interference-induced enhancement of ﬁeld entanglement in a...

Cent. Eur. J. Phys. • 12(10) • 2014 • 737-743DOI: 10.2478/s11534-014-0510-7

Central European Journal of Physics

Interference-induced enhancement of fieldentanglement in a microwave-driven V-typesingle-atom laser

Research Article

Wen-Xing Yang12∗, Ai-Xi Chen3, Ting-Ting Zha1, Yanfeng Bai1, Ray-Kuang Lee2

1 Department of Physics, Southeast University, Nanjing 210096, China,

2 Institute of Photonics Technologies, National Tsing-Hua University, Hsinchu 300, Taiwan

3 Department of Applied Physics, School of Basic Science, East China Jiaotong University, Nanchang 330013, China

Received 14 April 2014; accepted 18 June 2014

Abstract: We demonstrate the generation of two-mode continuous-variable (CV) entanglement in a V-type three-levelatom trapped in a doubly resonant cavity using a microwave field driving a hyperfine transition between twoupper excited states. By numerically simulating the dynamics of this system, our results show that the CVentanglement with large mean number of photons can be generated even in presence of the atomic relax-ation and cavity losses. More interestingly, it is found that the intensity and period of entanglement can beenhanced significantly with the increasing of the atomic relaxation due to the existence of the perfect spon-taneously generated interference between two atomic decay channels. Moreover, we also show that theentanglement can be controlled efficiently by tuning the intensity of spontaneously generated interferenceand the detuning of the cavity field.

PACS (2008): 42.50.Dv, 03.67.Mn

Keywords: continuous-variable entanglement • quantum interference • quantum information© Versita sp. z o.o.

1. Introduction

Entanglement is one of the most counter-intuitive aspectsof the quantum world. The concept has come to the forein recent years with the advent of quantum informationscience. The generation and manipulation of entangle-ment has attracted a great deal of interest owing to theirwide applications in quantum computation and quantum∗E-mail: [email protected]

communication [1–13]. In particular, due to the simplicityand high efficiency in their generation, manipulation, anddetection, optical continuous-variable (CV) states providean alternative but powerful source in quantum informa-tion processing [14–18]. In order to check whether astate is entangled or not, several inseparability criteriafor continuous-variable systems were proposed by Duan,Giedke, Cirac, and Zoller (DGCZ) [19], Simon [20], andHillery and Zubairy (HZ) [21] in the form of inequalities.A violation of these inequalities provides evidence of en-tanglement for a continuous-variable system. Recently,based on these entanglement conditions, more and more737

Interference-induced enhancement of field entanglement in a microwave-driven V-type single-atom laser

theoretical and experimental efforts have been devoted tothe generation and measurement of CV entanglement [22–29].Conventionally, CV entanglement has been producedby nondegenerate parametric down-conversion (NPD) innonlinear crystals [14, 30]. In order to improve the strengthof the NPD, engineering the NPD Hamiltonian withincavity QED has also attracted much attention [28, 29, 31–39]. For example, Xiong et al. [28] proposed a scheme foran entanglement amplifier based on two-mode correlatedspontaneous emission lasers in a nondegenerate three-level cascade atomic system, whereby the atom coherenceis induced by pumping the atom from the lower level tothe upper level. Li et al. [31] considered the genera-tion of two-mode entangled states in a cavity field via thefour-wave mixing process, by means of the interaction ofproperly driven V-type three-level atoms with two cav-ity modes. Subsequently, Tan et al. [32] extended aboveconcept and studied the generation and evolution of en-tangled light by taking into account the spontaneouslygenerated interference between two atomic decay chan-nels. In an earlier study, Qamar et al. [33] proposed ascheme to generate two-mode entangled states in a quan-tum beat laser [34], which consists of a V-type three-levelatom interacting with two modes of the cavity field in adoubly resonant cavity. Related properties of entangle-ment for different values of Rabi frequencies in the pres-ence of cavity losses are also investigated. Fang et al.[35] added into this scheme the effects of phase shift andRabi frequency in the classical driving field, cavity loss,and the purity and nonclassicality of the initial state ofthe cavity field [34].In the present paper, we propose an efficient scheme forthe generation of CV entanglement in a V-type three-levelatom trapped in a doubly resonant cavity. In our scheme,two transitions between the ground state and two excitedstates of the V-type atom are driven by the two cavitymodes independently, and the two excited states of theatom are driven by a strong microwave field. Using theentanglement criterion proposed by DGCZ [19], we showthat CV entanglement with large mean number of photonscan be obtained in the present system. By numericallysimulating the evolution of the CV entanglement, we alsoshow that the intensity and period of entanglement canbe amplified significantly by increasing the atomic relax-ation due to the existence of the perfect spontaneouslygenerated interference between two atomic decay chan-nels. We also analyze the effect of the strength of the

spontaneously generated interference and the detuning ofthe cavity field on the evolution of entanglement.

