Open access peer-reviewed chapter

# Cooperative Spontaneous Lasing and Possible Quantum Retardation Effects

By Nicolae A. Enaki

Submitted: July 31st 2018Reviewed: December 4th 2018Published: May 8th 2019

DOI: 10.5772/intechopen.83013

## Abstract

The collective decay effects between the dipole-active three-level subsystems in the nonlinear interaction with dipole-forbidden transitions, like 2 S − 1 S of hydrogen-like radiators, are proposed, taking into consideration the cooperative exchanges between two species of atoms through the vacuum field in the scattering and the two-photon resonance processes. One of them corresponds to the situation when the total energy of the emitted two photons by the three-level radiator in the cascade configuration enters into the two-photon resonance with another type of dipole-forbidden transitions of hydrogen-like (or helium-like) atoms. The similar situation appears in the cooperative scattering between two species of quantum emitters when the difference of the excited energies of the two dipole-active transitions of the three-level radiators is in the resonance with the dipole-forbidden transitions of the Hydrogen-like radiators. These effects are accompanied by the interference between single- and two-quantum collective transitions of the inverted radiators from the ensemble. The two-particle collective decay rate is defined in the description of the atomic correlation functions taking into consideration the phase retardation between them. The kinetic equations which describe the cooperative processes as the function of time and correlation are obtained. The behavior of the system of radiators at short and long time intervals in comparison with the retardation time between them is studied.

### Keywords

• 42.50.Fx Cooperative phenomena in quantum optical systems
• 32.80.Qk Coherent control of atomic interactions with photons
• 03.65.Ud Entanglement and quantum nonlocality
• 03.65.Yz Decoherence
• open systems
• quantum statistical methods

2000 AMS Subject Classification: Primary 82C10, 81Q15; Secondary 20G42, 81R15

## 1. Introduction

The single-photon cooperative emission of the inverted system of radiators proposed by Dicke [1] opens the new possibilities of this phenomenon in the description of decay processes in the multilevel system [2] and multi-photon interaction of radiators with EMF (see, e.g., [3, 4]). The experimental possibilities [3, 4] of nonlinear cooperative interaction of radiators with vacuum field remain in the center of attention of many theoretical models proposed in the last time [5, 6]. For example, using the classical and quantum approaches in Refs. [7, 8, 9, 10], it is given the quantitative description of two-color super-fluorescence, observed in [2]. In the recent experiment [11], the cooperative emission of excited atomic oxygen relatively the transition 3p3P3s3Sat wavelength 845nmas a result of two-photon photolysis of atmospheric O2followed by two-photon excitation of atomic oxygen by a laser pulse at 226nmis demonstrated.

Combining single- and two-photon processes, this chapter aims to investigate the cooperative emission of the inverted system of radiators taking into account the resonance between one- and two-photon cooperative transitions of two three-level atomic subsystems represented in Figure 1. In this approach, the two dipole-active species of radiators studied in Refs. [12, 13] are replaced with one three-level atomic subsystem Ξ(or V) inverted relative to the single-photon emission in the resonance with 2S- 1Sdipole-forbidden transitions of hydrogen (or He)-like sub-ensemble. This new cooperative effect between two species of radiators occurs when two three-level emitters enter into two-quantum resonances with other emitters of the second ensemble inverted relative dipole-forbidden transition. Similar collectivization processes can amplify (or inhibit) the collective spontaneous emission rate of each atomic sub-ensemble. The sign of exchange integral between the two atoms from different sub-ensembles depends on the retardation time and distance between them. This problem is connected to the possibilities of amplifying of entangled quanta and established the coherence between photon pairs. For this, the cooperative interaction of three-radiator subsystems is proposed in which one of them is inverted relative to the dipole-forbidden transitions, but another inverted dipole-active three-level system ignites this transition.

Taking into consideration the elementary acts of two-photon resonance between radiators, we have demonstrated the increasing of two-photon emission rate in one of the radiator subsystem comparison with traditional two-photon super-fluorescence [5]. The mutual influence of two- and single-photon super-fluorescent processes on the two-photon cooperative emission of the inverted subsystem relatively dipole-forbidden transition depends on the position of atoms in the exchange potential. Two possibilities of two- and three-particle exchanges through the vacuum field are represented in Figure 1A–C, taking into consideration the two-photon resonance and scattering processes between the dipole-forbidden subsystem Dand dipole-active subsystems of Ξ, Λ, and V, respectively. Here, the product of two vacuum polarizations of the atom Ξ(or V, Λ) comes into resonance with the polarization of the dipole-forbidden transitions of the Datom.

