## Abstract

This chapter examines the coherence properties of two modes of entangled photons and its application quantum communication and holography. It is proposed novel two-photon entangled sources which take into account the coherence and collective phenomena between the photon belonging to two different modes obtained in two-photon cooperative emission or Raman or lasing. The generation of the correlated bimodal entangled field in two-photon emission or Raman Pump, Stokes and anti-Stokes modes is proposed in the free space and cavity-induced emission. The application two-photon and Raman bimodal coherent field in communication and holography are given in accordance with the definition of amplitude and phase of such entangled states of light. At first, this method does not appear to be essentially different in comparison with the classical coherent state of information processing, but if we send this information in dispersive media, which separates the anti-Stokes and Stokes photons from coherent entanglement fields, the information is drastically destroyed, due to the quantum distribution of photons in the big number modes may be realized in the situation in which the mean value of strength of bimodal field tends to zero. The possibility of restoration of the signal after the propagation of the bimodal field through different fibers, we may restore the common square amplitude and phase.

### Keywords

- quantum bimodal field
- cooperative effects between blocks of photons
- quantum communication
- quantum holography

## 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 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, 2, 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

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,

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

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

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

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.

## 2. Generation of biharmonic strength operators and their coherent proprieties

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

*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,

Let us first discuss the three particle cooperative effects represented in Figure 1 described in Refs. [17, 18, 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,

In this situation is respect all resonance conditions between the pairs and equidistant D- ensemble:

Following this conception, we observe that for the big ensemble of radiators the first order correlation function

and two photon cooperative ignition by the

Here we consider the sums on the repeated indexes. It is observed, that such a sum is proportional to the number dipole-active pairs

In the second order of interaction of light with matter, these processes strongly connect two waves in the output detection schemes and they give us the possibility to distinguish the coherent effects between entangled photons. For traditional single-mode coherence, it is well known the possible lithographic limits in measurements

The coherent properties and entanglement between the photons, emitted in two-quantum lasers and parametric down conversion has a great impact on application in quantum information and communication. The possibility of induced two-photon generation per atomic transition was suggested by Sorokin, Braslau and Prohorov [27, 28]. The scattering effects in two-photon amplifier attenuate the possibility to realize two-photon lasing. The first experiments demonstrated that two-photon amplification and lasing in the presence of external sources are possible [16, 29]. These ideas open the new conception about the coherence. Indeed, introducing the amplitude of two-quantum field encapsulated in two-photon lasers we can observe that the generation amplitude is described by the field product

where

where

To decorrelate the coherence between the photons of the same mode, in Ref. [10, 11] we proposed the cooperative multi-mode operators with similar commutation relations in the cavities. Mediating the amplitude of the bimodal fields we can introduce the collective modes field operators

where

*Case B*. Another possibility to create a coherent field for a big number of photons distributed in the broadband spectrum represents the bimodal spectrum of scattered photons. Indeed if we represent superposition between the photons obtained from

We observe, that the Dicke cooperative effects between the sub-ensemble of atoms,

As follows from the expression (4) and the scattering generation of correlation photons in the cavity Figure 4 the scattered field into the blocs of two-modes

Let us study the interaction between the molecular systems and external Raman field prepared in the cooperative coherent process proposed in Refs. [12, 13, 30]. Following this Refs [19, 23], we can introduced the bimodal operators the product of which oscillates with the frequency

Here the annihilation (creation) operators,

where

Let us find the coherent phenomena which appears between two fields in Raman processes. If we study generation of Stokes light under the non-coherent pumping with anti-Stokes field, we can introduce the following representation of the bi-modal field

where

where

where

The lithographic limit between maximal and minimal values of amplitude of correlation function

## 3. Quantum communication and holographic proprieties of bi-boson coherent field

The main differences between this bimodal field and the classical coherent field consists in the aleatory distribution of energies and phases between the photons of each pair, which enter in the coherent ensemble of bi-photons. Passing through the dispersion media’s the common phrase of the ensemble may be drastically destroyed so, that the problem which appear consist in the restoration of common phase of the ensemble of photon pairs generated by the quantum sources. These phenomena of restoration of common phase of the ensemble have a quantum aspects and can be used in quantum communication and quantum holography.