∆2

Ω

∆1

|c⟩

|b⟩

|a⟩

g1

g2

Figure 1. Schematic diagram of the three-level atom system in aV-type configuration. Two (nondegenerate) cavity modeswith the coupling constant g1 and g2 interact with the tran-sition |a〉 ↔ |c〉 and |b〉 ↔ |c〉, respectively, while the atomtransition |a〉 ↔ |b〉 is driven by an assisting microwavedriving field with Lamor frequency Ω. Here, ∆1 and ∆2correspond to the frequency detunings.

2. Model and master equationLet us consider a three-level V-type atom trapped in adoubly resonant cavity. The atom level configuration isdepicted in Fig. 1. Assume the atom is pumped at a rateγa into the level |a〉. Two nondegenerate cavity modeswith frequencies ν1 and ν2 independently interact with thetransitions |a〉 ↔ |c〉 with the resonant frequency ωac and|b〉 ↔ |c〉 with the resonant frequency ωbc , respectively.An assisting microwave driving field with Lamor frequencyΩ is used to couple Zeeman sublevels |a〉 and |b〉 throughan allowed magnetic dipole transition and to further forma closed-loop configuration. The detuning of two cavitymodes from the atomic transitions |a〉 ↔ |c〉 and |b〉 ↔|c〉 are denoted as ∆1 = ωac − ν1 and ∆2 = ωbc − ν2,respectively. For simplicity, in the following calculationswe will note ∆1 = ∆2 = ∆.In the interaction picture, with the rotating-wave approxi-mation and the electric-dipole approximation, the Hamil-tonian describing the atom-field interaction for the systemunder investigation can be written as (taking h = 1) [40–43]

VI = ∆ |a〉〈a|+ ∆ |b〉〈b|+ (−Ω |a〉〈b|+ g1a1 |a〉〈c|+ g2a2 |b〉〈c|+H.c.), (1)738

Wen-Xing Yang, Ai-Xi Chen, Ting-Ting Zha, Yanfeng Bai, Ray-Kuang Lee

where the symbol H.c. denotes the Hermitian conju-gate. In obtaining the Hamiltonian (1), we have taken theground state |c〉 as the energy origin. Ω = |Ω| exp (iφ)describes the Lamor frequency of the microwave drivingfield with φ the phase of the field. The parameters g1 andg2 are used for describing the corresponding atom−fieldcoupling intensities. aj (a†j ) is the annihilation (creation)operator of the corresponding cavity modes.Considering the vacuum damping of the atom and the cav-ity modes, the reduced density equations of the cavityfields can be obtained from the Hamiltonian (1) by takinga trace over the atom degrees of freedom [44],ρf = −iT ratom[VI , ρ] + T ratom(Lfρ + Laρ)

= −ig1[a†1 , ρac ]− ig2[a†2 , ρbc ] + 2∑j=1 κj [aj , ρfa†j ] +H.c.,

(2)with

Lfρ = 2∑j=1 κj [aj , ρa†j ] +H.c., (3)

