**1. Introduction**

Modern physics deals with the consistent quantum concept of electromagnetic field. Creation and annihilation operators allow describing pure quantum states of the field as excited states of the vacuum one. The scale of its changes obliges to use statistical description of the field. Therefore the main object for full description of the field is a statistical operator (density matrix). Field evolution is reflected by operator equations. If the evolution equations are formulated in terms of field strength operators, their general structure coincides with the Maxwell equations. At the same time from the point of view of experiments only reduced description of electromagnetic fields is possible. In order to analyze certain physical situations and use numerical methods, we have the necessity of passing to observable quantities that can be measured in experiments. The problem of parameters, which are necessary for non-equilibrium electromagnetic field description, is a key one for building the field kinetics whenever it is under discussion. The field kinetics embraces a number of physical theories such as electrodynamics of continuous media, radiation transfer theory, magnetic hydrodynamics, and quantum optics. In all the cases it is necessary to choose physical quantities providing an adequate picture of non-equilibrium processes after transfer to averages. It has been shown that the minimal set of parameters to be taken into account in evolution equations included binary correlations of the field. The corresponding theory can be built in terms of one-particle density matrices, Wigner distribution functions, and conventional simultaneous correlation functions of field operators. Obviously, the choice depends on traditions and visibility of phenomenon description. Some methods can be connected due to relatively simple relations expressing their key quantities through one another. The famous Glauber's analysis (Glauber, 1966) of a quantum detector operation had resulted in using correlation functions including positiveand negative-frequency parts of field operator amplitudes in the quantum optics field. Herewith the most interesting properties of field states are described with non-simultaneous correlation functions. Various approaches in theoretical and experimental research into field correlations are compared in the present chapter.

Our starting point is investigation of the Dicke superfluorescence (Dicke, 1954) on the basis of the Bogolyubov reduced description method (Akhiezer & Peletminskii, 1981). It paves the way to constructing the field correlation functions. We can give a relaxation process picture in different orders of the perturbation theory. The set of correlation functions providing a

Description of Field States with Correlation Functions and Measurements in Quantum Optics 5

For a multi-mode field the statistical operator takes the form of a direct product of one-mode

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

describes the evolution of an arbitrary physical system. In the case when electromagnetic field interacting with matter is under consideration the problem is reduced to the correct account of the matter influence, so some kinds of effective Hamiltonians may appear in an analogue of (4) for the statistical operator of field. Evolution description in Heisenberg picture seems to be closer to the classical one. We come to operator Maxwell equations for field operators with terms corresponding to the matter influence and demanding some kind

More graphic way to describing the electromagnetic field, its states, and their evolution is using correlation functions of different types, i.e. averaged values of physical quantities

Conventional classical optics was very restricted in measuring the parameters of fields. All conclusions about properties of light including its polarization properties were drawn from measurements of light intensity, i.e. from values of some quadratic functions of the field (Landau & Lifshitz, 1988). Naturally, we speak now about transversal waves in vacuum. Regarding a wave, close to a monochromatic one, we use slowly varying complex amplitude

<sup>0</sup> *i t E E te n n*

 

\_\_\_\_\_\_\_\_ \*

where *m* and *n* corresponds to two possible directions of polarization and quick oscillations of field are neglected. Averaging is performed over time intervals or (in the case of statistically stable situation) in terms of probabilities. A sum of diagonal components of *mn J* is a real value that is proportional to the field intensity (the energy flux density in the wave in our case). Note that the discussion of field correlation functions by Landau in the earlier

A rather full analysis of the classical measurement picture is given in (Klauder & Sudarshan, 1968). It should be mentioned that real field parameters are obtained from complex conjugated values in this approach. Transition to the quantum electromagnetic theory (Scully & Zubairy, 1997) is connected with substitution of operator structures with creation and annihilation operators instead of complex conjugated functions and coming to positive- and negative-frequency parts of field operators. Such expressions will be

characterizing the field. The problem of choosing them will be discussed below.

**3. Correlation functions provided by methods of quantum optics** 

Partially polarized light is characterized with the tensor of polarization

edition of the mentioned book was one of the first in the literature.

 *t Ht*

(4)

. (5)

*mn m n* 0 0 *J EE* (6)

*<sup>i</sup>* 

statistical operators. In Schrödinger picture the Liouville equation

of material equations.

*E t* <sup>0</sup>*<sup>n</sup>* for its description:

shown later on.

rather full description of the superfluorescence phenomenon obeys the set of differential equations. The further research into the correlation properties of the radiated field requires establishing the connection with the behavior of Glauber functions of different orders.

