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 〈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 [21, 22], the quantum fluctuations in each mode may achieve zero value 〈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 photon is generated (anti-Stokes b<sup>þ</sup> <sup>a</sup> ) 〈E^rE^a〉 6¼ <sup>0</sup>,〈E^rE^s〉 6¼ 0. Below we consider below two situations.

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 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

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

ments 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

> 〈E^ð Þ � ksj ð Þ<sup>t</sup> <sup>E</sup>^ð Þ <sup>þ</sup> ks j

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

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

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

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

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

In the Section 2 we give the definition of bimodal coherent states in analogy with

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

kpj ð Þt . Here we notice that this new characteristic of the field in

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

ð Þ<sup>t</sup> 〉 � <sup>m</sup>2. Phase

ksl ð Þt 〉j6¼l. According to this only the diagonal ele-

ð Þt 〉 � m). The realization of

partitions gives zero contributions in the intensity correlation function

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

kpj ð Þt 〉 � m (or Is ¼ ∑<sup>j</sup>

equal to the square strength of each pumping (or Stokes) modes

are decomposed into the quantized states of the optical cavity.

already correlated for coherent excitation of molecular vibrations

proportional to the square number of adjacent bi-mods 〈Π^ �

〈E^ð Þ � kpj ð Þ<sup>t</sup> <sup>E</sup>^ð Þ <sup>þ</sup>

Ip ¼ ∑<sup>j</sup>

Π^ �

object.

42

ðÞ¼ t ∑<sup>j</sup>

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

photons from the pump field.

aspects of generation light.

〈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 �

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maximal contribution in two-photon decay of the ensemble. But in this case, the sums of the three-particle correlators on the indexes: j, l and m, containing nonequidistant two-level atoms, S, and R, becomes proportional to the number of pairs

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

In this situation is respect all resonance conditions between the pairs and equi-

sitions of D atomic subsystem and the pairs of the two dipole active atoms of R and S subsystems is represented in Figure 1. We can extend our attention to a big ensemble of three particles in such a cooperative process. Three atoms D, R and S are situated at relative small distances rds, rdr and rrs in comparison with emission wavelength. Such atomic may file up the volume with a dimension larger than the emission wavelengths. The exchange energies between the subsystems were analyzed in the literature and an attractive problem is connected with pencil shape atomic mixture described above. In this situation, the radiation can be observed

Following this conception, we observe that for the big ensemble of radiators the

IIð Þ¼ <sup>t</sup>; <sup>t</sup> <sup>þ</sup> <sup>τ</sup> 〈E^ð Þ <sup>þ</sup> ð Þ<sup>t</sup> <sup>E</sup>^ð Þ � ð Þ<sup>t</sup> <sup>E</sup>^ð Þ <sup>þ</sup> ð Þ <sup>t</sup> <sup>þ</sup> <sup>τ</sup> <sup>E</sup>^ð Þ � ð Þ <sup>t</sup> <sup>þ</sup> <sup>τ</sup> 〉. The first part describes the correlation between the photon pairs generated into the broadband interval and the second part describes the correlation between the bi-modes of scattered field. Here the positive and negative parts of the field strength <sup>E</sup>^ð Þ <sup>þ</sup> ðÞ¼ <sup>t</sup> <sup>∑</sup><sup>k</sup> gka^kð Þ<sup>t</sup> exp ½ � <sup>i</sup>ð Þ <sup>k</sup>; <sup>r</sup>

first order correlation function GIð Þ¼ <sup>t</sup>; <sup>t</sup> <sup>þ</sup> <sup>τ</sup> 〈E^ð Þ � ð Þ<sup>t</sup> <sup>E</sup>^ð Þ <sup>þ</sup> ð Þ <sup>t</sup> <sup>þ</sup> <sup>τ</sup> 〉 becomes smaller than second order one. The second order correlation functions between the photons

polarization λ respectively. Using the method of elimination operator developed in

between the bi-photons is proportional not only to the correlation function between the atoms of ensemble D but consists from the sum of two types of correlations, which contain the intrinsic correlation of D ensemble like in the Dicke process [24] and correlations between the D ensemble and dipole active sub-ensemble R and S

> 2 k<sup>2</sup> ∑ j, n

<sup>n</sup> ð Þ t þ τ D E,

<sup>n</sup> ð Þ t þ τ

Here we consider the sums on the repeated indexes. It is observed, that such a sum is proportional to the number dipole-active pairs Np and number of D radiators. In the degenerate case when all three sub-ensemble are equidistant the number of term increase in the system [19], but in the system substantially increase first

Ref. [19], we demonstrated, that the second order correlation function Gb

correlation between the atoms of D ensemble is proportional to the function

dk kð Þ � k<sup>0</sup>

<sup>j</sup> ð Þ<sup>t</sup> <sup>D</sup>^ �

and two photon cooperative ignition by the S ad R subensamble

D E

� exp ½ � ið Þ k<sup>1</sup> þ k; r<sup>n</sup> exp �i k<sup>0</sup>

. This resonance between the two-photon tran-

IIð Þ¼ <sup>t</sup>; <sup>t</sup> <sup>þ</sup> <sup>τ</sup> 〈E^ð Þ � ð Þ<sup>t</sup> <sup>E</sup>^ð Þ � ð Þ<sup>t</sup> <sup>E</sup>^ð Þ <sup>þ</sup> ð Þ <sup>t</sup> <sup>þ</sup> <sup>τ</sup> <sup>E</sup>^ð Þ <sup>þ</sup> ð Þ <sup>t</sup> <sup>þ</sup> <sup>τ</sup> 〉

<sup>k</sup>ð Þt field operators with wave vector k and

sin ð Þ k<sup>0</sup> � k rjn ð Þ k<sup>0</sup> � k rjn

> ; rl � � � � :

IIð Þ t; t þ τ

(1a)

(1b)

<sup>k</sup>ð Þt exp ½ � �ið Þ k; r are expressed through the superposition of

II ð Þ t; t þ τ according to the Refs. [17, 18] the

sin krjn krjn

; rj � � � � exp �<sup>i</sup> <sup>k</sup><sup>0</sup>

of this type of atoms and number of D atoms, NpNd.

along the pencil-shape atomic system (see Figure 1).

distant D- ensemble: 2ω<sup>0</sup> ¼ ωrj þ ωsl

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

can be divided into two parts: Gb

the annihilation <sup>a</sup>^kð Þ<sup>t</sup> and generation <sup>a</sup>^†

IIð Þþ <sup>t</sup>; <sup>t</sup> <sup>þ</sup> <sup>τ</sup> <sup>G</sup>b sr ð Þ

k ð0

0

<sup>l</sup> ð Þ<sup>t</sup> ^ S<sup>þ</sup> <sup>j</sup> ð Þ<sup>t</sup> <sup>D</sup>^ �

� <sup>D</sup>^ <sup>þ</sup>

II ð Þ� t; t þ τ

and <sup>E</sup>^ð Þ � ðÞ¼ <sup>t</sup> <sup>∑</sup><sup>k</sup> gka^†

IIð Þ¼ <sup>t</sup>; <sup>t</sup> <sup>þ</sup> <sup>τ</sup> Gbi

Gb rs ð Þ

Gb rs ð Þ

II ð Þ� <sup>t</sup>; <sup>t</sup> <sup>þ</sup> <sup>τ</sup> <sup>R</sup>^<sup>þ</sup>

and G<sup>s</sup>

Gb

45

### Figure 1.

The stimulation of two-photon cooperative effects of hydrogen-like (or helium-like) atomic transition 1S � 2S of D ensemble by dipole-active emission of Sj and Rj sub-ensembles in two-photon resonance described by the correlations (1a) and (1b). This two-photon cooperative decay can be prepared in pencil shape extended system of S, R and D radiators [19].

sub-ensemble (see Refs. [17–19]). The three-particles cooperative interaction through the vacuum of EMF is established taking into consideration the mutual influences between the single-photon polarization of S and R atomic subsystems and the non-linear polarization of the D atom (see Refs. [17–19]). To understand this effect it is necessary to examine the new correlation function which appears between the polarization of three different radiators from S, R and D subsystems: 〈D^ <sup>n</sup>ð Þ<sup>t</sup> <sup>R</sup>^� <sup>j</sup> ð Þ<sup>t</sup> ^ S� <sup>l</sup> ð Þ<sup>t</sup> 〉 and 〈R^lð Þ<sup>t</sup> ^ Sjð Þ<sup>t</sup> <sup>D</sup>^ � <sup>m</sup>ð Þt 〉. The ignition role of S and R atoms is observed in the third-order terms, which contain two-photon resonances between three-particles represented in Figure 1, described by the above correlation functions. In order to experimentally observe these correlations, we must maximally destroy the single photon Dicke superradiance between atoms of S and R subsystems. This is possible if we choose the broadband sub-ensemble of excited dipole active atoms S and R. In this case, the single photon correlations between the atoms of this type become oscillatory with detuning frequency ω<sup>s</sup> jlð Þr for j and l atoms. This corresponds to the situation when the correlations like 〈^ Sjð Þ<sup>t</sup> ^ S� <sup>l</sup> ð Þ<sup>t</sup> 〉 (or 〈R^jð Þ<sup>t</sup> <sup>R</sup>^� <sup>l</sup> ð Þt 〉) become proportional to a exp <sup>½</sup>iω<sup>s</sup> jlð Þr t� and rapidly oscillates during the cooperative decay process. But in this case in the sub-ensemble S, R we must have the big number of pairs Sj, Rj, j ¼ 1, 2, …, Np so that the established phase of each pair ^ S<sup>þ</sup> <sup>j</sup> and <sup>R</sup>^<sup>þ</sup> <sup>j</sup> will compensate the phase of D� <sup>n</sup> ð Þt atom from equidistant D ensemble so that above defined three-particle correlators becomes smooth functions, giving the

Coherence Proprieties of Entangled Bi-Modal Field and Its Application in Holography… DOI: http://dx.doi.org/10.5772/intechopen.85857

maximal contribution in two-photon decay of the ensemble. But in this case, the sums of the three-particle correlators on the indexes: j, l and m, containing nonequidistant two-level atoms, S, and R, becomes proportional to the number of pairs of this type of atoms and number of D atoms, NpNd.

In this situation is respect all resonance conditions between the pairs and equidistant D- ensemble: 2ω<sup>0</sup> ¼ ωrj þ ωsl . This resonance between the two-photon transitions of D atomic subsystem and the pairs of the two dipole active atoms of R and S subsystems is represented in Figure 1. We can extend our attention to a big ensemble of three particles in such a cooperative process. Three atoms D, R and S are situated at relative small distances rds, rdr and rrs in comparison with emission wavelength. Such atomic may file up the volume with a dimension larger than the emission wavelengths. The exchange energies between the subsystems were analyzed in the literature and an attractive problem is connected with pencil shape atomic mixture described above. In this situation, the radiation can be observed along the pencil-shape atomic system (see Figure 1).

Following this conception, we observe that for the big ensemble of radiators the first order correlation function GIð Þ¼ <sup>t</sup>; <sup>t</sup> <sup>þ</sup> <sup>τ</sup> 〈E^ð Þ � ð Þ<sup>t</sup> <sup>E</sup>^ð Þ <sup>þ</sup> ð Þ <sup>t</sup> <sup>þ</sup> <sup>τ</sup> 〉 becomes smaller than second order one. The second order correlation functions between the photons can be divided into two parts: Gb IIð Þ¼ <sup>t</sup>; <sup>t</sup> <sup>þ</sup> <sup>τ</sup> 〈E^ð Þ � ð Þ<sup>t</sup> <sup>E</sup>^ð Þ � ð Þ<sup>t</sup> <sup>E</sup>^ð Þ <sup>þ</sup> ð Þ <sup>t</sup> <sup>þ</sup> <sup>τ</sup> <sup>E</sup>^ð Þ <sup>þ</sup> ð Þ <sup>t</sup> <sup>þ</sup> <sup>τ</sup> 〉 and G<sup>s</sup> IIð Þ¼ <sup>t</sup>; <sup>t</sup> <sup>þ</sup> <sup>τ</sup> 〈E^ð Þ <sup>þ</sup> ð Þ<sup>t</sup> <sup>E</sup>^ð Þ � ð Þ<sup>t</sup> <sup>E</sup>^ð Þ <sup>þ</sup> ð Þ <sup>t</sup> <sup>þ</sup> <sup>τ</sup> <sup>E</sup>^ð Þ � ð Þ <sup>t</sup> <sup>þ</sup> <sup>τ</sup> 〉. The first part describes the correlation between the photon pairs generated into the broadband interval and the second part describes the correlation between the bi-modes of scattered field. Here the positive and negative parts of the field strength <sup>E</sup>^ð Þ <sup>þ</sup> ðÞ¼ <sup>t</sup> <sup>∑</sup><sup>k</sup> gka^kð Þ<sup>t</sup> exp ½ � <sup>i</sup>ð Þ <sup>k</sup>; <sup>r</sup> and <sup>E</sup>^ð Þ � ðÞ¼ <sup>t</sup> <sup>∑</sup><sup>k</sup> gka^† <sup>k</sup>ð Þt exp ½ � �ið Þ k; r are expressed through the superposition of the annihilation <sup>a</sup>^kð Þ<sup>t</sup> and generation <sup>a</sup>^† <sup>k</sup>ð Þt field operators with wave vector k and polarization λ respectively. Using the method of elimination operator developed in Ref. [19], we demonstrated, that the second order correlation function Gb IIð Þ t; t þ τ between the bi-photons is proportional not only to the correlation function between the atoms of ensemble D but consists from the sum of two types of correlations, which contain the intrinsic correlation of D ensemble like in the Dicke process [24] and correlations between the D ensemble and dipole active sub-ensemble R and S Gb IIð Þ¼ <sup>t</sup>; <sup>t</sup> <sup>þ</sup> <sup>τ</sup> Gbi IIð Þþ <sup>t</sup>; <sup>t</sup> <sup>þ</sup> <sup>τ</sup> <sup>G</sup>b sr ð Þ II ð Þ t; t þ τ according to the Refs. [17, 18] the correlation between the atoms of D ensemble is proportional to the function

