在两个耦合的量子点和腔QED系统中的双模激子的压缩性质_英文_杨

第24卷 第6期2007年12月

原 子 与 分 子 物 理 学 报

JOURNALOFATOMICANDMOLECULARPHYSICS

Vol.24 No.6

Dec.2007

文章编号:1000-0364(2007)06-1194-05

在两个耦合的量子点和腔QED系统中的

双模激子的压缩性质

杨 雄,向少华

1

2

(1.湖南师范大学物理与信息学院,长沙410081;2.怀化学院物理与信息科学系,怀化418008)

摘 要:本文研究了在两个耦合的量子点和腔QED系统中的双模激子的压缩性质.讨论了不同的初始光场对双模激子的正常压缩与和压缩的影响.计算表明,当初始态光场制备在相干态时,双模激子既不存在正常压缩,也不存在和压缩,这说明双模激子振辐的两个正交分量具有相同的量子涨落;然而,当初始腔场处于压缩真空态时,无论是正常压缩还是和压缩,双模激子振辐的两个正交分量总有一个存在压缩.这意味着量子噪声能被有效的得到抑制.此外,两种情形下的最大压缩都由初始腔场的压缩因子r决定.经过比较,我们还发现双模激子的正常压缩比和压缩大.关键词:量子点;双模激子;正常压缩;和压缩中图分类号:O413 文献标识码:A

Squeezingpropertiesoftwo-modeexcitonsinasystemoftwo

coupledquantumdotsandcavityQED

YANGXiong1,XIANGShao-hua2

(1.DepartmentofPhysicsandInformationScience,HunanNormalUniversity,Changsha410081,China;2.DepartmentofPhysicsandInformationScience,HuaihuaUniversity,Huaihua418008,China)

Abstract:Squeezingpropertiesoftwo-modeexcitonsinasystemoftwocoupledquantumdotsandcavityQEDisinvestigated.Theeffectofthedifferentinitialcavityfieldstatesontwo-modeexcitonsnormalsqueezingandsumsqueezinghasbeendiscussed.Aconclusionhasalsobeenobtainedthroughcalculating.Itisthatwhen

thecavityfieldisinitialpreparedincoherentstates,two-modeexcitonsnormalsqueezingdonotexistaswellasthesumsqueezing.Itindicatesthatthetwoquadraturecomponentsoftheexcitonsamplitudepossessthesamequantumfluctuation.However,inthecasethatcavityfieldisinthevacuumsqueezedstate,nomatterhownormalsqueezingorsumsqueezing,eitherofthetwoquadraturecomponentoftheexcitonsamplitudearealwaysofsqueezing,itimpliesthatquantumnoisecanbeeffectiverestrainedinthecase.Otherwise,themaximumsqueezinginbothsituationsisdeterminedbytheinitialcavityfieldsqueezingfactorr.Aftercom-paringnormalsqueezingandsumsqueezing,wecanfindthattwo-modeexcitonsnormalsqueezingislargerthansumsqueezing.

Keywords:quantumdots,two-modeexciton,normalsqueezing,sumsqueezing

收稿日期:2006-04-18

基金项目:湖南省教育厅科研项目(05C696)

作者简介:杨雄(1961-),湖南宁乡人,副教授.主要从事量子光学与量子信息方面的研究.E-mail:hhyxmm01@

第6期 杨雄等:在两个耦合的量子点和腔QED系统中的双模激子的压缩性质 1195

1 Introduction

ThesqueezingofquantumfluctuationisoneofthemostfundamentalmanifestationsoftheHeisen-berguncertaintyrelations,whichareamongthemostimportantprinciplesquantummechanics.Squeezing,inparticularthatoftheradiationfieldduetoitspotentialapplicationsintheopticalcom-munications[1]andweaksingledetections[2]havenbeenextensivelystudiedforthepastdecades.ThefieldandatomicsqueezinginavarietyofJaynes-Cummingsmodel(JCM)underdifferentinitialcon-ditionshavebeendiscussedinsomepapers

[7]

[3-6]

groundenergyoftheexcitonsineachdotsisas-sumedtobethesame.Thereisaresonantinterac-tionbethesingle-modecavityfieldandtheexcitons.

