**5. Quantum models for electromagnetic field in media**

The main problem of quantum optics is diagnostics of electromagnetic field ( f -system) interacting with a medium ( m -system). In this connection we have considered a number of models of medium and medium-field interaction. From various points of view the Dicke model of medium consisting of two-level emitters is very useful for such analysis. In the Coulomb gauge it is described by the Hamilton operator (Lyagushyn & Sokolovsky, 2010b)

$$
\hat{H} = \hat{H}\_{\text{f}} + \hat{H}\_{\text{m}} + \hat{H}\_{\text{mf}} \quad \hat{H}\_{\text{f}} \equiv \sum\_{\mathbf{k}\alpha} \hbar \alpha\_{\mathbf{k}} \mathbf{c}\_{\mathbf{k}\alpha}^{+} \mathbf{c}\_{\mathbf{k}\alpha} \quad \hat{H}\_{\text{m}} = \hbar \alpha \sum\_{1 \le \alpha \le N} \hat{r}\_{\alpha} \ , \tag{23}
$$

$$
\hat{H}\_{\text{mf}} = -\int d^3 \mathbf{x} \hat{E}\_n^{\dagger}(\mathbf{x}) \hat{P}\_n(\mathbf{x})
$$

Here ˆ *an r* is a quasispin operator, *a* is emitter's number, is polarization index, <sup>ˆ</sup> ( ) *P x <sup>n</sup>* is the density of electric dipole moment (polarization) of emitters

$$\hat{P}\_n(\mathbf{x}) = 2 \sum\_{1 \le a \le N} d\_{an} \hat{r}\_{ax} \delta\left(\mathbf{x} - \mathbf{x}\_a\right). \tag{24}$$

We neglect emitter-emitter interaction in (23). Operators of vector potential, transversal electric field and magnetic field are expressed via creation and annihilation boson operators , *k k c c* by formulas (9), (10) and commutation relations

$$\left[\hat{E}\_{m}^{t}(\mathbf{x}), \hat{E}\_{n}^{t}(\mathbf{x}')\right] = 0 \quad , \quad \left[\hat{B}\_{m}(\mathbf{x}), \hat{B}\_{n}(\mathbf{x}')\right] = 0 \quad , \quad \left[\hat{B}\_{m}(\mathbf{x}), \hat{E}\_{n}^{t}(\mathbf{x}')\right] = \varepsilon\_{mnl} 4\pi i \hbar c \frac{\hat{c}\delta(\mathbf{x} - \mathbf{x}')}{\hat{c}\mathbf{x}\_{l}} \tag{25}$$

are valid (we use the notation <sup>ˆ</sup> ( ) *<sup>t</sup> E x <sup>n</sup>* for electric field operator (10) in the discussion of the field-emitters system).

It is very convenient to use operator evolution equations for investigating the dynamics of the system (23). The Maxwell operator equations have a known form

$$
\hat{\vec{E}}\_n(\mathbf{x}) = \mathbf{c} \operatorname{rot}\_n \hat{B}(\mathbf{x}) - 4\pi \hat{f}\_n(\mathbf{x}) \,, \qquad \hat{\vec{B}}\_n(\mathbf{x}) = -\mathbf{c} \operatorname{rot}\_n \hat{E}(\mathbf{x}) \, \tag{26}
$$

where total electric field and electromagnetic current

$$
\hat{E}\_n(\mathbf{x}) \equiv \hat{E}\_n^t(\mathbf{x}) - 4\pi\hat{P}\_n(\mathbf{x}) \,, \quad \hat{f}\_n(\mathbf{x}) \equiv \hat{\bar{P}}\_n(\mathbf{x}) = -2a\nu \sum\_a d\_{an}\hat{r}\_{ay}\delta\left(\mathbf{x} - \mathbf{x}\_a\right) \tag{27}
$$

are introduced. Energy density of emitter medium

$$
\hat{\varepsilon}(\mathbf{x}) = \hbar \alpha \sum\_{1 \le a \le N} \hat{r}\_{az} \delta(\mathbf{x} - \boldsymbol{\pi}\_a),
$$