In the case A we proposed the new possibilities in decreasing of coherent proprieties between the photon pairs of two-photon beam. The application of coherent effect of the bimodal field of communication and holography opens the new perspectives in the transmission of information not only through entangled state of photons but also through the second order coherence. At the first glance one observes that such coherent registration of information may have nothing to do with the traditional method. But looking to the scheme of Figure 7 we observe that when the photon-pair pulses pass through a dispersive medium, the idler photons from the pair change their directions relative to signal photons. Focusing the signal and idler photons into different optical fibers, we are totally dropping the coherence among the photons. However, after a certain time interval, the idler and signal photons from the pairs could be mixed again (see Figure 7) and the coherence may be restored. The coherent state obtained in two-photon cooperative or laser emission takes into account not only entanglement between the pairs of photons, but the coherence between the bi-photons too, and can be used in mixed processing problems in which the quantum entanglement between the photon of each pair of photons is used simultaneously with classical coherence between the pairs.

Below we discuss how hologram can be constructed using the recording phase information of bimodal field on a medium sensitive to this phase, using two separate beams of bimodal field (one is the “usual” beam associated with the image to be recorded and the other is a known as the reference beam). Exploiting the interference pattern between these bi-boson fields described in the last section in principle this is possible. For example the Stokes and anti-Stokes fields can be regarded as a field with electromagnetic strength product (6), so that the common phases

Presently exist a lot of proposals in which is manifested holographic principals of processing of quantum information [5, 8, 9]. One of them is the model of Prof. Teich with co-authors [5]. According to this model the correlations between the entangled photons, obtained in parametric down conversion, can be used in quantum holography. The hologram in parametrical down conversion is realized in terms of the correlations between the entangled photon in the single pair. The coherence between the pairs is not taken into consideration.

Following the idea of classical holograms, we changed the conception of two-photon holograms using the second order interference described in Refs. [19, 30]. This new type of hologram registration is based on the coherent proprieties (3) and (8). As well known the holographic code in single photon coherent effects appears on mixing the original wave (hereafter called the “object wave”)

In the expression (9) the function

The same behavior has the bimodal field formed from Stokes Pump and anti-Stokes photons. In the case of scattering bimodal field the coherent proprieties of the vector

This type of Holography takes into consideration the coherent process at low frequency

The entanglement between each mode of the field can be detected by two-photon detector schemes, placed in the plan of hologram represented in Figure 9. This procedure may be in tangency with proposed experimental detections of vibration modes of biomolecules [8, 9].

In comparison with the spontaneous parametric down-conversion the super-radiance [21] or cooperative scattering processes [12, 13] represented generators of non-classical light source—the two-photon quantum entangled state with the coherent aspects between the two conjugate modes. Two-modes from such processes may become incoherent, but the coherence can be revived in the two-photon excitations of the detector which represents the photon pairs from adjacent modes. The two-photon detection scheme an interference connected to it is shown in Figure 6. The similar effect appears between stokes, pump, and anti-Stokes photon in induced scattering. In the pioneer theoretical work of two-photon optics, Belinskii and Klyshko [7] predicted three spooky schemes: two-photon diffraction, two-photon holography, and two-photon transformation of two-dimensional images. The first and last schemes have been demonstrated in the experiments, known as ghost interference [34] and ghost imaging [35], respectively. These experiments are connected to the original gedankenexperiment of EPR paradox and open the way to the detection of two-photon holography [8, 9]. In such holograms the signal photons play both roles of “object wave” and “reference wave” in holography, but are recorded by a point detector providing only encoding information, while the “idler” photons travel freely and are locally manipulated with spatial resolution along the fibers becomes possible.

## 4. Conclusions

The encrypted information, using the coherence of multi-mode bimodal field in quantum holography, opens the new perspective, in which the coherence proprieties between bi-photons are used together with non-local states of entangled photon pairs. The possibilities to use this coherence in the quantum communication and holographic registration of objects is described by the expressions (9) and (10) and is proposed for future developments. The main distinguish between the traditional holograms and such a hologram registration becomes attractive from physical points of view because it must take into consideration the common phase of two light modes described by the expressions (9)-(12). It also discusses the cooperative behavior of three cavity modes which corresponds to pump, Stokes and anti-Stokes photons stimulated by the atomic inversion. A new type of cooperative generation described by the correlations of the expressions (1) and (4) may be used in quantum nucleonics [36] as an ignition mechanism of coherence generation gamma photons by long-lived nuclear isomers in the single and two-quantum interaction with other species of excited radiators.

This method of recording of information affords the new perspectives in quantum cryptography and quantum information and has the tendency to improve the conception about quantum holograms observed in in literature [5, 7, 8, 9]. All these methods open new possibilities in the coding and decoding of data.