Tk h i Schrodingera rakhinimriln g tg i ithchnghchlEPR [1] n m 1935, c c nh khoahcs mc t ngvm t hth ngthngtinln g tm t r u n g tmcanc h nhlc cmytnhln g t. t ngv ht h n g t n h t o n l n g t v m y t nh ln g t cara l nu ti n b i Manin [2] v o n m 1980 vb i Feynman n m 1982 [3].Chonnay,c thn i c ct ngcb n trt h n h h i n th c,nhi u qu c gia m nhvkinhtvc n g n g h nhM ,TrungQuc,c,Ngax e m vi cphttrinht h ngmytnhln g tn iringvl nhvcthngtinln g tn ichu nglc h i nlcqucgianntonhng i uki n thu nl inh t vu tr tl n cho l nhv c ny T nhnn m2021, cn h i u h n g v t r u n g t m c n g n g h c a c c n c n y n g h i n cuvcht othnhcngccmhnhthnghimmytnhln g t, inhnhnhh ngIBMcQ SystemsOne,hngGooglecS y c a m o r e , ih c Khoa h c vC ng ngh Trung Qu c cZuchongzhi.l c c hth ng m y t nh ln g tl nlt c2 7 , 5 4 v6 6 q u b i t Ttcchnguthhi n nh ngui mvtxa c c si u m y t nh ci n m n h n h t h i n nay nhF u g a k u , S u m m i t v S i e r r a T u y n h i n c h n g v n l c c m h n h thn g h i m v c n b c lnhi u nhc i m v t n h n n h c n g n h t n h ch nh x c, nh tlc c t nh to nih i sq u b i t h o t n g l n V k h i ,t nh chtr i c a c c tr ng th i ln g tcsd ng trn n b t n n h , khkimsotvnhiuvnkthutkhc.
2 ngcaChcl ngtt r o n g ngchl r iln g t.Datrnvicmt chmlasern modeb ngtrngthikthpc xut biGlaubervS u d a r s h a n nm1963[4],
[5],hnglotcctrngthi phicincccnhk h o a hctrnthg i iaranht r ngth ikth p hai photon, tr ng th i k t h p c p [6], tr ng th i k t h p hai modeSU(1,1) th m photonn , t r n g t h i n n [ 7 ] , t r n g t h i c h n k h n g n n haimodethmvb t photon[8].Trongnc,nhmtcgiNguynB n,Trn g Minhc cngcccngscnga racctrngthimivc n g trnc ct pc h q u ct cuyt nnh t r ngt h ik thpbb a [9],trngthinndchchuynthmph otonhaimode[10]vn h i utrngthikhc.Cctnhchtphicincanhngt r ng thin h t nh chtn n t ng vhi u, t nh chtph n ktchm,t nh chtphi Gauss,tnhchta n ri, c ngc quantmnghincu[8],[10],[11],[12], [13].
Trong nhng t nh ch t phici nn u tr n, t nhan r ingvaitrquan tr ng trong vi c t o ra c c ngu n r i ph c vcho c c l nh v c th ngtin l ngtv vi nt i l ng t Do,c c nhV t l lthuytaraccmhnhtoracctrngthiphici n ct nhchtrinhm h nh t o ra tr ng th i k t h p c p c a Dong vc c c ng sn m 2008 [14],mh n h t o r a t r n g t h i kth pbba c a nh mt cgiNguynB nvc ccngsn m2013[15], V i ctoraccngunritrnlt h u y t ih i ph i c ki m tra th ng qua c c ti u chu n an r i phh p V y n n c c ti u chu n hayi u k i n k i m t r a t n h a n r i c n g c q u a n t m nghi nc u.N m 1996,Peresara ti uchuntchc ivimatrnmt[ 1 6 ] Nm1997,Horodeckia ratiuchuntchc v kh ngtch civitr ng th i pha tr n [17].Tm t lo t c cti u chu nctht ch civic ch l ngtcng c xu t nhti u chu ntchc choh cbi n li nt c c aSimon vc c c ngsn m2000 [18] vm t st i u c h u n k h c [ 1 9 ]
N m 2 0 0 3 H i l l e r y v Z u b a i r y a rai ukina n richohh a i mode[20].Cngnm2006,Nh avK i m aracctiuchuna n ridavoht h cb tn h trongi sS U ( 2 ) vS U ( 1 , 1 ) [21]tp h thinrac ctrngthia n riphiGauss.Tip ,c ci ukina n r ichohh a i mode,bamodeva modecngc a ra.i n h n h l n m 2 0 0 7 , n h m t c g i L i , F e i ,
W a n g v W u a rai u k i n a n r i c h o c c t r n g t h i a m o d e [ 2 2 ] ; n m 2 0 0 8 n h m t c giDuc, Noh vK i m arati uchun an r i d avo b t ng th cchotrngthikthpcpvb b a [8].Nhnchung,cctiuchunhay i uki na n r i c ara ch y u d a tr n ht h c b t n h g i a x u n g lngvt a,sv i phmbtngthcCauchy-Schwarz.
Hi n nay cc l nhv cth ng tinl ng t ,vi nt i l ng t ,m y t nhl ng tang c nghi n c ur ts ingtr n kh p th gi i Vihi v ngg p ph n v o sph t tri n chung, c h n g t i c h n K h o s t c c t n h ch t,x u t c c t i u c h u n a n r i v n g d n g c a m t s tr ng th i phi ci n h a i v b a m o d e m i l m t i n g h i n c u c alu nn C c ngg p m ic alu nn bao g m:thnh tla rahai tr ng th i phi ci n hai mode m i, thhai ll m r c c t n h c h t phicin c am tstr ng th i phici nhai vba mode m i.Thbalxu t m t ti u chu na n ri m i cho hh a i m o d e Th t l nhgim c t h n h c n g c aqu tr nh vi nt i l ng tvingu n r ilc ctrngthihaimodemi.Vinhngn g gp,lunngpphnhu ch cho sp h t t r i n c a l n h v c Q u a n g l n g t n i c h u n g v l n h v c Thngtinlngt,mytnhlngtniring.