Laρ =γ1[sca, ρsac ] + γ2[scb, ρsbc ] + γ12([sca, ρsbc ]+ [scb, ρsac ]) +H.c., (4)where Lfρ, Laρ are the damping of cavity modes andatomic excitation, respectively. In the above equations,γ1 and γ2 are used to describe the decay rates from theexcited states |a〉 and |b〉 to the ground state |c〉, respec-tively. The parameter κj (j = 1, 2) denotes the dampingconstants of two cavity modes. It should be noted thatγ12 = P√γ1γ2 describes the spontaneously generatedinterference effect, which comes from the cross couplingbetween the transitions |a〉 ↔ |c〉 and |b〉 ↔ |c〉. The pa-rameter P = −→bac ·−→bbc/(∣∣∣−→bac∣∣∣ · ∣∣∣−→bbc∣∣∣) = cosθ represents thestrength of spontaneously generated interference, wherethe values P = 0 and P = 1 correspond to no interfer-ence and perfect interference, respectively. Here, −→bac and−→bbc represent the atomic dipole polarizations and θ is theangle between the two dipole moments. From the aboveexpressions for the parameter P and γ12, one can findthat the spontaneously generated interference is depen-dent on the angle between the two dipole moments. Theinterference effect disappears (P = 0) when the two dipolemoments are perpendicular to each other, and reaches itsmaximum (P = 1) when they are parallel to each other.Based on the standard methods of laser theory [44], andconsidering the spontaneous decay of the atom, ρac andρbc can be evaluated to the first order in the couplingconstants g1 and g2 as

iρac =− iγ1ρac − iγ12ρbc + ∆ρac −Ωρbc

− g1ρ(0)aaa1 − g2ρ(0)

aba2 + g1a1ρ(0)cc , (5)

iρbc =− iγ2ρbc − iγ12ρac + ∆ρbc −Ω∗ρac− g1ρ(0)

baa1 − g2ρ(0)bba2 + g2a2ρ(0)

cc , (6)where the density matrix elements ρ(0)

ij can be obtained bythe corresponding zeroth-order equationsiρ(0)

aa=−2iγ1ρ(0)aa − iγ12(ρ(0)

ba + ρ(0)ab)−Ωρ(0)

ba + Ω∗ρ(0)ab + Ωaρf ,(7)

iρ(0)bb = −2iγ2ρ(0)

bb − iγ12(ρ(0)ba + ρ(0)

ab)−Ω∗ρ(0)ab + Ωρ(0)

ba, (8)iρ(0)

ab = −i(γ1 + γ2)ρ(0)ab − iγ12(ρ(0)

aa + ρ(0)bb)−Ωρ(0)

bb + Ωρ(0)aa,(9)

iρ(0)ba = −i(γ1 + γ2)ρ(0)

ba − iγ12(ρ(0)aa + ρ(0)

bb) + Ω∗ρ(0)bb −Ω∗ρ(0)

aa,(10)iρ(0)

cc = 0. (11)By inserting the steady-state solution of ρ(0)

ij into Eqs. (5-6), we can obtain the steady-state solutions for ρac andρbc , which have the form:

ig1ρac = α11ρfa1 + α12ρfa2, (12)ig2ρbc = α21ρfa1 + α22ρfa2, (13)

with the explicit expressions for the coefficients αij givenin Appendix A. By substituting Eqs.(12-13) into Eq. (2),the corresponding reduced master equation governing theevolution of the cavity field can be obtained asρf =− α11(a†1ρfa1 − ρfa1a†1 )− α12(a†1ρfa2 − ρfa2a†1 )

− α22(a†2ρfa2 − ρfa2a†2 )− α21(a†2ρfa1 − ρfa1a†2 )+ κ1(a1ρfa†1 − ρfa†1a1) + κ2(a2ρfa†2 − ρfa†2a2) +H.c..(14)Here we should note that we only consider the secondorder in the coupling constants g1, g2 and retain all ordersin the Lamor frequency of the microwave driving field Ω,because the coupling constants of the two cavity modesare smaller than other system parameters in our scheme.Thus we can ignore the saturation effects and operate inthe regime of linear amplification [29].3. Entanglement of the cavity fieldsAccording to Ref. [19], an inseparability criterion is pro-posed for continuous variable systems with the aid of thetotal variance of a pair of Einstein-Podolsky-Rosen (EPR)type operators, and the criterion provides a sufficient

739


condition for entanglement of any two-party continuous-variable states. In this section, we use the sufficient in-separability criterion [19] to study the entanglement prop-erties of the two cavity modes. Based on the spirit of thiscriterion, the system is said to be in an entangled state ifthe sum of the quantum fluctuations of the two EPR typeoperators u = x1 + x2 and v = p1 + p2 of the two modessatisfies the following inequality⟨(∆u)2 + (∆v )2⟩ < 2, (15)with xj = (aj +a†j )/√2 and pj = −i(aj−a†j )/√2 (j = 1, 2)the quadrature operators of the two cavity modes 1 and2. For a general state, this is a sufficient criterion forentanglement and furthermore, as demonstrated in Ref.[19], for all Gaussian states, this criterion turns out to bea necessary and sufficient condition for inseparability. Bysubstituting xj and pj into Eq. (15), we can obtain thetotal variance for the operators u and v in terms of theoperators aj and a†j . That is⟨(∆u)2 + (∆v )2⟩ =2[1 + 〈a†1a1〉+ 〈a†2a2〉+ 〈a1a2〉+ 〈a†1a†2 〉