obeys the evolution equation

10 Quantum Optics and Laser Experiments

presents some kind of a "dynamical phase transition" (Bogolyubov (Jr.) & Shumovsky, 1987). *N* excited atoms come to coordinated behavior without the mechanism of stimulated emission and a peak of intensity, proportional to <sup>2</sup> *N* , appeared for modes that were close to the resonant one in a direction determined by the geometry of the system (Banfi & Bonifacio, 1975). So we have a way of coherent generation that is alternative to the laser one. This way can be used hypothetically in X- and γ-ray generators opening wide possibilities for physics

Collective spontaneous emission in the Dicke quasispin model proved to be one of the most difficult for experimental observations collective quantum phenomena. That is why taking into account real conditions of the experiment is of great importance. Thus great quantity of Dicke model generalizations has been considered. There are two factors dependent of temperature, namely the own motion of emitters and their interaction with the media. The both factors are connected with additional chaotic motion, thus they worsen the prospects of self-organizing in a system. The last factor is discussed traditionally as an influence of a cavity (resonator) since experiments in superradiance use laser technology (Kadantseva et al., 1989). The corresponding theoretical analysis is based on modeling the cavity with a system of oscillators (Louisell, 1964). The problem of influence of emitter motion (which is of different nature in different media) can be solved with taking into account this motion via a nonuniform broadening of the working frequency of emitters (Bogolyubov (Jr.) & Shumovsky, 1987). The dispersion of emitter frequencies results in an additional fading in a

Traditional investigations obtain conclusions about a superfluorescent impulse generation on the basis of calculated behavior of the system of two-level emitters. The problem of light generation in the Dicke model can be investigated in the framework of the Bogolyubov method of eliminating boson variables (Bogolyubov (Jr.) & Shumovsky, 1987) with the suggestion of equilibrium state of field with a certain temperature. The correlation properties of light remain unknown in such picture. Good results can be obtained by applying the Bogolyubov reduced description method (Lyagushyn et al., 2005) to the model. The reduced description method eliminates some difficulties in the Dicke model investigations and allows both to take into account some additional factors (the orientation and motion of emitters, for instance) and to introduce more detailed description of the field.

A kind of correlation functions to be used in such approach will be of interest for us.

The main problem of quantum optics is diagnostics of electromagnetic field ( f -system) interacting with a medium ( m -system). In this connection we have considered a number of models of medium and medium-field interaction. From various points of view the Dicke model of medium consisting of two-level emitters is very useful for such analysis. In the Coulomb gauge it is described by the Hamilton operator (Lyagushyn & Sokolovsky, 2010b)

> <sup>ˆ</sup> *kk k k H cc*

, m

3

ˆ ˆˆ () () *<sup>t</sup> H d xE x P x n n*

mf

 

1 ˆ ˆ

*H r* 

*az a N*

, (23)

system and elimination of singularities in kinetic coefficients.

**5. Quantum models for electromagnetic field in media**

f m mf *HH H H* ˆˆ ˆ ˆ , f

and technology.

$$
\hat{\dot{\varepsilon}}(\mathbf{x}) = \hat{f}\_n(\mathbf{x}) \hat{E}\_n^t(\mathbf{x}) \tag{28}
$$

which describes the Joule heat exchange between the emitters and field. Since the field parameters are considered in different spatial points, we obtain the possibility of investigating the field correlation properties.

