}. \tag{44}$$

Simple relations between average field, correlations of the field, density matrices and Wigner distribution functions can be established by the formula

$$\text{tr}\_{\alpha k} = \left(8\pi a \rho\_k \hbar V\right)^{-1/2} e\_{\alpha k n}^\* \left[d^3 \mathbf{x}\{\hat{Z}\_n(\mathbf{x})/k - \mathrm{i}\hat{E}\_n^t(\mathbf{x})\} e^{-i\mathbf{k}\mathbf{x}}\right] \qquad \hat{Z}\_n(\mathbf{x}) \equiv \text{rot}\_n \hat{B}(\mathbf{x}) \,. \tag{45}$$

Further on kinetics of electromagnetic field in medium consisting of two-level emitters with the Hamilton operator (23) is considered in more detail. According to the general theory (Akhiezer & Peletminskii, 1981), an integral equation for the statistical operator (,) introduced by the functional hypothesis (31) can be obtained (Lyagushyn & Sokolovsky, 2010b)

$$
\rho(\xi,\varepsilon) = \rho\_{\text{f}}(\mathbf{Z}(\xi))\rho\_{\text{m}}(\mathbf{X}(\varepsilon)) + \int\_{-\varepsilon}^{0} d\tau e^{\frac{i}{\hbar}\tau \hat{H}\_{0}} \left\{ \frac{i}{\hbar} [\rho(\xi,\varepsilon),\hat{H}\_{\text{mf}}] - \right. \tag{46}
$$

$$
$$

where functions ( , ) *M* , *M*(,) are defined as right-hand sides of evolution equations for the reduced description parameters

$$\begin{aligned} \hat{\boldsymbol{\sigma}}\_t \boldsymbol{\xi}\_\mu(t) &= \mathrm{i} \sum\_{\mu'} \boldsymbol{c}\_{\mu\mu'} \boldsymbol{\xi}\_{\mu'}(t) + \boldsymbol{M}\_\mu(\xi(t), \boldsymbol{\varepsilon}(t)) \; , & \quad \hat{\boldsymbol{\sigma}}\_t \boldsymbol{\varepsilon}(\mathbf{x}, t) &= \boldsymbol{M}(\mathbf{x}, \boldsymbol{\varepsilon}(t), \boldsymbol{\varepsilon}(t)) \; ; \end{aligned} \tag{47}$$
 
$$\begin{aligned} \boldsymbol{M}\_\mu(\boldsymbol{\xi}, \boldsymbol{\varepsilon}) &= \frac{\mathrm{i}}{\hbar} \mathrm{Sp} \, \rho(\boldsymbol{\xi}, \boldsymbol{\varepsilon}) [\hat{H}\_{\mathrm{mi}}, \hat{\boldsymbol{\xi}}\_\mu] \; ; \end{aligned} \qquad \begin{aligned} \boldsymbol{\hat{\sigma}}\_t \boldsymbol{\varepsilon}(\mathbf{x}, t) &= \boldsymbol{M}(\mathbf{x}, \boldsymbol{\varepsilon}(t), \boldsymbol{\varepsilon}(t)) \; ; \end{aligned} \tag{48}$$

(see notations in (39)). Quasiequilibrium statistical operator of the emitters

$$\rho\_{\rm m}(X) = w\_d(d)w\_{\sigma}(o) \exp\{\Omega(X) - \int d^3x X(x)\widehat{z}(x)\}\tag{48}$$

describes a state of local equilibrium of the emitter medium with temperature <sup>1</sup> *Tx Xx* () () in the considered case. Function ( ) *w d <sup>d</sup>* describes distribution of orientations of emitter dipole moments (Lyagushyn et al., 2008). Further it is assumed for simplicity that 14 Quantum Optics and Laser Experiments

called an anomalous one-particle density matrix. However, average electromagnetic fields

 , ( ) <sup>ˆ</sup> f ( , ) Sp ( )f ( ) *k k xt t x* 

 , , /2 , /2 ˆ

Simple relations between average field, correlations of the field, density matrices and