Using two small parameters in Section 2, we propose the projection operator method of elimination of the EMF operators from the generalized equation of atomic subsystems in single- and two-photon resonances. The possibilities of two-photon cooperative resonance between three-level radiators situated at a distance compared with the emission wavelength are demonstrated. Following this description the resonance interaction of a dipole-forbidden atom and three-level dipole-active radiator in the cascade configuration is described by the cooperative rate and the exchange integral (13). The similar expression (16) is obtained in the scattering process of three-level system in Vor Λ—configurations with dipole-forbidden Dsubsystem represented in Figure 1. In Section 3 the spontaneous emission for the two radiators in the cascade or scattering resonances is given without the de-correlation of the atomic correlation functions between them.

## 2. Master equation of cooperative exchange between three-level radiators in two-quantum exchanges

Let us consider the interaction of three-level subsystems of radiators in Vand Ξconfiguration with Ddipole-forbidden two-level ensemble through the vacuum of EMF. The Ξthree-level subsystem in cascade configuration, prepared in excited state 2ξ, can pass into the Dicke super-radiance regime [1] relatively the dipole-active transitions 2ξιξ1ξat frequencies ω2and ω1(Figure 1A). According to Figure 1A, the excited Datom relatively the dipole-forbidden transition 2d1dpasses in the ground state 1dsimultaneously generating two quanta under the influence of cooperative decay of the Ξthree-level subsystem. Two-photon transition of the D-atom takes place through the virtual levels represented by the notations 3dwith opposite parity relative to the ground 1dand excited 2dstates, respectively. This case corresponds to the situation when the emission frequencies of the dipole-active Ξradiators and Ddipole-forbidden radiators satisfy the resonance condition ω1+ω2=2ω0. Here ω1and ω2are the transition frequencies of the Ξdipole-active radiators in Ξ, and ωd=2ω0is the energy distance between the ground nSand excited n+1Sstates of the dipole-forbidden transitions of Dradiator (see Figure 1A).

The similar cooperative emissions can be observed in the two-quantum resonance interactions between the V(or Λ) three-level radiator in two quanta scattering interactions and the dipole-forbidden transitions of Datoms through the vacuum field (see Figure 1B,C). In this situation, we consider that the dipole-active transitions of the three-level radiator in the V(or Λ) configuration satisfy the scattering condition ωaωs=ωdin interaction with the Dsubsystem. As it is represented in Figure 1B, the cone of the transition energies of the Vor Λdipole-active three-level atoms must be larger than the dipole-forbidden transition n+1SnSof atoms D, so that two-photon resonance between the two dipole-active transitions of Vatom enters in the exact scattering resonance, ωaωs=ωd, with Datom. This nonlinear transition increases with the decreasing of the detuning from resonance with virtual 3dstates of the Dtwo-level system.

The Hamiltonian of the system consists of the free and interaction parts H=Ĥ0+ĤI. Here the free part of this Hamiltonian is represented through the atomic and field operators:

Ĥ0=kωkâkâk+m=1NωdD̂zmα=s,a2l=1NλωαΛ̂αlα+α=s,a2l=1NvωαV̂αlα+α=1,2j=1Nξ1αωαΞ̂αjα,E1

where N, Nξ, Nλ, and Nvare the number of atoms in the D, Ξ, Λ, and Vsubsystems, respectively; the energies of first and second levels of the Ξ, Λ, and Vthree-level subsystems are measured from the third intermediate state ι. The operators Ξ1,j1, Ξι,jι, and Ξ2,j2describe the population of the ground, intermediary, and excited states of the Ξatom. The population operators of two excited and ground states V̂2,j2, V̂1,j1, and V̂ι,jιcan be introduced for the three-level atom in Vconfiguration too. The similar expressions for two ground and one excited state can be introduced for Λthree-level atomic configuration Λ̂2,j2, Λ̂1,j1, and Λ̂ι,jι, respectively. The Datoms are considered as a two-level system, the state energy positions of which are measured from the middle point between the excited and ground states, respectively, Dz,j=D2,j2D1,j1/2. The first term of the Hamiltonian describes the free energy of EMF, the kk,λmodes of which is initially considered in the vacuum state 0k. Here âkand âkare annihilation and creation operators of EMF photons with wave vector k, polarization ελ, and the frequency ωk, which satisfy the commutation relation âkâk'=δk,k'.

Taking into consideration the conservation energy laws, ω1+ω2=2ω0and ωaωs=2ω0(according to Figure 1A–C, respectively), we introduce the interaction Hamiltonian ĤI=ĤI1+ĤI2of the Ξ, Λ, V, and Dsubsystems with free EMF. Here ĤI1describes the single-photon interaction of three-level atoms in the Ξ, V, and Λconfigurations with a vacuum of EMF:

ĤI1=kj=1Nξμ1ιgkΞ̂1jι+μ2ιgkΞ̂ιj2âkexpikrjkl=1Nλμι1gkΛ̂1lι+μι2gkΛ̂2lιâkexpikrlkl=1Nvμι1gkV̂ιl1+μι2gkV̂ιl2âkexpikrl+H.c.,E2

where ε1ĤI1Ξ1Ξ̂1jιâkand ε1ĤI1Ξ2=Ξ̂ιj2âkrepresent the two-photon cascade excitation of Ξatom through the intermediary state ι; ε1ĤI1SΛ̂2,jιâk(or ε1ĤI1SV̂ι,j2âk) and ε1ĤI1AΛ̂1,jιâk(ε1ĤI1AV̂ι,j1âk) describe the excitation of Λ(or V) atom with the absorption of the photons with the energies ωsand ωa, respectively. μi,jis dipole momentum transitions between the iand jstates of the atoms. The second part of interaction Hamiltonian, ĤI2, describes the nonlinear interaction of the dipole-forbidden transition of Dtwo-level system with vacuum field:

ĤI2=k1,k2m=1N[qsk1k2D̂mâk2âk11δk1,k2expik1k2rmqbk1k2D̂m+âk2âk1expik1+k2rm]+H.c.E3

This interaction is expressed by two-photon emission terms ε2ĤI2b+D̂mâk2âk1and possible scattering of an emitted photon by the Ξand Vsubsystems ε2HI2s±D̂mâk2âk1. The excitation and lowering operators of V, Λ, and Ξdipole-active three-level subsystems are described by the operators of U3algebra, which satisfy the commutation relations ÛβjαÛαlβ=δl,jÛαjαδβ,β+Ûβjβδα,α. Here the operator Ûβjαis equivalent with Vand Ξoperators, V̂βjαand Ξ̂βjα, respectively. The inversion D̂lztogether with lowering and exciting D̂j±operators of Dsubsystem belongs to SU2algebra: D̂lzD̂j±=±D̂l±δl,jand D̂l+D̂m+=2δl,mD̂lz. In comparison with single-photon interaction of Ξand Vatoms with vacuum field μi,jgk, the nonlinear interaction of Dtwo-level subsystem with EMF in two-photon and scattering interaction is described by the interaction constants and second order:

qbk1k2=d31gk1d32gk22ω32+ωk1+d31gk2d32gk12ω31ωk2,
qsk1k2=d31gk1d32gk2ω32ωk1+d31gk2d32gk1ω31+ωk1,

where gk=ελ2πωk/Vand di,jis dipole momentum transitions between the levels of the Datom. In the definition of the interaction parts of the Hamiltonian (2) and (3), we introduced the fictive small parameters ε1and ε2which will help us to establish the contributions of the second and third orders in two-photon decay rates.

In this section the conditions for which the pure super-fluorescence of the small number of radiators [14, 15] in the subsystems Ξ, V, and Denters into interaction during the delay time of cooperative spontaneous emission of each subsystem are considered, so that inhomogeneous broadening of excited atomic states can be neglected, τiT2,i. Here τi=τ0/Niis the collective time for which the polarization of the isubsystem becomes macroscopic; T2,iis the de-phasing time of the subsystem i, which includes the reciprocal inhomogeneous and Doppler-broadened line-width, iΞ, V, and D(see, e.g., the papers [15, 16]). These conditions can be achieved using laser cooling method [17, 18] for three atomic ensembles represented in Figure 1A,B. Let us suppose that delay time of the super-radiant pulse is less than T2,i; we will drop the terms connected with de-phasing time T2,ifrom the kinetic equations. In order to estimate the three-particle cooperative interaction, we will examine the situation in which one- and two-quantum interactions with the EMF bath are taken into account simultaneously. In this case it is necessary to eliminate from the density matrix equation the boson operators of EMF in nonlinear interaction with atomic subsystem. In comparison with the paper [12], here we will take into consideration the two-quantum effects connected with the influence of three-level atomic systems Vand Ξon the two-photon spontaneous emission of dipole-forbidden Dsubsystem. In this case instead of two dipole-active atoms, we can take into consideration only one three-level atom in two-photon resonance with dipole-forbidden system.

Let Pbe the projection operator for the complete density matrix ρton the vector basis of a free EMF subsystem ρst=Pρtand ρbt=P¯ρt, where ρstand ρbtare slower and rapidly oscillating parts of the density matrix, respectively, P¯=1P. It can be shown that P2=Pand P¯P=0. Recognizing that for t=0an electronic subsystem does not interact with the EMF, we define the projection operator P=ρph0Trph, where the trace is taking over the photon states and ρph0=00represents the density matrix of the vacuum of EMF. In this case one can represent the slow part of density matrix through the density matrix Wt=Trphρtof the atomic subsystem ρst=ρphWt, where W0=Trphρ0=ρr0is the density matrix of the prepared state of the atomic subsystem. The equations for the matrix ρstand ρbtare

ρstt=iPLItρst+ρbt,E4
ρbtt=iP¯LItρst+ρbt,E5

where L̂It=ε1HI1t/+ ε2HI2t/is the interaction part of Liouville operator. Following the known procedure of elimination of the rapidly oscillating part of the density matrix, we integrate Eq. (5) with respect to ρbtand substitute the resulting solution in Eq. (4). After this procedure we obtain the expression