$$G\_{II}^{b(n)}(t, t + \tau) \sim \int\_0^{k\_0} dk (k - k\_0)^2 k^2 \sum\_{j \neq n} \frac{\sin k r\_{jn}}{k r\_{jn}} \frac{\sin (k\_0 - k) r\_{jn}}{(k\_0 - k) r\_{jn}} \tag{1a}$$

$$\times \left< \hat{D}\_j^+(t) \hat{D}\_n^-(t + \tau) \right>,$$

and two photon cooperative ignition by the S ad R subensamble

$$\begin{split} \mathbf{G}\_{\rm II}^{b(n)}(\mathbf{t}, \mathbf{t} + \boldsymbol{\tau}) &\sim \left< \hat{\mathbf{R}}\_{l}^{+}(\mathbf{t}) \hat{\mathbf{S}}\_{\dot{j}}^{+}(\mathbf{t}) \hat{\mathbf{D}}\_{n}^{-}(\mathbf{t} + \boldsymbol{\tau}) \right> \\ &\times \exp\left[i(\mathbf{k}\_{1} + \mathbf{k}, \mathbf{r}\_{\mathbf{t}})\right] \exp\left[-i\left(\mathbf{k}', \mathbf{r}\_{\dot{j}}\right)\right] \exp\left[-i\left(\mathbf{k}', \mathbf{r}\_{\mathbf{t}}\right)\right]. \end{split} \tag{1b}$$

Here we consider the sums on the repeated indexes. It is observed, that such a sum is proportional to the number dipole-active pairs Np and number of D radiators. In the degenerate case when all three sub-ensemble are equidistant the number of term increase in the system [19], but in the system substantially increase first

sub-ensemble (see Refs. [17–19]). The three-particles cooperative interaction through the vacuum of EMF is established taking into consideration the mutual influences between the single-photon polarization of S and R atomic subsystems and the non-linear polarization of the D atom (see Refs. [17–19]). To understand this effect it is necessary to examine the new correlation function which appears between the polarization of three different radiators from S, R and D subsystems:

The stimulation of two-photon cooperative effects of hydrogen-like (or helium-like) atomic transition 1S � 2S of D ensemble by dipole-active emission of Sj and Rj sub-ensembles in two-photon resonance described by the correlations (1a) and (1b). This two-photon cooperative decay can be prepared in pencil shape extended system

observed in the third-order terms, which contain two-photon resonances between three-particles represented in Figure 1, described by the above correlation functions. In order to experimentally observe these correlations, we must maximally destroy the single photon Dicke superradiance between atoms of S and R subsystems. This is possible if we choose the broadband sub-ensemble of excited dipole active atoms S and R. In this case, the single photon correlations between the atoms

decay process. But in this case in the sub-ensemble S, R we must have the big number of pairs Sj, Rj, j ¼ 1, 2, …, Np so that the established phase of each pair

that above defined three-particle correlators becomes smooth functions, giving the

<sup>m</sup>ð Þt 〉. The ignition role of S and R atoms is

Sjð Þ<sup>t</sup> ^ S�

jlð Þr t� and rapidly oscillates during the cooperative

<sup>n</sup> ð Þt atom from equidistant D ensemble so

jlð Þr for j and l atoms. This

<sup>l</sup> ð Þ<sup>t</sup> 〉 (or 〈R^jð Þ<sup>t</sup> <sup>R</sup>^�

<sup>l</sup> ð Þt 〉)

Sjð Þ<sup>t</sup> <sup>D</sup>^ �

of this type become oscillatory with detuning frequency ω<sup>s</sup>

corresponds to the situation when the correlations like 〈^

<sup>j</sup> will compensate the phase of D�

〈D^ <sup>n</sup>ð Þ<sup>t</sup> <sup>R</sup>^�

Figure 1.

^ S<sup>þ</sup> <sup>j</sup> and <sup>R</sup>^<sup>þ</sup>

44

<sup>j</sup> ð Þ<sup>t</sup> ^ S�

of S, R and D radiators [19].

<sup>l</sup> ð Þ<sup>t</sup> 〉 and 〈R^lð Þ<sup>t</sup> ^

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become proportional to a exp <sup>½</sup>iω<sup>s</sup>

order coherence between the same photons. In other words, the single photon process substantially ignites the generation of coherent photon 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 Δ ≥λ=2. Taking into account the concept about the dropped lithographic limit in two-quantum coherent processes, the authors of Ref. [20] proposed new lithographic limit in the two-photon processes with a magnitude two times smaller than traditional Δ ≥λ=4. This take place when frequencies of the signal and idler photons have the same value ω<sup>s</sup> ¼ ωi. This propriety is also contained and in two-photon super-radiance [21] but here λ ¼ 2λsiλri=ð Þ λsi þ λri . Here λri and λsi are the emitted wavelengths by S ad R from the pair i, i ¼ 1, 2, ::Np. An interesting effect of two photon cooperative emission is possible in micro-cavities. In this case the mode structure of the cavity stimulates the two-photon decay effect in comparison with cascade effect [19, 25, 26] (see Figure 2).

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 twophoton 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

$$\begin{split} \hat{\boldsymbol{P}}^{(+)} (\mathbf{t}, \mathbf{z}) &= \hat{\mathbf{E}}\_{\boldsymbol{r}}^{+} (\mathbf{z}, \mathbf{t}) \hat{\boldsymbol{E}}\_{\boldsymbol{r}}^{+} (\mathbf{z}, \mathbf{t}) \\ &= \mathbf{G} (\boldsymbol{k}\_{\boldsymbol{s}}, \boldsymbol{k}\_{\boldsymbol{r}}) \hat{\boldsymbol{a}}\_{\boldsymbol{s}} \hat{\boldsymbol{a}}\_{\boldsymbol{r}} \exp\left[2i\boldsymbol{a}\_{0}\boldsymbol{t} - \mathbf{i} (\mathbf{k}\_{\boldsymbol{s}} + \mathbf{k}\_{\boldsymbol{r}}) \mathbf{z}\right], \end{split} \tag{2}$$

order correlation function <sup>G</sup>2ð Þ¼ <sup>Δ</sup> 〈P^�

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

introduce the collective modes field operators ^I

ρmðÞ¼ t 2κð Þ m þ 1 ð Þ m þ 2j ρ<sup>m</sup>þ<sup>1</sup>

þ α<sup>2</sup>

Nσ0γ ð Þ <sup>ω</sup>�2ω<sup>0</sup> <sup>2</sup>

� <sup>2</sup>ð Þ <sup>m</sup> <sup>þ</sup> <sup>1</sup> ð Þ <sup>m</sup> <sup>þ</sup> <sup>2</sup><sup>j</sup>

<sup>1</sup> <sup>þ</sup> <sup>β</sup>ð Þ <sup>m</sup> <sup>þ</sup> <sup>1</sup> ð Þ <sup>m</sup> <sup>þ</sup> <sup>2</sup><sup>j</sup> <sup>α</sup><sup>1</sup> <sup>þ</sup> <sup>α</sup><sup>2</sup>

we used decomposition of density operator of bimodal field in the cavity

<sup>2</sup>m<sup>2</sup>ð Þ <sup>m</sup> <sup>þ</sup> <sup>2</sup><sup>j</sup> � <sup>1</sup> <sup>2</sup>

<sup>N</sup>σ<sup>0</sup> <sup>α</sup><sup>2</sup>

� <sup>¼</sup> <sup>∑</sup><sup>k</sup> <sup>∈</sup>ð Þ <sup>ω</sup>0;<sup>0</sup> <sup>a</sup>^2k0�ka^<sup>k</sup> and ^Iz <sup>¼</sup> <sup>∑</sup><sup>k</sup> <sup>∈</sup>ð Þ <sup>ω</sup>0;<sup>0</sup> <sup>ð</sup>a^†

increasing the number of bi-modes ^I

graphic limit as in the Ref. [20].

be observed in tow-photon.

^I

47

Figure 2.

was obtained

∂ ∂t

where <sup>α</sup><sup>1</sup> <sup>¼</sup> <sup>2</sup>j j <sup>g</sup> <sup>2</sup>

atomic inversion <sup>N</sup>σ0, <sup>α</sup><sup>2</sup> <sup>¼</sup> <sup>T</sup>

ð Þ<sup>z</sup> <sup>P</sup>^<sup>þ</sup>

(A) Sources of entangled photons in the two-photon bimodal processes. The horizontal blue lines represent the modes of the optical cavity modes formed between the vertical mirrors, Ml and Mr. (B) Two slits experiments with interference between pairs of modes which form the strengths product of EMF. The interference picture can

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

mal values for Δð Þ ks þ kr sin θ ¼ π . From this expression follows the some litho-

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

above description the absolute value of conserved Casimir operator increase with

2 ¼ ð^<sup>I</sup> z Þ

accompanied by the increasing of coherence between the bi-photons of each bi-modes relative the coherence which appears between the individual photons belonging to other modes. The similar coherent photon pairs may be generated in the broadband laser systems [11]. Following this idea the stationary solution of master equation for the bimodal cavity fled in the above multi-mode representation

ð Þ z þ Δ 〉 pass from maximal to mini-

<sup>2</sup>k0�<sup>k</sup>a^† k;

<sup>2</sup>k0�<sup>k</sup>Þ=2. As follows from

Þ. This effect is

Nσ0ð Þ ω�2ω<sup>0</sup> ð Þ <sup>ω</sup>�2ω<sup>0</sup> <sup>2</sup>

<sup>þ</sup>γ<sup>2</sup> . Here

<sup>þ</sup> <sup>¼</sup> <sup>∑</sup><sup>k</sup> <sup>∈</sup>ð Þ <sup>ω</sup>0;<sup>0</sup> <sup>a</sup>^†

þ^I � <sup>þ</sup> ^<sup>I</sup> �^I þ

ð Þ m þ 1 ð Þ m þ 2j 1 þ βð Þ m þ 1 ð Þ m þ 2j <sup>ρ</sup><sup>m</sup>

þγ<sup>2</sup>, and <sup>χ</sup> <sup>¼</sup> <sup>2</sup>j j <sup>g</sup> <sup>2</sup>

<sup>k</sup>a^<sup>k</sup> <sup>þ</sup> <sup>a</sup>^2k0�ka^†

<sup>2</sup> � <sup>1</sup>=2ð^<sup>I</sup>

½ � <sup>1</sup> <sup>þ</sup> <sup>β</sup>m mð Þ <sup>þ</sup> <sup>2</sup><sup>j</sup> � <sup>1</sup> <sup>2</sup> <sup>ρ</sup><sup>m</sup>�<sup>1</sup> � Ibid mf g ! <sup>m</sup> � <sup>1</sup> ,

<sup>1</sup> <sup>þ</sup> <sup>χ</sup><sup>2</sup> , <sup>β</sup> <sup>¼</sup> <sup>4</sup>j j <sup>g</sup> <sup>2</sup>

<sup>ρ</sup>ðÞ¼ <sup>t</sup> <sup>∑</sup><sup>m</sup>¼<sup>0</sup>ρm∣m, jihm, j∣. The evolution of correlation of the bi-photon intensity

<sup>þ</sup>γ<sup>2</sup> represents the generation rate of photon pairs for full

Tγ ð Þ <sup>ω</sup>�2ω<sup>0</sup> <sup>2</sup>

where 2ω<sup>0</sup> ¼ ω<sup>s</sup> þ ω<sup>r</sup> ¼ ω<sup>21</sup> is the total frequency of generated photons, 2k<sup>0</sup> ¼ ks þ kr. In this case we can introduce the following operators of bi-bosin field ^I <sup>þ</sup> <sup>¼</sup> <sup>a</sup>^† s a^† <sup>r</sup> and ^<sup>I</sup> � <sup>¼</sup> <sup>a</sup>^sa^r; ^Iz <sup>¼</sup> <sup>a</sup>^† <sup>s</sup> <sup>a</sup>^<sup>s</sup> <sup>þ</sup> <sup>a</sup>^† <sup>r</sup> a^<sup>r</sup> � �=2, which satisfy the commutation relations ½ ^I þ ,^I � �¼�2^Iz, <sup>½</sup> ^Iz,^I � �¼�^<sup>I</sup> � . Such a generation possibilities was proposed in Refs. [17, 18]. According to the representation of these operators, we may introduce the following coherent states for this field <sup>∣</sup>μi ¼ exp <sup>ð</sup>μ^<sup>I</sup> þ Þ∣ j, ji=1 � j j μ 2j , which belong to the suð Þ 1; 1 symmetry described in Refs. [11, 19]. Here μ is the coherent displacement of bi-photon oscillator. Following this conception, the function Pð Þ <sup>þ</sup> ð Þ t; z has the same behavior as the electrical component of single photon laser. For example, the mean value of this function on the coherent state can be represented through the harmonic functions with given phase and amplitude

$$\left\langle \hat{P}^{(+)}(t, \mathbf{z}) \right\rangle = P\_0 \exp\left[2i\alpha\_0 t - i(k\_s + k\_r)\mathbf{z} + \rho\right],\tag{3}$$

where P<sup>0</sup> ¼ G ks ð Þ ; kr h i a^sa^<sup>r</sup> j j is the amplitude and φ ¼ Argh i a^sa^<sup>r</sup> is the phase of electrical field strengths of two fields a and b . In the detection scheme represented in Figure 2B it is observe delay time through z-dependence of such functions. The lithographic limit follows from the difference between the maximum and minimum of two sit experiment represented in Figure 2. According to the expression (3) and the distinguish distance Δ between the slits follows, that the second