Further on kinetics of electromagnetic field in medium consisting of two-level emitters with the Hamilton operator (23) is considered in more detail. According to the general theory

introduced by the functional hypothesis (31) can be obtained (Lyagushyn & Sokolovsky,

0 ˆ f m mf ˆ ( , ) ( ( )) ( ( )) [ ,, ] *<sup>i</sup> <sup>H</sup> <sup>i</sup> Z X de <sup>H</sup>* 

 

<sup>ˆ</sup> <sup>3</sup> (,) (,) (,) (,,) ( ) *ic*

*M dx Mx e x*

 

> 

, ( , ) ( , ( ), ( )) *<sup>t</sup>*

, mf

m( ) ( ) ( )exp{ ( ) ( ) ( )} ˆ *X w dw <sup>d</sup> X d xX x x*

describes a state of local equilibrium of the emitter medium with temperature <sup>1</sup> *Tx Xx* () () in the considered case. Function ( ) *w d <sup>d</sup>* describes distribution of orientations of emitter dipole moments (Lyagushyn et al., 2008). Further it is assumed for simplicity that

(Akhiezer & Peletminskii, 1981), an integral equation for the statistical operator

  , ˆ ˆ ( ) rot ( ) *Z x Bx n n* . (45)

. Instead of density matrices Wigner distribution functions are

f () *iqx k kq kq q*

<sup>0</sup>

*e*

are defined as right-hand sides of evolution equations

3

 

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

 

 

(48)

 

 

*xc c e*

 

(43)

. (44)

(46)

0

*<sup>i</sup> <sup>H</sup>*

 *xt Mx t t* 

 *H x* 

 is

> (,)

; (47)

 

are not equal to zero, are considered as states with a broken symmetry. Therefore, ( ) *n t kk*

are expressed through ( ) *<sup>k</sup> x t*

where

2010b)

, /2 , /2 <sup>ˆ</sup> f () *iqx k kq kq q*

*xc c e*

( ) <sup>ˆ</sup> f ( , ) Sp ( )f ( ) *k k xt t x*

 

Wigner distribution functions can be established by the formula

  1 2 <sup>3</sup> \* ˆ ˆ (8 ) { ( ) / ( )} *t ikx k k kn n <sup>n</sup> c V e d x Z x k iE x e*

widely used (de Groot, S. & Suttorp L., 1972)

> 

> >

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

 

 

for the reduced description parameters

where functions ( , ) *M*

 

  

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

> 

 , *M*(,) 

 

> 

mf

 

(see notations in (39)). Quasiequilibrium statistical operator of the emitters

  correlations of dipole orientations are absent and their distribution is isotropic one. Function *w* ( ) is defined by formulas

$$ww\_{\sigma}(o) = c(\sigma) \frac{\sigma}{\left(o - a\_0\right)^2 + \sigma^2} \; \; \qquad \qquad \int\_0^{+\eta} dw w\_{\sigma}(o) = 1 \qquad (\sigma \ll 1) \tag{49}$$

and phenomenologically accounts for non-resonant interaction between the field and emitters.

The obtained integral equation is solved in perturbation theory in emitter-field interaction mf ˆ *H* ~ (1 ). Important convenience is provided by the structure of f ( ( )) *Z* allowing to use the Wick–-Bloch–-de Dominicis theorem. However, one needs this theorem for calculating contributions of the third and higher orders of the perturbation theory to the statistical operator (,) . Averages that are linear and bilinear in the field can be calculated on the basis of relations:

$$
\operatorname{Sp}\_{\mathbb{F}}\rho\_{\mathbb{F}}(\operatorname{Z}(\xi))\hat{\tilde{\xi}}\_{\operatorname{can}}(\mathbf{x}) = \underline{\mathfrak{z}}\_{\operatorname{can}}(\mathbf{x})\ ,\tag{50}
$$