Mc ti u chungc a t i lxu tc c c tr ng th i phic i nhai mode m i, x y d ngc t i u c h u n a n r i m i c h o h h a i m o d e , l m rc c c t nhchtc am tstr ng th i phic i nhaivb a modemi,nhgic mct h nhcngcaccqut r nhvintiln g tv ingunrilc ctrngthiphici nmi.Trncs ,mctiuc thcalunnl:
- arac hai tr ng th i phic i nhai mode m i b ng ph ngphpthm,btphotonnhxvk h ngnhx ;
- xu tcmttiuchuna n rimidt mvnhln g an ri chohh a i m o d e ;
- L m rc c c t n h c h t p h i c i n c a c c t r n g t h i h a i v b a modeminhtnhchtnn,tnhchtphnktchm,tnhchtphiGaussthngquah mWignervt nhchtanri;
- Sd n g c c trng th i hai mode m ixu t voc c qut r n h vintiln g tmt trngthikthp , ngthin h gim ct h nhcngcaccqutrnhthngquatrun gthctrungbnh.
3.itn g vphmvinghincu it ngnghi n c u c at i lc c tr ng th i phic i nhai vba mode; c c t nh ch t phi ci n b a o g m t n h c h t n n , t n h c h t p h n ktchm,tnhchtphiGauss,tnhchtanri;cctiuchunanrivccmhnh vintilngt.
Nidungnghincucaticgiihntrongccphmvisau y C c tr ng th i phi cc n g h i n c u t r o n g p h m v i h a i v b a m o d e c a trn g i n t Ti u chu na n r i m i c xu t chd n h c h o h haimode.Cctnhchtphici nc nghincubaogmtnhchtn n, t nh ch t ph n k t ch m, t nh ch t phi Gauss vt nh ch ta n r i Ngu na n r i c sd ng cho qut r n h v i n t i l ng tbao g m batr ng th i phici n hai modeltr ng th i kth p cpth m vbtphoton hai mode, tr ng th i kt h p c p c h n g c h t t h m p h o t o n v t r n g thikthpcpchngchtthmphotonvb tphoton.Giaothcvintics dngbaogmhailoilgiaothco ccthnhphntrc giaovg i a o thco tngsh tvh i upha.
Trongqutrnhnghincuc cnidungchnhcalunn chngti sdngccphn g phpsau:
- Ph ng ph p ln g t h a l n h a i c sd n g t r o n g q u t r n h t n h to n gi i t chx y d ng tr ng th i m i, x y d ng ti u chu na n r i m i vpdng,nghincuc ctnhchtphicincac ctrngthimi;
- Phn g p h p t h n g k l n g tcsd ng khixy d ng c c bi uth c gi i t chc a trung th c trung b nh nh m nh gim c t h n h cngcaqutrnhvintilngt;
- Phn g p h p t n h s v v t h cp d n g n h g i c c k t qug i it cht h u c t r o n g h uh tc cn id u n g n g h i nc uc h nhc a ti.Phnmmcsdngtnhsvv thl M a t h e m a t i c a
5.immicat i tilu nn cm tsi mmisau:
- arac hai tr ng th i phic i nhai mode m i b ng ph ngphpthmvb tphotonnhxvk h ngnhx;
- xutc tiuchuna n rimichohh a i modettonthiuphadng Hermitevt o nthiush tcatrn g int;
- L m rm t st nh ch t phi ci n c a c c t r n g t h i h a i v b a mode m i bao g m t nh ch t n n, t nh ch t ph n k t ch m, t nh ch t phiGaussvtnhchtanri;
- Chr a csth nh c ng c a c c qut r n h v i n t i l ng tvingunanric sdnglcctrngthiphici nhaimodemi.
Vica rahaitrngthihaimodem ivn g h i ncuchititmtst nh ch t phi ci n c a c h n g c n g v i m t s t r n g t h i h a i v b a m o d e m i kh c lc s q u a n t r n g c h o c c n h l t h u y t v t h c n g h i m n g h i n c u vpd ng ch ng voc c nhi mv l ng t Ph ng ph p th m, btphotonc s d n g t r o n g q u t r n h x y d n g t r n g t h i m i c n g h a quan tr ng trong vi c t ngcng c cct nh phic i n tg p ph nnngcaomcthnhcngkhip dngcctrngthimivothctin.
Thngquavicp dngcctrngthimiv oqut r nhvintiln g t ,lu nn n g g p c s l thuytcho vi cxy d ng v c i ti nc c mh n h l t h u y t c n g n h m h n h t h c n g h i m c a q u t r n h v i n tisdngcctrngthimitrongtnglai.
Vicxydngthnhcngtiuchunanrimidatrntonthiup ha cd n gHermite vt o nth i ush tcn g h atrongv i cdtmvnhlngan ric acctrngthihaimodecatrng i n t ,cbi tlc c tr ng th i my u tpha vs h t cvai trq u a n trng.
Bc c lu nn g m ba ph n: ph n mu,ph n n i dung v ph nk t lu n Ngo i ra c n cdanh m c kh i u v i t t t , d a n h s c h h n h v , danh m c c c c ng tr nh khoa h cc n g b l i n q u a n n c c k t q u nghincucalunn,tiliuthamkhovp h lc.
- Ph n mutr nh b yldo ch n t i, mcti u nghi n c u,itn g vphmvinghincu,phn g phpnghi ncu,i mmicati,nghakhoahcvt h ctincalunn,bc clunn.
- Phnnidungbaogmbnchn g Chn g mttrnhbyvc s lthuytc at i Chng haitr nhby vv i c ara hai tr ng th iphic i nhai mode m i v xu t ti u chu na n r i m i C h n g b a tr nhby vnghi n c u c c t nh ch t phici nc a c c tr ng th iamodemi.Chn g bntrnhbyngdngcctrngthimivovint ilngt.