Coherence Proprieties of Entangled Bi-Modal Field and Its Application in Holography… DOI: http://dx.doi.org/10.5772/intechopen.85857

### Figure 2.

order coherence between the same photons. In other words, the single photon

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 Δ ≥λ=2. Taking into account the concept about the dropped lithographic limit in two-quantum coherent processes, the authors of Ref. [20] proposed new lithographic limit in the two-photon processes with a magnitude two times smaller than traditional Δ ≥λ=4. This take place when frequencies of the signal and idler photons have the same value ω<sup>s</sup> ¼ ωi. This propriety is also contained and in two-photon super-radiance [21] but here λ ¼ 2λsiλri=ð Þ λsi þ λri . Here λri and λsi are the emitted wavelengths by S ad R from the pair i, i ¼ 1, 2, ::Np. An interesting effect of two photon cooperative emission is possible in micro-cavities. In this case the mode structure of the cavity stimulates the two-photon decay effect in

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 twophoton 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

process substantially ignites the generation of coherent photon pairs.

Quantum Cryptography in Advanced Networks

comparison with cascade effect [19, 25, 26] (see Figure 2).

<sup>P</sup>^ð Þ <sup>þ</sup> ð Þ¼ <sup>t</sup>; <sup>z</sup> <sup>E</sup>^<sup>þ</sup>

� <sup>¼</sup> <sup>a</sup>^sa^r; ^Iz <sup>¼</sup> <sup>a</sup>^†

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

^Iz,^I � �¼�^<sup>I</sup> �

�¼�2^Iz, <sup>½</sup>

^I <sup>þ</sup> <sup>¼</sup> <sup>a</sup>^† s a^† <sup>r</sup> and ^<sup>I</sup>

46

relations ½

^I þ ,^I � <sup>s</sup> ð Þ <sup>z</sup>; <sup>t</sup> <sup>E</sup>^<sup>þ</sup>

<sup>r</sup> ð Þ z; t

where 2ω<sup>0</sup> ¼ ω<sup>s</sup> þ ω<sup>r</sup> ¼ ω<sup>21</sup> is the total frequency of generated photons, 2k<sup>0</sup> ¼ ks þ kr. In this case we can introduce the following operators of bi-bosin field

posed in Refs. [17, 18]. According to the representation of these operators, we may

where P<sup>0</sup> ¼ G ks ð Þ ; kr h i a^sa^<sup>r</sup> j j is the amplitude and φ ¼ Argh i a^sa^<sup>r</sup> is the phase of electrical field strengths of two fields a and b . In the detection scheme represented in Figure 2B it is observe delay time through z-dependence of such functions. The lithographic limit follows from the difference between the maximum and minimum of two sit experiment represented in Figure 2. According to the expression (3) and the distinguish distance Δ between the slits follows, that the second

which belong to the suð Þ 1; 1 symmetry described in Refs. [11, 19]. Here μ is the coherent displacement of bi-photon oscillator. Following this conception, the function Pð Þ <sup>þ</sup> ð Þ t; z has the same behavior as the electrical component of single photon laser. For example, the mean value of this function on the coherent state can be represented through the harmonic functions with given phase and amplitude

<sup>s</sup> <sup>a</sup>^<sup>s</sup> <sup>þ</sup> <sup>a</sup>^† <sup>r</sup> a^<sup>r</sup>

introduce the following coherent states for this field <sup>∣</sup>μi ¼ exp <sup>ð</sup>μ^<sup>I</sup>

¼ G ks ð Þ ; kr a^sa^<sup>r</sup> exp 2½ � iω0t � i kð Þ <sup>s</sup> þ kr z ,

� �=2, which satisfy the commutation

¼ P<sup>0</sup> exp 2½ � iω0t � i kð Þ <sup>s</sup> þ kr z þ φ , (3)

. Such a generation possibilities was pro-

þ

Þ∣ j, ji=1 � j j μ

(2)

2j ,

(A) Sources of entangled photons in the two-photon bimodal processes. The horizontal blue lines represent the modes of the optical cavity modes formed between the vertical mirrors, Ml and Mr. (B) Two slits experiments with interference between pairs of modes which form the strengths product of EMF. The interference picture can be observed in tow-photon.

order correlation function <sup>G</sup>2ð Þ¼ <sup>Δ</sup> 〈P^� ð Þ<sup>z</sup> <sup>P</sup>^<sup>þ</sup> ð Þ z þ Δ 〉 pass from maximal to minimal values for Δð Þ ks þ kr sin θ ¼ π . From this expression follows the some lithographic limit as in the Ref. [20].

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 ^I <sup>þ</sup> <sup>¼</sup> <sup>∑</sup><sup>k</sup> <sup>∈</sup>ð Þ <sup>ω</sup>0;<sup>0</sup> <sup>a</sup>^† <sup>2</sup>k0�<sup>k</sup>a^† k; ^I � <sup>¼</sup> <sup>∑</sup><sup>k</sup> <sup>∈</sup>ð Þ <sup>ω</sup>0;<sup>0</sup> <sup>a</sup>^2k0�ka^<sup>k</sup> and ^Iz <sup>¼</sup> <sup>∑</sup><sup>k</sup> <sup>∈</sup>ð Þ <sup>ω</sup>0;<sup>0</sup> <sup>ð</sup>a^† <sup>k</sup>a^<sup>k</sup> <sup>þ</sup> <sup>a</sup>^2k0�ka^† <sup>2</sup>k0�<sup>k</sup>Þ=2. As follows from above description the absolute value of conserved Casimir operator increase with increasing the number of bi-modes ^I 2 ¼ ð^<sup>I</sup> z Þ <sup>2</sup> � <sup>1</sup>=2ð^<sup>I</sup> þ^I � <sup>þ</sup> ^<sup>I</sup> �^I þ Þ. This effect is accompanied by the increasing of coherence between the bi-photons of each bi-modes relative the coherence which appears between the individual photons belonging to other modes. The similar coherent photon pairs may be generated in the broadband laser systems [11]. Following this idea the stationary solution of master equation for the bimodal cavity fled in the above multi-mode representation was obtained

$$\begin{split} \frac{\partial}{\partial t} \rho\_m(t) &= 2\kappa (m+1)(m+2j)\rho\_{m+1} \\ &- \frac{2(m+1)(m+2j)}{1+\beta(m+1)(m+2j)} \left( a\_1 + a\_2 \frac{(m+1)(m+2j)}{1+\beta(m+1)(m+2j)} \right) \rho\_m \\ &+ a\_2 \frac{2m^2(m+2j-1)^2}{\left[1+\beta m(m+2j-1)\right]^2} \rho\_{m-1} - 1bid\{m \to m-1\}, \end{split}$$

where <sup>α</sup><sup>1</sup> <sup>¼</sup> <sup>2</sup>j j <sup>g</sup> <sup>2</sup> Nσ0γ ð Þ <sup>ω</sup>�2ω<sup>0</sup> <sup>2</sup> <sup>þ</sup>γ<sup>2</sup> represents the generation rate of photon pairs for full atomic inversion <sup>N</sup>σ0, <sup>α</sup><sup>2</sup> <sup>¼</sup> <sup>T</sup> <sup>N</sup>σ<sup>0</sup> <sup>α</sup><sup>2</sup> <sup>1</sup> <sup>þ</sup> <sup>χ</sup><sup>2</sup> , <sup>β</sup> <sup>¼</sup> <sup>4</sup>j j <sup>g</sup> <sup>2</sup> Tγ ð Þ <sup>ω</sup>�2ω<sup>0</sup> <sup>2</sup> þγ<sup>2</sup>, and <sup>χ</sup> <sup>¼</sup> <sup>2</sup>j j <sup>g</sup> <sup>2</sup> Nσ0ð Þ ω�2ω<sup>0</sup> ð Þ <sup>ω</sup>�2ω<sup>0</sup> <sup>2</sup> <sup>þ</sup>γ<sup>2</sup> . Here we used decomposition of density operator of bimodal field in the cavity <sup>ρ</sup>ðÞ¼ <sup>t</sup> <sup>∑</sup><sup>m</sup>¼<sup>0</sup>ρm∣m, jihm, j∣. The evolution of correlation of the bi-photon intensity

〈^I þ ð Þt ^I � ð Þt 〉, and the sum of the photon correlation functions in each mode d 〈 : n^<sup>2</sup> : 〉 to lasing phase transition point for the following values of the j—collective parameter: <sup>j</sup> <sup>¼</sup> <sup>0</sup>:5 and <sup>j</sup> <sup>¼</sup> 10. Here <sup>n</sup>^ <sup>¼</sup> <sup>2</sup> ^Iz � <sup>j</sup> � �. It is observed that with the increase of the number of the bi-modes the coherence between the bimodal field characteristic P^� ð Þ <sup>t</sup> <sup>þ</sup> <sup>τ</sup>; <sup>z</sup> and <sup>P</sup>^<sup>þ</sup> ð Þ <sup>t</sup>; <sup>z</sup> increases: <sup>i</sup>P^� ð Þ <sup>t</sup> <sup>þ</sup> <sup>τ</sup>; <sup>z</sup> <sup>P</sup>^<sup>þ</sup> ð Þ t; z 〈^I þ ð Þt ^I � ð Þt 〉 exp 2½ � iω0τ � 2ik0z . As it is observed from the behavior of parameter 〈 : n<sup>2</sup> : 〉, with increasing the number of modes, j, the coherence between the individual photons substantially decreases (see Figure 3a and b). This process of lasing stabilization is accompanied by the increasing the coherence between the photon pairs belonging to conjugate bi-modes and may be detected by the scheme represented in Figure 2B. The generation process of the coherent field in the some mode of the ensemble of the modes 2j is described by the sum of correlations <sup>∑</sup>k〈E^ð Þ � <sup>k</sup> ð Þ <sup>t</sup> <sup>þ</sup> <sup>τ</sup> <sup>E</sup>^ð Þ <sup>þ</sup> <sup>k</sup> ð Þt 〉, which become proportional to the sum of number of photons in each mode h i n^ exp ½ � iφ τð Þ . As follows from Figure 3 the amplitude of this function achieved the small all value with the increasing of the number of modes. In the single photon detection this correlations is described by the aleatory phase φ τð Þ and may be represented by the smooth function (see "red" line) on the screen F of interference scheme Figure 2B.

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 <sup>A</sup> and <sup>S</sup> atoms as a combination <sup>∣</sup>ψi � <sup>∣</sup>1ia∣0i<sup>s</sup> <sup>þ</sup> exp <sup>i</sup>ϕ∣0ia∣1is<sup>=</sup> ffiffi 2 p we may extrapolate such superposition for a big number of atoms from the dipole active sub-ensemble A and S belonging to suð Þ2 symmetry. Let us first discuss the three particle cooperative effects in the scattering interaction represented in Figure 4 [17–19]. In the free space, such field may be generated with pencil shape process described by three ensembles of atoms D, S and A . This description is devoted to this 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 sub ensemble of equidistant atoms, NsNrNd. In this situation only one possibility of resonance interaction between the dipole forbidden transition of D-Lambda atoms and ensemble of dipole active atoms S and A

2ω<sup>0</sup> ¼ ω<sup>a</sup> � ωs, which correspond to scattering resonance between the three particles, respectively. This resonance situation for the decorrelated ensemble of dipole

The similar mutual transitions between dipole-active Aj and Sj atoms in scattering resonance with hydrogen-

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

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

We observe, that the Dicke cooperative effects between the sub-ensemble of atoms, S, and A can be neglected if the atoms in the sub-ensembles S and A are not equidistant relative their excited energy. In such a situation the Dicke cooperative effect in sub-systems of dipole-active atoms is negligible due to the consideration that the frequency width of broadband emission Δω is large than the cooperative

<sup>l</sup> ð Þ<sup>t</sup> 〉 become proportional to the rapid oscillatory parts exp <sup>i</sup> <sup>ω</sup>aj � <sup>ω</sup>al <sup>t</sup> and exp <sup>i</sup> <sup>ω</sup>sj � <sup>ω</sup>sl <sup>t</sup> , and vanishes after an average procedure on the time interval less than the decay time 1=Γc. In such a situation, only the pairs of S, and A sub ensembles can excite the D-subsystem according to the third-order of perturbation decomposition [19]. It contains three particle scattering exchanges between the pairs of S, and A atoms and D-an ensemble of equidistant atomic represented by Figure 4. This exchange scheme of two S and A atomic pairs is described by the

<sup>m</sup>ð Þt 〉 described by master equation in Ref. [23]. It

<sup>j</sup> ð Þ<sup>t</sup> <sup>A</sup>^ �

<sup>l</sup> ð Þt 〉 and

emission rate Γc. In this situation the cooperative correlations like 〈A^ <sup>þ</sup>

correlations between the pairs of A and S scattering resonance with D:

corresponds to the scattering resonance between the pairs and D ensemble:

ωaj � ωsj � 2ω<sup>0</sup> ¼ 0. Here i ¼ 1, 2, Np, Nd is the number of atomic pairs of S and A sub-ensembles. In this situation, the second order coherence is also proportional to the product of two superposition of D-atoms and pairs of Si and Ai atoms in the scattering resonance with D-equidistant ensemble as in the two-photon resonance

active atoms is represented in Figure 4.

like (or helium like) ensemble of D atoms, described by the expression (4).