$$
\operatorname{Sp}\_{\mathbb{F}}\rho\_{\mathbb{F}}(\operatorname{Z}(\xi))\hat{\tilde{\xi}}\_{\operatorname{can}}(\mathbf{x})\hat{\tilde{\xi}}\_{\operatorname{can}}(\mathbf{x'}) = (\underline{\mathfrak{z}}\_{\operatorname{can}}^{\operatorname{X}}\underline{\mathfrak{z}}\_{\operatorname{an}}^{\operatorname{X}}) + \underline{\mathfrak{z}}\_{\operatorname{can}}(\mathbf{x})\underline{\mathfrak{z}}\_{\operatorname{an}}(\mathbf{x'}) + \frac{1}{2}[\underline{\hat{\tilde{\xi}}\_{\operatorname{can}}(\mathbf{x}), \hat{\tilde{\xi}}\_{\operatorname{an}}(\mathbf{x'})] \cdot
$$

Moreover, according to the general theory of the Peletminskii-Yatsenko model (Akhiezer & Peletminskii, 1981) the same formulas are valid for calculations with the statistical operator (,) :

$$
\operatorname{Sp}\rho(\xi,\varepsilon)\hat{\xi}\_{\alpha n}(\mathbf{x}) = \tilde{\varepsilon}\_{\alpha n}(\mathbf{x})\,. \tag{51}
$$

$$
\operatorname{Sp}\rho(\xi,\varepsilon)\hat{\xi}\_{\alpha m}(\mathbf{x})\hat{\xi}\_{\alpha' n}(\mathbf{x'}) = (\xi\_{\alpha m}^{x}\xi\_{\alpha' n}^{x'}) + \tilde{\varepsilon}\_{\alpha m}(\mathbf{x})\tilde{\varepsilon}\_{\alpha' n}(\mathbf{x'}) + \frac{1}{2}[\hat{\xi}\_{\alpha m}(\mathbf{x}), \hat{\xi}\_{\alpha' n}(\mathbf{x'})]\,. \tag{52}
$$

Averages with a quasiequilibrium statistical operator of the medium are calculated by the method developed for spin systems (Lyagushyn et al., 2005). It gives, for example, an expression for energy density of emitter medium via its temperature *T x*( ) and density *n x*( )

$$
\sigma(\mathbf{x}) = -\frac{\hbar\alpha\nu}{2} n(\mathbf{x}) \text{th}\frac{\hbar\alpha}{2T(\mathbf{x})} \qquad \text{ (}\ n(\mathbf{x}) \equiv \sum\_{1 \le a \le N} \delta(\mathbf{x} - \mathbf{x}\_a) \text{)}.\tag{52}
$$

Integral equation (46) solution gives evolution equations for all parameters of the reduced description. Average electric and magnetic fields satisfy the Maxwell equations

$$
\partial\_t E\_n(\mathbf{x}, t) = \mathbf{c} \operatorname{rot}\_n B(\mathbf{x}, t) - 4\pi I\_n(\mathbf{x}, \xi(t), \varepsilon(t)) \,, \qquad \partial\_t B\_n(\mathbf{x}, t) = -\mathbf{c} \operatorname{rot}\_n E(\mathbf{x}, t) \tag{53}
$$

where average current density in terms of the total electric field is given by the relation

$$I\_n(\mathbf{x}, \boldsymbol{\xi}, \boldsymbol{\varepsilon}) = \int d\mathbf{x}' \sigma(\mathbf{x} - \mathbf{x}', \boldsymbol{\varepsilon}(\mathbf{x})) E\_n(\mathbf{x}') + c \int d\mathbf{x}' \chi(\mathbf{x} - \mathbf{x}', \boldsymbol{\varepsilon}(\mathbf{x})) Z\_n(\mathbf{x}') + O(\boldsymbol{\lambda}^3) \tag{54}$$

$$E\_n(\mathbf{x}, t) = E\_n^t(\mathbf{x}, t) + O(\boldsymbol{\lambda}^2), \qquad \qquad Z\_n(\mathbf{x}, t) \equiv \text{rot}\_n B(\mathbf{x}, t)$$