− 〈a1〉〈a†1 〉 − 〈a2〉〈a†2 〉 − 〈a1〉〈a2〉− 〈a†1 〉〈a†2 〉]. (16)

With the help of Eq. (14), we can obtain the equationsof motion for the expectation values of the field operatorsshown in Eq. (16) as∂∂t 〈a1〉 = −(α11 + κ1)〈a1〉 − α12〈a2〉, (17)∂∂t 〈a2〉 = −(α22 + κ2)〈a2〉 − α21〈a1〉, (18)∂∂t 〈a

†1 〉 = −(α∗11 + κ1)〈a†1 〉 − α∗12〈a†2 〉, (19)∂∂t 〈a

†2 〉 = −(α∗22 + κ2)〈a†2 〉 − α∗21〈a†1 〉, (20)∂∂t 〈a

†1a1〉 = −(α11 + α∗11 + 2κ1)〈a†1a1〉 − α12〈a†1a2〉− α∗12〈a1a†2 〉 − (α11 + α∗11), (21)

∂∂t 〈a

†2a2〉 = −(α22 + α∗22 + 2κ2)〈a†2a2〉 − α∗21〈a†1a2〉− α21〈a1a†2 〉 − (α22 + α∗22), (22)

∂∂t 〈a

†1a2〉 = −α21〈a†1a1〉 − α∗12〈a†2a2〉− (α∗11 + α22 + κ1 + κ2)〈a†1a2〉 − (α∗12 + α21),(23)

∂∂t 〈a1a†2 〉 = −α∗21〈a†1a1〉 − α12〈a†2a2〉

− (α11 + α∗22 + κ1 + κ2)〈a1a†2 〉 − (α12 + α∗21),(24)

6 7 8 9 10 11 12 13 14−100

−90

−80

−70

−60

−50

−40

−30

−20

−10

0

gt

⟨(∆

u)2

+(∆

v)2

⟩

P=0

P=0.3

P=0.7

P=0.98

(a)

0 50 100 150 200 25010

2

103

104

105

gt

⟨N⟩

P=0

P=0.3

P=0.7

P=0.98

(b)

Figure 2. (a) The time evolution of 〈(∆u)2 + (∆v )2〉, and (b) the totalmean photon numbers 〈N〉, for different values ofP. Here,the cavity field is initially in the coherent state |10,−10〉.Other parameters used are κ1 = κ2 = 0.001g, γ1 = γ2 =γ = g, |Ω| = 10g, ∆1 = ∆2 = ∆ = g, γa = 5g, andφ = π/2.

∂∂t 〈a1a2〉 = −α21〈a1a1〉 − α12〈a2a2〉

− (α11 + α22 + κ1 + κ2)〈a1a2〉, (25)∂∂t 〈a1a1〉 = −2(α11 + κ1)〈a1a1〉 − 2α12〈a1a2〉, (26)∂∂t 〈a2a2〉 = −2(α22 + κ2)〈a2a2〉 − 2α21〈a1a2〉, (27)∂∂t 〈a

†1a†2 〉 = −α∗21〈a†1a†1 〉 − α∗12〈a†2a†2 〉− (α∗11 + α∗22 + κ1 + κ2)〈a†1a†2 〉, (28)

∂∂t 〈a

†1a†1 〉 = −2(α∗11 + κ1)〈a†1a†1 〉 − 2α∗12〈a†1a†2 〉, (29)∂∂t 〈a

†2a†2 〉 = −2(α∗22 + κ2)〈a†2a†2 〉 − 2α∗21〈a†1a†2 〉. (30)In order to make clear whether entanglement between twomodes of the field can be generated, we calculate the time