- Phnktluntrnhbyvnhngktqut c , n h gim c tc so vimcti ura, c cu nh ci mc at i,hn g khcphcvphttrincati.
0 6 c n g t r n h d id ng c c b i b o khoa h c, trong c0 1 b i c ng tr n t pchchuy n ng nh n m trong hth ng SCIE (Journal of ComputationalElectronics), 01 b ic n g t r n t p c h c h u y n n g n h n m t r o n g h th ng SCOPUS (Journal of Physics: Conference Series), c0 2 b i n g tr n t p ch chuy n ng nh trong nc thu c danh m c ACI (Hue UniversityJournal of Science: Natural Science vCommunicationsin Physics) vc02b ig in g t r nc ct pc h c h u y nn g nhq u ct n m t r o n g hth ngSCIE-SCOPUS (Journal of Physics B: Atomic, Molecular andOpticalPhysicsvInt er na tio na l JournalofTheoreticalPhysics).
Tr ngthiFock,trngthikthpvm tst r n g thiphi cin
Gi|n⟩ltrngthiringcaHamiltoniantrn g i ntn g vitr ri ng E n ,vto nt aˆ † ,aˆt ngnglto ntsinhvto nthuphoton Σ
Do aˆ † aˆ|n⟩= n|n⟩ (1.4) i ucngh a|n⟩ltr ngth iri ngc ato ntsh t N ˆ= aˆ † aˆ.Tr ng th i|n⟩c g iltr ng th i sh t h a y t r n g t h i F o c k , n g h a ltr ngth i cs h t x c n h v c kh i qu tttr ng th i ch n kh ng Tr ngth iFockcbiudindidng aˆ † n trong| 0⟩khiuchotrngthichnkhng,nc m tst nhcht aˆ|0⟩=0, trc chun aˆ
† |0⟩=|1⟩,⟨0|0⟩=1.Tr ngth iFockth am ni uki n
Trngthikthplnutinc Schrodingergiithiunm1926[23].Saut r ng thinycnghincu,phttrinvg i ithiubi Σ
Glauberv Sudarshan[ 4 ] , [ 5 ] von m1 9 6 3 , k hi k h os tt nhch tc achm s nglaser.T nhcht c bi t c achm laser lt n h chtkt h p,c ng ch m laser c ng cao th i h i t nh k t h p c ng nghi m ng t.Vt h t r ngthidngm t ncgiltrngthikthp.
Trngthikthp| α⟩cn h nghaltrngthiringcatonthuboson aˆ,ngh al aˆ|α⟩=α|α⟩, (1.8) trong α=r e (iϕ) lm tsph c vi r v ϕ lc csth c Khi khai tri nthngquacctrngthiFock|n⟩tht r ngthikthp|α⟩cbiudindidng
|α⟩= C n |n⟩, (1.9) n=0 trong C n l h s k h a i trin.Thay(1.9)vo(1.8)tac biuthccatrngthikthp biudintheohcs cacctrngthiFock
1 2 vi C 0 =e ( − 2 |α| ) lhs c h u nho.Khit r ngthikthpcvit li 1 2 ∞ α n
Trngthikthpcm tst nhchtquantrng.u tin,chn g lt r ngth ic chunho,nghal
Thh a i , c ct r ngt h ik th pk h ngt r cg i a o v in h a u , n g h al v i α̸ =β th
V|⟨α|β ⟩ | 2 =e−|α−β| 2n nchngc xeml tr cgiao khi|α−β| ≫ 1.T nhch t thbalt p h p c c tr ng th i|α⟩t o th nh m t t p h p,nghalp h ngiinvthom n
Tht ,ph n bs h t t r n g t h i k th p α̸=βt u n t h e o p h n b Poisson,lp h nbm s h ttrungbnhvp h n g saicatonts h t b ngnhau,ngh alD N ˆ E =∆N ˆ 2 X csu t p(n)tmthy n h t trngthikthp|α⟩l p(n)=⟨n|α⟩⟨α|n⟩=e −|α| 2 |α | , (1.15) n! trong p (n)lh mphnbP o i s s o n Dot r ngthikthplt r ngthic i n.Tnhchtthn m lt r ngthikthp| α⟩cbtnh ccti u,nghal
= , (1.16) trongt o nttav x u n g ln g cd ng xˆ= aˆ+aˆ †
Trngthikth pct h cbiudinthngquatontcbitkhi ul D ˆ a (α).thyd ngc ato ntny,uti ntathay(1.5)vo
Dd ng th yc (1.18) tha m n (1.8) Ch , do t nh ch t c a to n thynn e −α ∗ aˆ |
dob i uthc(1.18)ct h v i tlidi dng α=e − 1 | α| 2 e αaˆ † e −α ∗ aˆ |0=D ˆ (α)|0 ,
To nt D ˆ a (α)cm tst nhch tsau: a) To nt D ˆ a (α)cthcbi udi ntheonhi ucchkh cnhau,l
=e |α| 2 /2 e −α ∗ aˆ e αaˆ † b) To nt D ˆ a (α)lto ntchu nt c(unita),nghal
−1 ( α)D ˆ a (α )0 E =⟨0|0⟩=1 (1.23) c) To nt D ˆ a (α)ct nhchtdchchuyn,i unycthhi nbi
(1.25)Vphica(1.24)v(1.25)theothtb dchi mtlngbng α v α ∗ soviccvtr it ngng.Dovy,to nt D ˆ a (α)cg ilto nt dchchuyn.
