## 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

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

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

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

## 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

The similar cooperative emissions can be observed in the two-quantum resonance interactions between the

The Hamiltonian of the system consists of the free and interaction parts

where

Taking into consideration the conservation energy laws,

where

This interaction is expressed by two-photon emission terms

where

In this section the conditions for which the pure super-fluorescence of the small number of radiators [14, 15] in the subsystems

Let

where

where the two-time evolution operator is represented by the

Representing the Liouville operator,

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:

Here

The scattering resonance can be represented by the diagrams in which the conservation law

where

So that after the trace on the EMF variables, we obtain

We found the correlations between

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

All parameters and collective exchange integrals between the three-level radiators in

where

Following the two-parameter approach projection technique proposed in Ref. [13],

Here for

Here

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

where

The part of master Equation (10) for resonance scattering interaction between the absorbed and emitted photons by the dipole-active

where

First term in Eq. (15) describes the transition of

The similar expression is obtained for the interaction of

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

Here the wavelength

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

## 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

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

In this case one can introduce the expression exchange rate

where

Using this system of Eq. (20), we can numerically study the cooperative nonlinear exchanges through the vacuum field between the

Let us simplify the system of Eq. (20) in order to solve it exactly. Indeed, when dipole-active

The exact solution of this linear system of equation can be represented through solution of characteristic equation

The coefficients

The similar expression can be obtained for a

Here

As follows from the system (25), and numerical simulation plotted in Figure 5 the first

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

## 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