740


evolution of various moments 〈N〉 = 〈a†1a1〉+ 〈a†2a2〉 and⟨(∆u)2 + (∆v )2⟩ by use of the equations of motion givenin Eqs. (17-30). However, the exact analytical results arerather complicated in this situation. Since we mainly focuson examining the influence of the spontaneously gener-ated interference on the property of the CV entanglement,we show the typical numerical results for the time evo-lution of various moments 〈N〉 = 〈a†1a1〉 + 〈a†2a2〉 and⟨(∆u)2 + (∆v )2⟩ for different values of parameters P , γand ∆. Note that the two cavity modes are assumed to bein the coherent state |10,−10〉. For simplicity, all the pa-rameters used here are scaled with g, and we let φ = π/2during our numerical calculations.We first analyze the influences of the spontaneously gen-erated interference strength on the property of the CVentanglement when the cavity field is initially in the co-herent state |10,−10〉. Here the cavity loss rates, atomicrelaxation rates, intensity of the microwave driving field,the detuning, and the incoherent pumping are chosen asκ1 = κ2 = 0.001g, γ1 = γ2 = γ = g, |Ω| = 10g, ∆ = g,and γa = 5g. We plot in Fig. 2, the variable moment〈(∆u)2 + (∆v )2〉 and the mean photon number 〈N〉 as afunction of the time gt for different values of the spon-taneously generated interference strength P , i.e., P = 0(solid curve), P = 0.3 (dashed curve), P = 0.7 (dottedcurve), and P = 0.98 (dash-dotted curve). From Fig. 2(a),one can see that the intensity and period of entanglementincrease slightly with the increase of the value of P whenthe cavity field is initially in the coherent state. However,as shown in Fig. 2(b), the maximum mean photon num-ber 〈N〉 can be enlarged by increasing the spontaneouslygenerated interference strength P with the same set ofthe parameters. As a result, a perfect spontaneously gen-erated interference is needed for obtaining a long entan-glement time interval, entanglement intensity, and largemean number of photons.Choosing a certain strength of the spontaneously gen-erated interference (i.e., P = 0.5), we now analyze theeffects of different atomic relaxation rates on the prop-erties of the CV entanglement with the cavity field ini-tially in the same coherent state as Fig. 2. As shownin Fig. 3, we plot the time evolution of the variable mo-ment 〈(∆u)2 + (∆v )2〉 and the mean photon number 〈N〉for different values of the atomic relaxation rates γ, i.e.,γ = g (solid curve), γ = 1.5g (dashed curve), and γ = 2g(dotted curve). From Fig. 3, one finds that the intensityand period of entanglement can be enhanced significantlywithout decreasing the maximum mean photon numbers〈N〉 as the spontaneous emission decay rate of the atomincreases. The physical interpretation of these results israther clear. The spontaneous emission rates of the atomplay a constructive role in the spontaneously generated

0 5 10 15 20 25 30−500

−450

−400

−350

−300

−250

−200

−150

−100

−50

0

gt

⟨(∆

u)2

+(∆

v)2

⟩

γ=1g

γ=1.5g

γ=2g

(a)

0 50 100 150 20010

2

103

104

105

gt

⟨N⟩

γ=1g

γ=1.5g

γ=2g

(b)

Figure 3. (a) The time evolution of 〈(∆u)2 + (∆v )2〉, and (b) the totalmean photon numbers 〈N〉, with different atomic decayrates γ (γ1 = γ2 = γ). Here the cavity field is initially inthe coherent state |10,−10〉. Other parameters used areκ1 = κ2 = 0.001g, P = 0.5, |Ω| = 10g, ∆1 = ∆2 = ∆ = g,γa = 5g, and φ = π/2.

interference γ12. For a given value of the parameter P ,γ12 increases with increasing spontaneous emission ratesγj . As illustrated in Fig. 2, the entanglement intensityand time interval can be enhanced by the spontaneouslygenerated interference; as a result, the intensity and pe-riod of entanglement can be increased accordingly whenthe atom spontaneous emission rates increase. In otherwords, we can achieve the optimal combination of a longentanglement time interval and enhanced entanglementintensity by properly controlling the atom’s spontaneousemission rates.Up to now, we have investigated the effect of atomic spon-taneous emission and spontaneously generated interfer-ence on the properties of CV entanglement by consideringthe cavity field initially in coherent state. We now exam-ine the effect of the frequency detuning between the cavitymodes and the corresponding atomic transition on the timeevolution of entanglement. By considering that the initialstate of the cavity field is the coherent state |10,−10〉, we

