Controllable operation for distant qubits in a two-dimensional quantum network

0.26 MB·English

by  Z. -R. Zhong

Controllable operation for distant qubits in a two-dimensional quantum network
Zhi-Rong Zhong1,∗ Xiu Lin1,2, Bin Zhang1, Wan-Jun Su1, and Zhen-Biao Yang3†
1.Department of Physics, Fuzhou University,
Fuzhou 350002, P. R. China
2.School of Physics and Optoelectronics Technology, Fujian Normal
University, Fuzhou 350007, P. R. China
3.Key Laboratory of Quantum Information,
University of Science and
Technology of China, CAS, Hefei 230026, P. R.China
(Dated: October20, 2011)
Weproposeatheoreticalschemetorealize thecoherentcouplingofmultipleatomsinaquantum
1 network which is composed of a two-dimensional (2D) array of coupled cavities. In the scheme,
1 the pairing off-resonant Raman transitions of different atoms, induced by the cavity modes and
0 externalfields,canleadtoselectivecouplingbetweenarbitraryatomstrappedinseparatedcavities.
2 Basedonthisphysicalmechanism,quantumgatesbetweenanypairofqubitsandparalleltwo-qubit
t operationscanbeperformedinthe2Dsystem. Theschemeprovidesanewperspectiveforcoherent
c manipulation of quantum systems in 2D quantumnetworks.
O
PACSnumbers: 03.67.Bg;03.67.-a;42.50.Pq;
9
1

Coherent manipulation of quantum systems at a dis- ena [5–11] andfor distributed quantum informationpro-
]
h tance is one of the crucial ingredients in the upcoming cessing [13–30]. All such researchesfocus on the cases of
p area of quantum technologies. The realization of quan- either two-site [9, 12, 14–24, 27, 28] or one-dimensional
- tumnetworkscomposedofmanynodesandchannelspro- (1D) [5–8, 10, 11, 25, 26, 29, 30] coupled cavity arrays.
t
n vide opportunities for such purpose, and thus is of great Extendingsuchstudiestotwo-(2D)orthree-dimensional
a consequence to a series of frontiers, such as quantum (3D) coupled cavity arrays is in some sense of more sig-
u
computation, communication and metrology [1]. Fun- nificance, as it is shown that some kinds of 2D and 3D
q
damental to quantum networks are quantum intercon- quantumstates(say,clusterstates)areuniversalresource
[
nects, which achieve reversible quantum state transfor- for quantum computation [31]. There have been studies

1 mation among separate systems across network nodes. consideringthe 2D coupled cavityarrays,the correlative
v
Such quantum connectivity in quantum networkscan be examplescanbefoundinRef. [32]and[33],whichrespec-
1
realized by matter-light interaction in cavity quantum tively consider the realization of the fractional quantum
3
3 electrodynamicssetupsatnodes[2],combinedwithchan- Hall system and 2D one-way quantum computation. In
4 nels of light tunneling across the nodes [3]. As kinds these schemes [32, 33], the nearest-neighbor coupling is
. of matter particles (say, atoms, quantum dots, nitrogen- essential for the nonlocal quantum coherence across the
0
vacancy (NV) centres in diamond, etc), distributed in 2D network nodes.
1
1 network nodes, can act as stationary qubits for infor-
In this paper, we first propose a scheme to control
1 mation storage; while single photons are suitable to act
coherently the coupling between two arbitrary atomic
: as flying qubits, that are convenient for conveying from
v qubits at distant (not necessarily the nearest-neighbor)
andto the nodes the informationexchangedthroughthe
i nodes in a 2D array of coupled cavities. The scheme
X matter-light interaction.
follows a previous study by Zheng et al. [30], which
r Inrecentyears,muchattentionhas beenpaidto using
a deals with a 1D case considering the neighboring cavi-
the atom-light interaction in coupled cavity arrays for
ties that are linked by other resonators such as ’short
investigating novel physical phenomena and its possible
optical fibers’. In the scheme, the off-resonant Raman
applications [4]. Coupled cavity arrays possess advan-
transitions between two ground states of the atoms, in-
tagesoversomeothersystemssuchasJosephsonjunction
duced by the cavity modes and the external fields, can
arraysandopticallattices,inrespectthatincoupledcav-
lead to selective coupling between arbitrary two distant
itysystemstheneighboringsitesareusuallyseparatedby
atoms across the 2D network nodes, while with all the
dozens of micrometres thus it is convenient for allowing
atoms as well as with all the cavity modes only virtually
access to individual site.
excited. Quantum logic operations between any pair of
Manytheoreticalstudiesconcerningtheuseofcoupled
distant atomic qubits and parallel two-qubit operations
cavity arrays have been done, as to realize controllable
onselectivequbitpairscanbeimplementedbyappropri-
operationforquantumsimulationofmany-bodyphenom-
ately selecting the parameters of the external fields.