〈^ S<sup>þ</sup> <sup>j</sup> ð Þ<sup>t</sup> ^ S�

Figure 4.

〈^ S<sup>þ</sup> <sup>j</sup> ð Þ<sup>t</sup> <sup>D</sup>^ <sup>þ</sup>

49

<sup>m</sup>ð Þ<sup>t</sup> <sup>A</sup>^ �

<sup>l</sup> ð Þ<sup>t</sup> 〉, 〈A^ <sup>þ</sup>

<sup>j</sup> ð Þ<sup>t</sup> ^ S� <sup>l</sup> ð Þ<sup>t</sup> <sup>D</sup>^ �

### Figure 3.

The evolution of the photon correlations as function of the relative time tκ to the phase transition for following parameters of the system: α1=κ ¼ 0:4, α1=κ ¼ 0:01 and β=κ ¼ 0:001. Here it is represented: the square amplitudes of bimodal field 〈^I þ^I � 〉 (blue line), the correlation between the photons of each mode, 〈 : n^<sup>2</sup> : 〉 (red line) and the square of mean value of the photon number in each mode, h i <sup>n</sup>^ <sup>2</sup> (green line). Figure <sup>a</sup> corresponds to single mode two-photon emission, j ¼ 1=2, and the numerical representation in figure b corresponds to the number of the bimodal cavity field 2 j ¼ 20. As follows for the figures a and b the total photon correlations in each mode decreases with the increasing of the number of bi-modes (see the red and green lines).

Coherence Proprieties of Entangled Bi-Modal Field and Its Application in Holography… DOI: http://dx.doi.org/10.5772/intechopen.85857

Figure 4.

〈^I þ ð Þt ^I �

〈^I þ ð Þt ^I �

<sup>∑</sup>k〈E^ð Þ �

Figure 3.

48

amplitudes of bimodal field 〈^I

þ^I �

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

interference scheme Figure 2B.

characteristic P^�

ð Þt 〉, and the sum of the photon correlation functions in each mode d 〈 : n^<sup>2</sup> : 〉 to lasing phase transition point for the following values of the j—collective

ð Þ <sup>t</sup>; <sup>z</sup> increases: <sup>i</sup>P^�

represented in Figure 2B. The generation process of the coherent field in the some mode of the ensemble of the modes 2j is described by the sum of correlations

tons in each mode h i n^ exp ½ � iφ τð Þ . As follows from Figure 3 the amplitude of this function achieved the small all value with the increasing of the number of modes. In the single photon detection this correlations is described by the aleatory phase φ τð Þ and may be represented by the smooth function (see "red" line) on the screen F of

distributed in the broadband spectrum represents the bimodal spectrum of scattered photons. Indeed if we represent superposition between the photons obtained from <sup>A</sup> and <sup>S</sup> atoms as a combination <sup>∣</sup>ψi � <sup>∣</sup>1ia∣0i<sup>s</sup> <sup>þ</sup> exp <sup>i</sup>ϕ∣0ia∣1is<sup>=</sup> ffiffi

transition of D-Lambda atoms and ensemble of dipole active atoms S and A

The evolution of the photon correlations as function of the relative time tκ to the phase transition for following parameters of the system: α1=κ ¼ 0:4, α1=κ ¼ 0:01 and β=κ ¼ 0:001. Here it is represented: the square

line) and the square of mean value of the photon number in each mode, h i <sup>n</sup>^ <sup>2</sup> (green line). Figure <sup>a</sup> corresponds to single mode two-photon emission, j ¼ 1=2, and the numerical representation in figure b corresponds to the number of the bimodal cavity field 2 j ¼ 20. As follows for the figures a and b the total photon correlations in

each mode decreases with the increasing of the number of bi-modes (see the red and green lines).

〉 (blue line), the correlation between the photons of each mode, 〈 : n^<sup>2</sup> : 〉 (red

Case B. Another possibility to create a coherent field for a big number of photons

we may extrapolate such superposition for a big number of atoms from the dipole active sub-ensemble A and S belonging to suð Þ2 symmetry. Let us first discuss the three particle cooperative effects in the scattering interaction represented in Figure 4 [17–19]. In the free space, such field may be generated with pencil shape process described by three ensembles of atoms D, S and A . This description is devoted to this 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 sub ensemble of equidistant atoms, NsNrNd. In this situation only one possibility of resonance interaction between the dipole forbidden

ð Þt 〉 exp 2½ � iω0τ � 2ik0z . As it is observed from the behavior of parameter 〈 : n<sup>2</sup> : 〉, with increasing the number of modes, j, the coherence between the individual photons substantially decreases (see Figure 3a and b). This process of lasing stabilization is accompanied by the increasing the coherence between the photon pairs belonging to conjugate bi-modes and may be detected by the scheme

<sup>k</sup> ð Þt 〉, which become proportional to the sum of number of pho-

ð Þ <sup>t</sup> <sup>þ</sup> <sup>τ</sup>; <sup>z</sup> <sup>P</sup>^<sup>þ</sup>

ð Þ t; z

2 p

parameter: <sup>j</sup> <sup>¼</sup> <sup>0</sup>:5 and <sup>j</sup> <sup>¼</sup> 10. Here <sup>n</sup>^ <sup>¼</sup> <sup>2</sup> ^Iz � <sup>j</sup> � �. It is observed that with the increase of the number of the bi-modes the coherence between the bimodal field

ð Þ <sup>t</sup> <sup>þ</sup> <sup>τ</sup>; <sup>z</sup> and <sup>P</sup>^<sup>þ</sup>

Quantum Cryptography in Advanced Networks

The similar mutual transitions between dipole-active Aj and Sj atoms in scattering resonance with hydrogenlike (or helium like) ensemble of D atoms, described by the expression (4).

2ω<sup>0</sup> ¼ ω<sup>a</sup> � ωs, which correspond to scattering resonance between the three particles, respectively. This resonance situation for the decorrelated ensemble of dipole active atoms is represented in Figure 4.

We observe, that the Dicke cooperative effects between the sub-ensemble of atoms, S, and A can be neglected if the atoms in the sub-ensembles S and A are not equidistant relative their excited energy. In such a situation the Dicke cooperative effect in sub-systems of dipole-active atoms is negligible due to the consideration that the frequency width of broadband emission Δω is large than the cooperative emission rate Γc. In this situation the cooperative correlations like 〈A^ <sup>þ</sup> <sup>j</sup> ð Þ<sup>t</sup> <sup>A</sup>^ � <sup>l</sup> ð Þt 〉 and 〈^ S<sup>þ</sup> <sup>j</sup> ð Þ<sup>t</sup> ^ S� <sup>l</sup> ð Þ<sup>t</sup> 〉 become proportional to the rapid oscillatory parts exp <sup>i</sup> <sup>ω</sup>aj � <sup>ω</sup>al <sup>t</sup> and exp <sup>i</sup> <sup>ω</sup>sj � <sup>ω</sup>sl <sup>t</sup> , and vanishes after an average procedure on the time interval less than the decay time 1=Γc. In such a situation, only the pairs of S, and A sub ensembles can excite the D-subsystem according to the third-order of perturbation decomposition [19]. It contains three particle scattering exchanges between the pairs of S, and A atoms and D-an ensemble of equidistant atomic represented by Figure 4. This exchange scheme of two S and A atomic pairs is described by the correlations between the pairs of A and S scattering resonance with D: 〈^ S<sup>þ</sup> <sup>j</sup> ð Þ<sup>t</sup> <sup>D</sup>^ <sup>þ</sup> <sup>m</sup>ð Þ<sup>t</sup> <sup>A</sup>^ � <sup>l</sup> ð Þ<sup>t</sup> 〉, 〈A^ <sup>þ</sup> <sup>j</sup> ð Þ<sup>t</sup> ^ S� <sup>l</sup> ð Þ<sup>t</sup> <sup>D</sup>^ � <sup>m</sup>ð Þt 〉 described by master equation in Ref. [23]. It corresponds to the scattering resonance between the pairs and D ensemble: ωaj � ωsj � 2ω<sup>0</sup> ¼ 0. Here i ¼ 1, 2, Np, Nd is the number of atomic pairs of S and A sub-ensembles. In this situation, the second order coherence is also proportional to the product of two superposition of D-atoms and pairs of Si and Ai atoms in the scattering resonance with D-equidistant ensemble as in the two-photon resonance

$$\begin{split} \mathbf{G}\_{\rm II}^{s}(\mathbf{t}, \mathbf{t} + \tau) &\sim \left\langle \hat{\mathbf{A}}\_{l}^{+}(\mathbf{t}) \hat{\mathbf{S}}\_{\hat{j}}^{-}(\mathbf{t} + \tau) \hat{\mathbf{D}}\_{n}^{-}(\mathbf{t} + \tau) \right\rangle \\ &\times \exp\left[i(\mathbf{k}\_{1} + \mathbf{k}, \mathbf{r}\_{n})\right] \exp\left[-i\left(\mathbf{k}', \mathbf{r}\_{\hat{j}}\right)\right] \exp\left[-i\left(\mathbf{k}', \mathbf{r}\_{\hat{l}}\right)\right]. \end{split} \tag{4}$$

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 ωai; ωsi and ωal; ωsl can form the coherent suð Þ2 state, which corresponds to the generators of the superposition of collective discrete bi-modes of EMF ^J � <sup>¼</sup> <sup>∑</sup>ka^† <sup>k</sup>a^kþ2k<sup>0</sup> , ^J <sup>þ</sup> <sup>¼</sup> <sup>∑</sup>ka^† kþ2k<sup>0</sup> <sup>a</sup>^<sup>k</sup> and ^<sup>J</sup> <sup>z</sup> <sup>¼</sup> <sup>∑</sup>kfa^† kþ2k<sup>0</sup> <sup>a</sup>^kþ2k0� <sup>a</sup>^† ka^kg=2 which satisfies the commutation relations for suð Þ2 algebra described in Section 2. In this cooperative effect, due to large number non-equidistant atoms in each ensemble A or S, the frequency differences between the scattered modes ωai � ωsi ¼ 2ω<sup>0</sup> and ωal � ωsl have same wave vectors, Ki ¼ Kl, where K<sup>i</sup> ¼ kai � ksi and K<sup>l</sup> ¼ kal � ksl and can be used in the coherent phenomena like holograms, or optical processing. In such coherence it is manifested the correlations between the ensemble of bi-modes generated by the pairs of atoms Sl f g ; Al , l ¼ 1, …Np. These effects are accompanied with the interference between single- and two-quantum collective transitions of three inverted radiators from the ensemble. The three particle collective decay rate is defined in the description of the atomic correlation functions.

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 2ω<sup>0</sup> near the vibration frequency of the molecules (or bio-molecules) Ω

$$\hat{\Pi}^{(-)}(t,\mathbf{z}) = \lambda \hat{\boldsymbol{E}}\_{p}^{(+)}(\mathbf{z},t)\hat{\boldsymbol{E}}\_{a}^{(-)}(\mathbf{z},t) + \mathbf{g}\hat{\boldsymbol{E}}\_{s}^{(+)}(\mathbf{z},t)\hat{\boldsymbol{E}}\_{p}^{(-)}(\mathbf{z},t)$$

$$= \mathbf{G}(k\_{p},k\_{a})\hat{b}\hat{a}^{\dagger}\exp\left[2i\alpha\_{0}t - i\left(k\_{a} - k\_{p}\right)\mathbf{z}\right] \tag{5}$$

$$+ \mathbf{G}(k\_{s},k\_{p})\hat{\mathbf{s}}\hat{b}^{\dagger}\exp\left[2i\alpha\_{0}t - i\left(k\_{p} - k\_{t}\right)\mathbf{z}\right].$$

operators are similar to the commutators of the collective atomic operators:

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

small operator of the system, x^=ð Þ 1 þ x^ , we may estimate the quantum fluctuations and the correlation functions as a function of evolution time and photon number in

(a) The Raman excitation of radiators (atoms, molecules) in the four-level energetic scheme with the conversion of the photons in stokes and anti-Stokes modes. Such atoms may fling through the cavity in Figure 2A. (b) The

β=A ¼ 0:015, j ¼ 3 and j ¼ 6. (c) Evolution of von Neumann entropy for some parameters of the system. The substantial increase of the coherence between the bi-modal fields in comparison with coherence between the total numbers of photons belonging to each mode is observed. The entropy achieved minimal value in the lasing phase

<sup>2</sup> and n<sup>2</sup> � �=j

described in Refs. [12, 31] (γ�<sup>1</sup> <sup>¼</sup> <sup>Λ</sup>=<sup>v</sup> is main value of the flying time of atom, expressed through the atomic velocity, v, and cavity length, Λ). According to the projection operator method developed in Ref. [31] we start from the first order

. According to this description the new interac-

<sup>2</sup> for following parameters of the system

<sup>þ</sup> <sup>|</sup>e〉 <sup>g</sup>∣L^ � �� . Using the decomposition on the

þ H:c:,

Pmð Þt ∣ j, mihm, j∣. Here

<sup>L</sup>^† proportional to the small parameter <sup>ε</sup> <sup>¼</sup> <sup>g</sup><sup>2</sup>γ�<sup>2</sup>

<sup>1</sup> <sup>þ</sup> <sup>2</sup>εL^�L^† ; <sup>L</sup>^†

=2 is the conversion rate, β ¼ 2Aε is the attenuation of the con-

, <sup>W</sup>^ <sup>¼</sup> <sup>∑</sup><sup>j</sup>

m¼�j

� ð Þt ^J � ð Þ<sup>t</sup> � � exp ½ � <sup>i</sup>2ω0<sup>τ</sup> .