ThedistancebetweentwoquantumdotsisassumedtobemuchlargethantheopticalwavelengthKofthecavityfields,sotheinteractionbetweenquan-tumdotscanbeneglected.Undertherotatingwaveapproximation,theHamiltonianforthissystemcanbewrittenas[9]:

H0=ÜXa^a+ÜXE^b+^mbm+

+

m-1

m=1

2

.

Egm(^bm^a+

+

2

^+^bm)a

(1)

Hilleryhasgeneralizedsumsqueezingasaversion

ofhigher-ordersqueezingofthetwo-modefield.

Onotherhand,Nanotechnologyopenstechno-logicalpossibilitiestofabricatemesoscopicdevices.Semiconductornanostructures,especiallyquantumdotstructures,areverypromisingfortherealizationofquantumcomputationandthequantuminforma-tionprocessing.Therehavebeenmanypapersinves-tigatingthepreparationofquantumstateaswellastheentanglementofexcitonstatesusingquantumdots[8].Butuptonow,nolettersinvestigateprop-ertiesoftwo-modeexcitonssqueezing.Thispaperwillfocusonthistopic,especially,considerthein-fluenceofthedifferentinitialcavityfieldonsqueez-ing.

where^a+(^a)aretheoperatorsofthecavityfield

withfrequencyX,b^+-m(bm)denotetheexcitonoperatorswiththesamefrequencyXofthecavityfield,andm=1(2)representthefirstdotortheseconddot.Thecouplingconstantsbetweenquantumdotone(two)andcavityfieldarerepresentbygmwithm=1,2.Withoutlossofgenerality,wecantakethetwocouplingconstantsbetweenthecavityfieldandthequantumdotsasdifferent.TheHeisenbergequationsofmotionfortheoperatorsofthecavityfield^a+(^a)andtheexciton^b+m(bm)canbeeasilyobtainedas:

=-iXa-ig1^b1-ig2^b2^5^b1

=-iX^b1-ig1^a5^b2

=-iX^b2-ig2^a(2a)(2b)(2c)

2 Hamiltonianandsolution

Themodelthatisanalyzedinthisstudycon-sistsoftwoquantumdotswhichareplaceintoasin-gle-modecavity.ThedotshavelargesizessatisfyingtheconditionRmaB.Wealsoassumethatthereareafewelectronsexcitedfromvalence-bandtoconduc-tionband.ThentheexcitationdensityoftheCoulomb-correlatedelectron-holepairs,excitons,inthegroundstateforeachquantumdotislow.This,inturn,impliesthattheaveragenumberofexcitonsisnomorethanoneforaneffectiveareaoftheexc-itonicBohrradius.Thereforeexcitonoperatorscanbeapproximatedwithbosonoperators,andallnon-lineartermsincludingexciton-excitoninteractionsThesolutionoftheaboveoperatorequationsas:

-iXt

^a(t)=cos(gt)^a(0)e-g1

isin(gt)^b1(0)e-iXt-gg2

isin(gt)^b2(0)e-iXt(3a)gg1

^b1(t)=-isin(gt)^a(0)e-iXt+

g

g12cos(gt)+g22

b1(0)e-iXt+^2

g

g1g2[cos(gt)-1]

b2(0)e-iXt(3b)^gg2-iXt

^b2(t)=-igsin(gt)^a(0)e+

g1g2[cos(gt)-1]

b1(0)e-iXt+^2

1196 原 子 与 分 子 物 理 学 报

2

g22cos(gt)+g1-^b2(0)e2

g

第24卷

iXt

(3c)

denotingexcitonmodeone(two).Toachieveex-pectedresults,wewillcalculationthevaluesFiandRiunderdifferentinitialcavitystates.