WeconsideracoupledN N cavityarray,asshownin
×
FIG. 1. In each site (denoted here by jk) there are two
∗Electronicaddress: [email protected] cavity modes, which respectively couple to their neigh-
†Electronicaddress: [email protected] boring ones through the x and y directions with inter-
2
(cid:91)
(cid:92)
g
g
FIG. 1: (Color online) Schematic diagram of a two-
g
dimensional (2D) array of coupled cavities. Each atom is
confined in theintersection of two orthogonal cavity modes.
FIG. 2: The atom level scheme. The transitions of the jkth
atom |gi ↔ |ei of the jkth atom are coupled to the two
jk jk
cavity photon hopping. The interaction Hamiltonian for orthogonal cavity modes with detuning ∆xjk,1 and ∆yjk,1, the
the coupling between the cavities can be modeled by corresponding coupling rates are gx and gy , respectively.
jk jk
The transition |fi ↔ |ei of the jkth atom is driven by
N jk jk
H1 =X[vxaxjkaxjk++1+vyayjkayj++1k+h.c.], (1) twwitohctlahsesiccoarlrelaspseorndfiienldgsRwaibthi ftrheqeudeentcuiensinbgei∆ngxjkΩ,2xanadnd∆Ωyjky,2,,
j,k jk jk
respectively.
where au (u = x,y) denotes the annihilation operator
jk
for the mode of the jkth cavity, and vu is the coupling
where ωx = 2vxcos(2πn),ωy = 2vycos(2πm). We
ratebetweencavitiesintheuchannel. Eachsitecontains mn N mn N
now go into a new frame by defining H as a free Hamil-
a Λ-type atom, with two ground states g and f , 1
| ijk | ijk tonian, and obtain the interaction Hamiltonian for the
andoneexcitedstate e ,asshowninFIG.2. Theatom
| ijk whole system as
interacts with these cavity modes throughthe transition
g e , with coupling rates gx and gy , and de-
| ijk ↔ | ijk jk jk N
ftruenqiunegnsc∆iesxjkΩ,1xjkanadnd∆Ωyjkyj,k1.aTrewaoppcllaiesdsictaoldfireilvdestwheithatRomabici H2′ = Xj,k uX=x,y{Ωujkei△ujk,2t|eijkhf|
transition f g ,withdetunings∆x and∆y . gu
Theinterac|tiiojnk ↔Ham| iiljtkoniandescribingthejkin,2teractiojnk,o2f +X[ jke−i(2πNjm+2πNkn) ei(△ujk,1−ωmun)t
N
the atoms with the cavity modes and classical fields can m,n
be written as [30, 34] ×cum,n|eijkhg|+h.c.]}. (5)
N Considering the large detuning case with
H2 = jX,k=1uX=x,y [gjukaujk|eijkhg|ei△ujk,1t (cid:12)(cid:12)△ujk,2(cid:12)(cid:12) ≫ (cid:12)(cid:12)Ωujk(cid:12)(cid:12),(cid:12)(cid:12)△ujk,1−ωmun(cid:12)(cid:12) ≫ (cid:12)(cid:12)gNjuk(cid:12)(cid:12), and
+Ωujk|eijkhf|ei△ujk,2t+h.c.]. (2) (cid:12)(cid:12)(cid:12)(cid:12)x∆,yujk;,u1(cid:12)−=ωumu′)n, −w(cid:12)e∆cujak′(cid:12)n,2(cid:12)(cid:12)(cid:12)(cid:12)ad≫iab(cid:12)(cid:12)(cid:12)∆atujikc,a1l−ly(cid:12)ωemluimni−na∆teuj(cid:12)kt,2h(cid:12)(cid:12)(cid:12)e(cid:12)(uat,oum′ =ic
Weadoptperiodicboundaryconditionsauj1 =ajN and excited6 state e and turn H′ to
au1k = aNk, by introducing the nonlocal bosonic modes | ijk 2