### **2. Electromagnetic field as an object of quantum statistical theory**

A statistical operator of electromagnetic field should take into account the whole variety of field modes and statistical structure abundance for each of them. Proceeding from the calculation convenience provided by using coherent states *z* of field modes, the Glauber-Sudarshan representation for the statistical operator of field (Klauder & Sudarshan, 1968) footholds in physics. We refer to the following view of this diagonal representation

$$\rho = \int d^2 z P(z, z^\*) |z\rangle\langle z| \tag{1}$$

where \* *Pzz* (, ) is so called *P* -distribution ( { } *<sup>k</sup> z z* and these variables are numbered by polarization and wave vector *k* of the field modes). Since coherent states form an overcrowded basis in the state space of the mode with the completeness condition

<sup>1</sup> <sup>2</sup> <sup>ˆ</sup> *dz z z* | |1 , (2)

the most general representation for the statistical operator should include not only projection operators | | *z z* , but also more general operator products | | *z z* . Nevertheless it can be shown (Glauber, 1969; Kilin, 2003) that a *P*-distribution can be obtained as a twodimensional Fourier transformation of the generating functional

$$F\left(\mu\_{\prime}\mu^{\*}\right) = \mathbf{Sp}\,\rho\sigma^{ak}\stackrel{\sum u\_{ak}c\_{ak}^{+}}{\mathop{e}^{ak}}e^{-\sum\_{ak}^{\*}c\_{ak}^{-}}\tag{3}$$

which is a generating one for all normally ordered field moments and can be calculated directly with an arbitrary statistical operator . Here we use standard notation of quantum electrodynamics: *<sup>k</sup> c* , *<sup>k</sup> <sup>c</sup>* are Bose amplitudes (creation and annihilation operators) of the field.

So we can use the representation (1) in all cases when the Fourier integral for (3) exists. Such situation embraces a great variety of states that are interesting for physicists. More general cases reveal themselves in singularities of the *P*-distribution, the representation (1) still being prospective for using if the *P*-distribution can be expressed via generalized functions of slow growth, i.e. -function and its derivatives. The term " *P* -distribution" is relatively conventional: function \* *Pzz* (, )is a real but non-positive one. Nevertheless, the field state description with the Glauber-Sudarshan *P*-distribution remains the most demonstrative and consumable. For example, a proposed definition of non-classical states of electromagnetic field (Bogolyubov (Jr.) et al., 1988) uses the expression (1) for the statistical operator. A state is referred to as non-classical one if one of two requirements is obeyed: either average number of photons in a mode is less than 1, or *P*-function is not positively determined or has singularity that is higher than the -function.

4 Quantum Optics and Laser Experiments

rather full description of the superfluorescence phenomenon obeys the set of differential equations. The further research into the correlation properties of the radiated field requires establishing the connection with the behavior of Glauber functions of different orders.

of field modes and statistical structure abundance for each of them. Proceeding from the calculation convenience provided by using coherent states *z* of field modes, the Glauber-Sudarshan representation for the statistical operator of field (Klauder & Sudarshan, 1968)

<sup>2</sup> \*

<sup>1</sup> <sup>2</sup> <sup>ˆ</sup> *dz z z* | |1

the most general representation for the statistical operator should include not only projection operators | | *z z* , but also more general operator products | | *z z* . Nevertheless it can be shown (Glauber, 1969; Kilin, 2003) that a *P*-distribution can be obtained as a two-

> \* \* , Sp *kk kk k k uc uc*

which is a generating one for all normally ordered field moments and can be calculated

So we can use the representation (1) in all cases when the Fourier integral for (3) exists. Such situation embraces a great variety of states that are interesting for physicists. More general cases reveal themselves in singularities of the *P*-distribution, the representation (1) still being prospective for using if the *P*-distribution can be expressed via generalized functions

conventional: function \* *Pzz* (, )is a real but non-positive one. Nevertheless, the field state description with the Glauber-Sudarshan *P*-distribution remains the most demonstrative and consumable. For example, a proposed definition of non-classical states of electromagnetic field (Bogolyubov (Jr.) et al., 1988) uses the expression (1) for the statistical operator. A state is referred to as non-classical one if one of two requirements is obeyed: either average number of photons in a mode is less than 1, or *P*-function is not positively determined or has

 

 

are Bose amplitudes (creation and annihilation operators) of the


 

*F uu e e*

and wave vector *k* of the field modes). Since coherent states form an

of electromagnetic field should take into account the whole variety

*d zP z z z z* ,| | (1)

, (2)

(3)

. Here we use standard notation of quantum

and these variables are numbered by

**2. Electromagnetic field as an object of quantum statistical theory** 

footholds in physics. We refer to the following view of this diagonal representation

overcrowded basis in the state space of the mode with the completeness condition

dimensional Fourier transformation of the generating functional


A statistical operator

polarization

where \* *Pzz* (, ) is so called *P* -distribution ( { } *<sup>k</sup> z z*

directly with an arbitrary statistical operator

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

singularity that is higher than the

electrodynamics: *<sup>k</sup> c*

of slow growth, i.e.

field.

For a multi-mode field the statistical operator takes the form of a direct product of one-mode statistical operators. In Schrödinger picture the Liouville equation

$$
\hat{\sigma}\_t \rho(t) = -\frac{i}{\hbar} [\hat{H}, \rho(t)] \tag{4}
$$

describes the evolution of an arbitrary physical system. In the case when electromagnetic field interacting with matter is under consideration the problem is reduced to the correct account of the matter influence, so some kinds of effective Hamiltonians may appear in an analogue of (4) for the statistical operator of field. Evolution description in Heisenberg picture seems to be closer to the classical one. We come to operator Maxwell equations for field operators with terms corresponding to the matter influence and demanding some kind of material equations.