ρstt=P0tLItUttτLItτρstτ,E6

where the two-time evolution operator is represented by the Tproduct Uttτ=TexpiP¯tτtdτ1LIτ. In comparison with well-known procedure of the decomposition on the small parameter εof the right-hand site of expression (6), here we have two parameters ε1and ε2. The quantum correlation between the single- and two-photon interactions of atoms through the vacuum of the EMF can be found in the third order of the expansion on the small parameter product ε12ε2of the right-hand side of Eq. (6). Indeed considering the second and third order of the expansion on the small parameters ε1and ε2, we represent the evolution operators Uttτand ρstτin the following approximate form Uttτ1iP¯tτtdτ1Liτ1and ρstτ=ρst+P0τdτ1L̂itτ10tτ1dτ2L̂itτ1τ2ρstτ1τ2. Upon substitution of this expression in Eq. (6), in the third order of small parameter λ, the equation for ρstbecomes

tρst=P0tdτ1L̂itL̂itτ1itτ1tdτ2L̂iτ2L̂itτ1ρst.E7

Representing the Liouville operator, L̂It, through single-, LI1t=ε1HI1t/, and two-photon, λLI2t=ε2HI2t/, interaction parts, we can observe that in the third order on the decomposition on interaction Hamiltonian, the main contribution to the right-hand site of Eq. (7) gives the terms proportional to the ε12ε2. Indeed, taking into consideration that the trace of an odd number of boson operator is zero, Trphρ0ak1ak2ak3ak4ak5=0, it is not difficult to observe that the projection of the operator product ε22ε1PHI1HI2HI2takes the zero value too. In the third order of the small parameters εi, the contribution of Liouville operator L̂I1and L̂I2must be found from the terms like PL̂I1L̂I2L̂I1ρ̂st, which corresponds to two-photon resonances between the single- and two-photon transitions in the three-level atomic systems described by the Hamiltonian part (2) and (3), respectively. It is not difficult to observe that second-order decomposition on the interaction Hamiltonian gives zero contributions in the correlations between the Ξ, V, and Dsubsystems. This follows from the zero value of the trace of the odd number of boson operators, Trphρ0ak1ak2ak3=0, which corresponds to the projection of the operator product PHI1HI2P=0.

Following this procedure of calculation of mean value of boson operators, it is observed that the two-photon resonance represented in Figure 1A can be described by the following diagrams:

Δρ3b=iλ30tdτ10τ1dτ2PL̂I1Ξ1tL̂̂I1Ξ2tτ2L̂I2b+tτ1ρst+PL̂I1RtL̂I1Stτ2L̂I2b+tτ1ρst+PL̂I1Ξ2tL̂I2b+tτ2L̂I1Ξ1tτ1ρst+PL̂I1Ξ1tL̂I2b+tτ2L̂I1Ξ2tτ1ρst+PL̂I2b+tL̂I1Ξ1tτ2L̂I1Ξ2tτ1ρst+PL̂I2b+tL̂I1Ξ2tτ2L̂I1Ξ1tτ1ρst+H.c.E8

Here LI1Ξ1t=ε1HI1Ξ1t/, LI1Ξ2t=ε1HI2Ξ2t/, and LI2bt=ε2HI2bt/represent the Liouville operators of the interaction part of the Ξand Datoms expressed through EMF annihilation and atomic exciting operators in the single- and two-quantum interactions.

The scattering resonance can be represented by the diagrams in which the conservation law ωaωs=2ω0must take place as represented in Figure 1B:

Δρ3s=i0tdτ10τ1dτ2PLI1AtLI1S+tτ2LI2s+(tτ1)ρst+PLI1S+tLI1Atτ2LI2s+tτ1ρst+PLI1AtLI2s+tτ1LI1S+tτ2ρst+PLI1S+tLI2s+tτ1LI1Atτ2ρst+PLI2s+tLI1Atτ1LI1S+tτ2ρst+PLI2s+tLI1S+tτ1LI1Atτ2ρst}+H.c.,E9

where LI2st=ε2HI2st/is the Liouville parts for two-photon scarpering process of Datomic subsystem and LI1St=ε1HI1St/and LI1At=ε1HI1At/correspond to the single-photon transitions in Ξatomic subsystem described by the Hamiltonian parts (3) and (2), respectively.

So that after the trace on the EMF variables, we obtain Trρ̂phak1ak3ak2ak4=δk1k2δk3,k4+δk1,k4δk3,k2, Trρ̂phak1ak2=δk1k2, and Trρphak2ak4ak1ak3=0. We found the correlations between Ξ, V, and Datomic subsystem represented in Figure 1.