cu (u = x,y), and making the transformation au = ′′
1mn N exp[ i(2πjm + 2πkn)]cu . Thus we can rewjkrite H2 = −X X { εujk|fijkhf|+X[ζjuk|gijkhg|cum+ncumn
N Pm,n − N N mn j,k u=x,y m,n
the Hamiltonian H and H as
1 2 +λujke−i(2πNjm+2πNkn)ei(△ujk,1−ωmun−△ujk,2)tcumnSj+k
N +h.c.] , (6)
H1 = uX=x,yXm,nωmuncum+ncumn+h.c., (3) where εu =} (Ωujk)2, ζu = (gjuk)2 , λu =
jk ∆ujk,2 jk N2(∆ujk,1−ωmun) jk
and gjukΩujk( 1 + 1 ), and S+ = f g . The
H2 = X X [Ωujkei△ujk,2t|eijkhf|+XgNjuk firer2ssNpteacntid∆veujskl,ey1c−oinωnmudduncteerdm∆bsujyko,2tfhEeqc.la(s6si)cjadklesficerlidb|seiajSkntdhar|kbosshoinftics
j,k u=x,y m,n
modes, while the last two terms describe the multiple
×e−i(2πNjm+2πNkn)ei△ujk,1t|eijkhg|cumn+h.c.],(4) off-resonantRamantransitionsforeachatominducedby
3
theclassicalfieldsandthebosonicmodes. Underthecon- the state of the jkth qubit is now transferred to that
dition (cid:12) u ωu u (cid:12) λu , the bosonic modes of the pqth qubit, i.e., the two qubits are finally in
areonl(cid:12)(cid:12)y△vjikr,t1u−allymenxc−ite△dj.kT,2h(cid:12)(cid:12)i≫sthujksleadstothequanta- ψ st = g jk( ie−iς1χπc0 f pq+c1 g pq).
| i | i − | i | i
dependentStarkshiftsandeffectivecouplingbetweenthe We notice that the selective parallel two-qubit op-
atoms, and gives the effective Hamiltonian [35, 36] eration on different qubit pairs can also been imple-
mented in such a 2D model. Suppose that one wants
He = X X { −εujk|fijkhf|+X[−ζjuk|gijkhg|cum+ncumn Ttohpenerfwoermdrgivaeteseaocnhqoufbtithepsaeirqsu(bjikts,pwq)itahndtw(oj′ckl′a,sps′iqc′a).l
j,k u=x,y m,n
fields. The parameters of the system are suitably ad-
+ξu (cu cu+ f f cu+cu g g )]
jkmn mn mn| ijkh |− mn mn| ijkh | justed so that all such conditions gu = gu , Ωu =
jk pq jk
+ X χujkpqSp+qSj−k ΩΩupuq, △=ujΩk,1u =, △uupq,1,=△ujku,2 =, △uupq,2, =gju′k′u= ,gpua′nq′d,
p,q(pq6=jk) j′k′ p′q′ △j′k′,1 △p′q′,1 △j′k′,2 △p′q′,2
×e−i[(△upq,1−△ujk,1)−(△upq,2−△ujk,2)]t+h.c}, (7) (cid:12)(cid:12)∆µαβ,1−∆µαβ,2−∆µα′β′,1+∆µα′β′,2(cid:12)(cid:12) ≫ χµαβα′β′(αβ =
j(cid:12)k,pq;α′β′ =j′k′,p′q′) are also sat(cid:12)isfied. In such a case,
with
qubit jk (j′k′) only couples to qubit pq (p′q′), while it
(λu )2 decouplestoqubitsj′k′ (jk)andp′q′ (pq). Thereforethe
ξjukmn = u ωjuk u , (8) effective Hamiltonian is given by
△jk,1− mn−△jk,2
H = ς (f f + f f )+ χ (S+S− +h.c)
and e,par 1 | ijkh | | ipqh | 1 pq jk
1 1 + ς2(|fij′k′hf|+|fip′q′hf|)+ χ2(Sp+′q′Sj−′k′
χujkpq = X2λujkλupq( u ωu u +h.c), (12)
m,n △jk,1− mn−△jk,2
+△upq,1−ωmu1n−△upq,2)× wcPohhuee=rrexe,nyς2tuχlyuj=′kp′Ppe′rqfm′o,r,namnξjdut′wkj′om′k-nq′u6=−biεptuj′′qok′p′.,eTrςa2hti=isontPshuous=naxq,lyluoςbw2uist, uχps2aitr=os
e−i[2(j−p)mNπ+2(k−q)nNπ]. (9) (jk,pq) and (j′k′,p′q′) simultaneously.
