1. Introduction in the specific properties of correlated bimodal radiation field

Generated radiation in two- or multi-quantum processes opens new perspectives in studying new communication systems, holographic phase correlations, in the interaction of light with biomolecules and living systems. The specific attention is given to the new type of coherent emissions, which occurs not only between the quantum but between the photon groups generated in the non-linear interaction of the electromagnetic field (EMF) with emitters (atoms, molecules, biomolecules, etc.). This type of light generation supports the idea of coherent correlation that appears in the bi-modal field, in which it is generated the entangled photons.

A physical characteristic of field formed from the blocs of well-correlated bi-modes must be determined by the intensity of the electric field of each mode characteristic in such superposition. The applications of such a field characteristic can be fruitful both in quantum communication and holography. An attractive aspect of the problem consists in the selective two-quantum excitation of some atoms or molecules of the system, where it is necessary minimize the dipole active action of total photon flux over single-photon resonance of dipole-active transitions. The last idea can be applied in microbiology, where a selective dis-activation of some molecular structures (e.g. of viruses) in the tissue may become possible in two-quanta excitations. In this situation appears the necessity for a good description of both amplitude and phase of this new type of radiation formed from bimodal correlated photons.

the number of atomic sources, I ¼ ∑<sup>j</sup>

DOI: http://dx.doi.org/10.5772/intechopen.85857

through of each atom.

^ E � <sup>j</sup> ð Þ <sup>t</sup>; <sup>z</sup> (^ E <sup>þ</sup>

41

function between the strengths of different radiators is 〈^

using the two photon pairs with identical properties.

new studies of vibrational aspects of molecules.

〈^ E � <sup>j</sup> ð Þ <sup>t</sup>; <sup>z</sup> ^ E <sup>þ</sup>

Coherence Proprieties of Entangled Bi-Modal Field and Its Application in Holography…

a confined space which light enters, but from which it cannot escape. Light scattered from the object is detected in this confined space entirely without the benefit of spatial resolution. Quantum holography offers this possibility by virtue of the fourth-order quantum coherence inherent in entangled beams. This new conception is based on the application of second ordered coherence function, proposed firstly by Glauber [6], and intensively developed in the last years. The possibility to use of the fourth-order quantum coherence of entangled beams was studied in Ref. [7]. Here it is proposed a two-photon analog of classical holography. Not so far the authors of Refs. [8, 9] used an innovative equipment registered the behavior of pairs of distinguishable and non-distinguishable photons entering a beam splitter. When the photons are distinguishable, their behavior at the beam splitter is random: one or both photons can be transmitted or reflected. Non-distinguishable photons exhibit quantum interference, which alters their behavior: they join into pairs and are always transmitted or reflected together. This is known as two-photon interference or the Hong-Ou-Mandel effect. The visibility of the hologram of a single photon fringe, V, is defined by a spectral mode overlap which can be high and stable for photons generated by different sources such as two independent spontaneous parametric down-conversion. As the authors mentioned, in the registration of single photon hologram [8], the quantum interference can be observed by registering pairs of photons. The experiment needs to be repeated several times

The authors of the Refs. [10-13] have proposed to investigate the coherence which appears between undistinguished photon pairs and the possibility to generate such a pair in the two-photon quantum generators. The increased interest not only to two-photon generation, but to induce Raman microscopy in special medicine and biology opens the new perspective the coherent proprieties of bimodal fields. Compared to spontaneous Raman scattering, coherent Raman scattering techniques can produce much stronger vibrational sensitive signals. This excitation needs a strong phase correlation between the pump, Stokes, and anti-Stokes components of the induced Raman process. These difficulties have been overcome by recent advances in coherent Raman scattering microscopy, which is based on either coherent anti-Stokes Raman scattering or stimulated Raman scattering [14, 15]. Appear a possibility to use this type of coherent states of bimodal field [12, 13], and to propose a