Description of Field States with Correlation Functions and Measurements in Quantum Optics 17

Current-field correlation functions are defined analogously to (37). Material equations for

 

<sup>3</sup> ( , ( , ))( ) ( , ( )) ( ) *x x*

 

<sup>3</sup> ( , ( , ))( ) ( , ( )) ( ) *x x*

*mn t*

*mn t*

(62)

 

2 2

 

*k*

defined in (41). The problem for plasma

 : *nkk* , *<sup>k</sup> x* , \* *<sup>k</sup> x*and

. A statistical operator of the system

*ic*

   

> ,

*e*

one can use average Bose amplitudes

 

 

 

, (63)

these correlations are given by expressions in terms of the total electric field

where Fourier transformed functions *S xn mn*(,) , *T xn mn*(,) are given by expressions

 

2

*<sup>i</sup> T k n cd ne k d w*

<sup>4</sup> (,) ( )P <sup>3</sup> *mn mnl l*

*m n t mn c dx x x x t E Z S x x n x O*

*m n t mn c dx x x x t B Z T x x n x O*

,

<sup>2</sup> 2 2 (,) ( ) ( ) <sup>3</sup> *S kn dn k k w mn mn m n k k*

0

Quantities *S kn mn*(,) , *T kn mn*(,) determine equilibrium correlations of the electromagnetic field. Comparing relations (54) and (62) shows that the Onsager principle is valid for the

Hereafter we consider kinetics of electromagnetic field in plasma medium with the Hamiltonian (29) in more detail. We restrict ourselves by considering equilibrium plasma (Sokolovsky & Stupka, 2004) and states of the field described by average fields ( , ) *<sup>t</sup> E xt <sup>n</sup>* ,

medium in terms of hydrodynamic states has been investigated in (Sokolovsky & Stupka,

\* () () () () *kk kk k k g t n t x tx t*

introduced by the functional hypothesis depends in this case only on the field variables and

*<sup>i</sup> Z w de <sup>H</sup> <sup>M</sup> <sup>e</sup>*

 , (65)

 

. (64)

0 0 0 ˆ ˆ

( ) *Z* is given by formula (36) with 0 *Zkk*

*i i H H*

 

 

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

( ) <sup>ˆ</sup> ( ) ( ( )) [ ( ), ] ( )

   

,

( ) ( , ( , ))( ) *x x x x E J dx x x x t E E mn t*

( ) ( , ( , ))( ) *x x x x B J dx x x x t B E mn t*

( ,) *B xt <sup>n</sup>* and one-particle density matrix ( ) *n t kk*

2005). Instead of average fields and matrix ( ) *n t kk*

defined in (42) and correlation function

So, for this problem in above notations we have parameters

 : *k k c c* 

f m mf

 

where quasiequilibrium statistical operator f

*w*m is a statistical operator of equilibrium plasma

 

considered system.

corresponding operators ˆ

satisfies the integral equation

 

( ) *<sup>k</sup> x t* 

(for all parameters ˆ *A*( , ) Sp ( , ) *A* ). This material equation takes into account spatial dispersion and Fourier transformed functions (,) *x* , (,) *x* give conductivity (,) *k* and magnetic susceptibility (,) *k* of the emitter medium

$$\sigma(k,\varepsilon) = -\frac{2\pi}{3} \frac{\varepsilon d^2}{\hbar^2} w\_{\sigma}(o\_k) \,, \qquad \chi(k,\varepsilon) = -\frac{4}{3} \frac{\varepsilon d^2}{\hbar^2} \int\_0^{+\varepsilon} dw w\_{\sigma}(o) \mathcal{P} \frac{1}{o\rho^2 - o\_k^2} \,. \tag{55}$$