Firstconsideredthecasewherethecavityfieldispreparedinitiallyincoherentstate|A>.Then,thewholeinitialstatefortheexcitonsandthecavityfieldcanbeexpressedas:

|W(0)>=|A>á|0>

1

with g=

g1+g2

3 Theoryonexcitonssqueezing

Toconsiderthetwo-modeexcitonicsqueezing,

weusetwoslowly-varyingHermitianoperatorsde-finedas[6]:

X^1=2

-3/2

i=1

|0>

2

(14)

E[bi

2

2

+

exp(-iXt)+

(4)

+

accordingtoequations(3),makinguseofheisen-bergpicture,afterrigorouscalculation,somequan-titiescanbeobtainedasfollows:

<$X1>

(5)

or

A

FA1=F2=0

2

2

biexp(iXt)]X^2=i2

-3/2

i=1

E[biexp(-iXt)-and

=<$X1>

=4

(15)

2

biexp(iXt)]

withthecommutationrelation:

[X^1,X^2]=

2

Thevariances($X^i)2=<X^i2>-<X^i>

2

2

(6)

2

<$V1>

2

=<$V2>=

satisfy

or

R1=R2=0

A

A

theuncertaintyrelation($X1)($X2)\1/16,thenthefluctuationsinX^iaresaidtobesqueezed,if

<$X^i><4

2

(|A|2sin2gt+1)

4

(16)

(7)

or

<0(8)4

Wecallthissqueezingtwo-modeexcitonsnormalsqueezing(TMENS).

Fi=($xi)2-Forthesumsqueezing,thecorrespondingHer-mitianoperatorsaregivenby:

^1=V{b1+b2+exp[-i(X1+X2)t]+

2

b1b2exp[i(X1+X2)t]}(9)

^2={b1+b2+exp[-i(X1+X2)t]-V

2

b1b2exp[i(X1+X2)t]}(10)andtheysatisfythecommutationrelation:++

[V^1,V^2]=2(^b1b^1+^b2^b2+1)Sumsqueezingexistsif

<$Vi>or

Ri=($Vi)2-<b1+b1+b+(13)2b2+1><04

weassumethatthetwomodesofexcitonsareinitia-lly1|2

[7]

Forsimplicity,herewealreadyassumethattwoquantumdotsarecompletelythesamesothatinter-actionbetweencavityfieldandtwoquantumdotsareequaltoeachother,thatisg1=g2.

Secondly,weassumethatcavityfieldisinvac-uumsqueezingstate|F>=S^(N)|0>f,where|0>

f

expresscavityfield,S^(N)denotingsqueezing

operator,N=reiH,riscalledassqueezingfactorde-cidingsqueezingdegreeandHasphaseangledeter-miningsqueezingdirection.Thenwholesystemin-itialstatebecomes:

|W(0)>=|F>á|0>

^(NS)|0>

f

1

|0>

21

=|0>

2

á|0>

(11)

(17)

throughcalculating,wecanobtain:

2

F1=singtsinhr(sinhr+coshrcosU)

2

(18a)

F2=

2

singtsinhr(sinhr-coshrcosU)2

(18b)

and

RF1=

4

sin2gtsinh2r(cosh2r+<

<b^+b1+^b+b2+1>(12)1^2^4

w

第6期 杨雄等:在两个耦合的量子点和腔QED系统中的双模激子的压缩性质 1197

sinh2r-cosh2rcos2<+2

2coshrcos2U)2RF2=

4

sin2gtsinh2r(cosh2r+4222

sinhr-coshrsin<-2cosh2rcos2U)2

(19a)

(19)thatthereisalwaysacomponentofVicanbesqueezed.AsshowninFig.2,choosingdifferentU,adifferentcompooentofViwillbesqueezed.Forinstance,choosingU=0,thiswillresultinR1>0,R2<0.WhereaschoosingU=P/2,thenobtainR2>0,R1<0.Astotheinfluenceofsqueezingfactorronsumsqueezingisthesameasthecavityfieldin

(19b)

coherentstates.Otherwise,incontrasttotwo-modeexcitonsnormalsqueezing,wefindtwo-modeexc-itonsumsqueezingaresmallerundersamercond-ition.Forexample,forr=1,themaximumnormalsqueezingmayreachabout22%.Itisbecauseofsumsqueezingbeinghigh-ordersqueezing.