Following this idea let us discuss another effect related to the photon scattering processes into the pump, Stokes and anti-Stokes modes. Taking into consideration that in the Λ-type three-level system persists only pumping and Stokes modes when the atomic system is prepared in the ground state, or it may be reduced to the pump and anti-Stokes modes, when the atomic system is prepared in the excited state, we could reduce this cooperative scattering effect to the ensemble of correlated pairs of modes in the resonator. This is possible due to big detuning between the third level and pump modes when the system of atoms is prepared in the ground state. In this situation, it is possible to generate m pairs of correlated mods between

<sup>j</sup> ð Þ t; z ) is the negative (positive) frequency component of the EMF

The creation of entangled photons in two- and multi-quantum processes opens the new possibilities in quantum communication and quantum holography. For example in the paper of prof. Teich et al. [5] it is proposed to make use of quantum entanglement for extracting holographic information about a remote 3 � D object in

<sup>j</sup> ð Þ t; z 〉 ∝ N, because the correlation

<sup>l</sup>¼6 <sup>j</sup> ≈0. Here

E � <sup>j</sup> ð Þ <sup>t</sup>; <sup>z</sup> ^ E <sup>þ</sup> <sup>l</sup> ð Þ tz 〉j

The new concept of phase and amplitude correlations are important not only in interferometry but also in the holographic registration of information and are related to the conceptual aspects of physics, chemistry and microbiology for the recording of three and multi-dimensional images in cosmology [1–3]. According to the invention of Dennis Gabor [4] in 1947, the hologram is defined by the interference between two waves, the 'object wave' and the 'reference wave'. Like in laser experiments, this interference between the two waves requires to use the temporally and spatially consistent source, described by an intensity pattern, which represents the modulus squared of the sum of the two complex amplitudes. The reconstruction of the object field encoded within the hologram is based on the principle of light diffraction. This type of diffraction and interference can be keyed out by other coherent states, which can be an eigenstate of square parts of positive frequency strength of EMF. According to this description, the eigenvalue of vectors of square strength has the good amplitude and phase. For example, in the two photon cooperative emission by the pencil shape system of radiators (or by the cavity two-photon induced emission) the coherence is based between the photon pairs rather than between the individual photons. This effect is evident, when the pairs of photons are generated in the broadband spectral region of the EMF so, that the total energy of two photons in each pair is constant 2ℏω<sup>0</sup> ¼ ℏω<sup>k</sup><sup>1</sup> þ ℏω<sup>k</sup><sup>2</sup> ¼ Const. Considering that the frequencies of the photons in the pairs are aleatory distributed, ωki 6¼ ωkj, we conclude that such systems generates the higher the second order coherence relative to first order one. In this context appear the problem of the application of such field in communication and holography, using its good amplitude and phase of squared strength, generated by the nominated sources. This chapter discusses the problem associated with the possibilities to divide the wave front of the photon-pairs into two wave fronts. Studying the interference between each part, "object bimodal waves" E^<sup>þ</sup> ð Þ <sup>t</sup> <sup>þ</sup> <sup>τ</sup> <sup>E</sup>^<sup>þ</sup> ð Þ t þ τ and "reference bimodal waves" E^<sup>þ</sup> ð Þ<sup>t</sup> <sup>E</sup>^<sup>þ</sup> ð Þt , we may create the hologram image consisted of the interference and diffraction fringes between the bi-photons belonging to wave fronts of square vectors of field consisted from the ensemble of bi-modal field, 〈E^<sup>þ</sup> ð Þ <sup>t</sup> <sup>þ</sup> <sup>τ</sup> <sup>E</sup>^<sup>þ</sup> ð Þ <sup>t</sup> <sup>þ</sup> <sup>τ</sup> <sup>E</sup>^� ð Þ<sup>t</sup> <sup>E</sup>^� ð Þt 〉.