Average density of the dipole moment of emitters is given by expression

$$P\_n(\mathbf{x}, \boldsymbol{\varepsilon}, \boldsymbol{\gamma}) = \int d\mathbf{x}' \mathbf{x}(\mathbf{x} - \mathbf{x}', \boldsymbol{\varepsilon}(\mathbf{x})) E\_n(\mathbf{x}') + \epsilon \int d\mathbf{x}' a(\mathbf{x} - \mathbf{x}', \boldsymbol{\varepsilon}(\mathbf{x})) Z\_n(\mathbf{x}') + O(\boldsymbol{\lambda}^3) \tag{56}$$

where

$$\kappa(k,\varepsilon) = \chi(k,\varepsilon), \qquad \qquad \alpha(k,\varepsilon) = -\sigma(k,\varepsilon) / o\_k^2 \,. \tag{57}$$

Evolution equation for energy density ( ,) *x t* of emitters has the form

$$
\hat{\sigma}\_t \varepsilon(\mathbf{x}, t) = \mathcal{L}(\mathbf{x}, \xi(t), \varepsilon(t)) \; \tag{58}
$$

$$L(\mathbf{x}, \boldsymbol{\xi}, \boldsymbol{\varepsilon}) = \int d\mathbf{x}' \sigma(\mathbf{x} - \mathbf{x}', \boldsymbol{\varepsilon}(\mathbf{x})) \{ (E\_n^{\mathbf{x}} E\_n^{\mathbf{x}'}) + E\_n(\mathbf{x}) E\_n(\mathbf{x}') \} +$$

$$+ c \int d\mathbf{x}' \varrho(\mathbf{x} - \mathbf{x}', \boldsymbol{\varepsilon}(\mathbf{x})) \{ (E\_n^{\mathbf{x}} B\_n^{\mathbf{x}'}) + E\_n(\mathbf{x}) B\_n(\mathbf{x}') \} + R(n(\mathbf{x})) + O(\boldsymbol{\lambda}^3).$$

The last term describes dipole radiation of the emitters

$$R(n) \equiv -\frac{2d^2}{3\pi c^3} n \int\_0^\infty d\alpha \alpha o^4 w\_\sigma(\alpha) \tag{59}$$

and for small gives a known expression

$$R(n) = -\frac{2d^2\alpha\_0^4}{3\pi c^3}n\,. \tag{60}$$

Evolution equations for correlation functions of electromagnetic field in terms of the total electric field can be written in the form

( ) rot ( ) rot ( ) 4 ( ) 4 ( ) *x x x x x x x x xx t m n t m n t n m t m n t mn t E E c BE c E B J E E J* , (61) ( ) rot ( ) rot ( ) 4 ( ) *x x x x xx xx t mnt m nt n m t mnt E B c BB c EE J B* , ( ) rot ( ) rot ( ) 4 ( ) *x x x x x x xx t m n t m n t n m t mn t BE c EE c BB B J* , ( ) rot ( ) rot ( ) *x x x x x x t mnt m nt n m t BB c EB c BE* .

16 Quantum Optics and Laser Experiments

(,) *x* , (,) *x* 

of the emitter medium

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

( , ) ( , ( ), ( )) *<sup>t</sup>*

 

*nn n n c dx x x x E B E x B x R n x O*

2

3 0 <sup>2</sup> ( ) ( ) <sup>3</sup> *<sup>d</sup> Rn n d w c*

> <sup>2</sup> ( ) <sup>3</sup> *<sup>d</sup> Rn n*

.

 

<sup>3</sup> ( , ( )){( ) ( ) ( )} ( ( )) ( ) *x x*

4

 

, (61)

*c* 

Evolution equations for correlation functions of electromagnetic field in terms of the total

( ) rot ( ) rot ( ) 4 ( ) 4 ( ) *x x x x x x x x xx t m n t m n t n m t m n t mn t E E c BE c E B J E E J*

*t mnt m nt n m t mnt E B c BB c EE J B*

*t m n t m n t n m t mn t BE c EE c BB B J*

,

*t mnt m nt n m t BB c EB c BE* .