More graphic way to describing the electromagnetic field, its states, and their evolution is using correlation functions of different types, i.e. averaged values of physical quantities characterizing the field. The problem of choosing them will be discussed below.

#### **3. Correlation functions provided by methods of quantum optics**

Conventional classical optics was very restricted in measuring the parameters of fields. All conclusions about properties of light including its polarization properties were drawn from measurements of light intensity, i.e. from values of some quadratic functions of the field (Landau & Lifshitz, 1988). Naturally, we speak now about transversal waves in vacuum. Regarding a wave, close to a monochromatic one, we use slowly varying complex amplitude *E t* <sup>0</sup>*<sup>n</sup>* for its description:

$$E\_n = E\_{0n} \left( t \right) e^{-i\alpha t} \,. \tag{5}$$

Partially polarized light is characterized with the tensor of polarization

$$J\_{mm} = \overline{E\_{0m}\overline{E\_{0n}}} \tag{6}$$

where *m* and *n* corresponds to two possible directions of polarization and quick oscillations of field are neglected. Averaging is performed over time intervals or (in the case of statistically stable situation) in terms of probabilities. A sum of diagonal components of *mn J* is a real value that is proportional to the field intensity (the energy flux density in the wave in our case). Note that the discussion of field correlation functions by Landau in the earlier edition of the mentioned book was one of the first in the literature.

A rather full analysis of the classical measurement picture is given in (Klauder & Sudarshan, 1968). It should be mentioned that real field parameters are obtained from complex conjugated values in this approach. Transition to the quantum electromagnetic theory (Scully & Zubairy, 1997) is connected with substitution of operator structures with creation and annihilation operators instead of complex conjugated functions and coming to positive- and negative-frequency parts of field operators. Such expressions will be shown later on.

Description of Field States with Correlation Functions and Measurements in Quantum Optics 7

The correlation function of detector sensitivity in the suggestion that matrix elements of the dipole moment operator between the ground and excited states (so called dipole moment of

\* *Rmn* <sup>2</sup> *m n mn pp s*

 

to notice that the dependence of matrix elements of electric dipole moment on time in the interaction picture results in positive- and negative-frequency parts of field operators

It follows from (7) and (12) that the rate of counting for the considered model of an ideal

(1,1) ( ) *mn mn* ( ,; ,) *mn*

The problem of correlation of modes with different polarizations is a complicated one from the point of view of quantum measurements. So in most cases theoretical consideration goes

confirming that an ideal detector measures a correlation function of the first order with coinciding space-time arguments, i.e. field intensity in a fixed point ( *ne* is polarization

Correlation properties of radiation manifest themselves in interference experiments. The well-known Young scheme with signals from two apertures interfering can be analyzed in quantum terms. Schematically, we regard (in accordance with Huygens-Fresnel principle) a field value in an observation point *x* at some time *t* as a linear combination of field parameters in aperture points 1 *x* and 2 *x* at proper time moments. Using our previous considerations concerning quantum detectors, we put down, for example, for negative-

 

(15)

2 are determined by the system

1 11 2 22 ˆˆ ˆ *E xt E x t E x t* ,, ,

1 and

geometry. Thus for readings of an ideal detector placed in *x* we obtain an expression

 \* () () 12 1 1 2 2 ˆ ˆ 2Re

*E xtE xt* , , .

(1,1) *<sup>p</sup>*() ( ,; ,) , , *t sG x t x t s E x t E x t* , ˆ ˆ ( ,) ( ,) *Ext E xte n n* (14)

to the presence of polarization filter. For such case the correlation (13) takes the form

*p t s G xtxt*

stands for the spectral density of states in the continuous spectrum. It is expedient

 

(12)

*dt* (13)

, ˆ ˆ () () ( ,) ( ,) *E xt E xt n n*

.

1 2 ( ) ( ) 1 2 (2 ) <sup>ˆ</sup> , *<sup>k</sup> <sup>k</sup> i kx t n kn k k*

transition) | | ˆ*n n ep g p* are independent of a final state takes the form

 

*dw*

frequency part of the electric field strength for a fixed field polarization

 

where *t ts c* 1,2 1,2 / and 1,2 1,2 *s xx* ;

including an interference term

*E xt i e ce V*

 

 

appearing in calculated averages.

vector depending on the filter).