To understood this type of coherence let us look at the light that consists of distinctive photons, which belong to broadband spectrum energy. Since the number of modes is relatively large, it is virtually impossible to find the two photons in the same mode and to create the coherent states from them, 〈E^� <sup>k</sup> ð Þ <sup>t</sup>; <sup>z</sup> <sup>E</sup>^<sup>þ</sup> <sup>k</sup>0ð Þ t; z 〉≈0, where E^� <sup>k</sup> ð Þ <sup>t</sup>; <sup>z</sup> and <sup>E</sup>^<sup>þ</sup> <sup>k</sup>0ð Þ t; z is the Fourier transform the negative or positive defined EMF strength components of the radiation modes k 6¼ k<sup>0</sup> obtained from the inverted atomic ensemble in z direction. Of course, the total intensity of such a light, obtained from individual sources (nuclei, atoms, molecules) becomes proportional to Coherence Proprieties of Entangled Bi-Modal Field and Its Application in Holography… DOI: http://dx.doi.org/10.5772/intechopen.85857

the number of atomic sources, I ¼ ∑<sup>j</sup> 〈^ E � <sup>j</sup> ð Þ <sup>t</sup>; <sup>z</sup> ^ E <sup>þ</sup> <sup>j</sup> ð Þ t; z 〉 ∝ N, because the correlation function between the strengths of different radiators is 〈^ E � <sup>j</sup> ð Þ <sup>t</sup>; <sup>z</sup> ^ E <sup>þ</sup> <sup>l</sup> ð Þ tz 〉j <sup>l</sup>¼6 <sup>j</sup> ≈0. Here ^ E � <sup>j</sup> ð Þ <sup>t</sup>; <sup>z</sup> (^ E <sup>þ</sup> <sup>j</sup> ð Þ t; z ) is the negative (positive) frequency component of the EMF through of each atom.

The creation of entangled photons in two- and multi-quantum processes opens the new possibilities in quantum communication and quantum holography. For example in the paper of prof. Teich et al. [5] it is proposed to make use of quantum entanglement for extracting holographic information about a remote 3 � D object in a confined space which light enters, but from which it cannot escape. Light scattered from the object is detected in this confined space entirely without the benefit of spatial resolution. Quantum holography offers this possibility by virtue of the fourth-order quantum coherence inherent in entangled beams. This new conception is based on the application of second ordered coherence function, proposed firstly by Glauber [6], and intensively developed in the last years. The possibility to use of the fourth-order quantum coherence of entangled beams was studied in Ref. [7]. Here it is proposed a two-photon analog of classical holography. Not so far the authors of Refs. [8, 9] used an innovative equipment registered the behavior of pairs of distinguishable and non-distinguishable photons entering a beam splitter. When the photons are distinguishable, their behavior at the beam splitter is random: one or both photons can be transmitted or reflected. Non-distinguishable photons exhibit quantum interference, which alters their behavior: they join into pairs and are always transmitted or reflected together. This is known as two-photon interference or the Hong-Ou-Mandel effect. The visibility of the hologram of a single photon fringe, V, is defined by a spectral mode overlap which can be high and stable for photons generated by different sources such as two independent spontaneous parametric down-conversion. As the authors mentioned, in the registration of single photon hologram [8], the quantum interference can be observed by registering pairs of photons. The experiment needs to be repeated several times using the two photon pairs with identical properties.

The authors of the Refs. [10-13] have proposed to investigate the coherence which appears between undistinguished photon pairs and the possibility to generate such a pair in the two-photon quantum generators. The increased interest not only to two-photon generation, but to induce Raman microscopy in special medicine and biology opens the new perspective the coherent proprieties of bimodal fields. Compared to spontaneous Raman scattering, coherent Raman scattering techniques can produce much stronger vibrational sensitive signals. This excitation needs a strong phase correlation between the pump, Stokes, and anti-Stokes components of the induced Raman process. These difficulties have been overcome by recent advances in coherent Raman scattering microscopy, which is based on either coherent anti-Stokes Raman scattering or stimulated Raman scattering [14, 15]. Appear a possibility to use this type of coherent states of bimodal field [12, 13], and to propose a new studies of vibrational aspects of molecules.