,

( ) rot ( ) rot ( ) 4 ( ) *x x x x xx xx*

( ) rot ( ) rot ( ) 4 ( ) *x x x x x x xx*

( ) rot ( ) rot ( ) *x x x x x x*

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

*A* ). This material equation takes into account spatial

2

*<sup>d</sup> <sup>k</sup> d w*

 

 *k k* 

( ,) *x t* of emitters has the form

(56)

 

0 4 1 (,) ( )P <sup>3</sup> *<sup>k</sup>*

 

give conductivity

. (55)

 

. (57)

 

2 2 2

 

*xt Lx t t* , (58)

(59)

. (60)

 

(,) *k* and

(for all parameters ˆ *A*( , ) Sp ( , ) 

magnetic susceptibility

where

and for small

electric field can be written in the form

  

Evolution equation for energy density

dispersion and Fourier transformed functions

(,) *k* 

   

(,) (,) *k k* 

 

Average density of the dipole moment of emitters is given by expression

( , , ) ( , ( )){( ) ( ) ( )} *x x L x dx x x x E E E x E x nn n n*

 

2 2 <sup>2</sup> (,) ( ) <sup>3</sup> *<sup>k</sup> <sup>d</sup> k w*

,

 

The last term describes dipole radiation of the emitters

gives a known expression

 Current-field correlation functions are defined analogously to (37). Material equations for these correlations are given by expressions in terms of the total electric field

 ( ) ( , ( , ))( ) *x x x x E J dx x x x t E E mn t mn t* (62) <sup>3</sup> ( , ( , ))( ) ( , ( )) ( ) *x x m n t mn c dx x x x t E Z S x x n x O* , ( ) ( , ( , ))( ) *x x x x B J dx x x x t B E mn t mn t* <sup>3</sup> ( , ( , ))( ) ( , ( )) ( ) *x x m n t mn c dx x x x t B Z T x x n x O* ,

where Fourier transformed functions *S xn mn*(,) , *T xn mn*(,) are given by expressions

$$S\_{mn}(k,n) = -\frac{2\pi}{3}d^2n(\mathcal{S}\_{mn} - \tilde{k}\_m\tilde{k}\_n)o\rho\_k^2w\_\sigma(o\_k)\,. \tag{63}$$

$$T\_{mn}(k,n) = \frac{4\pi i}{3}cd^2n\,e\_{mn}k\_l\int\_0^{+\alpha}d\alpha\,w\_\sigma(o)\mathcal{P}\frac{\alpha}{\alpha^2 - o\_k^2}$$

Quantities *S kn mn*(,) , *T kn mn*(,) determine equilibrium correlations of the electromagnetic field. Comparing relations (54) and (62) shows that the Onsager principle is valid for the considered system.

Hereafter we consider kinetics of electromagnetic field in plasma medium with the Hamiltonian (29) in more detail. We restrict ourselves by considering equilibrium plasma (Sokolovsky & Stupka, 2004) and states of the field described by average fields ( , ) *<sup>t</sup> E xt <sup>n</sup>* , ( ,) *B xt <sup>n</sup>* and one-particle density matrix ( ) *n t kk* defined in (41). The problem for plasma medium in terms of hydrodynamic states has been investigated in (Sokolovsky & Stupka, 2005). Instead of average fields and matrix ( ) *n t kk* one can use average Bose amplitudes ( ) *<sup>k</sup> x t* defined in (42) and correlation function

$$\mathbf{g}\_{kk'}^{\alpha\alpha'}(t) = n\_{kk'}^{\alpha\alpha'}(t) - \mathbf{x}\_{\alpha k}^\*(t)\mathbf{x}\_{\alpha'k'}(t) \tag{64}$$