Nhv y trngthikthp|α ct h cbiudintheonhiucchkhcnhau.u tinnl t r ng thiring catonth yht bosonnh
Trngthikthpthmphotonc Agarwalvc ngsa ranm1991[24]cd n gnhsau
(1.26) trong| α⟩lt r ngthi kthp, ml s n g u y nkhngm , v
Ngo i ra, tr ng th i k t h p th m photon v i mm c n g c x u t v cdng(biudintheotrngthiFock)
2 a trongNlhschunha,|α⟩ a v | β⟩ b l nlt ltrngthikth pmode avb
Trngthihaimodekthpthm,btphotonc torabngcchtcdngccto ntsinh (t h m)vhy(b t)lnccmodecatr ng th i hai modek t h p S a u y c h n g t i g i i t h i u m t t r n g t h i th m (b t)cthltr ng th i kth p hai mode th m photon d ngchngchtnhsau
|Ψ⟩ ab = N α,β aˆ † +ˆb † |α⟩ a |β⟩ b , (1.33) trongN α,β l hs chu nha.Cthv i t trngthi|Ψ ab ⟩didng
Tr ng th i kt h p c p ( P C S ) [ 6 ] c A g a r w a l p h t h i n v g i i t h i u n m1988,nltr ngth iri ngc ato nthyc pboson aˆˆbv to nthi usphotongi ahaimode Q ˆ=ˆ b † ˆ b−aˆ † aˆvitrri ngt ngng ξvq nhsau
" Σ Σ q n!(n+q)! a b n n=0 trong ξ=r e (iφ) , φl c csth c b tkv ql m tsnguy n chs chnhlchsp h o t o n h a i mode.Trongkhnggiancctr ngthiFock,trngthikthpc pcbiudindidng
Chonnay,c nhi umh nhlthuytvt h c nghi mt oraPCS [14], [25],[26],[27],y l t r n g t h i p h i G a u s s v c n h i u t n h c h t phic i n[6],[28],[29] Don c quant mnghi ncu vc n g d ngt r o n g m t s l n h v c nhv i n t i l n g t [ 3 0 ] , [ 3 1 ] V i c nghi nc u c c ph ngph pt o ra tr ngth im itPCSc c ct nh chtphicinnichungvtnhchtanrinirin g ctngcn g v angc q u a n t m n g h i n c u M t t r o n g n h n g p h n g p h p r t c quan t mlphng ph p th m, b t c c photon v o hai mode c a PCS[32],
[33].y c ngc h nhl p h n g p h pc s d ngt or a t r ngthiphici nmichn ghai.
1.2.4.2 TrngthikthpcpthmhocbtphotonnmodeTrngt h ik th pc pt h mp h o t o n nm od e v kth pc pb t
2 n photonn modec a rabiChunqingvH o n g nm2000[34].Hai Σ
(1.42) trong A mqA vA mqS l c c hs c h u n h a K h i m c t r n g t h inyquyvP C S
Trngthikth pcpthmphoton(PAPCS)c a rabiHongvc n g s n m 1 9 9 9 [ 3 2 ] T r n g t h i n y c t o r a b n g c c h t h m mphoton vohaimodecaPCSvcvitdi dng
(1.43) trong A qm l hs c h u nha,ncxcn h cd ng
, (1.44) qm [n!(n+q)!] 2 vc h rngkhi m=0thP A P C S quyvP C S
N m 2009, Yuan v c c c ngsxu t tr ng th i kth p c pthm photon tng qut (GPAPCS),tcltrng thic thm kphoton lnmode av thm lphoton lnmode b[33] Theo, GPAPCScvit Σ
|ξ,q,k,l⟩= Cq′aˆ†kb†l|ξ,q⟩=Aq Dn|n+k,n+q+l⟩, (1.45) n=0 trong C q ′ lhs c h u nha, D n v A q cxcn h nhs a u ξ n
Kh c v i PAPCS [32] vGPAPCS [33], tr ng th i k t h p c p ch ngch t th m photon (SPAPCS)c t o ra b ng c ch th m ch ng ch t c cphoton v o hai mode c a PCS Tr ng th i n yc a r a b i t c g i Thanh vc c c n g s [ 3 5 ] T r o n g k h n g g i a n c c t r n g t h i
= (C n,qkl |n+k,n+q⟩+D n,qkl |n,n+q+l⟩), (1.48) n=0 aˆ +k v ˆ b +l lc cto ntsinhb c k v l ivimode a v b , k v l l ccsn g u y nkhngm , εl st h ckhngm , vh s A qkl c xcnhtiukin chunh anhsau
Hs C n,qkl v D n,qkl phn g trnh(1.48)cxcnhbi ξ n (n+k)!
(n+q+l)! (1.51) thS P A P C S c xcnhphn g trnh(1.48)c vitlidngn ginsau
MtstnhchtphicincaSPAPCSsc tigiithiuchn g ba.c h n g bn,ch ngtitrnhbyn g dngcatrngthinyvoqutrnhVintilngt.
1.2.5 Trngthikthpbb a vk thpbb a chngchtthmphoton 1.2.5.1 Trngthikthpbb a
Trngthikthpbb a c n h nghalt r ngthiringcabbato nthy aˆˆbcˆvhi uc cto ntsh t N ˆ b −N ˆ a v N ˆ c −N ˆ b [36] aˆˆbcˆ|Ψ p,q ⟩ abc =ξ |Ψ p,q ⟩ abc , (1.53)
N ˆ c −N ˆ b |Ψ p,q ⟩ abc =q |Ψ p,q ⟩ abc , (1.55) trong ξ = r e iφ vi r,φl n h ngst h c, pv q l c csn g u y n.Khngmttnhtng qut,gis p vq kh ngm , trongbiudintrngthis
|Ψ p,q ⟩ abc = c n (ξ)|n,n+p,n+p+q⟩ abc , (1.56) n=0 trongh s k h a i trin c n (ξ)l c n (ξ)= N p,q (r)ξ n n!(n+p)!(n+p+q)! (1.57) vihs chunhaN p,q (r)cxcn h bi ∞
St h c nghim t ora trng thi nylan truyn td o t r o n g k h nggian mphh pchoccnhimvln g tcchngtia ratrong[15]. yc x e m lm t tr ng th ic tr ng phi Gauss ba mode.C ct nhchtphici n c atr ng th iny nht h n g k s u b - P o i s s o n , t n h c h tph n ktchm , t n h c h tn n, t nh chtan ri,c c h mt ngquan phic i n, c nghi n c u trong [8],[36],[37].M t str ng th ic phtt r i nt t r ngt h i k thpb ba l t r ngt h ik thpb b a p h i tuy n [38], tr ng th i kth p bb a b c h a i [ 3 9 ] , t r n g t h i k th p bb a huh nchiu[40],trngthikth pbba K -chiu[41].