� �

version rate, which increases with the increasing the mean value of the lifetime of the excited N-atoms flings through the cavity. The numerical solution of this equation is obtained, decomposing the density matrix on the angular momentum states,

the Hilbert vectors j i j; m belong to the three mode states in the resonator (Pump, Stokes and anti-Stokes), Pmð Þt is the population probability of the j i j; m state. As in the two-photon emission (Case A) we are interested in the developing of the quantum between the photons belonging to scattered bi-modes. As a simple representation, we consider the situation when the non-correlated photons from the pump mode "b" is converted into the anti-Stokes mode "b". This process is possible for big detuning from resonance Δ<sup>1</sup> ≫ Δ<sup>2</sup> (see Figure 5a). As follows from the interaction Hamiltonian in this situation the coherent function G2ð Þ¼ τ

From Figure 5 follows that in the process of conversion of the un-correlated pump photons into the anti-Stokes one na <sup>¼</sup> <sup>j</sup> <sup>þ</sup> ^<sup>J</sup> <sup>z</sup>ð Þ<sup>t</sup> � � the cooperative phase of these two modes is established. The process achieved the saturation phase like in the single photon lasers. With the increasing the number of uncorrelated pump photons in broadband of the modes, this process is accompanied with the decreasing of relative coherence of the photons in each mode, so that the sum of total converted photons in anti-Stokes modes remain smaller than second order coherent function G2ð Þ 0

�¼�L^�

with increasing the number of uncorrelated modes in the pump and anti-Stokes field.

� ð Þ<sup>t</sup> ^<sup>J</sup> � ð Þ<sup>t</sup> � �=<sup>j</sup>

dt W t ^ ðÞ¼�<sup>A</sup> W t ^ ð ÞL^� <sup>1</sup>

<sup>Π</sup>�ð Þ <sup>t</sup>; <sup>z</sup> <sup>Π</sup><sup>þ</sup> h i ð Þ <sup>t</sup> <sup>þ</sup> <sup>τ</sup>; <sup>z</sup> becomes proportional to the expression ^<sup>J</sup>

ffiffi 2 <sup>p</sup> f i <sup>∣</sup> <sup>g</sup> 〈e|L^†

½L^þ , L^�

51

Figure 5.

� ¼ <sup>2</sup>L^<sup>z</sup> and <sup>½</sup>L^z, <sup>L</sup>^�

time evolution of the relative correlations ^J

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

the applied field. Here <sup>x</sup>^ <sup>¼</sup> <sup>2</sup>εL^�

approximation of the master equation d

described by the eigenstates of the operator <sup>L</sup>^<sup>2</sup>

tion Hamiltonian, <sup>H</sup>^ <sup>I</sup> <sup>¼</sup> <sup>ℏ</sup><sup>g</sup>

where <sup>A</sup> <sup>¼</sup> Ng<sup>2</sup>

Here the annihilation (creation) operators, ^ bð^ b † Þ, ^s ^s † � � and a^ a^† � �, correspond to the pump, Stokes and anti-Stokes modes, respectively, which satisfy the Bose commutation rules: <sup>½</sup>a^i, <sup>a</sup>^† <sup>j</sup> � ¼ δj,i, and ½a^i, a^j� ¼ 0, j � a, b, s). The interaction Hamiltonian of molecules (bio-molecules) with bimodal field is described by the Hamiltonian <sup>H</sup>^ <sup>I</sup> ¼ �P t ^ð Þ ; <sup>z</sup> <sup>Π</sup>^ � ð Þþ <sup>t</sup>; <sup>z</sup> <sup>H</sup>:c:, where the vector <sup>P</sup>^ is proportional to the displacement of the molecular oscillator P t ^ð Þ� ; <sup>z</sup> Q t ^ ð Þ� ; <sup>z</sup> <sup>∣</sup>eih i <sup>g</sup>jþj <sup>g</sup> <sup>h</sup>e∣. In the interaction with atomic sub-system (for example, four level system represented in Figure 5) in many situations λ � ℏ=Δ<sup>a</sup> and g � ℏ=Δs. From commutation of the bi-photon field operators between them <sup>½</sup> <sup>g</sup>^s^ <sup>b</sup>† <sup>þ</sup> <sup>λ</sup>^ ba^†, g^ b^s † <sup>þ</sup> <sup>λ</sup>a^^ <sup>b</sup>†� ¼ <sup>g</sup><sup>2</sup>ð^ b†^ b� ^s^s † <sup>þ</sup> <sup>1</sup><sup>Þ</sup> <sup>þ</sup> <sup>λ</sup><sup>2</sup> <sup>ð</sup>a^†a^ � ^ b^ <sup>b</sup>† <sup>þ</sup> <sup>1</sup>Þ, it is not difficult to observe that, when the interaction constant of atoms with Stokes and anti-Stokes modes coincide g ¼ λ, the new operators, belonging to angular momentum SUð Þ2 symmetry, can be easily introduced: <sup>L</sup>^<sup>z</sup> <sup>¼</sup> <sup>a</sup>^† a^ �^s † ^s, , <sup>L</sup>^� <sup>¼</sup> ffiffi 2 <sup>p</sup> <sup>ð</sup>^ <sup>b</sup>^s† <sup>þ</sup> <sup>a</sup>^^ <sup>b</sup>†Þ, <sup>L</sup>^<sup>þ</sup> <sup>¼</sup> ffiffi 2 <sup>p</sup> <sup>ð</sup>^s^ <sup>b</sup>† <sup>þ</sup> ^ ba^†Þ. The similar commutation relation can be obtained in the case, when the relations λ ≫ g or λ ≪ g are satisfied. In the last two cases we may neglect the Stokes or anti-Stokes scattering process so, that the similar operators may be defined for this two special situations: (a) ^J � <sup>¼</sup> <sup>a</sup>^^ b† ; ^J <sup>þ</sup> <sup>¼</sup> ^ ba^†, ^<sup>J</sup> <sup>z</sup> ¼ ða^†a^ � ^ b†^ <sup>b</sup>Þ=2 for <sup>λ</sup> <sup>≫</sup> <sup>g</sup> and (b) ^<sup>J</sup> � <sup>¼</sup> ^ b^s†, ^J <sup>þ</sup> <sup>¼</sup> ^s^ <sup>b</sup>†, ^<sup>J</sup> <sup>z</sup> ¼ ð^ b†^ b �^s † ^sÞ=2 for λ ≪ g. The commutation relations between these

Coherence Proprieties of Entangled Bi-Modal Field and Its Application in Holography… DOI: http://dx.doi.org/10.5772/intechopen.85857

### Figure 5.

Gs

^J

<sup>þ</sup> <sup>¼</sup> <sup>∑</sup>ka^†

kþ2k<sup>0</sup>

molecules (or bio-molecules) Ω

commutation rules: <sup>½</sup>a^i, <sup>a</sup>^†

Hamiltonian <sup>H</sup>^ <sup>I</sup> ¼ �P t

^s^s

^J <sup>þ</sup> <sup>¼</sup> ^s^

50

† <sup>þ</sup> <sup>1</sup><sup>Þ</sup> <sup>þ</sup> <sup>λ</sup><sup>2</sup>

duced: <sup>L</sup>^<sup>z</sup> <sup>¼</sup> <sup>a</sup>^†

situations: (a) ^J

<sup>b</sup>†, ^<sup>J</sup> <sup>z</sup> ¼ ð^

IIð Þ� <sup>t</sup>; <sup>t</sup> <sup>þ</sup> <sup>τ</sup> <sup>A</sup>^ <sup>þ</sup>

Quantum Cryptography in Advanced Networks

<sup>l</sup> ð Þ<sup>t</sup> ^ S�

<sup>a</sup>^<sup>k</sup> and ^<sup>J</sup> <sup>z</sup> <sup>¼</sup> <sup>∑</sup>kfa^†

the description of the atomic correlation functions.

<sup>Π</sup>^ ð Þ � ð Þ¼ <sup>t</sup>; <sup>z</sup> <sup>λ</sup>E^ð Þ <sup>þ</sup>

Here the annihilation (creation) operators, ^

^ð Þ ; <sup>z</sup> <sup>Π</sup>^ �

the bi-photon field operators between them <sup>½</sup> <sup>g</sup>^s^

^s, , <sup>L</sup>^� <sup>¼</sup> ffiffi

2 <sup>p</sup> <sup>ð</sup>^

displacement of the molecular oscillator P t

<sup>ð</sup>a^†a^ � ^ b^

> a^ �^s †

> > � <sup>¼</sup> <sup>a</sup>^^ b† ; ^J <sup>þ</sup> <sup>¼</sup> ^

b†^ b �^s † <sup>j</sup> ð Þ <sup>t</sup> <sup>þ</sup> <sup>τ</sup> <sup>D</sup>^ �

� exp ½ � ið Þ k<sup>1</sup> þ k; r<sup>n</sup> exp �i k<sup>0</sup>

tors of the superposition of collective discrete bi-modes of EMF ^J

kþ2k<sup>0</sup>

D E

<sup>n</sup> ð Þ t þ τ

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 ωai; ωsi and ωal; ωsl can form the coherent suð Þ2 state, which corresponds to the genera-

<sup>a</sup>^kþ2k0� <sup>a</sup>^†

tion relations for suð Þ2 algebra described in Section 2. In this cooperative effect, due to large number non-equidistant atoms in each ensemble A or S, the frequency differences between the scattered modes ωai � ωsi ¼ 2ω<sup>0</sup> and ωal � ωsl have same wave vectors, Ki ¼ Kl, where K<sup>i</sup> ¼ kai � ksi and K<sup>l</sup> ¼ kal � ksl and can be used in the coherent phenomena like holograms, or optical processing. In such coherence it is manifested the correlations between the ensemble of bi-modes generated by the pairs of atoms Sl f g ; Al , l ¼ 1, …Np. These effects are accompanied with the interference between single- and two-quantum collective transitions of three inverted radiators from the ensemble. The three particle collective decay rate is defined in

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 2ω<sup>0</sup> near the vibration frequency of the

<sup>a</sup> ð Þþ <sup>z</sup>; <sup>t</sup> gE^ð Þ <sup>þ</sup>

ba^† exp 2iω0<sup>t</sup> � i ka � kp

bð^ b † Þ, ^s ^s

^ð Þ� ; <sup>z</sup> Q t

<sup>j</sup> � ¼ δj,i, and ½a^i, a^j� ¼ 0, j � a, b, s). The interaction

<sup>b</sup>† <sup>þ</sup> <sup>λ</sup>^

<sup>b</sup>†Þ, <sup>L</sup>^<sup>þ</sup> <sup>¼</sup> ffiffi

b†^

^sÞ=2 for λ ≪ g. The commutation relations between these

<sup>b</sup>† <sup>þ</sup> <sup>1</sup>Þ, it is not difficult to observe that, when the interaction

<sup>s</sup> ð Þ <sup>z</sup>; <sup>t</sup> <sup>E</sup>^ð Þ �

� �z � �

ð Þþ <sup>t</sup>; <sup>z</sup> <sup>H</sup>:c:, where the vector <sup>P</sup>^ is proportional to the

ba^†, g^ b^s

2 <sup>p</sup> <sup>ð</sup>^s^

<sup>b</sup>† <sup>þ</sup> ^

<sup>b</sup>Þ=2 for <sup>λ</sup> <sup>≫</sup> <sup>g</sup> and (b) ^<sup>J</sup>

exp 2iω0t � i kp � ks � �z � �:

<sup>p</sup> ð Þ z; t

† � � and a^ a^† � �, correspond

^ ð Þ� ; <sup>z</sup> <sup>∣</sup>eih i <sup>g</sup>jþj <sup>g</sup> <sup>h</sup>e∣. In the

