Optics Communications 285 (2012) 4446–4452

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Optics Communications

0030-40

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n Corr

E-m

journal homepage: www.elsevier.com/locate/optcom

Continuous-variable multimode entanglement in multi-wave mixing

Wenxing Shi a,n, Fei Wang b, Lihui Zhang a, Zhiming Zhan a, Xing Li a

a School of Physics & Information Engineering, Jianghan University, Wuhan 430056, People’s Republic of Chinab College of Science, New Energy Research Institute, China Three, Gorges University, Yichang 443002, People’s Republic of China

a r t i c l e i n f o

Article history:

Received 28 February 2012

Received in revised form

19 May 2012

Accepted 19 June 2012Available online 3 July 2012

Keywords:

Continuous-variable entanglement

Quantum-beat

Multi-wave mixing

18/$ - see front matter & 2012 Elsevier B.V. A

x.doi.org/10.1016/j.optcom.2012.06.047

esponding author. Tel.: þ86 13135698821.

ail address: SWXing@163.com (W. Shi).

a b s t r a c t

We present a scheme to obtain continuous-variable multimode entanglement via multi-wave mixing.

For a four-level atomic system in a diamond configuration, four strong coherence fields are applied to

the four dipole-allowed transitions to amplify eight sidebands as cavity fields, respectively. Due to the

atomic coherence, the eight side modes constitute a pair of double quantum-beats and combine into

two quantum-beat collective modes, which are in a four-wave mixing parametric interaction. As a

result, the entanglement occurs between these two collective modes. Correspondingly, any two

individual modes from two different collective modes are entangled with each other. This gives rise

to continuous-variable eight-mode entanglement with different frequencies, which has extensively

application in quantum communication and quantum network.

& 2012 Elsevier B.V. All rights reserved.

1. Introduction

Quantum entanglement is a unique property of quantummechanics. In recent years, quantum entanglement has attractedgreat interest due to its potential applications in quantuminformation processing, such as quantum teleportation [1,2],quantum telecloning [3–5], and quantum dense coding [6–9]. Inparticular, continuous-variable (CV) entanglement, as an impor-tant part of quantum information theory [10], has been exten-sively studied because of the advantages of the relative simplicityand high efficiency in the generation and manipulation. Theoptical nondegenerate parametric down-conversion [11–15] andcorrelated spontaneous emission lasers [16–23] have been provedto be the two different effective ways to produce CV entangle-ment. Recent studies have shown that mixing parametric inter-actions provide another alternative effective way for generatingCV entanglement [24–30]. For a two-level atomic system, twoside modes combine into a quantum-beat and are involved in thefour-wave mixing parametric interaction with each other. Thevariance sum of a pair of Einstein–Podolsky–Rosen (EPR) likequadrature operators for the intracavity fields is reduced by about50% below the standard quantum limit [24,25]. Essence of thequantum correlation is attributed to the parametric interaction ofthe cavity fields. In order to enhance the quantum correlation,various schemes have been proposed by using a two-step atomicreservoir engineering [26], or a two-channel dissipation mechan-ism [27], or an auxiliary channel for transfer of dressed state

ll rights reserved.

population [28,29]. These works are mainly confined to two-mode entanglement. More recently, some schemes based on themixing parametric interaction have been presented for generatingCV three-mode or four-mode entanglement [31,32]. For example,in the three-level V-type atomic system [32], due to coherence-induced depopulation, four side modes are pulled into a pair ofquantum-beats and enter into two collective modes. The twocollective modes mediate into the mixing parametric interaction.This yields the entanglement between the collective modes.Correspondingly, any two individual modes from two differentcollective modes are entangled with each other. By the means ofthis manner, the CV four-mode square entanglement is obtained.

Here we extend to the eight-mode case and explore theentanglements between them. It is well known that CV multi-mode entangled light with different frequencies will be necessaryfor the storage and communication of quantum information inthe nodes of quantum teleportation and networks [10]. Contin-uous-variable multimode entanglements have three differentclasses depending on the partition of the system density matrix[33,34]. The first one is extreme case where the density matrix isnot separable for any grouping of the modes, which correspondsto perfect multimode entanglement. The second is the otherextreme case where all modes are not entangled in any way,and the third classes lie between these two extreme cases.