Following this idea let us discuss another effect related to the photon scattering processes into the pump, Stokes and anti-Stokes modes. Taking into consideration that in the Λ-type three-level system persists only pumping and Stokes modes when the atomic system is prepared in the ground state, or it may be reduced to the pump and anti-Stokes modes, when the atomic system is prepared in the excited state, we could reduce this cooperative scattering effect to the ensemble of correlated pairs of modes in the resonator. This is possible due to big detuning between the third level and pump modes when the system of atoms is prepared in the ground state. In this situation, it is possible to generate m pairs of correlated mods between

A physical characteristic of field formed from the blocs of well-correlated bi-modes must be determined by the intensity of the electric field of each mode characteristic in such superposition. The applications of such a field characteristic can be fruitful both in quantum communication and holography. An attractive aspect of the problem consists in the selective two-quantum excitation of some atoms or molecules of the system, where it is necessary minimize the dipole active action of total photon flux over single-photon resonance of dipole-active transitions. The last idea can be applied in microbiology, where a selective dis-activation of some molecular structures (e.g. of viruses) in the tissue may become possible in two-quanta excitations. In this situation appears the necessity for a good description of both amplitude and phase of this new type of radiation formed from bimodal correlated photons.

Quantum Cryptography in Advanced Networks

The new concept of phase and amplitude correlations are important not only in

ð Þ <sup>t</sup> <sup>þ</sup> <sup>τ</sup> <sup>E</sup>^<sup>þ</sup>

interference and diffraction fringes between the bi-photons belonging to wave fronts of square vectors of field consisted from the ensemble of bi-modal field,

To understood this type of coherence let us look at the light that consists of distinctive photons, which belong to broadband spectrum energy. Since the number of modes is relatively large, it is virtually impossible to find the two photons in the

EMF strength components of the radiation modes k 6¼ k<sup>0</sup> obtained from the inverted atomic ensemble in z direction. Of course, the total intensity of such a light,

obtained from individual sources (nuclei, atoms, molecules) becomes proportional to

ð Þt , we may create the hologram image consisted of the

<sup>k</sup>0ð Þ t; z is the Fourier transform the negative or positive defined

ð Þ t þ τ and "reference

<sup>k</sup> ð Þ <sup>t</sup>; <sup>z</sup> <sup>E</sup>^<sup>þ</sup>

<sup>k</sup>0ð Þ t; z 〉≈0,

interferometry but also in the holographic registration of information and are related to the conceptual aspects of physics, chemistry and microbiology for the recording of three and multi-dimensional images in cosmology [1–3]. According to the invention of Dennis Gabor [4] in 1947, the hologram is defined by the interference between two waves, the 'object wave' and the 'reference wave'. Like in laser experiments, this interference between the two waves requires to use the temporally and spatially consistent source, described by an intensity pattern, which represents the modulus squared of the sum of the two complex amplitudes. The reconstruction of the object field encoded within the hologram is based on the principle of light diffraction. This type of diffraction and interference can be keyed out by other coherent states, which can be an eigenstate of square parts of positive frequency strength of EMF. According to this description, the eigenvalue of vectors of square strength has the good amplitude and phase. For example, in the two photon cooperative emission by the pencil shape system of radiators (or by the cavity two-photon induced emission) the coherence is based between the photon pairs rather than between the individual photons. This effect is evident, when the pairs of photons are generated in the broadband spectral region of the EMF so, that the total energy of two photons in each pair is constant 2ℏω<sup>0</sup> ¼ ℏω<sup>k</sup><sup>1</sup> þ ℏω<sup>k</sup><sup>2</sup> ¼ Const. Considering that the frequencies of the photons in the pairs are aleatory distributed, ωki 6¼ ωkj, we conclude that such systems generates the higher the second order coherence relative to first order one. In this context appear the problem of the application of such field in communication and holography, using its good amplitude and phase of squared strength, generated by the nominated sources. This chapter discusses the problem associated with the possibilities to divide the wave front of the photon-pairs into two wave fronts. Studying the interference

between each part, "object bimodal waves" E^<sup>þ</sup>

ð Þ<sup>t</sup> <sup>E</sup>^�

ð Þt 〉.