† <sup>þ</sup> <sup>λ</sup>a^^

<sup>b</sup>†� ¼ <sup>g</sup><sup>2</sup>ð^

ba^†Þ. The similar

� <sup>¼</sup> ^ b^s†,

b†^ b�

<sup>p</sup> ð Þ <sup>z</sup>; <sup>t</sup> <sup>E</sup>^ð Þ �

to the pump, Stokes and anti-Stokes modes, respectively, which satisfy the Bose

Hamiltonian of molecules (bio-molecules) with bimodal field is described by the

interaction with atomic sub-system (for example, four level system represented in Figure 5) in many situations λ � ℏ=Δ<sup>a</sup> and g � ℏ=Δs. From commutation of

constant of atoms with Stokes and anti-Stokes modes coincide g ¼ λ, the new operators, belonging to angular momentum SUð Þ2 symmetry, can be easily intro-

<sup>b</sup>^s† <sup>þ</sup> <sup>a</sup>^^

ba^†, ^<sup>J</sup> <sup>z</sup> ¼ ða^†a^ � ^

commutation relation can be obtained in the case, when the relations λ ≫ g or λ ≪ g are satisfied. In the last two cases we may neglect the Stokes or anti-Stokes scattering process so, that the similar operators may be defined for this two special

¼ G kp; ka � �^

> þ G ks; kp � �^s^ b †

; rj � � � � exp �<sup>i</sup> <sup>k</sup><sup>0</sup>

; rl � � � � :

ka^kg=2 which satisfies the commuta-

� <sup>¼</sup> <sup>∑</sup>ka^†

(4)

<sup>k</sup>a^kþ2k<sup>0</sup> ,

(5)

(a) The Raman excitation of radiators (atoms, molecules) in the four-level energetic scheme with the conversion of the photons in stokes and anti-Stokes modes. Such atoms may fling through the cavity in Figure 2A. (b) The time evolution of the relative correlations ^J � ð Þ<sup>t</sup> ^<sup>J</sup> � ð Þ<sup>t</sup> � �=<sup>j</sup> <sup>2</sup> and n<sup>2</sup> � �=j <sup>2</sup> for following parameters of the system β=A ¼ 0:015, j ¼ 3 and j ¼ 6. (c) Evolution of von Neumann entropy for some parameters of the system. The substantial increase of the coherence between the bi-modal fields in comparison with coherence between the total numbers of photons belonging to each mode is observed. The entropy achieved minimal value in the lasing phase with increasing the number of uncorrelated modes in the pump and anti-Stokes field.

operators are similar to the commutators of the collective atomic operators: ½L^þ , L^� � ¼ <sup>2</sup>L^<sup>z</sup> and <sup>½</sup>L^z, <sup>L</sup>^� �¼�L^� . According to this description the new interaction Hamiltonian, <sup>H</sup>^ <sup>I</sup> <sup>¼</sup> <sup>ℏ</sup><sup>g</sup> ffiffi 2 <sup>p</sup> f i <sup>∣</sup> <sup>g</sup> 〈e|L^† <sup>þ</sup> <sup>|</sup>e〉 <sup>g</sup>∣L^ � �� . Using the decomposition on the small operator of the system, x^=ð Þ 1 þ x^ , we may estimate the quantum fluctuations and the correlation functions as a function of evolution time and photon number in the applied field. Here <sup>x</sup>^ <sup>¼</sup> <sup>2</sup>εL^� <sup>L</sup>^† proportional to the small parameter <sup>ε</sup> <sup>¼</sup> <sup>g</sup><sup>2</sup>γ�<sup>2</sup> described in Refs. [12, 31] (γ�<sup>1</sup> <sup>¼</sup> <sup>Λ</sup>=<sup>v</sup> is main value of the flying time of atom, expressed through the atomic velocity, v, and cavity length, Λ). According to the projection operator method developed in Ref. [31] we start from the first order approximation of the master equation

$$\frac{d}{dt}\hat{W}(t) = -A\left[\hat{W}(t)\hat{L}^-\frac{\mathbf{1}}{\mathbf{1} + 2\varepsilon\hat{L}^-\hat{L}^\dagger}, \hat{L}^\dagger\right] + H.c.,$$

where <sup>A</sup> <sup>¼</sup> Ng<sup>2</sup> =2 is the conversion rate, β ¼ 2Aε is the attenuation of the conversion rate, which increases with the increasing the mean value of the lifetime of the excited N-atoms flings through the cavity. The numerical solution of this equation is obtained, decomposing the density matrix on the angular momentum states, described by the eigenstates of the operator <sup>L</sup>^<sup>2</sup> , <sup>W</sup>^ <sup>¼</sup> <sup>∑</sup><sup>j</sup> m¼�j Pmð Þt ∣ j, mihm, j∣. Here the Hilbert vectors j i j; m belong to the three mode states in the resonator (Pump, Stokes and anti-Stokes), Pmð Þt is the population probability of the j i j; m state. As in the two-photon emission (Case A) we are interested in the developing of the quantum between the photons belonging to scattered bi-modes. As a simple representation, we consider the situation when the non-correlated photons from the pump mode "b" is converted into the anti-Stokes mode "b". This process is possible for big detuning from resonance Δ<sup>1</sup> ≫ Δ<sup>2</sup> (see Figure 5a). As follows from the interaction Hamiltonian in this situation the coherent function G2ð Þ¼ τ <sup>Π</sup>�ð Þ <sup>t</sup>; <sup>z</sup> <sup>Π</sup><sup>þ</sup> h i ð Þ <sup>t</sup> <sup>þ</sup> <sup>τ</sup>; <sup>z</sup> becomes proportional to the expression ^<sup>J</sup> � ð Þt ^J � ð Þ<sup>t</sup> � � exp ½ � <sup>i</sup>2ω0<sup>τ</sup> . From Figure 5 follows that in the process of conversion of the un-correlated pump photons into the anti-Stokes one na <sup>¼</sup> <sup>j</sup> <sup>þ</sup> ^<sup>J</sup> <sup>z</sup>ð Þ<sup>t</sup> � � the cooperative phase of these two modes is established. The process achieved the saturation phase like in the single photon lasers. With the increasing the number of uncorrelated pump photons in broadband of the modes, this process is accompanied with the decreasing of relative coherence of the photons in each mode, so that the sum of total converted photons in anti-Stokes modes remain smaller than second order coherent function G2ð Þ 0

(see Figure 5). The von Neumann entropy of the system, obtained from the representation St <sup>¼</sup> <sup>∑</sup><sup>j</sup> m¼�j Pm log ½ � Pm achieves the maximal value at the initial stage of conversion after that when it is established the coherence between the pump photons and converted one like in a similar way like in the super-radiance. After that, it decreases correspond to the established a new coherent phase described above.

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 bimodal field

$$
\hat{\Pi}^{-}(t,\mathbf{z}) = \hat{E}^{(+)}\_{\mathbf{z}}(\mathbf{z},t)\hat{E}^{-}\_{\mathbf{z}}(\mathbf{z},\mathbf{t}) = \mathbf{G}(\mathbf{k}\_{\mathbf{}},\mathbf{k}\_{\mathbf{z}})\hat{a}\hat{b}^{\dagger}\exp\left[2i\alpha\_{0}t - i(\mathbf{k}\_{\mathbf{z}} - \mathbf{k}\_{\mathbf{t}})\mathbf{z}\right],\tag{6}
$$

where E^ð Þ <sup>þ</sup> <sup>s</sup> ð Þ <sup>z</sup>; <sup>t</sup> and <sup>E</sup>^� <sup>a</sup> ð Þ z; t are positive and negative defined strength of Stokes and anti-Stokes field (see Figure 5a), a^† s , ^ b† <sup>a</sup> and <sup>a</sup>^s, ^ ba are the annihilation and creation operators of electromagnetic field at Stokes, ωs, and anti-Stokes, ωa, frequencies respectively; ω<sup>a</sup> � ω<sup>s</sup> ¼ ω<sup>0</sup> is the fixed frequency of bi-modal field according to transition diagram represented in Figure 5. Following this definition one can introduced the new bi-quantum operators ^J � <sup>¼</sup> ^ b † a^; ^J <sup>þ</sup> <sup>¼</sup> <sup>a</sup>^†^ b; and ^<sup>J</sup> <sup>z</sup> ¼ ða^<sup>þ</sup>a^ � ^ b †^ bÞ=2. In this case for constant number of photons in resonator the conservation of Kasimir vector is possible, J <sup>2</sup> <sup>¼</sup> ^<sup>J</sup> <sup>2</sup> <sup>z</sup> <sup>þ</sup> ^<sup>J</sup> <sup>2</sup> <sup>x</sup> <sup>þ</sup> ^<sup>J</sup> <sup>2</sup> <sup>y</sup>, where

^Jx ¼ ð^<sup>J</sup> þ þ ^J � <sup>Þ</sup>=2, ^<sup>J</sup> <sup>y</sup> ¼ ð^<sup>J</sup> <sup>þ</sup> � ^<sup>J</sup> � Þ=2i. Considering that initially the photons are prepared in anti-Stokes mode of cavity N ¼ 2j, one can describe the two photon scattering lasing processes by coherent state for this bi-boson field, belonging to suð Þ2 algebra.

$$|a\rangle = \exp\left\{a\hat{\mathbf{J}}^{+}\right\}|-j,j\rangle\left\{\mathbf{1}+|a|^{2}\right\}^{-j},\tag{7}$$

scheme represented in Figure 9. The atoms situated in the evanescent zone of nanofiber stimulate the cooperative conversion of the photons from anti-Stokes pulse

(a) The non-coherent pump of the bimodal field in the cavity (see Figure 5a) and scattering double slit interference on the frequency, <sup>2</sup>ω<sup>0</sup> <sup>¼</sup> <sup>ω</sup><sup>p</sup> � <sup>ω</sup>s. (b) The time dependence of the mean value of vector <sup>Π</sup>^ ð Þ <sup>z</sup>; <sup>t</sup> (black line) and phase aliasing distribution in Raman components of the field E zð Þ ; t . For the detection of the

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

coherent properties (6)-(8) of <sup>Π</sup>^ ð Þ <sup>z</sup>; <sup>t</sup> , it is possible to use a two-photon detection scheme.

3. Quantum communication and holographic proprieties of bi-boson

used in quantum communication and quantum holography.

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

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 ϕ ¼ 2ω0t � k0z of

into the pump and Stokes pulses.

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

coherent field

Figure 6.

53

where α ¼ tan ð Þ θ=2 is the amplitude of this bi-boson field obtained in the Raman lasing processes. Taking into account the coherent state (7) one can found the mean value of strength product 〈Π^ ð Þ <sup>t</sup>; <sup>z</sup> 〉 ¼ ½〈Π^ � ð Þ <sup>t</sup>; <sup>z</sup> 〉 <sup>þ</sup> 〈Π^ <sup>þ</sup> ð Þ t; z 〉�=2

$$
\langle \langle \hat{\Pi}(t, x) \rangle \rangle = \Pi\_0 \cos \left[ \alpha\_0 t - (k\_a - k\_s) \mathbf{z} + \rho \right], \tag{8}
$$

where Π<sup>0</sup> ¼ G ks ð Þ ; ka |〈a^<sup>s</sup> ^ b<sup>þ</sup> <sup>a</sup> 〉| and φ ¼ arg〈a^<sup>s</sup> ^ b<sup>þ</sup> <sup>a</sup> 〉 are the amplitude and phase of bimodal field formed from Stokes and anti-Stokes photons.