STMPATCS[13]c x u tbngcchchngchtcctontv cˆ †l l n cbamodec aTCS.Ncvi td id ng aˆ
0,u+p,0 0,0,u+p+q trong| Ψ p,q ⟩ abc lt r n g t h i k t h p b b a c x c n h ( 1 5 6 ) , h, k, l lc c s n g u y n d n g , ε, λ, σ lc c s t h c v n m t r o n g o n [−1,1]vhs chunha N p,q;h,k,l (r)cxcn h cdngsau
A i,j,m (r)= P F Q (1+i,1+j,1+m;1+p,1+p+q,1+t,1+u,1+v;r 2 ) ×p !(p+q)!t!u!v! , (1.62) vi P F Q l k h i uc ah ms i ub i.C ct r t r u n g b nhc at o nt aˆ i aˆ †i ˆ b j ˆ b †j cˆ w cˆ †w v ( aˆ † ˆ b † cˆ † ) u iviSTMPATCScx cnhl
(r), (1.64) vi i , j,w,ul c cs n g u y nk h ngm C ct nhc h tp h i c i nc atrngthin yckhostchititchngba.
Mtst nhchtcacctrngthiphicin
1.3.1.1 Khinimvtrngthinn ivihaii ln g vtlkhngo c ngthi AvB t r o n g khnggiantrng thi|ψ⟩,hthcbtnhcach ngcxcnhl
Trong btn g t h c (1.65), nu du bng xy ra
4 thtanihthcb tnhtnbtnhtithiu.Khicci ln g C A v C B thamn
VA/i>;VB= CB;CA.CB= 1 ⟨ψ|hA ˆ ,B ˆ i |ψ⟩., (1.67)
lc cgi ihnln g tc h u ncailn g AvB Taxttrngthi|ψ⟩ tronghaitrn g hpsau: a) Trn g hp|ψ⟩lt r ngthikthp(|ψ⟩=|α⟩).
Trongtr ngh tBoson,x t A ˆv B ˆlto ntbi ntr cgiaonhsau
Tt a thyrnght h cbtnh cachngt nbtnh tithiu.
4 b) Trnghp|ψ⟩lt r ngthiFock(|ψ⟩=|n⟩vi n>0 ).Tac
Dophn g saica Ab ngphn g saica Bn h ngl nhngiihnlngtchun.
Trong hai trng h p tr n ta th yivic c tr ng th i ln g t ,ph ngsaic amtiln g vtlungthil nh nh ocb nggiih nln g tchu n.
Vntralli uctht n t i m t tr ng th i vt l|ψ⟩n oc atrn g htboso ncho hai i ln g A , Bm t r o n g V A( h o c V B )bh ngitr gi i h nl ng tchu n (vt t nhi n VB (ho c VA )ph il nhngiihn)saochonguynlb tn h khngbv i phmkhng?
Nuctht a cthg i |ψ⟩ltr ng th i n niviil ng A (ho c B ).Ta hi u n nlvp h n g s a i c a A c n n xu ng nh h ncgi i h nl ng tchu n. Trn g h pcbi t n u tr ng th i n nc a A (ho c B )c n th a m ni u ki n VAV B b ngb tnht ithi u thn cg iltr ng th i n nl t ng.Hai ti u mcti p theosc pnc cht oravm ttonhccctrngthinnthamnyucutrn.
S ˆ( z)=e (z ∗ aˆ 2 −zaˆ +2 ) , (1.73) a 2 2 trong z =s e iχ vi s v χ lccst h c.Rrngtac
Vy S ˆ a (z)lto ntchu nt cvto ntli nh pvinc ngltont nghcho p dngcngthc(1.73)v( 1 7 4 ) tartrac cccngthcbin iquantrngsau:
S ˆ a (z)aˆS ˆ † (z)= aˆcoshs+aˆ † e (iχ) sinhs,S ˆ a (z)aˆ † S ˆ † (z)=aˆ † coshs+aˆe (−iχ) sinhs,S ˆ † (z)aˆS ˆ a (z)=aˆc oshs−aˆ † e (iχ) sinhs,S ˆ † (z)aˆ † S ˆ a (z)= aˆ † coshs−aˆe (
M t tr ng th inmode c tht r t h n h t r n g t h i n n n u t c d n g tontn n(1 73)lntrngthi.
Hi un g n n t n g c q u a n s t t h y t r o n g q u t r n h t o t n s t n g Tclt nsc atr ngth in nu rab ng t ngc ctnsc ac ctrn g nnnmodeuvo.Nun h sngvomitrn g phitu ynl nhsngnnthngthn g thn h sngtothnhct nhch tnntng. a a a nhnghatontn ntnghaimodec Hillery a ratrong[7]nh sau
2 trong ϕl g cxcnhhn g ca V ϕ trong mtphngphc.Giaohontgiahaitonttr c giaophal h V ˆ( ϕ),V ˆ( ϕ+π/2) i = iN ˆ+ Nˆ+1, (1.77) b trong N ˆ a =a ˆ † aˆv N ˆ b = ˆ b † ˆ b lc cto ntsh tc ahaimode a v b Theoht h cbtn h giahaitontnycdng
Tr ngt h ih a i m o d e c t nhn nt ngh a i m o d e n u− 1≤ S < 0 K h i S=−1,tr ng th i bn n t i a,nh ng vi S≥0tht r n g t h i k h n g b nn. b Nntngbamodebccao
Ti ukinnntngbamodebcnht,tcgiN g u y nBn ara ti u chu n x cnh n n t ng ba mode b c cao [37].iviti uchunny,haitontbccaoc xtnhsau:
V ˆ= i(aˆ †m ˆ b †m cˆ †m −aˆ m ˆ b m cˆ m )2 , (1.82) vi ml snguynkhngm Haitontnytuntheoquanhg i a o hon trong
|⟨ Wˆ | ⟩ g (2) (0), x c su t ph t hi n raphoton thh a i t n g t h e o t h i g i a n t r , c c p h o t o n c x e m l p h n k t chmhaycngilt nhphnktchmcaphoton.cnglm thiu ngphicin.
