Cooperative Spontaneous Lasing and Possible Quantum Retardation Effects

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 3p3P→3s3Sat 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 2ξ→ιξ→1ξat frequencies ω2and ω1(Figure 1A). According to Figure 1A, the excited Datom relatively the dipole-forbidden transition 2d→1dpasses in the ground state ∣1d⟩simultaneously 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 ∣3d⟩with opposite parity relative to the ground ∣1d⟩and excited ∣2d⟩states, 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 ℏωd=2ℏω0is the energy distance between the ground ∣nS⟩and excited ∣n+1S⟩states 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 ωa−ωs=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+1S⟩−∣nS⟩of atoms D, so that two-photon resonance between the two dipole-active transitions of Vatom enters in the exact scattering resonance, ωa−ωs=ωd, with Datom. This nonlinear transition increases with the decreasing of the detuning from resonance with virtual ∣3d⟩states 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:

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,j2−D1,j1/2. The first term of the Hamiltonian describes the free energy of EMF, the k≡k,λmodes of which is initially considered in the vacuum state ∣0k⟩. Here âkand âk†are annihilation and creation operators of EMF photons with wave vector k, polarization ελ, and the frequency ωk, which satisfy the commutation relation âk†âk'†=δk,k'.

Taking into consideration the conservation energy laws, ℏω1+ω2=2ℏω0and ℏωa−ωs=2ℏω0(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:

where ε1ĤI1Ξ1−∼Ξ̂1jιâkand ε1ĤI1Ξ2=∼Ξ̂ιj2âkrepresent the two-photon cascade excitation of Ξatom through the intermediary state ∣ι⟩; ε1ĤI1S−∼Λ̂2,jιâk(or ε1ĤI1S−∼V̂ι,j2âk) and ε1ĤI1A−∼Λ̂1,jιâk(ε1ĤI1A−∼V̂ι,j1âk) describe the excitation of Λ(or V) atom with the absorption of the photons with the energies ℏωsand ℏωa, 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:

This interaction is expressed by two-photon emission terms ε2ĤI2b+∼D̂m−âk2†âk1†and possible scattering of an emitted photon by the Ξand Vsubsystems ε2H⌣I2s±∼D̂m∓âk2†âk1. 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 ÛβjαÛα′lβ′=δl,jÛα′jαδβ,β′+Ûβ′jβδα,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:

where gk=ελ2πℏωk/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, τi≪T2,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, i≡Ξ, 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 ρ⌣ton the vector basis of a free EMF subsystem ρst=Pρ⌣tand ρ⌣bt=P¯ρ⌣t, where ρ⌣stand ρ⌣btare slower and rapidly oscillating parts of the density matrix, respectively, P¯=1−P. 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=ρ⌣ph0⊗Trph⋯, where the trace is taking over the photon states and ρ⌣ph0=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 W⌣t=Trphρ⌣tof the atomic subsystem ρ⌣st=ρ⌣ph⊗W⌣t, where W⌣0=Trphρ⌣0=ρ⌣r0is the density matrix of the prepared state of the atomic subsystem. The equations for the matrix ρ⌣stand ρ⌣btare

where L̂It=ε1H⌣I1t…/ℏ+ ε2H⌣I2t…/ℏis 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 ρ⌣btand substitute the resulting solution in Eq. (4). After this procedure we obtain the expression

where the two-time evolution operator is represented by the Tproduct U⌣tt−τ=Texp−iP¯∫t−τtdτ1LIτ. 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 U⌣tt−τand ρ⌣st−τin the following approximate form U⌣tt−τ≈1−iP¯∫t−τtdτ1Liτ1and ρ⌣st−τ=ρ⌣st+P∫0τdτ1L̂it−τ1∫0t−τ1dτ2L̂it−τ1−τ2ρ⌣st−τ1−τ2. Upon substitution of this expression in Eq. (6), in the third order of small parameter λ, the equation for ρstbecomes

Representing the Liouville operator, L̂It, through single-, LI1t=ε1H⌣I1t…/ℏ, and two-photon, λLI2t=ε2H⌣I2t…/ℏ, 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, Trphρ0a⌣k1†a⌣k2†a⌣k3a⌣k4a⌣k5=0, it is not difficult to observe that the projection of the operator product ε22ε1PH⌣I1H⌣I2H⌣I2takes 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 PL̂I1L̂I2L̂I1ρ⌣̂st, 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, Trphρ⌣0a⌣k1†a⌣k2a⌣k3†=0, which corresponds to the projection of the operator product PH⌣I1H⌣I2P=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:

Here LI1Ξ1−t=ε1H⌣I1Ξ1−t…/ℏ, LI1Ξ2−t=ε1H⌣I2Ξ2−t…/ℏ, and LI2b−t=ε2H⌣I2b−t…/ℏrepresent 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 ωa−ωs=2ω0must take place as represented in Figure 1B:

where LI2s−t=ε2H⌣I2s−t…/ℏis the Liouville parts for two-photon scarpering process of Datomic subsystem and LI1S−t=ε1H⌣I1S−t…/ℏand LI1A−t=ε1H⌣I1A−t…/ℏcorrespond 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 Trρ̂pha⌣k1a⌣k3a⌣k2†a⌣k4†=δk1k2δk3,k4+δk1,k4δk3,k2, Trρ̂pha⌣k1a⌣k2†=δk1k2, and Trρpha⌣k2†a⌣k4†a⌣k1a⌣k3=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

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]:

where τι,α=3ℏc3/4μα,ι2ωα3is the spontaneous emission time of the dipole-active transitions ∣α⟩→∣ι⟩of three-level atom in Ξand Vconfigurations and τd=π32ℏ2c6/42ω07d232d312qb2ω0ω0is 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 Reχdjl∼sin2ω0rj,l/c/ω0rj,l/c2.

Following the two-parameter approach projection technique proposed in Ref. [13], HI1∼ε1and HI2∼ε2, we easily found the three-particle exchanges between the radiators represented in Figure 1A described by master equation

Here for ωs≃ωr, we have found the following integrals:

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 rjl≪λsr. 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):

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=ω1−ω0/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 V⌣ι,j1→Λ̂1jι: V⌣2,lι→Λ⌣ι,l2and their Hermit conjugated operators.

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

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 ε2N2∼ε3N3. 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 ε1∼μ1ιgkand ε2∼qbk1k2or 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:

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 ω1≃ω2≃ω0, the dependence (14) becomes real and positive defined function Ξand Dradiators. Here expiω0r/c−1[exp−iω0r/c−1=21−cosω0r/c. According to this expression, the exchange integrals become

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:

where λ0=c/2πω0. Taking into account the above definitions and introducing the correlation functions between the polarizations of Ξand Datoms F̂btr=iΞ̂12tD̂−t−D̂+tΞ̂21tand Êbtx=Ξ̂12tD̂−t+D̂+tΞ̂21t, we obtain the closed system of equations from expression (18):