In this paper, we show that the CV eight-mode entanglementcan be obtained in a four-level diamond atomic system, anddiscuss the entanglement between any two individual modes.Four external strong coherent fields are applied to four dipole-allowed transitions to induce the atomic coherence. Eight sidemodes from these four transitions are amplified as cavity fields,and are respectively, pulled into a pair of double quantum-beats

W. Shi et al. / Optics Communications 285 (2012) 4446–4452 4447

and combine into two quantum-beat collective modes. These twocollective modes are involved in a four-wave mixing parametricinteractions. This leads that entanglement occurs between the twocollective modes, the best achievable state approaches the EPRentangled state [35]. Correspondingly, entanglement exists betweenany two individual modes from different collective modes. However,no entanglement appears between any two individual modes in onecollective mode. Essentially, the entanglement of the collectivemodes corresponds to multimode entanglement, which belongs tothe third case between the above two extremes. This paper isorganized as follows. In Section 2, we describe the system modeland analyze the physics mechanism of the system, and then derivethe master equation for the cavity fields. In Section 3, we present theresults, and analyze the entanglement between any two individualmodes. Our conclusion is given in Section 4.

2. Model and master equation

An ensemble of N four-level diamond atoms is placed in amultimode resonant optical cavity. Four external coherent fieldswith complex Rabi frequencies Ol (l¼1, 2, 3, 4) are coupled to thefour different dipole-allowed transitions 90S�91;2S and 91;2S�93S, respectively, as shown in Fig. 1. From which four highsidebands and four low sidebands are amplified as cavity fields. Asusual, we treat the driving fields classically and the cavity fieldsquantum mechanically. The master equation for the density opera-tor r of the atom-field system is derived in the dipole approxima-tion and in a frame rotating at the driving field frequencies as [36]

_r ¼� i

_½H0þV1þV2,r�þLarþLcr, ð1Þ

where H0 ¼P

l ¼ 1;2_Dlsll describes the free term of the atoms, V1,2

denote the interactions of the atoms with driving fields and thecavity fields, respectively

V1 ¼�X

l ¼ 1;2

_

2ðOlsl0þOlþ2s3lÞþH:c:, ð2Þ

V2 ¼X

l ¼ 1;2

_ðgale�iðDlþdlÞtalþgblþ 2

eiðDlþdlÞtblþ2Þsl0

þX

l ¼ 1;2

_ðgalþ 2eiðDlþdlÞtalþ2þgbl

e�iðDl þdlÞtblÞs3lþH:c: ð3Þ

a1a2

a3a4 b1b2

b34b

Ω2 Ω1

Ω3Ω4

Fig. 1. Schematic level diagram for four-level diamond atomic system. Four

coherent fields with Rabi frequencies Ol (l¼1, 2, 3, 4) are applied to four different

dipole-allowed transitions 90S�91;2S and 91;2S�93S, respectively, from which

four high sidebands al and four low sidebands bl (l¼1, 2, 3, 4) are amplified as

cavity fields.

Here H.c. denotes the Hermitian conjugate. sjk ¼PN

m ¼ 1 9jmS/km9

(j,k¼0,1,2,3) are the projection operators for j¼k and the spin-flipoperators for jak. al,lþ2 (bl,lþ2) and ayl,lþ2 (byl,lþ2) are the annihila-tion and creation operators for the cavity fields, where a (b)represent high (low) sidebands, respectively. Dl ¼�ðol�ol0Þ ¼

olþ2�o3l are the corresponding detunings between the atomictransitions and driving fields, ol,lþ2 are the frequencies of thedriving fields, ol0 and o3l are the atomic resonance frequencies.dl ¼ nal

�ol0 ¼�ðnbl�o3lÞ ¼ nalþ 2

�o3l ¼�ðnblþ 2�ol0Þ are the fre-

quency detunings between the cavity fields and driving fields,nal,lþ 2

and nbl,lþ 2are the frequencies of the cavity fields. gal,lþ 2

andgbl,lþ 2

are the atom-cavity field coupling constants. Lar representsthe atomic damping terms as

Lar¼L01rþL02rþL13rþL23r,

Lklr¼gkl

2ð2sklrslk�rslkskl�slksklrÞ, ð4Þ

where gkl (kl¼ 01;02,12;13) are the atomic decay rates. Lcrdenotes the cavity loss of the modes and takes the form:

Lcr¼X4

l ¼ 1

kal

2ð2alrayl�ayl alr�rayl alÞþ

kbl

2ð2blrbyl�byl blr�rbyl blÞ

� �,

ð5Þ

with the cavity field loss rates kaland kbl

.