We found the correlations between Ξ, V, and Datomic subsystem represented in the Figure 1. Following projection technique procedures developed in Refs. [5, 13, 19], we find the terms of in the right-hand side of the master equation (7)–(9) for three species of radiators in interaction

dWtdt=dW0tdt+dW21btdt+dW21stdt.E10

First term describes the cooperative single- and two-photon effects in each subsystem, respectively. Second term describes the exchanges between the single-photon processes of Ξthree-level subsystem and the two-photon transitions of the Dradiators as this is represented in Figure 1A. The third term describes the scattering effect of the two radiators represented in Figure 1B.

All parameters and collective exchange integrals between the three-level radiators in Vconfiguration and dipole-forbidden two-level system Dare defined in the literature [1–12]:

dW0tdt=12τι,1l,j=1Nξχ1jlΞι,j1WtΞ1,lι+12τι,2l,j=1Nξχ2jlΞ2,jιWtΞι,l2+12τι,al,j=1NvχajlV1,jιWtVι,l1+12τι,sl,j=1NvχsjlV2,jιWtVι,l2+12τι,sl,j=1NλχsjlΛι,j2WtΛ2,lι+12τι,al,j=1NλχajlΛι,j2WtΛ2,lι+12τdl,j=1NχdjlDjWtDl++H.c.,E11

where τι,α=3c3/4μα,ι2ωα3is the spontaneous emission time of the dipole-active transitions αιof three-level atom in Ξand Vconfigurations and τd=π322c6/42ω07d232d312qb2ω0ω0is the two-photon spontaneous emission rate in the Datomic subsystem. This equation can be used for the description of interaction between the dipole-forbidden and dipole-active subsystems of radiators. For comparison of the real parts of the single- and two-photon exchange integrals, we can observe that the second decreases inversely proportional to the square distance rJlbetween two Dradiators: Reχαjl= sinωαrj,l/c/ωαrj,l/cand Reχdjlsin2ω0rj,l/c/ω0rj,l/c2.

Following the two-parameter approach projection technique proposed in Ref. [13], HI1ε1and HI2ε2, we easily found the three-particle exchanges between the radiators represented in Figure 1A described by master equation

dW21btdt=i4τ12dbm=1Nl=1Nξj=1NξUbml×DmWtΞι,l2Ξ1,jι+DmWtΞ1,jιΞι,l2+Ubjlm{Ξι,l2Ξ1,jιDmWt+[Ξ1,jι[Ξι,l2DmWt]]}i2τ12dbm=1Nj=1Nξl=1NξVbjml×DmWtΞιl2Ξ1,jι+DmWtΞ1,jιΞι,l2+H.c.E12

Here for ωsωr, we have found the following integrals:

1τ12db=432ωs3ωr3μι2μι1d23d3122c61ω32+ω2+1ω31+ω1,Vbjmlc2expiω2rml/c1expiω1rjm/c1ω1ω2rjmrml,Ubjml=expiω1rmj/cVbjlm.E13

Here 1/τ12dbis the three-particle cooperative emission rate of two atoms from Ξsubsystems and one atom from Densemble situated at the relatively small distance rjlλsr. Vbjmlis the exchange integral which describes the influence of the matom from Densemble on the single-photon transitions of the jand lradiators from the Ξsubsystem. Ubjmlis the inverse process of the cooperative action of jand lradiators from the Ξensemble on the two-photon transitions of mradiator from the Dsubsystem.

For two atoms represented in Figure 1A, the simple exchange integral between these radiators can be obtained from expression (13):

Vb=λ1λ2exp2iπr/λ21exp2iπr/λ112πr2,E14

where λ2and λ1are the emission wavelengths in cascade transition of the dipole-active three radiators in Ξconfiguration, situated at distance r. The real part of this function describes the three-particle decay rate of the system. The dependence of exchange integral (14) on the relative distance between the Ξand Datoms (14), X=ω0r/cand the displacement, Δ=ω1ω0/ω0relatively the degenerate frequency ω0, is plotted in Figure 2. As follows from this dependence, the exchange integral achieved the maximal radius, when ω1=ω2, which corresponds to the situation Δ=0.

The part of master Equation (10) for resonance scattering interaction between the absorbed and emitted photons by the dipole-active Λand Vsubsystems and Ddipole-forbidden radiators can be obtained from the third-order expansion on the smallest parameter λ. In this situation, the scattering part of the master equation represented by the scheme 1 Bbecomes

where

First term in Eq. (15) describes the transition of Datom under the influence of the scattering process of emitted photons of the atoms from Vsubsystem. This process is described by exchange integral Vsjm.l. The last two terms in master Eq. (15) describe the scattering process of emitted photons by the Vatoms under the influence of Dsubsystem.

The similar expression is obtained for the interaction of Λthree-level radiator with Datom represented in Figure 1C. In this case we must replace the operators of Vsubsystem in expression (15) with corresponding transition operators of Λsystem Vι,j1Λ̂1jι: V2,lιΛι,l2and their Hermit conjugated operators.