So, for this problem in above notations we have parameters : *nkk* , *<sup>k</sup> x* , \* *<sup>k</sup> x* and corresponding operators ˆ : *k k c c* , *<sup>k</sup> <sup>c</sup>* , *<sup>k</sup> c* . A statistical operator of the system introduced by the functional hypothesis depends in this case only on the field variables and satisfies the integral equation

$$\rho(\xi) = \rho\_{\text{i}}(Z(\xi))w\_{\text{m}} + \int\_{-\mathfrak{s}}^{0} d\tau e^{\frac{i}{\hbar}\tau\hat{H}\_{0}} \left\{ \frac{\text{i}}{\hbar} [\rho(\xi), \hat{H}\_{\text{m}}] - \sum\_{\mu} \frac{\partial \rho(\xi)}{\partial \xi\_{\mu}} M\_{\mu}(\xi) \right\}\_{\xi \to \epsilon^{-i\pi}\xi} e^{-\frac{i}{\hbar}\tau\hat{H}\_{0}},\tag{65}$$

where quasiequilibrium statistical operator f ( ) *Z* is given by formula (36) with 0 *Zkk* , *w*m is a statistical operator of equilibrium plasma

Description of Field States with Correlation Functions and Measurements in Quantum Optics 19

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

f m f m ˆ ˆ ( , ) ( ( )) ( ( )) [ ( ( )) ( ( )), ( , ) ( , )] ( ) *<sup>t</sup>*

fm fm ˆ ˆ ( ) ( ( )) [ ( ( )) , ( , ) ( , )] ( ) *n n <sup>i</sup> Z w d dx Z w A x j x O*

general theory of the Peletminskii-Yatsenko model (Akhiezer & Peletminskii, 1981), the

<sup>f</sup> Sp ( , ) Sp ( ( )) *k k k k cc Z cc*

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

( ) <sup>1</sup> ˆ ˆˆ ( , ) { ( , ) ( ) ( , ) ( )} *<sup>t</sup> E x t i dx D x x t Z x D x x t E x n nn <sup>c</sup>*

 

*Z X d dx Z X E x P x O*

. (73)

 

nonequilibrium statistical operator for the field-plasma system is given by the formula

0

0

 

 

*c*

 

*i*

  .

2

 

*n n*

 

 

 

(77)

 

2

 

, (75)

(76)

 

 

> 

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

are operators <sup>ˆ</sup> ( ) *A x <sup>n</sup>* , <sup>ˆ</sup> ( ) *nj <sup>x</sup>* in the interaction picture. According to

(74)

 , Sp ( , ) Sp ( ( )) *k k* <sup>f</sup> *k k cc Z cc* 

> 

 , Sp ( ) Sp ( ( )) *k k* f f *k k cc Z cc* 

**7. Connection between correlation functions of different nature and some** 

**suitable representations for them** 

statistical operator has the form

 

 , <sup>ˆ</sup> (,) *P x <sup>l</sup>* 

> , <sup>ˆ</sup> (,) *nj <sup>x</sup>*

and for the field-plasma system

the following formula

 

following relations for the field-emitters system

<sup>f</sup> Sp ( , ) Sp ( ( )) *k k c Zc*

f f Sp ( ) Sp ( ( )) *k k c Zc*

 

 

 

where <sup>ˆ</sup> (,) *<sup>t</sup> E x <sup>n</sup>*

where <sup>ˆ</sup> (,) *A x <sup>n</sup>*

$$w\_{\mathbf{m}} = e^{\left(\Omega - \hat{H}\_{\mathbf{m}} + \sum\_{a} \mu\_{a} \hat{\mathcal{N}}\_{a}\right) \dagger} \qquad \left(\hat{\mathcal{N}}\_{a} \equiv \int d\mathbf{x} \hat{n}\_{a}(\mathbf{x})\right). \tag{66}$$

Functions ( ) *M* define the right-hand sides of evolution equations for the reduced description parameters

$$\left(\hat{\boldsymbol{\sigma}}\_{t}\boldsymbol{\xi}\_{\mu}(t) = \mathrm{i}\sum\_{\mu'} \boldsymbol{c}\_{\mu\mu'}\boldsymbol{\xi}\_{\mu'}(t) + \boldsymbol{M}\_{\mu}(\boldsymbol{\xi}(t)) \right) , \qquad \boldsymbol{M}\_{\mu}(\boldsymbol{\xi}) \equiv \frac{\mathrm{i}}{\hbar} \mathrm{Sp}\,\rho(\boldsymbol{\xi}) [\hat{H}\_{\mathrm{mf}}, \hat{\boldsymbol{\xi}}\_{\mu}] \, \mathrm{.} \tag{67}$$