To confirm the validity of all our above arguments,
As the quantum number of the bosonic modes is con-
we numerically simulate the dynamics governed by the
served during the interaction, they will remain in the
derived effective model in Eq. (11), and compare it to
vacuum state if they are initially in the vacuum state.
the dynamics governedby the full Hamiltonian
Then H reduces to
e
N
He = X X { ς1u|fijkhf|+ X [χujkpqSp+qSj−k Hf = X{ωgf|fijkhf|+ωge|eijkhe|+ X [ωcu,jkaujk+aujk
j,k u=x,y p,q(pq6=jk) j,k=1 u=x,y
×e−i[(△upq,1−△ujk,1)−(△upq,2−△ujk,2)]t+h.c]}, (10) +gjukaujk|eijkhg|+Ωujk|eijkhf|e−iωlu,jkt
where ς1u =Pm,nξjukmn−εujk. + vxaxjkaxj++1k+vyayjkayjk++1+h.c.]}, (13)
The Hamiltonian in Eq. (10) allows for the coherent
whereω andω arefrequenciesforthestate g and
gf ge jk
operation of two arbitrary distant qubits across the 2D e (g is assumedto be null energylevel), a|nidωu
quantum network. In order to do so, we apply classical a|nidjkωu| ijakrefrequenciesforthecavityandclassicalfielcd,jsk.
fieldstothejkthandpqthqubits,andselectthefrequen- l,jk
WeconsiderthecasewithN =2,andsettheparameters
cies for the cavity and classical fields in such a way that in the following way (u = x,y): vu = g, Ωu = Ωu =
t△tohnupeqic,a2on−nHd△ietujiokre,n2dsaugrcjueeksa=ltsoogpufqu,lfiΩllujekd=. TΩhupuq,satnhde e△ffupeqc,t1i−ve△Hujak,m1i=l- 1△2.35x2.g29,g,2,g=a1u1n1d=3.Ω9ggu2u,2△==y110,1g(,j=k△△x1=1y2,121,21=,=21△)2.x202g,T1,h△=ey1v11,a20l1ig1d=,it△y△x1y2o12f2,,222th==e
jk
H = ς (f f + f f )+ χ (S+S− +h.c), effective model is numerically simulated by taking the
e,jkpq 1 | ijkh | | ipqh | 1 pq jk evolutionoftheoccupationprobabilityP = ψ ψ(t) 2of
(11) |h | i|
with ς1 = Pu=x,yς1u, χ1 = Pu=x,yχujkpq, and jk 6= pq. tahsseusmtainteg|iψniiti=all|yfit1h1e|gai1t2o|mgis21a|rgei2i2nasstaatne eψxamanpdle,alwlhthilee
We assume the jkth and pqth qubits are first in state | i
cavity modes are in vacuum state. FIG. 3 illustrates
ψ(0) = f g , the time evolution can be expressed
jk pq
|as ψ(it) =| ie−i|Hie,jkpqt ψ(0) =e−iς1t[cos(χ1t)f jk g pq the numerical results obtained from both the effective
isi|n(χ ti)g f ]. A| fterian interaction tim|ei π| ,ith−e (red-solidline) and full (blue-dashed line) Hamiltonians.
1 | ijk| ipq 4χ1 Discrepancies between the two curves are due to higher
two qubits evolve to a maximal entangled state ψ =
e−i4ς1χπ1(f jk g pq ig jk f pq)/√2. Or,ifthe two| qiuenbits toebrvmiosusfotrhatthethpeaerffaemcteitveersmςo1dealnisdvχal1i.d,iEtsvednevniaotwio,nitcains
are in |stiate| iψ′(−0)| i=|(ic0 f jk + c1 g jk)g pq, with be madeto be smallerassoonasthe relativeparameters
| i | i | i | i
|c0|2 + |c1|2 = 1. After an interaction time t = χπ1, are appropriately fixed.
4
The probability that the atoms undergo a transition
1 to the excited state due to the off-resonant interaction
0.9 with the classical fields is p1 = (∆2x2Ω2,21)2 = 2.25×10−2.