For the two atoms, expression (16) was reduced to the simple representation

Vs=λsλa1exp2iπr/λs1exp2iπr/λa2πr2.E17

Here the wavelength λs(λa) corresponds to the emitted photons at Stokes or anti-Stokes frequencies represented in Figure 1. The numerical representation of the real part of the exchange integral (17) as the function of the relive distance between the atoms X=ω0r/cand the relative Stokes frequency ωs/ω0is plotted in the Figure 3. It is observing the nonsignificant dependence of this exchange integral on the frequency ωs. The significant dependence on the detuning from resonance can be observed in the dependence of cooperative rate 1/τsadsrepresented in expressions (16).

In this section we obtained the correlations between dipole-active and dipole-forbidden subsystems of radiators, where the two-quantum exchange integral has the same magnitude as the two-photon quantum interaction between atoms of Dsubsystem. In the case of the big number of radiators in each subsystem, the correlated terms, expressions (12) and (15), give the cubic contribution in the cooperative diagrams of the kinetic equation ε12ε2NNξ2. When N=Nξthese terms can archived the value proportional to the Dicke super-radiance [1] even for the same small parameters of each subsystem ε1=ε2. In this case the number of atoms in each subsystem must achieve the value for which the third order has the same magnitude as the second order ε2N2ε3N3. In conclusion we observe that the decomposition on the small parameter εcan be regarded as a sum of single- and the two-photon transition amplitudes proportional to ε1and ε2, where ε1μ1ιgkand ε2qbk1k2or qsk1k2. Considering the situation when the two-photon amplitude is smaller than the single-photon amplitude ε2<ε1, we conclude that beginning with the third-order term, the correlation diagrams (12) and (15), proportional to ε12ε2, can play an important role in the two-quantum decay process even for the two-atomic system consisted from one atom of each subsystems: Dand Ξ(or Dand V). For example, in the situation when ε1=0.7and ε2=0.25, the magnitude of two-photon emission, ε22=0.0625, becomes smaller than the cooperative magnitude ε12ε2=0.1225). In other words we can find the condition for which we can neglect the decay rate of two-photon emission of the Datom in comparison with the cooperative effect described by expressions (12) and (15). This possibility to control the two-photon decay process of Datom with the decay process of Ξor Vexcited three-level atom is given in the next section.

## 3. Two-photon energy transfer between the two three-level radiators

Master Eq. (10) can be used for the description of cooperative interaction between the dipole-forbidden and dipole-active radiators in two-photon exchanges. Indeed passing again from Schrodinger to Heisenberg pictures TrŴtÔ0=TrŴ0Ôt, we can obtain from this expression the equation of the arbitrary atomic operator Ôt. Let us firstly discuss the nonlinear interaction in which Ξand Datoms enter in two-photon resonance as represented in Figure 1A. Studying the cooperative interaction between the dipole-forbidden and dipole-active radiators, the closed system of equations for the correlation functions can be found in such approach. Considering that the numbers of atoms in the each subsystem are relatively small, we can obtain the following generalized equation for the arbitrary operator Ob:

dObtdt=12τι,1l,j=1Nξχ1jlΞ̂1,lιtÔbtΞ̂ι,j1t+12τι,2l,j=1Nξχ2jlΞ̂ι,l2tÔbtΞ̂2,jιt+12τdl,j=1NχdjlD̂l+tÔbtD̂jti4τ12dbm=1Nl=1Nξj=1Nξ{Ubmlj[Ξ̂ι,l2tΞ̂1,jιtÔbtD̂mt+Ξ̂1jιtΞ̂ιl2tÔbtD̂mt]+Ubjlm{ÔbtΞ̂1,jιtΞ̂ι,l2t+ÔbtΞ̂ι,l2tΞ̂1,jιt}D̂m}i2τ12dbm=1Nj=1Nξl=1NξVbjmlΞ̂1,jιt[Ξ̂ι,l2tÔbt]D̂mt+Ξ̂ι,l2t[Ξ̂1,jιtÔbt]D̂mt+H.c.E18

In order to simplify this problem, we analyze below the situation in which we have only a single atom in each subsystem. In this case we can replace the operator Obwith the excitation numbers operators N̂α=Ξ̂ααtand N̂d=D̂z+0.5of Ξand Datoms, respectively. Here α=1,2and ι. When emission frequencies of the one-photon radiators coincide with ω1ω2ω0, the dependence (14) becomes real and positive defined function Ξand Dradiators. Here expiω0r/c1[expiω0r/c1=21cosω0r/c. According to this expression, the exchange integrals become

V12db=2(1cos2πr/λ02πr/λ02,U12db=expiω0r/cVsrd.