Theo(1.92)th g (2) (τ)k hngpht h u cvothigiantr τ , nnkhngcb tkphoton ph nktchmhayphotonktchm n oivitr ngkth pnmode. Trong trn g h p t n g q u t , v im t tr ng th ic i
[37],[48].Theo Nguyn Bnvc c c n g s[37],[39], m t tr ng th i thhintnhphnktchmbc mtrong mode xn u
R x (m)cngm thm cphnktchmnmodebccaocngln.Mcphnktch mnmodebccaolc cikhi R x (m)=−1.
Ph nktchmlm ttrongnhngtnhchtphici nquantrngvcn g d ng trongc cnhi mv l ng tnht o rac ctr ng th ith m photon b ng bt c h c h m [ 4 9 ] i u k i n p h n k t c h m h a i m o d e b c cao c Lee [28]aral n uti n n m 1990 vs a u c ph ttri n vmr ng bi t c giNguyn Bn[9] n m 2002 Theot c giLee,hmphnktchmhaimodebccaocd ng
Mttrngthichaimode av b c t nhchtphnktchmhaimodebccaonuh m R ab (u,v)thamniukin
R ab (u,v)0 Tr ng th ic E Vc ng l n, tr ngth ic n g r i V i c s d n g e n t r o p y v o n N e u m a n n t n h t o n r i c h o cctrngthihaimodenhk thpcphaytrngthinnchnkhnghai modec s d n g t r o n g [ 6 2 ] , t r n g t h i n n c h n k h n g h a i m o d e chngchtkthpthmvh yphoton[65]. b Tiuchunentropytuyntnh
HnchcatiuchunanrientropyvonNeumannlk h ngcgit r c n t r n V v y r t k h ara c nh ngh a cho m t tr ngthiriccihocltn g Dov i cchu nhar ilcnthi t.Ti u chu n entropy tuy n t nh kh c ph cc nhci m c a ti u chu nentropy von Neumann Theo nhA g a r w a l - B i s w a s [ 6 2 ] , e n t r o p y t u y n t n h cdng
Theo,trngthibr ikhi00)c ng l mchomcp h nktchmcaPAASTMPCScaohnPCS.
Nhvy,i vi t nhcht ph nktchm haimodeb c cao, c cktqukhostchoth ytnhphnktchmcaPAASTMPCSpht h u cvovicchnbc( u,v)cah mphnktchm R ab (u,v).Cth,khi u c ng l n ho c v c ng nht h p h n k tch m c a PAASTMPCS t ng.B n c nh, k h i g i m s p h o t o n t h m v o m o d e ah a yψ b t i m o d e bth m cph n ktch m hai mode b c caoc t ng c ng.cbi tlkhithmvb tphotonlnhaimodesaochotng k+hk h ngi , nusp h o t o n h a i m o d e c n b n g (k= h ), thp h n k t c h m h a i m o d e b c caoll n nh t K t quk h o s t c n c h o t h y r n g P A A S T M P C S c tnhchtphnktchmcaohnsovitrngthigcPCS.
Trongquanghcln g t,hmWignercsd ngxcn h cc ctnhcatrngthivtlt r o n g khnggianpha.i vimttrngth ib t k,h mW i g n e r c a n c t h nh nc c g i trt y Th ngthn g , hmWignerdn g cthxcnhnm ttrngthilln g tho cl ng tphi Gauss [87] Tuy nhi n,m t str ngth i,h m Wignerc achng cthnh n m ts gitrm T r o n g tr ngh p, c ctr ngth inycx cnh nlc ctr ng th i phi Gauss [88],[89], [90].iviPAASTMPCSc chop h n g trnh(2.4),hmWignerc giithiu (1.101)ct h cvitlidi dngsau
4e 2(|α a | 2 +|α b | 2 ) ∫ π 4 × ba ⟨ − γ b ,−γ a |ρˆ ab |γ a ,γ b ⟩ ab , (3.11) trong α a =|α a |e iφ a v α b =|α b | e iφ b lc csph ctrongkh nggianpha,|γ a ⟩ a v |γ b ⟩ b khi uchoc ctr ngth ikth pv ρˆ ab lto ntm t c aPAASTMPCScx cnh(2.6).Thayto ntm t ρˆ ab phngtrnh(2.6)vophngtrnh(3.11)saut nhcctchphnphc(xem phl c), chng ta thuc hm Wigner ca PAASTMPCS nhs a u
Ch ng t i sd n g b i u t h c k h a i t r i n p h n g t r n h ( 3 1 2 ) k h o s t c cc t n h p h i G a u s s v p h i c i n c a P A A S T M P C S T r o n g H n h 3.8,chngtivt h s p h t h u ccahmWigner W v ophnthcvp h n o c a α a v i|ξ|= 0.2, α b = 0.5, φ= 0v h= k= 1.K t qucho th y r ng h m Wigner c a PAASTMPCS nh n c c gitrmtrong m t sv ng cakh ng gian pha Do,ch ng t i ktlu n r ngPAASTMPCS lm t t r n g t h i p h i c i n v p h i G a u s s N g o i r a , t r o n g H nh 3.9, ch ng t ic h n m t v n g n h t r o n g k h n g g i a n p h a c h r a r ng s u c a h m Wignercth c t ng cn g n u s p h o t o n c th m v o k vs p h o t o n b t i l c t ng l n.ngli n n t m u xanhdn g(k=0, h=q= 6)tn g n g v i
P C S cim c c t i u t m n h t Ccngkhctn g n g viPAASTMPCScimcctium nhiuh n so viPCS.i u n y ch ngtr ng vi c th m vb tc c photon v oc c modeng vai trquan tr ng trong vi c n ng cao t nh phi Gauss c aPAASTMPCSsovitrngth igcPCS.