same mode and to create the coherent states from them, 〈E^�

ð Þ<sup>t</sup> <sup>E</sup>^<sup>þ</sup>

ð Þ <sup>t</sup> <sup>þ</sup> <sup>τ</sup> <sup>E</sup>^�

<sup>k</sup> ð Þ <sup>t</sup>; <sup>z</sup> and <sup>E</sup>^<sup>þ</sup>

bimodal waves" E^<sup>þ</sup>

ð Þ <sup>t</sup> <sup>þ</sup> <sup>τ</sup> <sup>E</sup>^<sup>þ</sup>

〈E^<sup>þ</sup>

where E^�

40

the pump and Stokes components, so that the frequency difference between the modes of each pair, ωkp<sup>1</sup> � ωks<sup>1</sup> , ωks<sup>2</sup> � ωks<sup>2</sup> , ωkpj � ωksj, ωkpm � ωksm is equal to the transition energy between the ground and excited levels 2ω0. In this description, the pump, and Stokes fields are considered to be incoherent due to the fact that the correlation functions between the pump and Stokes modes, belonging to different partitions gives zero contributions in the intensity correlation function 〈E^ð Þ � kpj ð Þ<sup>t</sup> <sup>E</sup>^ð Þ <sup>þ</sup> kpl ð Þ<sup>t</sup> 〉j6¼<sup>l</sup> and 〈E^ð Þ � ksj ð Þ<sup>t</sup> <sup>E</sup>^ð Þ <sup>þ</sup> ksl ð Þt 〉j6¼l. According to this only the diagonal elements belonging to the same modes remain non-zero so that the field intensity is proportional to this number "m". The resulting intensity of the pumping light is equal to the square strength of each pumping (or Stokes) modes Ip ¼ ∑<sup>j</sup> 〈E^ð Þ � kpj ð Þ<sup>t</sup> <sup>E</sup>^ð Þ <sup>þ</sup> kpj ð Þt 〉 � m (or Is ¼ ∑<sup>j</sup> 〈E^ð Þ � ksj ð Þ<sup>t</sup> <sup>E</sup>^ð Þ <sup>þ</sup> ks j ð Þt 〉 � m). The realization of such cooperative effects between the incoherent bi-modes can be obtained exactly as in the case of the two-photon generation in a wide spectrum at Raman emission (e.g., in the multimodal cavity or crossing the pumping pulse through a multimodal optical fiber). The total pumping field strength and Stokes is a multi-mode superposition for both pumping and Stokes mods, where the pumping field and Stokes are decomposed into the quantized states of the optical cavity.

2. Generation of biharmonic strength operators and their coherent

Coherence Proprieties of Entangled Bi-Modal Field and Its Application in Holography…

[21, 22], the quantum fluctuations in each mode may achieve zero value

the individual photons (first-order coherence) becomes inhibited.

Let us first discuss the three particle cooperative effects represented in Figure 1 described in Refs. [17–19]. This interaction is focused on a new type of three particle collective spontaneous emission, in which the decay rate of three atomic subsystems is proportional to the product of the numbers of atoms in each subsystem, NsNrNd in the case all three sub-ensemble are equidistant. The quantum master equations take into consideration the correlations between three subsystems S, R and D in the single and two-photons cooperative exchanges between the atoms of each

photon is generated (anti-Stokes b<sup>þ</sup>

below two situations.

43

〈E^r〉 <sup>¼</sup> 〈E^s〉 <sup>¼</sup> 0. Following this conception, the similar coherence between the photons we introduced in the Raman emission processes [12, 13, 23]. In this case one photon from non-coherent driving field is absorbed (Stokes photon as) and other

Case A. correspond to the generation of the coherent bi-photons along the axes of the pencil shape system of an inverted atomic system relative a dipole forbidden transition [19] together with two dipole active subsystems of radiators, S and R (see Ref. [23]). Here we propose another effect, in which the two-quantum cooperative emissions are ignited by single-photon decay process. As one-photon decay process of an exciting ensemble of atoms passes into Dicke super-fluorescence [24], we propose the situation in which this effect can be inhibited and the new cooperative interaction of this ensemble with dipole forbidden transitions of other atomic subsystems will stimulate another cooperative decay process, in which the coherence is established between the photon pairs. Indeed, if we consider an ensemble of excited atoms with non-equidistant transition energy, we may observe that in this situation the phase correlations between the atoms may be neglected. The non-equidistant dipole-active ensemble may be divided into two sub-ensembles of excited atoms, so that the pair of excited radiators from each sub-ensemble enters in resonance dipole-forbidden transition ð Þ n þ 1 S � nS of D of sub-ensemble D. In other words, we are interested in cooperative interaction between two dipole-active subensembles and dipole-forbidden one as this is represented in Figure 1. Such superradiance has the coherence between the photon-pairs and the coherence between