In this case one can introduce the expression exchange rate 1/τsrdbas a function of the distance between the dipole-active and dipole-forbidden subsystems:

1τ12dbr=1τsrdb2(1cos2πr/λs2πr/λs2,E19

where λ0=c/2πω0. Taking into account the above definitions and introducing the correlation functions between the polarizations of Ξand Datoms F̂btr=iΞ̂12tD̂tD̂+tΞ̂21tand Êbtx=Ξ̂12tD̂t+D̂+tΞ̂21t, we obtain the closed system of equations from expression (18):

ddtN̂2tx=N̂2tτι,214τ12dbxcosxF̂btxsinxÊtx,ddtN̂ιtx=N̂2txτι,2N̂ιtxτι,1+12τ12dbx[cosxF̂btxsinxÊtx]+F̂btx2τ12dbx,ddtN̂1tx=N̂ιtτι,114τ12dbxcosxF̂btxsinxÊbtx12τ12dbxF̂btx;ddtN̂dtx=N̂dtxτd+14τ12dbxcosxF̂btx+sinxÊbtx,ddtF̂btx=F̂btx21τd+1τι,212τ12dbx{cosx[2N̂2txN̂dtN2txN̂dtN̂itN̂2t+N̂dt(1N̂2t2N̂it]2N̂dtN̂it+2N̂dtN̂2t};dÊbtxdt=Êbtx21τd+1τι,212τ12dbx{sinx[2N̂2tN̂dtN̂2t+N̂dtN̂itN̂2tN̂dt1N̂2t2N̂it];ddtN̂2tN̂dt=N̂2tN̂dt1τι,2+1τd,ddtN̂ιtN̂dt=N̂2tN̂dtτι,2N̂ιtN̂dt1τι,1+1τd.E20

Using this system of Eq. (20), we can numerically study the cooperative nonlinear exchanges through the vacuum field between the Ξand Dradiators situated at relative distance x. One can observe that the spontaneous generation of photon pair by the Datom is drastically modified by the time increase of the cooperative correlation between the radiators. Indeed considering that the decay rate of the Datom 1/τdis smaller than similar rates of the cascade transition in the Ξatom (τd/τξ,i6; τd/4τ12d=2), we can numerically represent this dependence as a function of the relative time, t/τd, and the relative distance between the radiators, x=2πr/λ0. As shown in Figure 4A, the decay rate of Datom is drastically modified at small distances between the radiators which is in accordance with the analytic expressions (19). Considering that both atoms Ξand Dare prepared in the excited state, we observe the significant enhancement of the two-photon emission rate of the Dradiator under the influence of the Ξdecay process.

Let us simplify the system of Eq. (20) in order to solve it exactly. Indeed, when dipole-active Ξatom is situated at small distance relative to the Dradiator (x1), the system of Eq. (20) is drastically simplified:

ddtN̂2t=N̂2tτι,2F̂bt4τ12db,ddtN̂dt=N̂dtτd+14τ12dbF̂bt,ddtF̂bt=F̂bt21τd+1τι,212τ12db[4N̂2tN̂dt+N̂dtN̂2t5N̂dN̂i],ddtN̂2tN̂dt=N̂2tN̂dt1τι,2+1τd,ddtN̂ιtN̂dt=N̂2tN̂dtτι,2N̂ιtN̂dt1τι,1+1τd.E21

The exact solution of this linear system of equation can be represented through solution of characteristic equation Yα=j=15CαjexpΘjt, where α=1,2,3,4,5and Yαare the atomic functions, Y1t=N̂dt, Y2t=N̂2t, Y3t=F̂bt, Y4t=N̂2tN̂dt, and Y5t=N̂ιtN̂dt; the solution of characteristic equation is

Θ1=1τ2+1τd;Θ2=1τ1+1τd;Θ3=121τd+1τι,2;Θ4,5=121τι,2+1τd±1τd1τι,221τ12b2.E22

The coefficients Cαjare determined from the initial conditions. As follows from the numerical estimation plotted in Figure 4B and solutions of characteristic in Eq. (22), the oscillatory decay of the atomic inversion is possible, when 1/τd=1/τι,2. In this case the solutions Θ4,5become complex. We observe such an oscillation of the atoms inversion of Ξradiator prepared initially in the excited state. In this process the rate of energy transfer from Ξto Datoms represented in Figure 4B has the oscillator behavior. In the case of the excitation of D, the coupling between the radiators becomes more effective, when the virtual level of the Datom is situated between the excited and ground states (see Figure 4B). As the virtual states of the Dradiator is off from the resonance with the dipole-active transitions of the Ξradiators, the excitation of Datom takes place only with the absorption of both emitted photons by the Ξatom. The cooperative effects between the Ξand Dradiators are described by second-order correlation function G2=ÊtÊtÊ+tÊ+t=G20+αF̂bt. Here G20was derived in Ref. [5]. The contribution to the second-order correlation function remains larger than the square value of the first-order correlation function G1=EtE+t, so that we can conclude that new cooperative effects between single- and two-photon transitions of Dand Ξsubsystems play an important role in the two-photon decay process. Let us now return to the Vthree-level system in scattering interaction with the Dsystem as this is represented in Figure 1B. In accordance with master Eq. (10) and its analytic representation (15), we can obtain the following expression for arbitrary atomic operators Ôst.