Integral equation (65) is solvable in a perturbation theory in plasma-field interaction based on estimations <sup>1</sup> 1 ˆ *H* ~ , <sup>2</sup> 2 ˆ *H* ~ (see (29)). As a result, evolution equations for the reduced description parameters take the form (Sokolovsky & Stupka, 2004)

$$
\hat{\boldsymbol{\sigma}}\_{t} \mathbf{g}\_{kk'}^{\alpha\alpha'} = \mathbf{i} (\boldsymbol{\Omega}\_{k} - \boldsymbol{\Omega}\_{k'}) \mathbf{g}\_{kk'}^{\alpha\alpha'} - (\boldsymbol{\nu}\_{k} + \boldsymbol{\nu}\_{k'}) (\mathbf{g}\_{kk'}^{\alpha\alpha'} - \boldsymbol{n}\_{k} \boldsymbol{\delta}\_{\alpha\alpha'} \boldsymbol{\delta}\_{kk'}) + O(\boldsymbol{\lambda}^{3}), \tag{68}
$$

$$
\hat{\boldsymbol{\sigma}}\_{t} \mathbf{x}\_{\alpha k} = -(\mathbf{i}\boldsymbol{\Omega}\_{k} + \boldsymbol{\nu}\_{k}) \mathbf{x}\_{\alpha k} + (\boldsymbol{\nu}\_{k} + \boldsymbol{i}\boldsymbol{o}\_{k} \boldsymbol{\chi}\_{k}) \mathbf{x}\_{\alpha, -k}^{\*} + O(\boldsymbol{\lambda}^{3})
$$

where *k* is photon spectrum in the plasma, *nk* is the Planck distribution with the plasma temperature, *<sup>k</sup>* is a frequency of photon emission and absorption. These quantities are given by formulas

$$
\Omega\_k \equiv \alpha\_k \{ 1 - 2\pi \chi(k) \} \,, \quad \nu\_k \equiv 2\pi \sigma(k) \,. \tag{69}
$$

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

$$J\_n(\mathbf{x}, \xi) = \left[ d\mathbf{x}' \sigma(\mathbf{x} - \mathbf{x}') E\_n(\mathbf{x}') + c \right] d\mathbf{x}' \mathcal{Z}(\mathbf{x} - \mathbf{x}') Z\_n(\mathbf{x}') + O(\mathcal{X}^3) \,. \tag{70}$$

This material equation takes into account spatial dispersion and Fourier transformed functions ( ) *x* , ( ) *x* give conductivity ( ) *k* and magnetic susceptibility ( ) *k* of the plasma medium. Their values are given by relations

$$\sigma(k) = -\frac{\operatorname{Im} G(k, o\_k)}{o o\_k} \qquad\qquad \qquad \mathbb{X}(k) = -\frac{\operatorname{Re} G(k, o\_k) + \mathbb{X}}{\operatorname{c.o} o\_k} \tag{71}$$

where *G k*(, ) is a transversal part of current-current Green function:

$$\mathbf{G}(k,\alpha) \equiv \frac{1}{2} \mathbf{G}\_{nm}(k,\alpha)(\delta\_{nm} - \tilde{k}\_{m}\tilde{k}\_{n}), \qquad \mathbf{G}\_{mn}(\mathbf{x},t) = -\frac{i}{\hbar} \theta(t) \mathbf{S} \mathbf{p}\_{m} w\_{m} [\hat{j}\_{m}(\mathbf{x},t), \hat{j}\_{n}(0)]; \tag{72}$$
 
$$\gamma\_{\alpha} \quad \text{ }\_{-2}$$

$$\mathcal{X} \equiv \sum\_{a} \frac{m\_a e\_a^2}{m\_a} \equiv \frac{\Omega^2}{4\pi} \cdot \frac{1}{4}$$

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