2.1. Interaction of atoms with two collective modes

In order to show the underlying physics more clearly, wetransform to the dressed state picture. For simplicity, we assumethat the complex Rabi frequencies of the external driving fieldssatisfy the matching conditions 9O19=9O39¼ 9O29=9O49 andc2�c1 ¼�ðc4�c3Þ, cl are the corresponding phase angles, andintroduce the orthonormal superposition states of the bare atomicstates

910S¼ sin y91Sþcos ye�iðc2�c1Þ92S,

920S¼ cos yeiðc2�c1Þ91S�sin y92S, ð6Þ

900S¼ sin b90Sþcos be�iðc3þc1Þ93S,

930S¼ cos b eiðc3þc1Þ90S�sin b93S, ð7Þ

where tan y¼ 9O19=9O29¼ 9O39=9O49, tan b¼ 9O19=9O39¼ 9O29=9O49. Then the interaction V1 is rewritten as

V1 ¼�_

2O0s1000 þH:c:, ð8Þ

with O0 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðtan2bþ1Þð9O39

2þ9O49

2Þ

qe�ic1 . It is seen from Eq. (8)

that only the superposition states 900S and 910S mediate into theinteraction, the other superposition states 920S and 930S aredecoupled from the driving field O0. But the levels 920S and930S still have the corresponding atomic populations and sponta-neous emission, as will be shown in Eq. (19).

By taking D1 ¼D2 ¼D and diagonalizing the Hamiltonian_Ds1010 þV1, the dressed atomic states can be written as

9þS¼ sin feic1 900S�cos f910S,

9�S¼ cos f900Sþsin fe�ic1 910S, ð9Þ

where tan f¼ 9O09=ðDþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD2þ9O09

2q

Þ. The dressed states have their

eigenvalues l7 ¼ ðD7ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD2þ9O09

2q

Þ=2. We tune the cavity fields

such that d1 ¼ d2 ¼ d and Dþd¼ lþ�l�. Making a rotating-wavetransformation and neglecting the rapidly oscillating terms, the

W. Shi et al. / Optics Communications 285 (2012) 4446–44524448

Hamiltonian V2 is expressed as

V2 ¼�_g sin b cos2fðsin ya1þcos yeiðc2�c1Þa2Þsþ�þ_g sin b sin2fðsin ye2ic1 b3þcos yeiðc2þc1Þb4Þs�þ�_g cos b cos2fðsin yeiðc3þc1Þb1þcos yeiðc4þc1Þb2Þs�þþ_g cos b sin2fðsin yeiðc3�c1Þa3þcos yeiðc4�c1Þa4Þsþ�þH:c:

ð10Þ

Here we have assumed equal coupling constants gal¼ gbl

¼ g. Note

that the modes a1 and a2, b1 and b2, a3 and a4, b3 and b4 mediate intointeractions in sum mode forms. By using the combination modes

A1 ¼ sin ya1þcos yeiðc2�c1Þa2, A3 ¼ sin ya3þcos yeiðc4�c3Þa4,

A2 ¼ cos yeiðc1�c2Þa1�sin ya2, A4 ¼ cos yeiðc3�c4Þa3�sin ya4, ð11Þ

B1 ¼ sin yb1þcos yeiðc4�c3Þb2, B3 ¼ sin yb3þcos yeiðc2�c1Þb4,

B2 ¼ cos yeiðc3�c4Þb1�sin yb2, B4 ¼ cos yeiðc1�c2Þb3�sin yb4: ð12Þ

The Hamiltonian V2 is rewritten as

V2 ¼�_gðsin b cos2fA1�cos b sin2fA3eiðc3�c1ÞÞsþ��_gei2c1 ðcos b cos2feiðc3�c1ÞB1�sin b sin2fB3Þs�þ þH:c:

ð13Þ

It is seen from Eq. (13) that the modes A2;4 and B2;4 are decoupled

from the Hamiltonian V2, and stay in their vacuum states. This

indicates that the cavity modes a1;2, b1;2, a3;4, b3;4, respectively, are

in quantum-beat interactions with atoms [37–40], i.e., eightsidebands are pulled into a pair of double quantum-beats. Forconvenience we define the phase angles of the cavity modes

/alS¼ 9/alS9e�ijal , /blS¼ 9/blS9e�ijbl (l¼1, 2, 3, 4). Using this

definition and the fact that the A2;4 and B2;4 modes stay in their

vacuum states, we have the relative phase locking

ja2�ja1

¼jb4�jb3

¼c2�c1,

jb2�jb1

¼ja4�ja3

¼c4�c3: ð14Þ

In terms of the form of the Hamiltonian V2 in Eq. (13), we againdefine two pairs of orthonormal combination modes