The similar expression can be obtained for a Λthree-level system in interaction with Dradiators, doing the substitution V̂α,jβΛβjα. For two atoms in each subsystem, an attractive peculiarity follows from this substitution. If Ostis the inversion of the Datom, the direct modification of the Datomic excitation by Λthree-level atom is equal to zero Λ̂1,lιtΛ̂ι,l2tN̂dtD̂mt=0due to the operator product Λ̂1,lιtΛ̂ι,l2t=0for the same atom. In order to obtain the closed system of equation from master Eqs. (15)and (23), we consider the simple interaction of two atoms in the scattering process represented by the analytical scheme of Figure 1B. In this case we introduce the new indexes "s"and "a"instead of "1"and "2", which correspond to the Stokes and anti-Stokes scattering frequencies ωsand ωa. Considering that the anti-Stokes frequency ωais larger than Stokes ωs, one can approximate the exchange integrals (17) with expression

Vssinxaxa+i1cosxaxa.E24

Here xa=ωar/c. The mean values of the operators N̂s=V̂22, N̂a=V̂11, and N̂dare considered the populations of excited states of Vand Dradiators, respectively. The functions F̂stxa=iV̂21tD̂tD̂+V̂12t, Êstxa=V̂21tD̂tD̂+tV̂12t, N̂dN̂s, and N̂dN̂adescribe the polarization and population correlations between the atoms Ξand D. For this two-atom system, we can obtain the following closed system of equations from generalized equation (23).

As follows from the system (25), and numerical simulation plotted in Figure 5 the first N̂d/τdand second terms 1/τsadsF̂sdescribe the generation rate of entangled photon pairs and scattering rate with absorption of Stokes photon and generation of two anti-Stokes photons by the system formed from Vand Datoms. When the time tends to infinity, all excited atomic energies E0=ωa+ωs+ωdof three-level Vand two-level Datoms are emitted by the system. Taking into account the conservation law in the scattering process ωaωsωd=0, we observe that this cooperation between the atoms becomes predominant, when the collective scattering rate 1/τsadsincreases. In other words, the probability of absorption of Stokes photon ωswhich is accompanied with the generation of the new anti-Stokes photon ωaby Datom becomes possible. In this case two atoms represented in the Figure 1B can generate an entangled anti-Stokes photons with energy E0=2ωa. The possibility of the excitation transfer between the atoms Ξand Drepresented in Figure 4B can be found in the special preparation of the system.

We can conclude that it is possible to study all cooperations two-photon process between single atoms in each system represented in Figure 1A–C. For example, the system of Eqs. (20) and (25) can be solved simultaneously taking into consideration scattering and two-photon transitions. In this case the effective energy transfer of the excitation between the atoms Ξ, V, and Dradiator prepared in the special initial states can open the new possibilities of non-resonance interaction between the atomic subsystems.

## 4. Conclusions

This chapter proposed the cooperative effects between three-level system and dipole-forbidden two-level systems in nonlinear interaction through the vacuum field during the spontaneous emission time. The possibility of cooperative migration of energy from one excited dipole-active three-level atom to another takes place with phase retardation effects and depends on the position of atoms in the system. This excitation transfer from dipole-active to dipole-forbidden subsystems takes place with phase dependence amplitudes, so that the cooperative excitation of the system consisted from two species of atoms depends on the retardation of radiation along the sample and geometry of the system. This follows from the excited or ground state of one of the radiators represented in Figures 4 and 5. As in Ref. [20], the exchanges between the Ξ(or V) three-level atom and Dtake place with the absorption and emission of two quanta, but in this chapter, we take into consideration the real and imaginary parts of exchange integrals. In this case, two correlation functions introduced functions F̂bstxaand Êbstxa, which modify the dynamics of possible excitation of Datoms by Ξand Vradiators. The scattering transfer of the energy between the excited state of Vthree-level radiator and dipole-forbidden transitions of Dtwo-level atoms are effective when the dipole-forbidden atom enters in the two-photon resonance with the energy difference between the two dipole transitions (Figures 1A and 5A). When the atom Dis in the excited state, the emitted Stokes photon by one atom of the Vsystems can be absorbed by another radiator from the Dsubsystem, so that two radiators pass into the ground state generating two anti-Stokes photons with energies E0=2ωa. The opposite situation can be observed when Datom is prepared in the ground state.

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Nicolae A. Enaki (May 8th 2019). Cooperative Spontaneous Lasing and Possible Quantum Retardation Effects, Panorama of Contemporary Quantum Mechanics - Concepts and Applications, Tuong T. Truong, IntechOpen, DOI: 10.5772/intechopen.83013. Available from:

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