Nh v y, ch ng t i k t lu n r ng PAASTMPCS lm t tr ng th i phiGauss M c phi Gauss c a PAASTMPCS cao h n PCS v c t ngcn g bngcchtngsphotonthmvb tlnccmodecaPCS. k
Hn h 3.8: th h mWigner ( W) t h e o phnthcvp h no ca α a v i | ξ|
3.2.4.1.nhl n g a n ribngtiuchunentropytuyntnh ivicctrngthihaimode,mtst i uchunct h csd ng dt m s a n r i [ 1 6 ] , [ 2 0 ] , [ 5 9 ] , [ 9 1 ] v n h l n g a n r i [ 6 2 ] , [ 6 4 ] , [ 9 2 ] c ach ng.k h o s tm ca n r i caPAASTMPCS,ch ng tich ntiuchunentropytuyntnh[62].Theo, hma n ri E lincchod ng
W ρˆ=[Tr (ρ^ Σ )]= |C(ξ)||n+k⟩ b ab n;k,h b ^ Σ Σ Σ − Σ a lyc entropytuyntnh E lincaPAASTMPCSnhs a u b trongT r kh i uchophptonly vtcamatrn.Trngthibr inu E lin > 0 K h i E lin = 1,tr ngth ic mca n ritia.ivi
PAASTMPCSc chop h n g trnh(2.4)vto ntm t ca rap h n g trnh(2.6),chngtithuc
Ch ng t i kh o s t m can r i c a PAASTMPCS b ng c chsd ngphn g trnh(3.16).TrongHnh3.10,chngtivt h s phthucc a E linv o| ξ|ivim t sg i t r k v h=q−l Trong,trn g h p k=l=0v h =q=8( ngli n n t m u xanh lam) tn g n g v iPCSvn h n g n g c n l ilc a
PAASTMPCS.c ng git r| ξ|, c cngcongchothygit r c a E liniviPAASTMPCSluncaohnPCS.B n c nh, c c n g c o n g t r o n g H n h 3 1 0 ( a ) v ( b ) c h o t h y g i t r E lint ng n u sp h o t o n c t h m v o kv s p h o t o n b t i l t ng (t cl hg i m).i uc nghalmca n ricaPAASTMPCSctngcn g bngcch ngthitngsphotonthmvomode av b t imode b c a PCS Th m vo,trong trng h p vi c th m vbtphotonngth i t ng l n, thgitrc a E linc ng t ng theo,c bi tltrongtrnghphiu k−lc ngln(xemHnh3.10(c)). b
H nh3.10: thhm E lin ph thucv obi n |ξ| vcpthams (k,h) Trongcch nh (a), (b) v(c)ngn t li n (k, h)=(0,8) lc a PCS, ccngn tt lc aPAASTMPCS.Hnh(a)ltrnghp h=8 v k t ng.h nh(b),ccng ntttn g n g k=8 v h gi m ( l t n g ) h n h ( c ) , c c ngn ttng v i k t n g v h gim( l t ng),ngthihiu k−l t ng. photonc th m vb tc ng t ng, th m c an r i c a PAASTMPCScngcao.M cr ict h t i ntilt n g khibi nk th pvslngccphotoncthmvbtrtln.
3.2.4.2 nhln g a n ribngtiuchuna n rimi xcnhmcr icaPAASTMPCSbngtiuchuna n rimi,ch ngti tnhc ci ln g v p h icaphn g trnh(2.46).
ξξξ ξξξ Σ b Σ a b m;k, h utinlcci ln g E 1, E 2 v E 3,mtlnnachngtathychngbt r i ttiudocctr t r u n g bnhlinquann tontp h a bngkhng, ct h l
C cil ng N 1 vN 2 vp h i c aphn g t r n h ( 2 4 6 ) c x c nhquacctrt r u n g bnhlinquann tonts htcxcnhphn g t r n h
Th m r iℜiviPAASTMPCSphn g t r n h ( 3 1 8 ) chng t ivthn h gim ca n ricaPAASTMPCSdavotiuchun nhln g a n rimiH nh3.11.Ccn g congtrongHnh3.11l thc a h m ℜ pht h u c v o b i n k t h p|ξ|v i bt h a m s ( k, h)thayi v q=8.ngcong(k, l) =(0,8)m u xanh dn g n t l i n n g viPCS, c cngcong c n l ing viPAASTMPCS.H n h
( a ) ccngnttn g vicctrn g hp k t ngdntrongkhicnh h=8(ha y l=0).Trong khi,H nh 3.11 (b), c cngn ttngvicctrn g hp hgi mdn(hay lt ngdn)trongkhicnh k=8. a
P A A S T M - PCSbtu br ivng|ξ|cg i t r b ,r itngnhanhvdd ng tgit r l n(trn0.7)ngaykhigit r | ξ|tkhongmtn v.Mt i m ng ch lr i c a PAASTMPCS lu n cao h n PCS tr n to nminca| ξ|.Cct h c ngchothykhitngsp h o t o n c thmvo
H nh3 1 1 : t h h m r i ℜ c a P A A S T M P C S t h e o b i n | ξ| v b t h a m s ( k, h) v i q= 8 Hnh(a)ngvitrn g hp h= 8 v k t ngdn.Hnh(b)n g vitrn g hp k=8 v h gi md n( l t ngdn). mode a (t ng k ) hay t ng sp h o t o n b b t i m o d e b (t ng l hay gi m h ) th an r i c a PAASTMPCS t ng cao ng k so viPCS, nh tlv ng| ξ|n h.Khi|ξ| rtln,r icaPAASTMPCStimcnmtnv( g i t r l tn g ) v n h h n g k h n g n g k v o s p h o t o n c t h m vohaybtic cmode. so s nh viktquan c a PAASTMPCS khis dng ti uchu n m i vt i u c h u n e n t r o p y t u y n t n h , c h n g t a s o s n h c c t h H nh 3.11 vH n h