Ad ¼�sin b cos2fA1þcos b sin2feiðc3�c1ÞA3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

sin2b cos4fþcos2b sin4fq ,

As ¼cos b sin2fe�iðc3�c1ÞA1þsin b cos2fA3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

sin2b cos4fþcos2b sin4fq , ð15Þ

Bd ¼�cos b cos2feiðc3�c1ÞB1þsin b sin2fB3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

cos2b cos4fþsin2b sin4fq ,

Bs ¼sin b sin2fB1þcos b cos2fe�iðc3�c1ÞB3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

cos2b cos4fþsin2b sin4fq : ð16Þ

Then the Hamiltonian V2 takes the ultimately form

V2 ¼ _ðgAAdþgBe�2ic1 BydÞsþ�þH:c:, ð17Þ

where

gA ¼ gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisin2b cos4fþcos2b sin4f

q,

gB ¼ gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficos2b cos4fþsin2b sin4f

q. Obviously, the modes As and Bs

are also decoupled from the system Hamiltonian V2. It means that thecombination modes A1 and A3, B1 and B3 are once again in twoquantum-beat interactions, respectively, i.e., quantum-beat of quan-tum-beat. It is found that four high sidebands and four low sidebandscombine into a high frequency quantum-beat collective mode and alow frequency quantum-beat collective mode, respectively. Since the

quantum-beat collective modes As and Bs are decoupled from thesystem, the relative phases of the collective modes /AlS¼

9/AlS9e�ijAl and /BlS¼ 9/BlS9e�ijBl (l¼1,3) locking are

jA3�jA1

¼jB1�jB3

¼c3�c1: ð18Þ

It is seen from Eq. (17) that the absorption of photons from thecollective mode Ad is accompanied with the emission of photons into

the collective mode Bd through the transition from 9�S to 9þS and

vice versa. Essentially, the collective modes Ad and Bd are in thesqueeze-transformed interaction [24–28,41–47]. It is for the veryreason that entanglement happens between the collective modes Ad

and Bd. This will be discussed in detail in Section 3. Equivalently,entanglements exist between high sidebands a1;2,3;4 and low side-

bands b1;2,3;4. However, no entanglements occur between any two of

high sidebands a1;2,3;4 or between any two of low sidebands b1;2,3;4

due to that they are in quantum-beat interactions with each other.

2.2. Master equation for the quantum-beat collective modes

We derive the master equation for the reduced densityoperator of the cavity fields in the linear laser theory. Assumingthe atoms decay much more rapidly than the cavity fields. In theabsence of the cavity fields, we obtain the steady state popula-tions as

/s2020Sþ/s3030S¼N cos2b, ð19Þ

/sþ þS¼ Z�1N½cos2bðcos2fþsin2f sin2bÞð1þcos2fÞ

þsin4f sin2bð1þsin2bÞ�,

/s��S¼ Z�1N½cos2bðsin2fþcos2f sin2bÞð1þsin2fÞ

þcos4f sin2bð1þsin2bÞ�, ð20Þ

with Z¼ sin�2b½sin22f sin22b=8þð1þsin4bÞðsin4fþ cos4fÞþð1þsin2bÞ cos2b�. Here we have assumed equal atomic decayrates gkl ¼ g. Note that the steady state populations on the levels920S and 930S are not zero as shown in Eq. (19). As stated on theabove, the levels 920S and 930S are only decoupled from thedriving field O0 due to atomic coherence but not decoupled fromthe system. Therefore, they have spontaneous emission contribu-tion to the system. This is different from the coherent depopula-tion in Refs. [31,32]. In their scheme, the states are not onlydecoupled from the driving field but also from the atomic system.It leads to no population in these decoupled levels. By tracing outthe atomic variables, one can obtain the master equation of thereduced density operator r for the cavity fields as

_r ¼ wAðAy

drAd�rAdAydÞþw0AðAdrAyd�rAydAdÞ

þwBðBy

drBd�rBdBydÞþw0BðBdrByd�rBydBdÞ

þwABe2ic1 ð2AdrBd�rBdAd�BdAdrÞþwBAe2ic1 ð2BdrAd�rAdBd�AdBdrÞþH:c:, ð21Þ

where the parameters w read as

wA ¼ g2A/sþ þS=G, w0A ¼ g2

A/s��S=Gþka=2,

wB ¼ g2B/s��S=G, w0B ¼ g2

B/sþ þS=Gþkb=2,

wAB ¼ gAgB/s��S=G, wBA ¼ gBgA/sþ þS=G, ð22Þ

with G¼ gð1þ2 sin2f cos2fþ2 cos2bÞ=2. In the derivation pro-cess of the master Eq. (21), we have assumed cavity field lossrates kal

¼ ka, kbl¼ kb, respectively, and separated the decay

terms of the collective modes As and Bs. Since the collectivemodes As and Bs are decoupled from the system, stay intheir vacuum state, /AsS¼/BsS¼/Ays AsS¼/Bys BsS¼ 0. Theyundergo only absorption but no gain. This allows us to separate