photon detector makes

where 

Physical picture of field parameter registration in the quantum case can be reduced to the problem of photon detection. An ideal detector should have response that is independent of radiation frequency and be small enough in comparison with the scale of field changes. Generally accepted analysis of quantum photon detector (Glauber, 1965; Kilin, 2003) is based on using an atom in this role and regarding the operator of field-atom interaction in the electric dipole approximation

$$
\hat{V} = -\hat{p}\_n \cdot \hat{E}\_n \left( \mathbf{x} \right),
$$

with ˆ*<sup>n</sup> p* standing for the operator of the electric dipole moment of an atom localized in a point with a radius-vector *x* (we shall denote in such a simple way a three-dimensional spatial vector). The quantum theory derives the total probability *w* of atom transition from a definite initial ground state |g to an arbitrary final excited one |e belonging to the continuous spectrum during the time interval from 0*t* to *t* on the basis of Dirac's nonstationary perturbation theory in the interaction picture (Kilin, 2003)

$$d\sigma = \int\_{t\_0} d\tau \int\_{t\_0} d\tau' \sum\_{mn} R\_{mn} \left(\tau - \tau'\right) G\_{mn}^{(1,1)}\left(\mathbf{x}, \tau'; \mathbf{x}, \tau\right) \tag{7}$$

where *Rmn* is a function of detector sensitivity and

$$\mathbf{G}\_{mn}^{(1,1)}(\mathbf{x}\_1, t\_1; \mathbf{x}\_1', t\_1') \equiv \langle \hat{E}\_m^{(-)}(\mathbf{x}\_1, t\_1) \hat{E}\_n^{(+)}(\mathbf{x}\_1', t\_1') \rangle \tag{8}$$

is field correlation function of the first order (we use the notation ˆ ˆ *A* Sp*A* for an arbitrary operator *A*ˆ ). Here and further we use standard expressions for operators of the vector potential, electric and magnetic field in the Coulomb gauge (Akhiezer A. & Berestetsky V., 1969)

$$\hat{A}\_n(\infty) = c \sum\_{k\alpha} \left(\frac{2\pi\hbar}{\alpha\_k V}\right)^{1/2} e\_{akn} (c\_{ak} + c\_{a,-k}^+) e^{ikx} \tag{9}$$

$$\hat{E}\_n(\mathbf{x}) = i \sum\_{k\alpha} \frac{(2\pi\hbar o\_k)^{1/2}}{V^{1/2}} e\_{akn} (\mathbf{c}\_{ak} - \mathbf{c}\_{a,-k}^+) e^{ik\mathbf{x}} \tag{10}$$

$$\hat{B}\_n(\infty) = i \sum\_{k\alpha} \frac{\left(2\pi\hbar\alpha\_k\right)^{1/2}}{V^{1/2}} \varepsilon\_{nlm} \tilde{k}\_l e\_{akm} (\varepsilon\_{ak} + \varepsilon\_{\alpha,-k}^+) e^{ik\infty}$$

In these formulas *kn e* are vectors of the circular polarization ( 0 *kn n e k* ), / *l l kkk* , *k ck* , *V* is field volume. Field operators in (8) are the positive- and negative-frequency parts of electric field operator in the picture of interaction

$$
\hat{E}\_n(\mathbf{x}, t) = \hat{E}\_n^{(+)}(\mathbf{x}, t) + \hat{E}\_n^{(-)}(\mathbf{x}, t) \; \; \; \tag{11}
$$

6 Quantum Optics and Laser Experiments

Physical picture of field parameter registration in the quantum case can be reduced to the problem of photon detection. An ideal detector should have response that is independent of radiation frequency and be small enough in comparison with the scale of field changes. Generally accepted analysis of quantum photon detector (Glauber, 1965; Kilin, 2003) is based on using an atom in this role and regarding the operator of field-atom interaction in

ˆ ˆ <sup>ˆ</sup> *V pEx n n*

with ˆ*<sup>n</sup> p* standing for the operator of the electric dipole moment of an atom localized in a point with a radius-vector *x* (we shall denote in such a simple way a three-dimensional spatial vector). The quantum theory derives the total probability *w* of atom transition from a definite initial ground state |g to an arbitrary final excited one |e belonging to the continuous spectrum during the time interval from 0*t* to *t* on the basis of Dirac's

*mn mn*

*w dd R G x x*

 

 1,1 11 11 11 11 ˆ ˆ *G xtxt E xtE xt mn* ,; , , , *<sup>m</sup> <sup>n</sup>*

arbitrary operator *A*ˆ ). Here and further we use standard expressions for operators of the vector potential, electric and magnetic field in the Coulomb gauge (Akhiezer A. &

1 2

<sup>2</sup> <sup>ˆ</sup> ( ) *ikx n kn k k*

1 2

(2 ) <sup>ˆ</sup> ( ) *<sup>k</sup> ikx n kn k k*

(2 ) <sup>ˆ</sup> ( ) *<sup>k</sup> ikx n nlm l km k k*

*ck* , *V* is field volume. Field operators in (8) are the positive- and negative-frequency

ˆˆ ˆ () () ( ,) ( ,) ( ,) *E xt E xt E xt nn n*

*Bx i ke c c e*

*Ex i ec c e V*

1 2

1 2 ,

1 2 ,

are vectors of the circular polarization ( 0 *kn n e k*

*Ax c e c c e V*

1,1 , ;,

 

,

 

 

> 

> >

(7)

(8)

, (10)

, (11)

; (9)

), / *l l kkk* ,

*A* for an

nonstationary perturbation theory in the interaction picture (Kilin, 2003)

0 0

parts of electric field operator in the picture of interaction

 