The lithographic limit between maximal and minimal values of amplitude of correlation function G2ð Þ¼ Δ Π�ð Þz Π<sup>þ</sup> h i ð Þ z þ Δ in the two-slit experiments observed with two-photon detectors corresponds to the lithographic limit of this conjugate entangled bimodal field (see Figure 6). In this case, this limit is larger than in two-photon coherent emission Δ ≥λpλs= 2 λ<sup>s</sup> � λ<sup>p</sup> � � � � . The frequency of this coherent field achieves the frequency of the vibration states of the molecule (biomolecules) when the difference of wavelengths between the Stokes, pump, and anti-Stokes have the same magnitude. This coherent phenomenon between the Stokes and anti-Stokes fields can be used in Holographic representation of molecular vibrations and other coherent processes with phase memory. For holography, we propose the generation of new coherent states between Stokes pump and anti-Stokes field using nano-fiber systems [32]. In comparison with the cavity field, this type of generation permits to use the correlated bi modes out-site of generation

Coherence Proprieties of Entangled Bi-Modal Field and Its Application in Holography… DOI: http://dx.doi.org/10.5772/intechopen.85857

### Figure 6.

(see Figure 5). The von Neumann entropy of the system, obtained from the

stage of conversion after that when it is established the coherence between the pump photons and converted one like in a similar way like in the super-radiance. After that, it decreases correspond to the established a new coherent phase

<sup>a</sup> ð Þ¼ <sup>z</sup>; <sup>t</sup> G ks ð Þ ; ka <sup>a</sup>^^

s , ^ b†

creation operators of electromagnetic field at Stokes, ωs, and anti-Stokes, ωa, frequencies respectively; ω<sup>a</sup> � ω<sup>s</sup> ¼ ω<sup>0</sup> is the fixed frequency of bi-modal field according to transition diagram represented in Figure 5. Following this definition

prepared in anti-Stokes mode of cavity N ¼ 2j, one can describe the two photon scattering lasing processes by coherent state for this bi-boson field, belonging to

where α ¼ tan ð Þ θ=2 is the amplitude of this bi-boson field obtained in the Raman lasing processes. Taking into account the coherent state (7) one can found

<sup>þ</sup> n o

<sup>a</sup> 〉| and φ ¼ arg〈a^<sup>s</sup>

correlation function G2ð Þ¼ Δ Π�ð Þz Π<sup>þ</sup> h i ð Þ z þ Δ in the two-slit experiments observed with two-photon detectors corresponds to the lithographic limit of this conjugate entangled bimodal field (see Figure 6). In this case, this limit is larger

The lithographic limit between maximal and minimal values of amplitude of

coherent field achieves the frequency of the vibration states of the molecule (biomolecules) when the difference of wavelengths between the Stokes, pump, and anti-Stokes have the same magnitude. This coherent phenomenon between the Stokes and anti-Stokes fields can be used in Holographic representation of molecular vibrations and other coherent processes with phase memory. For holography, we propose the generation of new coherent states between Stokes pump and anti-Stokes field using nano-fiber systems [32]. In comparison with the cavity field, this type of generation permits to use the correlated bi modes out-site of generation

<sup>a</sup> and <sup>a</sup>^s, ^

bÞ=2. In this case for constant number of photons in resonator the

<sup>z</sup> <sup>þ</sup> ^<sup>J</sup> <sup>2</sup>

j i �j; <sup>j</sup> <sup>1</sup> <sup>þ</sup> j j <sup>α</sup> <sup>2</sup> n o�<sup>j</sup>

<sup>Π</sup>^ ð Þ <sup>t</sup>; <sup>z</sup> � � <sup>¼</sup> <sup>Π</sup><sup>0</sup> cos½ � <sup>ω</sup>0<sup>t</sup> � ð Þ ka � ks <sup>z</sup> <sup>þ</sup> <sup>φ</sup> , (8)

^ b<sup>þ</sup>

<sup>2</sup> <sup>¼</sup> ^<sup>J</sup> <sup>2</sup>

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-

Pm log ½ � Pm achieves the maximal value at the initial

<sup>a</sup> ð Þ z; t are positive and negative defined strength of Stokes

� <sup>¼</sup> ^ b † a^; ^J

<sup>x</sup> <sup>þ</sup> ^<sup>J</sup> <sup>2</sup>

Þ=2i. Considering that initially the photons are

ð Þ <sup>t</sup>; <sup>z</sup> 〉 <sup>þ</sup> 〈Π^ <sup>þ</sup>

<sup>b</sup>† exp 2½ � <sup>i</sup>ω0<sup>t</sup> � i kð Þ <sup>a</sup> � ks <sup>z</sup> , (6)

ba are the annihilation and

<sup>þ</sup> <sup>¼</sup> <sup>a</sup>^†^

<sup>y</sup>, where

b; and

, (7)

ð Þ t; z 〉�=2

<sup>a</sup> 〉 are the amplitude and phase of

� � � � . The frequency of this

representation St <sup>¼</sup> <sup>∑</sup><sup>j</sup>

described above.

modal field

Π^ �

^<sup>J</sup> <sup>z</sup> ¼ ða^<sup>þ</sup>a^ � ^

suð Þ2 algebra.

^Jx ¼ ð^<sup>J</sup> þ þ ^J �

52

where E^ð Þ <sup>þ</sup>

ð Þ¼ <sup>t</sup>; <sup>z</sup> <sup>E</sup>^ð Þ <sup>þ</sup>

b †^

where Π<sup>0</sup> ¼ G ks ð Þ ; ka |〈a^<sup>s</sup>

m¼�j

Quantum Cryptography in Advanced Networks

<sup>s</sup> ð Þ <sup>z</sup>; <sup>t</sup> <sup>E</sup>^�

one can introduced the new bi-quantum operators ^J

<sup>þ</sup> � ^<sup>J</sup> �

j i <sup>α</sup> <sup>¼</sup> exp <sup>α</sup>^<sup>J</sup>

the mean value of strength product 〈Π^ ð Þ <sup>t</sup>; <sup>z</sup> 〉 ¼ ½〈Π^ �

^ b<sup>þ</sup>

than in two-photon coherent emission Δ ≥λpλs= 2 λ<sup>s</sup> � λ<sup>p</sup>

bimodal field formed from Stokes and anti-Stokes photons.

<sup>s</sup> ð Þ <sup>z</sup>; <sup>t</sup> and <sup>E</sup>^�

and anti-Stokes field (see Figure 5a), a^†

conservation of Kasimir vector is possible, J

<sup>Þ</sup>=2, ^<sup>J</sup> <sup>y</sup> ¼ ð^<sup>J</sup>

(a) The non-coherent pump of the bimodal field in the cavity (see Figure 5a) and scattering double slit interference on the frequency, <sup>2</sup>ω<sup>0</sup> <sup>¼</sup> <sup>ω</sup><sup>p</sup> � <sup>ω</sup>s. (b) The time dependence of the mean value of vector <sup>Π</sup>^ ð Þ <sup>z</sup>; <sup>t</sup> (black line) and phase aliasing distribution in Raman components of the field E zð Þ ; t . For the detection of the coherent properties (6)-(8) of <sup>Π</sup>^ ð Þ <sup>z</sup>; <sup>t</sup> , it is possible to use a two-photon detection scheme.

scheme represented in Figure 9. The atoms situated in the evanescent zone of nanofiber stimulate the cooperative conversion of the photons from anti-Stokes pulse into the pump and Stokes pulses.

## 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 ϕ ¼ 2ω0t � k0z of

In the expression (9) the function GO<sup>2</sup> <sup>¼</sup> <sup>P</sup>^�

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

described by argument of complex number P�

Tb <sup>¼</sup> GO<sup>2</sup> <sup>þ</sup> Gr<sup>2</sup> <sup>þ</sup> <sup>2</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

¼ Π�

þ Π�

the vector ΠOð Þ¼ z; t Π<sup>þ</sup>

Figure 8.

55

(5)–(8) so that the transmittance

Gr<sup>2</sup> <sup>¼</sup> <sup>P</sup>^�

argðP^<sup>þ</sup>

<sup>r</sup> ð Þ <sup>0</sup>; <sup>t</sup> <sup>þ</sup> <sup>τ</sup> <sup>P</sup>^<sup>þ</sup>

<sup>O</sup>ð ÞÞ � <sup>0</sup>; <sup>t</sup> argðP^<sup>þ</sup>

intensity of detecting bi-photons from the original bimodal field;

the interpretation of image can be expressed in classical terms

Gr2GO<sup>2</sup>

<sup>O</sup>ð Þþ z; t Π� <sup>O</sup>ð Þ <sup>z</sup>; <sup>t</sup> � �<sup>=</sup> ffiffi

Ts <sup>¼</sup> : ð Þ <sup>Π</sup>Oð Þþ <sup>z</sup>; <sup>t</sup> <sup>Π</sup>rð Þ <sup>z</sup>; <sup>t</sup> <sup>þ</sup> <sup>τ</sup> <sup>2</sup> : D E

<sup>O</sup>ð Þ <sup>0</sup>; <sup>t</sup> � � <sup>þ</sup> <sup>Π</sup>�

<sup>r</sup> ð Þ <sup>0</sup>; <sup>t</sup> <sup>þ</sup> <sup>τ</sup> � � <sup>þ</sup> <sup>Π</sup>�

(1) Two-photon coherent light described in Section 2 and the registration of hologram taking into consideration the phase and amplitude of bi-photon field of the "object" and "reference" waves. (2) The possibilities to detect

the "signal" (ω<sup>i</sup> ∈ ð Þ ω0; 2ω<sup>0</sup> ) and "idler" (ω<sup>i</sup> ∈ð Þ 0;ω<sup>0</sup> ) photons on separate screens A and B.

<sup>O</sup>ð Þ 0; t Π<sup>þ</sup>

<sup>O</sup>ð Þ 0; t Π<sup>þ</sup>

p cos arg P<sup>þ</sup>

<sup>O</sup>ð Þ <sup>0</sup>; <sup>t</sup> <sup>P</sup>^<sup>þ</sup>

<sup>O</sup>ð Þ 0; t P<sup>þ</sup>

<sup>O</sup>ð Þ <sup>0</sup>; <sup>t</sup> � � � arg <sup>P</sup><sup>þ</sup>

<sup>r</sup> ð Þ 0; t þ τ Π<sup>þ</sup> <sup>r</sup> ð Þ <sup>0</sup>; <sup>t</sup> <sup>þ</sup> <sup>τ</sup> � �

> <sup>r</sup> ð Þ 0; t þ τ Π<sup>þ</sup> <sup>O</sup>ð Þ <sup>0</sup>; <sup>t</sup> � �:

<sup>r</sup> ð Þ <sup>r</sup>; <sup>t</sup> <sup>þ</sup> <sup>τ</sup> � � � � (10)

<sup>p</sup> can be found from the expressions

(11)

<sup>r</sup> ð ÞÞ r; t þ τ . The propriety of two-photon bimodal field is

<sup>r</sup> ð Þ <sup>0</sup>; <sup>t</sup> <sup>þ</sup> <sup>τ</sup> � � or

<sup>r</sup> ð Þ <sup>0</sup>; <sup>t</sup> <sup>þ</sup> <sup>τ</sup> � � is the intensity of detecting bi-photons from refer-

ence bimodal wave from the object. The phase dependence of the image can be

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

described by the expressions (2)–(3) of the last Section 2. The detection scheme on the plan z ¼ 0 is described in Figure 8 and as in the scheme 7, we may take into consideration the fact, that the coherent blocks of bi-photons can be separated into idler and signal photons. This entangled effect may be registered by two separate detector screens represented in the Figure 8. For example the A screen, it is used for the registration of 'idler' photons while the screen B can be used for the registration of amplitude an fluctuations phase of 'signal' photons. The problem consists in the restoration of this bimodal field with coherent proprieties between the bi-photons. The transmittance can be detected by two photon detectors on the plane ð Þ x; y and

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

2

<sup>O</sup>ð Þ <sup>0</sup>; <sup>t</sup> � � represents the

### Figure 7.

The two-quanta coherence and its possible experimental observations: a, two photon coherent generator; b, dispersive media; c, lenses; d, fibers; e, signal restoration.

these two fields amplitude as and b<sup>þ</sup> <sup>a</sup> has similar behavior as the phase of single mode coherent field, here 2ω<sup>0</sup> ¼ ω<sup>a</sup> � ω<sup>s</sup> and k<sup>0</sup> ¼ ka � ks are the frequencies and wave vector difference between the Stokes and anti-Stokes fields respectively. The coherent propriety of this product of the electric field components is proposed to apply in possibilities to construct the time, space holograms of real objects, taking in to account the conservation of phase of amplitude product in the propagating and interference processes. The quantum phase between the radiators can also be used in holography.

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 twophoton 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") I<sup>0</sup> with a known "reference wave" Ir and recording their interference pattern in the z ¼ 0 plane. According to transmittance conception T of single-mode holograms, the correlations are proposed in the strength product of "object wave" and the "reference wave" waves <sup>E</sup>^Oð Þ <sup>z</sup>; <sup>t</sup> <sup>E</sup>^rð Þ <sup>z</sup>; <sup>t</sup> <sup>þ</sup> <sup>τ</sup> � �, where <sup>E</sup>^Oð Þ¼ð <sup>z</sup>; <sup>t</sup> <sup>E</sup>^<sup>þ</sup> <sup>O</sup>ð Þþ <sup>z</sup>; <sup>t</sup> <sup>E</sup>^� <sup>O</sup>ð ÞÞ <sup>z</sup>; <sup>t</sup> <sup>=</sup> ffiffi 2 <sup>p</sup> and <sup>E</sup>^rð Þ¼ð <sup>z</sup>; <sup>t</sup> <sup>þ</sup> <sup>τ</sup> <sup>E</sup>^<sup>þ</sup> <sup>r</sup> ð Þþ <sup>z</sup>; <sup>t</sup> <sup>þ</sup> <sup>τ</sup> <sup>E</sup>^� <sup>r</sup> ð ÞÞ <sup>z</sup>; <sup>t</sup> <sup>þ</sup> <sup>τ</sup> <sup>=</sup> ffiffi 2 <sup>p</sup> the scheme of interference pattern is represented in Figure 8 and has many analogies with classical holograms. The transmittance is given by interference between the original and reference bimodal waves at t (see for example [33]). Extending this conception we construct such a hologram, replacing the EMF strength the two-photon coherence using the field vector (2)-(3). According to the classical definition one can represent correlations between the original bimodal field through <sup>P</sup>^Oð Þ¼ð <sup>z</sup>; <sup>t</sup> <sup>P</sup>^<sup>þ</sup> <sup>O</sup>ð Þþ <sup>z</sup>; <sup>t</sup> <sup>P</sup>^� <sup>O</sup>ð ÞÞ <sup>z</sup>; <sup>t</sup> <sup>=</sup> ffiffi 2 <sup>p</sup> and reference bimodal wave vector, described by the expression <sup>P</sup>^rð Þ¼ð <sup>z</sup>; <sup>t</sup> <sup>þ</sup> <sup>τ</sup> <sup>P</sup>^<sup>þ</sup> <sup>r</sup> ð Þþ <sup>z</sup>; <sup>t</sup> <sup>þ</sup> <sup>τ</sup> <sup>P</sup>^� <sup>r</sup> ð ÞÞ <sup>z</sup>; <sup>t</sup> <sup>þ</sup> <sup>τ</sup> <sup>=</sup> ffiffi 2 <sup>p</sup> . In this case, the points on the plan of hologram, <sup>z</sup> <sup>¼</sup> 0, we have the transmittance

$$\begin{split} T\_{b} &= \left\langle : \left( \hat{P}\_{O}(\mathbf{z},t) + \hat{P}\_{r}(\mathbf{z},t+\tau) \right)^{2} : \right\rangle \\ &= \left\langle \hat{P}\_{O}^{-}(\mathbf{0},t)\hat{P}\_{O}^{+}(\mathbf{0},t) \right\rangle + \left\langle \hat{P}\_{r}^{-}(\mathbf{0},t+\tau)(\hat{P}\_{r}^{+}(\mathbf{0},t+\tau)) \right\rangle \\ &\quad + \left\langle \hat{P}\_{O}^{-}(\mathbf{0},t)\hat{P}\_{r}^{+}(\mathbf{0},t+\tau) \right\rangle + \left\langle \hat{P}\_{r}^{-}(\mathbf{0},t+\tau)\hat{P}\_{O}^{+}(\mathbf{0},t) \right\rangle. \end{split} \tag{9}$$