Meanwhile, the probability that the field modes
0.8
are excited due to off-resonant Raman couplings is
0.7 p = (λujk)2 = 4.96 10−4. The ef-
0.6 2 PuPjk △1jkx+ωmxn−△u2jkx ×
p(f11) fective decoherence rates due to the atomic spontaneous
0.5 emission and the field decay are γ =p γ and κ =p κ,
e 1 e 2
0.4 where γ andκ are the decayratesfor the atomic excited
state and the field modes, respectively. The parameter
0.3
in a strongly coupled single quantum dot-cavity system
0.2 reported in Ref. [37] is κ 5 10−4g(κ = 1 ),
∼ × 1800
0.1 and γ 3 10−3g(κ = 1 ). This corresponds to a
∼ × 300
cooperativity factor g2/2γκ 105, which is assumed to
0 ∼
0 100 200 300 400 500 be available[38]. Then the effective decoherentrates are
gt γ 6.75 10−5g and κ 2 10−7g . This leads a
e e
∼ × ∼ ×
gate fidelity with F 1 (γ +κ )t 99%.
e e
≃ − ≃
FIG. 3: (Color online) Numerical illustration for the time
2 In conclusion, we propose a theoretical scheme to re-
evolution of the occupation probability P = |hψ|ψ(t)i| gov-
alize the coherentcoupling between two arbitraryqubits
erned respectively by the effective (red-solid line) and full
(blue-dashed line) models, the atoms are initially in state in a 2D quantum network. We consider the scheme in
|ψi = |fi11|gi12|gi21|gi22, while all the cavity modes are a 2D coupled cavity system. In such a case, the pairing
in vacuum state. The relative parameters are set to be off-resonantRamantransitionsofdistantatoms,induced
(u=x,y): vu =g,Ωu11 =Ωu22 =1.5g, g1u1 =g2u2 =g,△x11,1 = by the cavity modes and external fields, can lead to se-
△x22,1 = 10g, △x11,2 = △x22,2 = 13.9g, △y11,1 = △y22,1 = 20g, lective coupling between two arbitrary atoms trapped in
△y11,2 =△y22,2 =23.9g, and Ωujk =0 (jk =12,21). separated cavities; quantum gates between any pair of
qubits and parallel two-qubit operations in the network
can be performed. The scheme can also be applied to
It is necessary to give a brief discussion of the N-V ensemble-based quantum network [39, 40].
experimental feasibility of the proposed scheme.
We thank Zhang-Qi Yin for helpful discussion. This
For N = 2 and for the parameters introduced
in FIG. (3), we get χ = χu = work is supported from the Natural Science Founda-
1 Pu=x,y jkpq tion of Fujian Province under Grant No. 2009J06002.
1λu λu ( 1 +
Pu=x,yPm,n 2 11 22 △111x−ωmxn−△211x Z.B.Y is funded by the National Basic Research Pro-
△122x−ωmx1n−△222x)e−i[2mNπ+2nNπ] = 7.67 × 10−3g, and gram of China under Grants No. 2011CB921200 and
the time needed to complete the entangling operation No. 2011CBA00200,andtheChinaPostdoctoralScience
between qubit 11 and qubit 22 is t = 0.52 102/g. Foundation under Grant No. 20110490828.
×
[1] H.J. Kimble, Nature(London)453, 1023 (2008). Mol. Opt. Phys42, 065502 (2009).
[2] S.HarocheandJ. M.Raimond,ExploringtheQuantum [11] Z. X. Chen, Z. W. Zhou, X. X. Zhou, X. F. Zhou, and
(Oxford University Press, Oxford,2006). G.C Guo, Phys. Rev.A 81, 022303 (2010)
[3] D.P. DiVincenzo, Fortschr. Phys. 48, 771 (2000). [12] A.C.Ji,Q.Sun,X.C.Xie,W.M.Liu,Phys.Rev.Lett.
[4] M.J.Hartmann,F.G.S.L.Brandao,andM.B.Plenio, 102, 023602 (2009).
Laser Photon. Rev.2, 527 (2008). [13] J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi,
[5] M.J.Hartmann,F.G.S.L.Brandao,andM.B.Plenio, Phys. Rev.Lett. 78, 3221 (1997).
Nat. Phys. 2, 849 (2006); Phys. Rev. Lett. 99, 160501 [14] T. Pellizzari, Phys.Rev.Lett.79,5242 (1997); S.J.van
(2007). Enk, H. J. Kimble, J. I. Cirac and P. Zoller, Phys. Rev.