*t t mn*

is a function of detector sensitivity and

*k k*

 

*V*

 

*k*

*k*

is field correlation function of the first order (we use the notation ˆ ˆ *A* Sp

*t t*

the electric dipole approximation

where *Rmn*

 

Berestetsky V., 1969)

In these formulas *kn e*

*k* 

$$\hat{E}\_{n}^{(+)}\left(\mathbf{x},t\right) \equiv i \sum\_{k\alpha} \frac{\left(2\pi\hbar\alpha\_{k}\right)^{1/2}}{V^{1/2}} e\_{\alpha k n} c\_{\alpha k} e^{i\left(k\mathbf{x}-\alpha\_{k}t\right)}\,, \qquad \hat{E}\_{n}^{(-)}\left(\mathbf{x},t\right) = \hat{E}\_{n}^{(+)}\left(\mathbf{x},t\right)^{+}\;.$$

The correlation function of detector sensitivity in the suggestion that matrix elements of the dipole moment operator between the ground and excited states (so called dipole moment of transition) | | ˆ*n n ep g p* are independent of a final state takes the form

$$R\_{mn}\left(\tau-\tau'\right)=\pi\frac{\nu}{\hbar^2}p\_{m}^\*p\_{n}\delta\left(\tau-\tau'\right)\equiv s\_{mn}\delta\left(\tau-\tau'\right)\tag{12}$$

where stands for the spectral density of states in the continuous spectrum. It is expedient to notice that the dependence of matrix elements of electric dipole moment on time in the interaction picture results in positive- and negative-frequency parts of field operators appearing in calculated averages.

It follows from (7) and (12) that the rate of counting for the considered model of an ideal photon detector makes

$$p(t) = \frac{dw}{dt} = \sum\_{mn} \mathbf{s}\_{mn} \mathbf{G}\_{mn}^{(1,1)}(\mathbf{x}, t; \mathbf{x}, t) \tag{13}$$

The problem of correlation of modes with different polarizations is a complicated one from the point of view of quantum measurements. So in most cases theoretical consideration goes to the presence of polarization filter. For such case the correlation (13) takes the form

$$p(\mathbf{t}) = \mathbf{s} \cdot \mathbf{G}^{(1,1)}(\mathbf{x}, t; \mathbf{x}, t) \equiv \mathbf{s} \langle \mathbf{E}^{(-)}(\mathbf{x}, t) \mathbf{E}^{(+)}(\mathbf{x}, t) \rangle \,, \qquad \hat{E}(\mathbf{x}, t) \equiv \hat{E}\_n(\mathbf{x}, t) \mathbf{e}\_n \tag{14}$$

confirming that an ideal detector measures a correlation function of the first order with coinciding space-time arguments, i.e. field intensity in a fixed point ( *ne* is polarization vector depending on the filter).

Correlation properties of radiation manifest themselves in interference experiments. The well-known Young scheme with signals from two apertures interfering can be analyzed in quantum terms. Schematically, we regard (in accordance with Huygens-Fresnel principle) a field value in an observation point *x* at some time *t* as a linear combination of field parameters in aperture points 1 *x* and 2 *x* at proper time moments. Using our previous considerations concerning quantum detectors, we put down, for example, for negativefrequency part of the electric field strength for a fixed field polarization

$$
\hat{E}^{(-)}(\mathbf{x},t) = a\_1 \hat{E}^{(-)}(\mathbf{x}\_1, t\_1) + a\_2 \hat{E}^{(-)}(\mathbf{x}\_2, t\_2) \tag{15}
$$

where *t ts c* 1,2 1,2 / and 1,2 1,2 *s xx* ; 1 and 2 are determined by the system geometry. Thus for readings of an ideal detector placed in *x* we obtain an expression including an interference term

$$2\operatorname{Re}\alpha\_1\alpha\_2^\*\langle \hat{E}^{(-)}(x\_1,t\_1)\hat{E}^{(+)}(x\_2,t\_2)\rangle \ .$$

Description of Field States with Correlation Functions and Measurements in Quantum Optics 9

in the case of *M* detectors. At last, the most general set of normally ordered correlation

Functions (20) equal to zero usually at *M N* except very special states with broken symmetry (Glauber, 1969). Such function complex provides the most full description of the field correlation properties. In this picture taking into account magnetic field amplitudes is not necessary since they are simply connected with electric field amplitudes for each mode of electromagnetic field. Notice that the electric-dipole mechanism of absorption really

Method of photon counting corresponds to the general ideas of statistical approach; in its terms a number of quantum optics phenomena is described adequately, so the term "quantum optics" is used mainly as "statistical optics". Traditional terminology concerning correlation properties of light is based on the notion "coherence". In scientific literature coherences of the first and second orders are distinguished. It can be substantiated that, for example, the visibility of interference fringes in the Young scheme is determined by the coherence function of the first order that is a normalized correlation function of the first

. (20)

( ,) () () () () ... , ... 11 1 <sup>1</sup> ˆ ˆˆ ˆ ( ... , ... ) ( )... ( ) ( )... ( ) *M N <sup>M</sup> <sup>N</sup> M N G yyyy E y E yE y E y mmnn M N m mn n <sup>M</sup> <sup>N</sup>*

1 1 1 1

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

1 2 () () () ( )

<sup>2</sup> ˆˆ ˆˆ ,, ,, , ˆˆ ˆ ˆ ,, , ,

measurements, functions of lower orders are really urgent for experimental work.