4 Resultsanddiscussion

Inthissectionwewilldiscussvarioussqueezingoftwo-modeexcitonsunderdifferentinitialcavity

fields.Whencavityfieldbeingincoherentstate,ac-cordingtoEqs.(15)and(16),thetwo-modeexc-itonsnormalsqueezingdonotexistaswellasthesumsqueezing.Itindicatesthatthetwoquadraturecom-ponentsoftheexcitonsamplitudepossessthesamequantumfluctuation.

Nextwefocusonthecasethatcavityfieldisinitialpreparedinvacuumsqueezedstate.Abouttwo-modeexcitonsnormalsqueezing(TMENS),fromEq.(18),onlywhenU=P/2,componentX2canbesqueezed.AtgtXnP(n=0,1,2,,),theresultforthesqueezingfunctionF2inEq.(18b),withdifferentr,areshowninFig.1.Wealsono-ticethateffectofsqueezingfactorrontheTMENS.Itisobviousthatthesqueezingisincreasingwithinitialsqueezingfactorrto

beincrease.

Fg.2 TimeevolutionofR1(R2)asfunctionS=gtfor

r=0.5(solidline),r=1(dashedlined)

5 Conclusion

Thetwo-modeexcitonsqueezingpropertiesun-derdifferentinitialcavityfieldarediscussed.Whencavityfieldbeingincoherentstates,itisnoticeablethattwotypesofsqueezingcannotappear.Itind-icatesthatthetwoquadraturecomponentoftheexc-itonsamplitudepossessthesamequantumfluctua-tion.However,inthecasethatcavityfieldisinthevacuumsqueezedstatenormalsqueezingandsumsqueezingareallexisted,itimpliesthatquantumnoisemayberestrained.Otherwise,themaximumsqueezinginbothcasesisdeterminedbytheinitialcavityfieldsqueezingfactorr.Aftercomparingnor-malsqueezingandsumsqueezing,wecanfindthattwo-modeexcitonsnormalsqueezingislargerthanFg.1 TimeevolutionofF2asfunctionsofS=gtforr

=0.5(solidline),r=1(dashedlined)

,

1198 原 子 与 分 子 物 理 学 报 第24卷

References:

[1] GiacobinoE,FabreC.Quantumnvisereductioninop-ticalsystems)experiments[J].Appl.Phys.B,1992,55:189.

[2] CavesCM.Quantum-mechanicalnoiseinaninterfer-ometer[J].Phys.Rev.D.1981,23:1693

[3] ZhouP,PengJS.Dipolesqueezinginthetwo-photon

Jaynes-Cummingsmodelwithsuperpositionstateprepa-ration[J].Phys.Rev.A,1991,44:3331

[4] HuangCJ,WanJY,HuangZH.Dynamicsofsingle

modefieldRamaninteractingwithtwocoupledatoms[J].J.At.Mol.Phys.,2001,18:452(inChinese).[5] WangCC,WangXC,CaoZL.Radationsqueezing

effectofthesystemofthesqueezedvacuumfieldinterat-ingwithamoving.-typethree-levelatom[J].J.At.

Mol.Phys.,2006,23:287(inChinese)

[6] Abde-lHafezAM.Degenerateandnondegeneratetwo-modenormalsqueezinginatwo-levelatomandtwo-modesystem[J].Phys.Rev.A,1992,45:6610[7] HilleryM.Sumanddifferencesqueezingoftheelectro-magneticfield[J].Phys.Rev.A,1989,40:3147[8] JohnsonNF.EntangledbellandGreenberger-Horne-Zeilingerstatesofexcitonsincoupledquantumdots[J].Phys.

Rev.

Lett.,

1999,

83:2270;ReinaJH,

QuirogaL,JohnsonNF.Quantumentanglementandinformationprocessingviaexcitonsinopticallydrivenquantumdots[J].Phys.Rev.A,2000,62:012305[9] BelleguieL,BanyaiL.Theoryofexciton-population-in-ducednonlinearabsorptioninlargemicrocrystallites[J].Phys.Rev.B,199144:8785;CaoH,PauS,Ya-maotoY,etal.Exciton-polaritonladderinasemicon-ductormicrocavity[J].Phys.Rev.B,1996,54:8083

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