<sup>a</sup> ) 〈E^rE^a〉 6¼ <sup>0</sup>,〈E^rE^s〉 6¼ 0. Below we consider

Let us consider two nonlinear processes of light generation in laser [16, 13] and collective decay phenomena [17, 18]. In the second order of interaction of light with matter, these processes strongly connect the quantum fluctuations of two waves. In the output detection region, this effect gives us the possibility to obtain the coherent effects between the bimodal fields as this is proposed in Section 1. In this nonlinear generation of light, the new signal at another frequency has a common coherent phase with impute mode in the nonlinear medium. We discuss the situation when the phases of the emitted waves are random relative to one another so that the total field average of EMF strength takes zero value 〈E x ^ð Þ ; <sup>t</sup> 〉 <sup>¼</sup> 0. In such a situation the emission is considered non-coherent. An opposite conception appears in quantum optics in which it is proposed a lot of effects connected with quantum entanglement and coherent proprieties of bimodal field [19]. Here as we discuss in Section 1 we have the possibility to introduce another type of coherence [12, 19, 20], which appear not between the photons of the same mode, but between the biphotons from the ensemble of pairs of modes or correlations of photons belonging to scattered modes (bi-modes) 〈E^rE^s〉 6¼ 0. When the number of bi-modes with the same energy of the photons in the pair increase

proprieties

DOI: http://dx.doi.org/10.5772/intechopen.85857

The possibilities of correlations between the anti-Stokes, Stokes and pump modes have been overcome by recent advances in coherent Raman scattering microscopy, which is based on either coherent coherent anti-Stokes Raman scattering or coherent Raman scattering [14, 15]. In many cases, the phase correlations between these components become not so simple in the experimental realization. Appear the possibility to apply here the coherent states of bimodal field proposed in this chapter and a possibility to use holographic aspects of such bimodal field in biology and medicine where the phase and amplitude of Raman component are already correlated for coherent excitation of molecular vibrations Π^ � ðÞ¼ t ∑<sup>j</sup> E^ð Þ � ksj ð Þ<sup>t</sup> <sup>E</sup>^ð Þ <sup>þ</sup> kpj ð Þt . Here we notice that this new characteristic of the field in induced Raman process may have a good phase and amplitude as the traditional coherent field, Π�ðÞ¼ t Π<sup>0</sup> exp ½ � �iϕ , the correlatives between the adjacent modes is proportional to the square number of adjacent bi-mods 〈Π^ � ð Þ<sup>t</sup> <sup>Π</sup>^ <sup>þ</sup> ð Þ<sup>t</sup> 〉 � <sup>m</sup>2. Phase ϕ ¼ 2ω0t � Kz contains a fixed frequency 2ω<sup>0</sup> ¼ ωpki � ωski (here i ¼ 1, 2, …m) and

a well-defined wave vector, K ¼ kpi � ksi in the collinear cavity conversion of the photons from the pump field.

In the Section 2 we give the definition of bimodal coherent states in analogy with single photon coherent states. The definition of phase and amplitude of this bimodal field is also granted, taking into account the coherent states of bimodal superposition of entangled photon pairs and bimodal superposition of Stokes pump and anti-Stokes modes in the Raman scattering process. The lithographic proprieties of such bimodal field are given, taking into consideration multi-mode aspects of generation light.

The Section 3 is devoted to applications of coherent emission of two subgroups of photons, the total (or difference) energies of which can be reckoned as a constant, so that coherence appear between the vectors formed from the product of two electromagnetic field strengths. As it is shown that the coherence between such vectors is manifested if the emitted bi-photons belonging to broadband spectrum, hence that the coherence between individual photons can be neglected. The application of product strength amplitudes and phases in holographic registration is advised. The superposition of two vectors of bimodal field obtained in two-photon or Raman lasing effect is estimated for construction of the holographic image of the object.