**4. Superfluorescence in Dicke model as an important example of collective** 

The Dicke model of a system of great quantity of two-level emitters interacting via electromagnetic field (Dicke, 1954) is a noticeable case of synergetics in statistical system behavior during the relaxation processes. Its research history is very informative. R. Dicke came to the conclusion about superradiant state formation proceeding from the analysis of symmetry of quantum states of emitters described with quasispin operators. For long time equilibrium properties of the Dicke model were under discussion and the possibility of phase transition has been established; it was associated with field states in lasers. At the next step it has become clear that self-organizing takes place in the dynamical process and

ˆ ˆ ( ,) ( , ) , , ˆˆ ˆ ˆ ( ,) ( ,) ( , ) ( , ) *E x tE x t*

Similarly to (21), the photon grouping effect is determined by the coherence function of the

Coherences of higher orders (Bogolyubov (Jr.) et al., 1988) can be introduced in the same way. We shall refer to Glauber functions (20) as the main means of field description in quantum optics. Differences between time arguments play the decisive role in the physical interpretation of functions. Taking into account all difficulties and conditions for

 

11 2 2

 . (21)

(22)

*E x tE x t E x t E x t*

 

*E xt E xt E xt E xt*

 

*E xt E xt E xt E xt* 

functions introduced by Glauber (Glauber, 1963) looks like

dominates in experiments.

order (Scully & Zubairy, 1997)

second order

**quantum phenomena**

*g x*

*g xx*

The most important conclusion at this stage is possibility of measuring a correlation function of the first order defined by (8) with arbitrary arguments on the basis of the Young scheme and one photon detector. The stability of the statistical situation is suggested, thus function (8) is transformed into the function of 1 1 *t t* . So, using polarization filters after apertures, we obtain a scheme for measuring a correlation function (8) in the most general form.

We see that optical measurements with one quantum detector lead to considering a correlation function of the first order (8) with necessity. In order to obtain information about more complex correlation properties of electromagnetic fields, we should consider a more complicated model problem corresponding to the scheme of the famous pioneer experiments of Hanbury Brown and Twiss (Hanbury Brown & Twiss, 1956). We suppose that two ideal detectors of photons are located in points 1 *x* and 2 *x* ; optical shutters are placed in front of the detectors. The shutters are opened at the time moment 0*t* and closed at the moments 1*t* and 2*t* . Calculation of probability of photon absorption in each detector gives the following result

$$\boldsymbol{w}^{(2)} = \bigwedge\_{t\_0}^{t\_1} d\tau\_1 \bigwedge\_{t\_0}^{t\_2} d\tau\_2 \bigwedge\_{t\_0}^{t\_1} d\tau\_2' R\_{m\_1 n\_1} \left(\tau\_1 - \tau\_1'\right) R\_{m\_2 n\_2} \left(\tau\_2 - \tau\_2'\right) G\_{m\_1 m\_2, n\_1 n\_2}^{(2,2)} \left(\mathbf{x}\_1, \tau\_1; \mathbf{x}\_2, \tau\_2; \mathbf{x}\_1, \tau\_1'; \mathbf{x}\_2, \tau\_2'\right) (16)$$

where *Rmn* is a sensitivity correlation function determined by (12) and a correlation function of the second order

$$
\langle G\_{m\_1 m\_2, n\_1 n\_2}^{(2,2)}(y\_1, y\_2; y\_1', y\_2') \rangle = \langle \hat{E}\_{m\_1}^{(-)}(y\_1) \hat{E}\_{m\_2}^{(-)}(y\_2) \hat{E}\_{n\_1}^{(+)}(y\_1') \hat{E}\_{n\_2}^{(+)}(y\_2') \rangle \tag{17}
$$

is introduced (we use here an abbreviated notation *y* ( ,) *x t* ) . In the above-considered case of a broadband detector the rate of coinciding of photon registrations by two detectors makes

$$p^{(2)} = \frac{\partial^2 \mathbf{u} v^{(2)}}{\partial t\_1 \partial t\_2} = \mathbf{s}\_{m\_1 n\_1} \mathbf{s}\_{m\_2 n\_2} \mathbf{G}\_{m\_1 m\_2, n\_1 n\_2}^{(2,2)} \left( \mathbf{x}\_1, t\_1; \mathbf{x}\_2, t\_2; \mathbf{x}\_1, t\_1; \mathbf{x}\_2, t\_2 \right) \tag{18}$$

with detector parameters *mn s* introduced in (12). Therefore the Hanbury Brown–Twiss experimental scheme with registering the coincidence of photon absorption by two detectors obtaining signals from the divided light beam with a delay line in front of one of detectors provides measuring of the correlation function of the second order (17) if each detector operates with a certain polarization of the wave.