[6] A. D. Greentree, C. Tahan, J. H. Cole, L. C. L. Hollen- A 59, 2659 (1999).
berg, Nat. Phys.2, 856 (2006). [15] A.Serafini,S.Mancini,andS.Bose,Phys.Rev.Lett.96,
[7] D. G. Angelakis, M. F. Santos, and S. Bose, Phys. Rev. 010503 (2006).
A 76, 031805(R) (2007). [16] Z.Q.YinandF.L.Li,Phys.Rev.A75, 012324 (2007);
[8] J. Cho, D. G. Angelakis, and S. Bose, Phys. Rev. A 78, Z.Q.Yin,F.L.Li,andP.Peng,Phys.Rev.A76,062311
062338 (2008). (2007).
[9] E.K.Irish,Phys.Rev.A80,043825(2009).C.D.Ogden, [17] J. Song, Y. Xia, H. S. Song, J. L. Guo, and J. Nie, Eur.
E. K. Irish, and M. S. Kim, Phys. Rev. A 78, 063805 Phys. Lett. 80, 60001 (2007).
(2008). [18] S. Y. Ye, Z. R. Zhong, and S. B. Zheng, Phys. Rev. A
[10] L. Zhou, W. B. Yan, and X. Y. Zhao, J. Phys. B: At. 77, 014303 (2008).
5
[19] X. Y. Lu¨, J. B. Liu, C. L. Ding, J. H. Li, Phys. Rev. A [30] S. B. Zheng, C. P. Yang, and F. Nori, Phys. Rev. A 82,
78, 032305 (2008). 042327 (2010).
[20] S.B. Zheng,Appl. Phys.Lett. 94, 154101 (2009). [31] R. Raussendorf and H. J. Briegel, Phys. Rev. Lett. 86,
[21] S. B. Zheng, Z. B. Yang, and Y. Xia, Phys. Rev. A 81, 5188 (2001).
015804 (2010). [32] J. Cho, D. G. Angelakis, and S. Bose, Phys. Rev. Lett.
[22] Z.B. Yang,H.Z. Wu,W.J. Su,and S.B. Zheng,Phys. 101, 246809 (2008).
Rev.A 80, 012305 (2009); Z. B. Yang, S.Y. Ye,A. Ser- [33] G. W. Lin, X. B. Zou, X. M. Lin, and G. C. Guo, Appl.
afini, and S. B. Zheng, J. Phys. B: At. Mol. Opt. Phys. Phys. Lett. 95, 224102 (2009).
43, 085506 (2010). [34] Y. Wu,Phys. Rev.A 54, 1586 (1996).
[23] J. Song, X. D. Sun, Y. Xia, and H. S. Song, Phys. Rev. [35] A.Imamoglu,D.D.Awschalom,G.Burkard,etal.,Phys.
A.83, 052309 (2011). Rev. Lett.83, 4204 (1999).
[24] J. Cho, D. G. Angelakis, S. Bose, Phys. Rev. A 78, [36] S. B. Zheng and G. C. Guo, Phys. Rev. Lett. 85, 2392
022323 (2008). (2000).
[25] L. Zhou,Z. R. Gong, Y. X.Liu, C. P.Sun,and F. Nori, [37] K.Hennessy,A.Badolato, M.Winger,et.al., Nature445
Phys.Rev.Lett. 101, 100510 (2008). , 896 (2007).
[26] P.B.Li,Y.Gu,Q.H.Gong, andG.C.Guo,Eur.Phys. [38] S. M. Spillane, T. J. Kippenberg, K. J. Vahala, et al.,
J. D 55, 205 (2009); Phys.Rev.A 79, 042339 (2009). Phys. Rev.A 71, 013817 (2005).
[27] S. M. Giampaolo and F. Illuminati, Phys. Rev. A 80, [39] S. C. Benjamin, B. W. Lovett, and J. M. Smith, Laser
050301 (2009). Photo. Rev.3, 556 (2009).
[28] K.Zhang and Z. Y. Li, Phys.Rev. A 81, 033843 (2010) [40] W. L. Yang, Y. Hu, Z. Q. Yin, et al., Phys. Rev. A 83,
[29] J. Song, Y. Xia, and H. S. Song, Appl. Phys. Lett. 96, 022302 (2011).
071102 (2010).