W. Shi et al. / Optics Communications 285 (2012) 4446–4452 4449

the density operator [37–40]. In Eq. (21), the terms proportionalto wA (wB) are the gain for the collective modes Ad (Bd), which aredue to the stimulated emission from level 9þS ð9�SÞ. The termsproportional to w0A ðw0BÞ are the absorption for the collective modesAd (Bd), which originate from atomic population in level9�S ð9þSÞ. The terms proportional to wAB and wBA arise fromatomic coherence between levels 9�S and 9þS, and describe thecorrelations between the collective modes Ad and Bd. It is seenfrom the terms that the collective modes Ad and Bd are in a four-wave mixing parametric interaction [41–47]. This determines theexistence of the entanglement between Ad and Bd, which will beshown below.

-30-2

-1

0

R = 0.5

R = 1

R = 2

R = 4

S(0

)

Δ-15 0 15 30

Fig. 2. Zero-frequency output correlation spectra S(0) as a function of D for the

different ratio R¼0.5 (dash dotted), 1 (dotted), 2 (dashed), 4 (solid). The other

parameters are g2N ¼ 50, ka ¼ kb ¼ 0:1, 9O19¼ 9O29¼ 10.

3. Entanglement analysis

In this section, we explore the correlation between thequantum-beat collective modes Ad and Bd, and analyze theentanglement between any two individual sidebands.

3.1. Correlation between the quantum-beat collective modes

Choosing the normal order of the operators Ayd, Byd, Bd, Ad anddefining the corresponding c-numbers an

Ad, an

Bd, aBd

, aAd, respec-

tively, we derive the Langevin equations for the intracavity fieldsfrom Eq. (21) as

/ _aAdS¼ ðwA�w0AÞ/aAd

SþðwBA�wABÞe�2ic1/an

BdSþFaAd

,

/ _aBdS¼ ðwB�w0BÞ/aBd

S�ðwBA�wABÞe�2ic1/an

AdSþFaBd

, ð23Þ

with those for the variables an

Adand an

Bd. At the steady state we have

/alS¼ 0 (l¼ Ad,Bd). For the phases of the collective modes/AdS¼ 9/AdS9e�ijAd and /BdS¼ 9/BdS9e�ijBd , we have the sumphase jAd

þjBd¼ 2c1. Combining with the phase-matching condi-

tions (14) and (18), the phases of the cavity modes are locked asja1þjb3

¼ 2c1, ja2þjb4

¼ 2c2, ja3þjb1

¼ 2c3, ja4þjb2

¼ 2c4.In Eq. (23), FaAd

ðtÞ and FaBdðtÞ are d-correlated noise terms with the

zero averages /FaAdðtÞS¼/FaBd

ðtÞS¼ 0 and correlations/Fxðt

0ÞFyðtÞS¼Dxydðt�t0Þ. The nonzero diffusion coefficientsDyx ¼Dxy are

Dan

AdaAd¼ 2wA, Dan

BdaBd¼ 2wB,

DaAdaBd¼�ðwBAþwABÞe

�2ic1 , Dan

Adan

Bd

¼�ðwBAþwABÞe2ic1 : ð24Þ

Here we investigate the quantum correlation spectrum of thequantum-beat collective modes Ad and Bd, which is derived byfollowing the standard techniques [48–50]. The differential equa-tion (23) can be written in a compact matrix form

d

dtQ ðtÞ ¼�MQ ðtÞþFðtÞ, ð25Þ

where the column vectors Q ðtÞ ¼ ðan

Ad, an

Bd, aBd

, aAdÞT and

FðtÞ ¼ ðFn

aAd, Fn

aBd, FaBd

, FaAdÞT . The drift matrix M can be obtained

from Eq. (23). Since /QS¼ 0, we have /dQdQTS¼/QQTS�/QS/QTS¼/QQTS. The normally ordered intracavitycorrelation spectrum SðoÞ ¼

R1�1

eiot/dQ ðtÞdQTðtþtÞSdt is

derived as

SðoÞ ¼ ðM�ioIÞ�1DðMTþ ioIÞ�1, ð26Þ

where D is the diffusion matrix, /FðoÞFTðo0ÞS¼Ddðoþo0Þ and I

is a unit matrix.In order to show the correlation between the modes Ad and Bd,

we consider a sufficient inseparability criterion for the contin-uous-variable bipartite entanglement as that proposed by Duanet al. [51]. The two quantum-beat collective modes Ad and Bd areentangled if the variance sum V of the operators u and u satisfies

the inequality:

V ¼/ðduÞ2Sþ/ðduÞ2S�c2�1

c2o0, ð27Þ

where u¼ cxoutA �ð1=cÞxout

B , u¼ cpoutA þð1=cÞpout

B . The quadrature

operators for the collective modes Ad and Bd are defined as

xoutA ¼ ðA

outd eijAd þAyout

d e�ijAd Þ=ffiffiffi2p

, poutA ¼ ðA

outd eijAd�Ayout

d e�ijAd Þ=ffiffiffi2p

i

and xoutB ¼ xout

A (A-B), poutB ¼ pout

A (A-B). By choosing

c2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikb/BydBdS=ka/AydAdS

q, we have /ðduÞ2S¼/ðduÞ2S. In the

circumstances we can denote the degree of entanglement by V.According to the input–output relation [52], the entanglementcriterion (27) is reduced to the inequality:

V ¼ 4ffiffiffiffiffiffiffiffiffiffiffikakbp

ð

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi/AydAdS/BydBdS

q�/AdBdSÞo0, ð28Þ

which indicates that the nonclassical correlation between thequantum-beat collective modes Ad and Bd corresponds to thequantum entanglement between them. We note that V¼�2corresponds to the original EPR entanglement [35]. Using theintracavity spectrum (26), we can obtain the output correlationspectrum:

SðoÞ ¼ k1c2½S14ðoÞþS41ðoÞ�þk2

c2½S23ðoÞþS32ðoÞ�

�ffiffiffiffiffiffiffiffiffiffiffik1k2p

½S12ðoÞþS21ðoÞþS34ðoÞþS43ðoÞ�: ð29Þ

The spectrum holds when the system is stable, i.e., when the realparts of all eigenvalues of the matrix M are positive. If SðoÞo0,the quantum-beat collective modes Ad and Bd are entangled witheach other.

In our numerical calculation, we scale detunings, Rabi fre-quencies, coupling constants and decay rates in units of g. Forconvenience we define the Rabi frequency ratio R¼ 9O19=9O39. InFig. 2, we plot the zero-frequency correlation spectrum Sð0Þ as afunction of D for the equal rates of cavity loss ka ¼ kb ¼ 0:1. Theother parameters are chosen as g2N¼ 50, 9O19¼ 9O29¼ 10, R¼0.5(dash dotted), 1 (dotted), 2 (dashed), 4 (solid). The results showthat the output correlation spectra are symmetrical with respectto D¼ 0 and Sð0Þo0 except at D¼ 0 or R¼1. That is to say, theentanglement criterion is satisfied except at D¼ 0 or R¼1. Thisindicates the existence of the entanglement between the twoquantum-beat collective modes Ad and Bd. The strongest entan-glement appears in two symmetric regions about D¼ 0. WhenD¼ 0, S(0)¼0, which means that no entanglement occursbetween the collective modes Ad and Bd. In the case of D¼ 0, wehave sin f¼ cos f¼ 1=

ffiffiffi2p

, /sþ þS¼/s��S¼N sin2b=2 and

-30-1.0

-0.5

0.0

R = 0.5

R = 1

R = 2

R = 4V

Δ-15 0 15 30

Fig. 5. The variance sum V for the intracavity fields as a function of D. The

parameters are the same as in Fig. 2.

W. Shi et al. / Optics Communications 285 (2012) 4446–44524450

wA ¼ w0A, wB ¼ w0B, wBA ¼ wAB. As seen from Eq. (23), there is neithergain nor absorption for the collective modes Ad and Bd, and theparametric interaction vanishes. This leads that the entanglementis absent between the modes Ad and Bd. At the same time, theentanglement between the two collective modes is stronglydependent on the Rabi frequencies ratio R. As the ratio R

increases, the output spectrum becomes small, i.e., the correlationbecomes stronger. The best achievable value approachesSð0Þ ��2. This indicates that the collective modes Ad and Bd areapproximately in an EPR entangled state. And we note thatR¼ 9O19=9O39¼ 1, there is also no entanglement between thecollective modes. The reason is that when R¼1, we havesin b¼ cos b¼ 1=

ffiffiffi2p

, /sþ þS¼/s��S¼N=4, and wA ¼ w0A,wB ¼ w0B, wBA ¼ wAB. This is similar to the case of D¼ 0. At this time,the parametric interaction vanishes, no entanglement appearsbetween the collective modes Ad and Bd. Fig. 3 shows that thezero-frequency output correlation spectrum S(0) as a function ofD for the different rates of cavity damping ka ¼ 0:05, kb ¼ 0:1. Theother parameters are chosen as g2N¼ 50, 9O19¼ 9O29¼ 10,R¼0.5. It is obvious that the curve is no longer symmetric withrespect to D¼ 0. The correlation is more weaker than in the equal

-30-2

-1

0

S(0

)

∆

R = 0.5

-15 0 15 30

Fig. 3. Zero-frequency output correlation spectra S(0) as a function of D for the

different rates of cavity loss ka ¼ 0:05, kb ¼ 0:1. The other parameters are

g2N¼ 50, 9O19¼ 9O29¼ 10, R¼0.5.