Generalizations of the Hanbury Brown–Twiss coincidence scheme for the case of *N* detectors are considered as obvious. The rate of *N*-fold coincidences is connected with a correlation function of *N*th order. The analysis of ideal quantum photon detector operation and coincidence scheme by Glauber has elucidated the nature of field functions measured via using the noted schemes – they are functions built with the set of normally ordered operators

$$
\langle \hat{E}\_{m\_1}^{(-)}(y\_1)...\hat{E}\_{m\_M}^{(-)}(y\_M)\hat{E}\_{n\_1}^{(+)}(y\_1)...\hat{E}\_{n\_M}^{(+)}(y\_M) \rangle \tag{19}
$$

8 Quantum Optics and Laser Experiments

The most important conclusion at this stage is possibility of measuring a correlation function of the first order defined by (8) with arbitrary arguments on the basis of the Young scheme and one photon detector. The stability of the statistical situation is suggested, thus function (8) is transformed into the function of 1 1 *t t* . So, using polarization filters after apertures,

We see that optical measurements with one quantum detector lead to considering a correlation function of the first order (8) with necessity. In order to obtain information about more complex correlation properties of electromagnetic fields, we should consider a more complicated model problem corresponding to the scheme of the famous pioneer experiments of Hanbury Brown and Twiss (Hanbury Brown & Twiss, 1956). We suppose that two ideal detectors of photons are located in points 1 *x* and 2 *x* ; optical shutters are placed in front of the detectors. The shutters are opened at the time moment 0*t* and closed at the moments 1*t* and 2*t* . Calculation of probability of photon absorption in each detector

 12 12 1 1 2 2 1 2 12

 

*m n m n mm nn*

1 2 12 <sup>1</sup> <sup>2</sup> <sup>1</sup> <sup>2</sup>

is introduced (we use here an abbreviated notation *y* ( ,) *x t* ) . In the above-considered case of a broadband detector the rate of coinciding of photon registrations by two detectors

11 22 1 2 12

with detector parameters *mn s* introduced in (12). Therefore the Hanbury Brown–Twiss experimental scheme with registering the coincidence of photon absorption by two detectors obtaining signals from the divided light beam with a delay line in front of one of detectors provides measuring of the correlation function of the second order (17) if each detector

Generalizations of the Hanbury Brown–Twiss coincidence scheme for the case of *N* detectors are considered as obvious. The rate of *N*-fold coincidences is connected with a correlation function of *N*th order. The analysis of ideal quantum photon detector operation and coincidence scheme by Glauber has elucidated the nature of field functions measured via using the noted schemes – they are functions built with the set of normally ordered operators

1 1

() () () () 1 1 ˆ ˆˆ ˆ ( )... ( ) ( )... ( ) *E y E yE y E y m mn n M M M M*

*p s s G xtxtxtxt*

*mn mn* ,; ,; ,; , *mm nn*

, <sup>1212</sup> <sup>1212</sup> ˆˆˆˆ ,;, *G yyyy E yE yE yE y mm nn mmn n*

*w d d d dR R G x x x x*

1 2 1 2 11 22 , 11 22 11 22 ,; ,;,; ,

(16)

is a sensitivity correlation function determined by (12) and a correlation

 

(17)

, 11 22 11 22

(19)

(18)

     

2 2,2

 

we obtain a scheme for measuring a correlation function (8) in the most general form.

gives the following result

> 

function of the second order

where *Rmn*

makes

0000

2,2

2 2

*w*

*t t*

operates with a certain polarization of the wave.

1 2

2 2,2

 

*tttt*

*tttt*

in the case of *M* detectors. At last, the most general set of normally ordered correlation functions introduced by Glauber (Glauber, 1963) looks like

$$\left(G\_{m\_1\ldots m\_M, n\_1\ldots n\_N}^{(M,N)}(y\_1\ldots y\_M, y\_1'\ldots y\_N') \equiv \langle \hat{E}\_{m\_1}^{(-)}(y\_1)\ldots \hat{E}\_{m\_M}^{(-)}(y\_M)\hat{E}\_{n\_1}^{(+)}(y\_1')\ldots \hat{E}\_{n\_N}^{(+)}(y\_N')\rangle\right). \tag{20}$$

Functions (20) equal to zero usually at *M N* except very special states with broken symmetry (Glauber, 1969). Such function complex provides the most full description of the field correlation properties. In this picture taking into account magnetic field amplitudes is not necessary since they are simply connected with electric field amplitudes for each mode of electromagnetic field. Notice that the electric-dipole mechanism of absorption really dominates in experiments.