Coherence Proprieties of Entangled Bi-Modal Field and Its Application in Holography… DOI: http://dx.doi.org/10.5772/intechopen.85857

In the expression (9) the function GO<sup>2</sup> <sup>¼</sup> <sup>P</sup>^� <sup>O</sup>ð Þ <sup>0</sup>; <sup>t</sup> <sup>P</sup>^<sup>þ</sup> <sup>O</sup>ð Þ <sup>0</sup>; <sup>t</sup> � � represents the intensity of detecting bi-photons from the original bimodal field; Gr<sup>2</sup> <sup>¼</sup> <sup>P</sup>^� <sup>r</sup> ð Þ <sup>0</sup>; <sup>t</sup> <sup>þ</sup> <sup>τ</sup> <sup>P</sup>^<sup>þ</sup> <sup>r</sup> ð Þ <sup>0</sup>; <sup>t</sup> <sup>þ</sup> <sup>τ</sup> � � is the intensity of detecting bi-photons from reference bimodal wave from the object. The phase dependence of the image can be described by argument of complex number P� <sup>O</sup>ð Þ 0; t P<sup>þ</sup> <sup>r</sup> ð Þ <sup>0</sup>; <sup>t</sup> <sup>þ</sup> <sup>τ</sup> � � or argðP^<sup>þ</sup> <sup>O</sup>ð ÞÞ � <sup>0</sup>; <sup>t</sup> argðP^<sup>þ</sup> <sup>r</sup> ð ÞÞ r; t þ τ . The propriety of two-photon bimodal field is described by the expressions (2)–(3) of the last Section 2. The detection scheme on the plan z ¼ 0 is described in Figure 8 and as in the scheme 7, we may take into consideration the fact, that the coherent blocks of bi-photons can be separated into idler and signal photons. This entangled effect may be registered by two separate detector screens represented in the Figure 8. For example the A screen, it is used for the registration of 'idler' photons while the screen B can be used for the registration of amplitude an fluctuations phase of 'signal' photons. The problem consists in the restoration of this bimodal field with coherent proprieties between the bi-photons. The transmittance can be detected by two photon detectors on the plane ð Þ x; y and the interpretation of image can be expressed in classical terms

$$T\_b = G\_{O2} + G\_{r2} + 2\sqrt{G\_{r2}G\_{O2}}\cos\left[\arg\left(P\_O^+(0,t)\right) - \arg\left(P\_r^+(r,t+\tau)\right)\right] \tag{10}$$

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 ΠOð Þ¼ z; t Π<sup>þ</sup> <sup>O</sup>ð Þþ z; t Π� <sup>O</sup>ð Þ <sup>z</sup>; <sup>t</sup> � �<sup>=</sup> ffiffi 2 <sup>p</sup> can be found from the expressions (5)–(8) so that the transmittance

$$\begin{split} T\_{\boldsymbol{s}} &= \left\langle : \left( \Pi\_{O}(\mathbf{z}, \mathbf{t}) + \Pi\_{r}(\mathbf{z}, \mathbf{t} + \boldsymbol{\tau}) \right)^{2} : \right\rangle \\ &= \left\langle \Pi\_{O}^{-}(\mathbf{0}, \mathbf{t}) \Pi\_{O}^{+}(\mathbf{0}, \mathbf{t}) \right\rangle + \left\langle \Pi\_{r}^{-}(\mathbf{0}, \mathbf{t} + \boldsymbol{\tau}) \Pi\_{r}^{+}(\mathbf{0}, \mathbf{t} + \boldsymbol{\tau}) \right\rangle \\ &\quad + \left\langle \Pi\_{O}^{-}(\mathbf{0}, \mathbf{t}) \Pi\_{r}^{+}(\mathbf{0}, \mathbf{t} + \boldsymbol{\tau}) \right\rangle + \left\langle \Pi\_{r}^{-}(\mathbf{0}, \mathbf{t} + \boldsymbol{\tau}) \Pi\_{O}^{+}(\mathbf{0}, \mathbf{t}) \right\rangle. \end{split} \tag{11}$$

### Figure 8.

(1) Two-photon coherent light described in Section 2 and the registration of hologram taking into consideration the phase and amplitude of bi-photon field of the "object" and "reference" waves. (2) The possibilities to detect the "signal" (ω<sup>i</sup> ∈ ð Þ ω0; 2ω<sup>0</sup> ) and "idler" (ω<sup>i</sup> ∈ð Þ 0;ω<sup>0</sup> ) photons on separate screens A and B.

these two fields amplitude as and b<sup>þ</sup>

dispersive media; c, lenses; d, fibers; e, signal restoration.

Quantum Cryptography in Advanced Networks

Figure 7.

<sup>a</sup> has similar behavior as the phase of single mode

coherent field, here 2ω<sup>0</sup> ¼ ω<sup>a</sup> � ω<sup>s</sup> and k<sup>0</sup> ¼ ka � ks are the frequencies and wave vector difference between the Stokes and anti-Stokes fields respectively. The coherent propriety of this product of the electric field components is proposed to apply in possibilities to construct the time, space holograms of real objects, taking in to account the conservation of phase of amplitude product in the propagating and interference processes. The quantum phase between the radiators can also be used in holography. Presently exist a lot of proposals in which is manifested holographic principals of

The two-quanta coherence and its possible experimental observations: a, two photon coherent generator; b,

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

Following the idea of classical holograms, we changed the conception of twophoton 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") I<sup>0</sup> with a known "reference wave" Ir and recording their interference pattern in the z ¼ 0 plane. According to transmittance conception T of single-mode holograms, the correlations are proposed in the strength product of "object wave" and the "reference wave"

<sup>r</sup> ð ÞÞ <sup>z</sup>; <sup>t</sup> <sup>þ</sup> <sup>τ</sup> <sup>=</sup> ffiffi

2

<sup>O</sup>ð Þþ <sup>z</sup>; <sup>t</sup> <sup>P</sup>^�

<sup>r</sup> ð ÞÞ <sup>z</sup>; <sup>t</sup> <sup>þ</sup> <sup>τ</sup> <sup>=</sup> ffiffi

<sup>p</sup> . In this case, the points on the plan of hologram, <sup>z</sup> <sup>¼</sup> 0, we have the transmittance

<sup>þ</sup> 〈P^�

<sup>r</sup> ð Þ 0; t þ τ

<sup>r</sup> ð Þð <sup>0</sup>; <sup>t</sup> <sup>þ</sup> <sup>τ</sup> <sup>P</sup>^<sup>þ</sup>

<sup>þ</sup> <sup>P</sup>^�

represented in Figure 8 and has many analogies with classical holograms. The transmittance is given by interference between the original and reference bimodal waves at t (see for example [33]). Extending this conception we construct such a hologram, replacing the EMF strength the two-photon coherence using the field vector (2)-(3). According to the classical definition one can represent correlations between the orig-

<sup>O</sup>ð Þþ <sup>z</sup>; <sup>t</sup> <sup>E</sup>^�

<sup>O</sup>ð ÞÞ <sup>z</sup>; <sup>t</sup> <sup>=</sup> ffiffi

<sup>O</sup>ð ÞÞ <sup>z</sup>; <sup>t</sup> <sup>=</sup> ffiffi

<sup>p</sup> the scheme of interference pattern is

2

<sup>r</sup> ð Þ 0; t þ τ 〉

<sup>r</sup> ð Þ <sup>0</sup>; <sup>t</sup> <sup>þ</sup> <sup>τ</sup> <sup>P</sup>^<sup>þ</sup>

D E

<sup>r</sup> ð Þþ <sup>z</sup>; <sup>t</sup> <sup>þ</sup> <sup>τ</sup> <sup>P</sup>^�

<sup>O</sup>ð Þ 0; t

:

(9)

2 <sup>p</sup> and

<sup>p</sup> and reference bimodal

coherence between the pairs is not taken into consideration.

waves <sup>E</sup>^Oð Þ <sup>z</sup>; <sup>t</sup> <sup>E</sup>^rð Þ <sup>z</sup>; <sup>t</sup> <sup>þ</sup> <sup>τ</sup> � �, where <sup>E</sup>^Oð Þ¼ð <sup>z</sup>; <sup>t</sup> <sup>E</sup>^<sup>þ</sup>

wave vector, described by the expression <sup>P</sup>^rð Þ¼ð <sup>z</sup>; <sup>t</sup> <sup>þ</sup> <sup>τ</sup> <sup>P</sup>^<sup>þ</sup>

Tb <sup>¼</sup> : <sup>P</sup>^Oð Þþ <sup>z</sup>; <sup>t</sup> <sup>P</sup>^rð Þ <sup>z</sup>; <sup>t</sup> <sup>þ</sup> <sup>τ</sup> � �<sup>2</sup> : D E

<sup>O</sup>ð Þ 0; t

D E

<sup>O</sup>ð Þ <sup>0</sup>; <sup>t</sup> <sup>P</sup>^<sup>þ</sup>

D E

<sup>O</sup>ð Þ <sup>0</sup>; <sup>t</sup> <sup>P</sup>^<sup>þ</sup>

<sup>r</sup> ð Þþ <sup>z</sup>; <sup>t</sup> <sup>þ</sup> <sup>τ</sup> <sup>E</sup>^�

inal bimodal field through <sup>P</sup>^Oð Þ¼ð <sup>z</sup>; <sup>t</sup> <sup>P</sup>^<sup>þ</sup>

<sup>¼</sup> <sup>P</sup>^�

<sup>þ</sup> <sup>P</sup>^�

<sup>E</sup>^rð Þ¼ð <sup>z</sup>; <sup>t</sup> <sup>þ</sup> <sup>τ</sup> <sup>E</sup>^<sup>þ</sup>

2

54

This type of Holography takes into consideration the coherent process at low frequency ω<sup>p</sup> � ω<sup>s</sup> (or ω<sup>a</sup> � ω<sup>p</sup> which may coincide with the vibration frequencies of biomolecules. The popularity of coherent Raman scattering techniques in optical microscopy increases and it may be developed using another type of coherence described in the section. The holography developed on the bases of coherence proprieties between the two- (or three) conjugate modes of the scattering field opens this possibility not only for the description of the spectral diapason and time dependence of scattered field intensity, but the topological aspects of the molecular structures manifested in holographic representations of the vibrational modes of molecules. The coherence proposed in the Section 2 B needs the low intensity of each mode component in comparison with traditional Raman diagnostic proposed in Refs. [14, 15]. Using the coherent proprieties, described at the point B of the last section, we can estimate a lot of peculiarities connected with geometrical structures of biomolecules for lower intensities of each mode component of Raman process described by Refs. [14, 15]. In this case the transmission can be detected by the scheme proposed in Figure 9 on the plan ð Þ x; y , where the interpretation of Hologram imaging can be expressed in classical terms.

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

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

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

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, 7–9]. All these

methods open new possibilities in the coding and decoding of data.

Quantum Optics and Kinetic Processes Lab, Institute of Applied Physics of

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

resolution along the fibers becomes possible.

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

with other species of excited radiators.

Moldova, Chisinau, MD, Republic of Moldova

provided the original work is properly cited.

\*Address all correspondence to: enakinicolae@yahoo.com

4. Conclusions

Author details

Nicolae A. Enaki

57

$$T\_s = G\_{O2} + G\_{r2} + 2\sqrt{G\_{r2}G\_{O2}}\cos\left[\arg\left(\Pi\_O^+(0,t)\right) - \arg\left(\Pi\_r^+(r,t+\tau)\right)\right] \tag{12}$$

The entanglement between each mode of the field can be detected by twophoton 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 superradiance [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,

### Figure 9.

Two-photon coherent light and principle of hologram registration taking into consideration the phase and amplitude of the three modes of Raman scattered field Stokes, pump and anti-Stokes modes.

Coherence Proprieties of Entangled Bi-Modal Field and Its Application in Holography… DOI: http://dx.doi.org/10.5772/intechopen.85857

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.