-30-2

-1

0

κ = 0.01

κ = 0.1

κ = 0.5

S(0

)

∆-15 0 15 30

Fig. 4. Zero-frequency output correlation spectra S(0) as a function of D for the

equal rates of cavity loss, ka ¼ kb ¼ 0:01 (solid), 0.1 (dashed), 0.5 (dotted). The

other parameters are g2N¼50, 9O19¼ 9O29¼ 10, R¼0.5.

rate case ka ¼ kb ¼ 0:1, although the rate ka is more smaller. Forthe chosen parameters, there is no entanglement in the regime�4:895oDo16:35. Here we focus mainly on the equal rate case.In Fig. 4, We show the dependence of the zero-frequencycorrelation spectrum S(0) on the equal rates of the cavity loss,ka ¼ kb ¼ 0:01 (solid), 0.1 (dashed), 0.5 (dotted), g2N¼50,9O19¼ 9O29¼ 10, and R¼0.5. It is seen from this figure that theoutput correlation spectrum S(0) is strongly dependent on therates of the cavity case. As the rates of the cavity loss decrease, thespectrum S(0) becomes more and more smaller, i.e., the entangle-ment between the two collective modes Ad and Bd becomes moreand more stronger. The best achievable state approaches theoriginal EPR state with the improvement of the quality factor ofthe cavity. In order to compare the correlations between theoutput fields and the intracavity fields, we plot the variance sumV for the intracavity fields in Fig. 5. The parameters are chosen thesame as those in Fig. 2. Comparing Fig. 5 with Fig. 2, the curves aresimilar. The entanglement criterion (27) is satisfied except toD¼ 0 or R¼1, i.e., the entanglement appears between the twocollective modes Ad and Bd. However, the output fields havestronger correlations than the intracavity fields. For the bestcorrelation, the variance sum V approaches �1. Correspondingly,the best achievable value of the output fields approachesSð0Þ � �2, as shown in Fig. 2. This is because that the input–output relation leads to quantum interference between two out-put fields.

As stated in Section 2, this bipartite entanglement between thecollective modes Ad and Bd corresponds to the entanglement ofmultiple sidebands with different frequencies. Any one highsideband in the collective mode Ad and any one low sideband inthe Bd mode are entangled with each other. However, no entan-glement appears between any two sidebands in one collectivemode, as will be shown below.

3.2. Entanglement between any two individual sidebands

Here we present the entanglement between any two indivi-dual sidebands. Applying the sufficient inseparability criterion(27), we can denote the degree of the entanglement between anytwo individual sidebands by the relations:

Va1b1¼ G�1 sin2y sin b cos b cos4fSðoÞ,

Va1b2¼ Va2b1

¼ G�1 sin y cos y sin b cos b cos4fSðoÞ,

Va2b2¼ G�1 cos2y sin b cos b cos4fSðoÞ, ð30Þ

a1

a2

a3

a4

b1

b2

b3

b4

a1

a2

b1

b2

a3

a4

b3

b4 a4

b1

b2

a1

a2

a3 b3

b4

Fig. 6. Graphical representation of entanglements between sidebands. Each dot represents a corresponding sideband and the lines represent the entanglements.

(A) Entanglements between any two individual sidebands. (B) Entanglements between any two sidebands for Rb1. (C) Entanglements between any two sidebands for R51.