Method of photon counting corresponds to the general ideas of statistical approach; in its terms a number of quantum optics phenomena is described adequately, so the term "quantum optics" is used mainly as "statistical optics". Traditional terminology concerning correlation properties of light is based on the notion "coherence". In scientific literature coherences of the first and second orders are distinguished. It can be substantiated that, for example, the visibility of interference fringes in the Young scheme is determined by the coherence function of the first order that is a normalized correlation function of the first order (Scully & Zubairy, 1997)

$$\log^{(1)}(\mathbf{x}\_1, \mathbf{x}\_2, \tau) = \frac{\langle \hat{E}^{(-)}(\mathbf{x}\_1, t) \hat{E}^{(+)}(\mathbf{x}\_2, t + \tau) \rangle}{\sqrt{\langle \hat{E}^{(-)}(\mathbf{x}\_1, t) \hat{E}^{(+)}(\mathbf{x}\_1, t) \rangle \langle \hat{E}^{(-)}(\mathbf{x}\_2, t + \tau) \hat{E}^{(+)}(\mathbf{x}\_2, t + \tau) \rangle}}. \tag{21}$$

Similarly to (21), the photon grouping effect is determined by the coherence function of the second order

$$\log^{(2)}\left(\mathbf{x},\tau\right) = \frac{\langle \hat{E}^{(-)}\left(\mathbf{x},t\right)\hat{E}^{(-)}\left(\mathbf{x},t+\tau\right)\hat{E}^{(+)}\left(\mathbf{x},t\right)\hat{E}^{(+)}\left(\mathbf{x},t+\tau\right)\rangle}{\langle \hat{E}^{(-)}\left(\mathbf{x},t\right)\hat{E}^{(+)}\left(\mathbf{x},t\right)\rangle\langle \hat{E}^{(-)}\left(\mathbf{x},t+\tau\right)\hat{E}^{(+)}\left(\mathbf{x},t+\tau\right)\rangle}\tag{22}$$

Coherences of higher orders (Bogolyubov (Jr.) et al., 1988) can be introduced in the same way. We shall refer to Glauber functions (20) as the main means of field description in quantum optics. Differences between time arguments play the decisive role in the physical interpretation of functions. Taking into account all difficulties and conditions for measurements, functions of lower orders are really urgent for experimental work.

## **4. Superfluorescence in Dicke model as an important example of collective quantum phenomena**

The Dicke model of a system of great quantity of two-level emitters interacting via electromagnetic field (Dicke, 1954) is a noticeable case of synergetics in statistical system behavior during the relaxation processes. Its research history is very informative. R. Dicke came to the conclusion about superradiant state formation proceeding from the analysis of symmetry of quantum states of emitters described with quasispin operators. For long time equilibrium properties of the Dicke model were under discussion and the possibility of phase transition has been established; it was associated with field states in lasers. At the next step it has become clear that self-organizing takes place in the dynamical process and

Description of Field States with Correlation Functions and Measurements in Quantum Optics 11

<sup>ˆ</sup> () 2 <sup>ˆ</sup> *<sup>n</sup> an ax a a N Px dr x x*

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

ˆ ˆ [ ( ), ( )] 0 *t t E xEx m n* , ˆ ˆ [ ( ), ( )] 0 *B xBx m n* , ( ) ˆ ˆ [ ( ), ( )] 4 *<sup>t</sup>*

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

It is very convenient to use operator evolution equations for investigating the dynamics of

1

*m n mnl*

, <sup>ˆ</sup> <sup>ˆ</sup> ( ) rot *B x c Ex n n* (26)

, <sup>ˆ</sup> <sup>ˆ</sup> () 2 <sup>ˆ</sup> *n n an ay <sup>a</sup>*

 *x r xx* 

*a J x P x dr x x* 

 

*x J xE x* (28)

, mf 1 2 *H HH* ˆ ˆˆ , (29)

2 ˆ ˆ () () *<sup>a</sup> <sup>a</sup> a a e x nx m*

).

(27)

. (24)

*x x B x E x ic*

 

is polarization index, <sup>ˆ</sup> ( ) *P x <sup>n</sup>* is

*l*

(25)

*x* 

*an r* is a quasispin operator, *a* is emitter's number,

the density of electric dipole moment (polarization) of emitters

by formulas (9), (10) and commutation relations

the system (23). The Maxwell operator equations have a known form

1 ˆ ˆ ( ) ( ) *az a a N*

> ˆ ˆˆ ( ) *<sup>t</sup> n n*

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

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

> <sup>ˆ</sup> *kk k k H cc*

<sup>1</sup> <sup>ˆ</sup> <sup>ˆ</sup> ()() <sup>ˆ</sup> <sup>2</sup> *H dxA x x*

2 2

*c*

 

(

<sup>ˆ</sup> ˆ ˆ rot 4 *E x c Bx J x nn n*

where total electric field and electromagnetic current

ˆˆ ˆ () () 4 () *<sup>t</sup> Ex Ex Px nn n*

are introduced. Energy density of emitter medium

investigating the field correlation properties.

f m mf *HH H H* ˆˆ ˆ ˆ , f

<sup>1</sup> <sup>1</sup> <sup>ˆ</sup> ˆ ˆ *H dxA x j x n n <sup>c</sup>* , <sup>2</sup>

(Sokolovsky & Stupka, 2004) in the form

Here ˆ

, *k k c c* 

field-emitters system).

obeys the evolution equation

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 and technology.

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 system and elimination of singularities in kinetic coefficients.

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.