741


4 6 8 10 12 14−120

−100

−80

−60

−40

−20

0

gt

⟨(∆

u)2

+(∆

v)2

⟩

∆=1g

∆=2g

∆=3g

(a)

0 50 100 150 20010

2

103

104

105

gt

⟨N⟩

∆=1g

∆=2g

∆=3g

(b)

Figure 4. (a) The time evolution of 〈(∆u)2 + (∆v )2〉, and (b) the totalmean photon numbers 〈N〉, for different detuning frequen-cies ∆(∆1 = ∆2 = ∆). Here, the cavity field is initially inthe coherent state |10,−10〉. Other other parameters usedare γ1 = γ2 = g, κ1 = κ2 = 0.001g, P = 0.5, |Ω| = 10g,γa = 5g, and φ = π/2.

plot in Fig. 4 the time evolution of the variable moment〈(∆u)2 + (∆v )2〉 and the mean photon number 〈N〉 for dif-ferent values of the detunings ∆, i.e., ∆ = g (solid curve),∆ = 2g (dashed curve), and ∆ = 3g (dotted curve). It canbe easily seen from Fig. 4(a) that the intensity and periodof entanglement between the two cavity modes can be de-creased markedly by augmenting the frequency detuning∆. On the other hand, it can be found from Fig. 4(b) thatthe total mean photon numbers 〈N〉 increase significantlywith the increasing of the frequency detuning ∆ and themaximum mean number of photons for which the entangle-ment criterion (15) is still satisfied. These results indicatethat one can obtain the entanglement of cavity modes witha greater magnitude and a longer period by choosing thenear-resonant condition. Based on this interesting result,

it is possible to obtain the entanglement of cavity modeswith a low-Q cavity.4. ConclusionIn conclusion, we have demonstrated the generation oftwo-mode continuous-variable (CV) entanglement and an-alyzed the properties of the entanglement in a V-typethree-level atom trapped in a doubly resonant cavity usinga microwave field driving a hyperfine transition betweentwo upper excited states. Our numerical results have indi-cated that the CV entanglement with large mean numberof photons can be generated even in the presence of atomicrelaxation and cavity losses. Importantly, we have shownthat the intensity and period of entanglement can be en-hanced significantly with increasing atomic relaxation dueto the existence of the perfect spontaneously generated in-terference between two atomic decay channels. Moreover,we have also shown that CV entanglement of cavity modeswith large mean photon numbers can be generated evenunder near-resonant conditions, which implies that it ispossible to obtain the entanglement of cavity modes evenwith a low-Q cavity.AcknowledgementWe would like to thank Prof. Peng Xue for her enlighten-ing discussions. The research is supported in part by Na-tional Natural Science Foundation of China under GrantNos. 11374050 and 61372102, by Qing Lan project ofJiangsu, and by the Fundamental Research Funds for theCentral Universities under Grant No. 2242012R30011.Appendix A: COEFFICIENTSHere, we give the explicit expressions for the coefficientsAij and Bij (i, j = 1, 2) in Eqs. (12-14),

α11 = −g21[(R + i∆)βaa + (iΩ− γ12)βba]/D1, (A1)α12 = −g1g2[(R + i∆)βab + (iΩ− γ12)βbb]/D1, (A2)α22 = −g22[(R + i∆)βbb + (iΩ∗ − γ12)βab]/D1, (A3)α21 = −g1g2[(R + i∆)βba + (iΩ∗ − γ12)βaa]/D1, (A4)βaa = (2R2 − γ212 + |Ω|2)Rγa/D2, (A5)βbb = (γ12 + iΩ)(γ12 − iΩ∗)Rγa/D2, (A6)βab = −(γ12 + iΩ)(2R2 − iγ12Ω− iγ12Ω∗)Rγa/(2D2),(A7)

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βba = −(γ12 − iΩ∗)(2R2 + iγ12Ω + iγ12Ω∗)Rγa/(2D2),(A8)withD1 = (R + i∆)2 − (iΩ− γ12)(iΩ∗ − γ12), (A9)D2 = 4R4 + 4R2|Ω|2 − 4R2γ212 − γ212(Ω + Ω∗)2, (A10)

where we have assumed γ1 = γ2 = γ = R .References

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