W. Shi et al. / Optics Communications 285 (2012) 4446–4452 4451

Va1b3¼ G�1 sin2y sin2b sin2f cos2fSðoÞ,

Va1b4¼ Va2b3

¼ G�1 sin y cos y sin2b sin2f cos2fSðoÞ,

Va2b4¼ G�1 cos2y sin2b sin2f cos2fSðoÞ, ð31Þ

Va3b1¼ G�1 sin2y cos2b sin2f cos2fSðoÞ,

Va3b2¼ Va4b1

¼ G�1 sin y cos y cos2b sin2f cos2fSðoÞ,

Va4b2¼ G�1 cos2y cos2b sin2f cos2fSðoÞ, ð32Þ

Va3b3¼ G�1 sin2y sin b cos b sin4fSðoÞ,

Va3b4¼ Va4b3

¼ G�1 sin y cos y sin bcos b sin4fSðoÞ,

Va4b4¼ G�1 cos2 y sin b cos b sin4fSðoÞ, ð33Þ

Va1a2¼ Va1a3

¼ Va1a4¼ 0, Vb1b2

¼ Vb1b3¼ Vb1b4

¼ 0, ð34Þ

with G¼ ½sin2 b cos2bðsin8fþcos8fÞþsin4f cos4fðsin4bþcos4

bÞ�1=2. It is seen from Eq. (34) that no entanglements occurbetween any two high sidebands or between any two low side-bands. This is due to the fact that high sidebands a1;2,3;4 or lowsidebands b1;2,3;4 are in quantum-beat interactions, respectively,which are analyzed in Section 2. However, so long as thequantum-beat collective modes Ad and Bd are entangled, i.e.,SðoÞo0, the entanglement appears between any one high side-band and any one low sideband. In other words, the entanglementbetween the Ad and Bd modes corresponds to the entanglementsbetween high sidebands a1;2,3;4 and low sidebands b1;2,3;4, asshown in Fig. 6(A). The degree of entanglement between indivi-dual high sideband and individual low sideband depends stronglyon the ratio R. When Rb1, i.e., sin b� 1, cos b� 0, we have thenonzero correlation Va1b3

¼ G�1 sin2y sin2f cos2fSðoÞ, Va2b4¼

G�1 cos2y sin2f cos2fSðoÞ, Va1b4¼ Va2b3

¼ G�1sin y cos ysin2f cos2fSðoÞ. The entanglements between high sidebandsand low sidebands rest on the entanglements of sidebands a1;2

with sidebands b3;4 (Fig. 6(B)). And when 9O19=9O29¼9O39=9O49¼ 1, sin y¼ cos y¼ 1=

ffiffiffi2p

, we have Va1b3¼ Va2b4

¼

Va1b4¼ Va2b3

¼ G�1 sin2f cos2fSðoÞ=2. This means that the entan-glements between sidebands a1;2 and sidebands b3;4 are equiva-lent. Similarly, for R51, i.e., sin b� 0, cos b� 1, we haveVa3b1

¼ G�1sin2y sin2f cos2fSðoÞ, Va4b2¼ G�1cos2y sin2f cos2f

SðoÞ, Va3b2¼ Va4b1

¼ G�1 sin y cos y sin2f cos2fSðoÞ. The entan-glements between sidebands a3;4 and sidebands b1;2 determinethe entanglements between high sidebands and low sidebands

(Fig. 6(C)). For 9O19=9O29¼ 9O39=9O49¼ 1, Va3b1¼ Va4b2

¼ Va3b2¼

Va4b1¼ G�1 sin2f cos2fSðoÞ=2, this indicates that the entangle-

ments between sidebands a3;4 and sidebands b1;2 have the sameentangled degree.

So far, we have shown the entanglements of multiple side-bands in a four-level diamond atomic system. Experimentalrealization can be performed by using a gas of cold atoms. Agreat number of atomic structures are suitable for the present thescheme. For example, we can use the atom 87Rb, 95S1=2,F ¼ 1S,95P3=2,F 0 ¼ 2S, 95P3=2,F 0 ¼ 3S and 95D5=2,F ¼ 2S corresponding tothe levels 90S, 91S, 92S, and 93S, respectively.

4. Conclusion

In conclusion, we have shown that the eight-mode entanglementcan be obtained in a four-level diamond atomic system. Four strongcoherent fields drive the four different dipole-allowed transitions,respectively, from which four high sidebands and four low sidebandsare amplified as cavity fields. Due to atomic coherence, these highsidebands and low sidebands are pulled into a pair of doublequantum-beats, and combine into one high frequency quantum-beatcollective mode and one low frequency quantum-beat collectivemode, respectively. The two quantum-beat collective modes are inthe four-wave mixing parametric interaction with each other. It leadsto the entanglement between them. Correspondingly, any one highsideband is entangled with any one of low sidebands, but noentanglements occur between any two high sidebands or betweenany two low sidebands. Such entanglement corresponds to multi-mode entanglement. This provides an efficient way for generatingcontinuous-variable multimode entangled light with different fre-quencies, which has potential applications in quantum communica-tion and quantum network.

Acknowledgments

We would like to thank Prof. Xiangming Hu for helpful discus-sion. This work is supported by National Natural Science Foundationof China under Grant No. 11104163 and Natural Science Foundationof Hubei Province under Grant No. 2011CDC155.

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