Also the model of electromagnetic field in plasma medium plays a significant role. The Hamilton operator of such system in the Coulomb gauge was taken in the paper (Sokolovsky & Stupka, 2004) in the form

$$
\hat{H} = \hat{H}\_{\text{f}} + \hat{H}\_{\text{m}} + \hat{H}\_{\text{mf}} \prime \prime \qquad \hat{H}\_{\text{f}} \equiv \sum\_{\text{ka}} \hbar \alpha\_{\text{k}} \mathbf{c}\_{\text{ka}}^{+} \mathbf{c}\_{\text{ka}} \prime \prime \qquad \hat{H}\_{\text{mf}} = \hat{H}\_{\text{1}} + \hat{H}\_{\text{2}} \prime \prime \tag{29}
$$

$$
\hat{H}\_1 = -\frac{1}{c} \Big[ d\mathbf{x} \hat{A}\_n(\mathbf{x}) \hat{f}\_n(\mathbf{x}) \, \right], \qquad \hat{H}\_2 = \frac{1}{2c^2} \Big[ d\mathbf{x} \hat{A}^2(\mathbf{x}) \hat{\mathcal{Z}}(\mathbf{x}) \, \quad \qquad \left( \hat{\mathcal{Z}}(\mathbf{x}) \equiv \sum\_a \frac{e\_a^2}{m\_a} \hat{n}\_a(\mathbf{x}) \right).
$$

Description of Field States with Correlation Functions and Measurements in Quantum Optics 13

ˆ ˆ () () *<sup>t</sup> n n*

least, exponent in (34) should contain quadratic terms. So the simplest quasiequilibrium

( ) exp{ ( ) ( . .)} *kk k k kk k k k k k k k k k Z Z Z c c Z c c Z c hc*

Kinetics of the field based on this statistical operator describes states with zero average

 *m nt* 

In other words, the quasiequilibrium statistical operator (34) corresponds to field

The theory can be significantly simplified in the Peletminskii-Yatsenko model (Akhiezer &

 , <sup>1</sup> ˆ ˆ { ( ), ( )} <sup>2</sup> *m n x x* 

, *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

 

*x Ex* , 2

ˆ ˆ () () *n n*

*x Bx* . However, in this case

) as additional reduced description

 

(37)

. (38)

 

 

 

*m nt m nt m n xt x t*

, ( ) ( ,) ( ,) *xx xx*

 

   

 

<sup>1</sup> <sup>ˆ</sup> [ ,] ˆ ˆ *a aa a aa*

. (41)

(42)

(39)

ˆ ˆ <sup>ˆ</sup> [ , ( )] rot ( ) *<sup>t</sup> H B x ic E x n n* , (40)

*H c* 

(36)

. Quadratic terms in (36) correspond to binary fluctuation of the field

 

in conformity with the observation made in (Peletminkii et al., 1975). At

( ) *Z* does not exist (its exponent contains only linear in Bose

( ) *Z* is non-normalized). Therefore, one has to use a wider set of

, ,

 

 

*x x*

 

 

 : <sup>ˆ</sup> ( ) *<sup>n</sup> <sup>x</sup>* 

> 

> m ˆ [ , ( )] 0 *H x* ˆ ,

In usual kinetic theory nonequilibrium states of quantum system are described by one-

( ) ( ) Sp ( ) *n t tc c kk k k*

, ( ) ( ) Sp ( ) *k k x t tc*

 

, m

choose operators ˆ

parameters ˆ

the statistical operator f

 

amplitudes form and f

f

fields at 0 *Zk* 

*x x m nt* 

where *c*

operator

parameters

 

 in (32) as <sup>ˆ</sup> ( ) *<sup>n</sup> <sup>x</sup>* : 1

> 

statistical operator of the field can be written as

(or two binary correlations ( ) *x x*

*m n m nt t xx*

ˆ 

<sup>1</sup> ˆˆ ˆ [ ,] *H c* 

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

 

 ˆ ˆ <sup>ˆ</sup> [ , ( )] rot ( ) *<sup>t</sup> H E x ic B x n n* , f

> 

( ) ( ) Sp ( ) *n t tc c kk k k*

f

ˆ( ) *x* satisfy these conditions

f

particle density matrix ( ) *n t kk*

States, for which parameters

 

description by average values of operators

 

( ) <sup>1</sup> ˆ ˆ Sp ( ) { ( ), ( )} <sup>2</sup>

 

Peletminskii, 1981) in which

Here ˆ *Hm* is the Hamilton operator of plasma particles with account of Coulomb interaction, <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.