3 1 0 ( a ) v ( b ) T c c t h t a t h y c n h i u i m tngng khisd ng ti u chu n m i vti u chu n entropy tuyn t nh.Cthl khi k t ng vc nh l (c nh h ) c c H nh 3.11 (a) v3 1 0 (a),hockhitng l(gi m h )vc n h kc cHnh3.11(b)v3 1 0 (b),
ξξξ ξξξ ricaPAASTMPCStngn g kvng|ξ|b.PAASTMPCSbt uc xemlr ivt r c g i t r | ξ|cngrtgnnhau.Cthiv i ng(k, h)= ( 1 6 ,8)H nh 3.11 (a), PAASTMPCS b t u r i khi ℜ≈0 t i|ξ|
3.11v3 1 0 u tngnhanhvm tn vk h i | ξ|tng,tuynhintheo
D E D E ti u chu n m i, r i c a PAASTMPCS t ng nhanh h n v ng c| ξ|bv cao h n v ng c|ξ|l n V d cho ng(k, h)=(4,8)t i|ξ|
Nhv y vi cp d n g t i u c h u n m i i viPAASTMPCS cho ktqut t , n g t i n c y v c n h i u i m t n g n g v ivi cp d n g t i u chunentropytuyntnh.ricatrngth itheotiuchunmithngcaohntiuchunentropytuy ntnhkhibinkthpkhngb.
Tnhcht phi ci nca trng thi kt hp cp chngcht thmphoton
S a u , t h u n t i n cho vi c nghi n c u, h m n n t ng hai mode S(ϕ)[10]c n h n g h a nhsau
⟨N a +N b +1⟩ , (3.19) trong phn g s a i (∆V(ϕ)) 2 =V(ϕ) 2 − ⟨V(ϕ)⟩ 2 vi V(ϕ)lto ntn n t ng hai mode cd n g V(ϕ)=a † b † e iϕ +ab e −iϕ 2 , ϕ lth c, ⟨N a ⟩v ⟨ N b ⟩ls p h o t o n t r u n g b n h c a c c m o d e a v b Theo [10], m t tr ngthict nhchtnntnghaimodenuhmnn S(ϕ)catrngthithamn−1≤S(ϕ)
Tt r ngt h ik thpcp( P C S ) c n h n g h ap h ngt r nh(1.38)vh m n ntnghaimode( 3 1 9 ) , dd ngthyrng S(ϕ)=0.Do
PCSkhngct nhchtnn tnghaimode.Tuynhin,i viSPAPCS cn h nghap h n g trnh(1.48)vi ukinnntnghaimode
Hn h 3 1 2 : th h mn n S (ϕ) p h t h u cvoc cbin | ξ| , ϕ v b thams ( k,l) v i ε= 1 ,q= 2 h nh(a),h m S (ϕ) p h t h u cvocbi n | ξ| v ϕ v i ( k,l)= ( 4 ,2) hnh(b), S (ϕ) p h thucvo | ξ| vi k= 2 v l t ngdn.h nh(c), S(ϕ) p h thucvo
(3.19), ch ng t i t nh to n v thuc h m n n S(ϕ)[93] i v i m isn g u y ndng kvlnh sau
+ ε ( n + q + l )!(2 n + q + l ) , trongh s A qkl cchob iphn g trnh(1.49), φ v ϕ c chnsaocho φ=ϕ
Chngtisd ngbiuthckhaitrinp h n g trnh(3.20)l mrc tnhnnt nghaimodecaSPAPCS.Hnh3.12,chngtiv
∞ 2 ths p h t h u c c a h m S(ϕ)ph thu c v o bi n|ξ| vicho m t sgit r c a ϕ , k,qv l Hnh3.12(a) chothyrngkhi ϕ = m π /
4vi m l m tsn g u y n,thS P A P C S st h h i ntnhchtnnt nghaimodettnht.B ncnh, Hnh3.12(b)v(c)chothyrngnuthams k ccnhv lc tnglnhocthams l c cnhv kc t ng l n, thm c n n t n g s h a i m o d e c a S P A P C S c n g c t n g cng Ngo i ra, trong nh ng tr ng h p,c c t nh ch t n n t ng haimodepht h u cvothams km nhhnthams l
Nhv y , tc c ktqun u t r n c h o t h y S P A P C S ct nh ch t n nt ng hai mode trong khi PCS thk h n g i m n g c h l c t n h n n t ng hai mode c a SPAPCS c t ng cn g n u sl n g p h o t o n c thmvochaimodecaPCStng.
3.3.2 TnhchtphiGauss iviSPAPCS, ch ng t i c ngsd ng phng ph p kh o s th mWigner vd a v o t nhm c a n , nhiviPAASTMPCSti u mc3.2.3,n h g i t n h c h t p h i G a u s s H m W i g n e r c h o t r n g t h i h a i modec n h n g h a p h n g t r n h
( 1 1 0 1 ) x c n h c h m n y cho SPAPCS, ch ng ta x t to ntma tr n m tiviSPAPCS [35]nhsau ρˆ ab =|Θ q;k,l ⟩ ab ⟨Θ q;k,l |
( 1 4 8 ) v (1.52).Bngcchthay ρˆ ab ph ngtr nh(3.21)voph ngtr nh(1.101)vt nhtchphnphcchngtithuc
|β|= 0 3 , φ a = φ b = φ= 0 Trong(a),hm W p h t h u cvophnthcvp h no ca α v i ξ= 4 v ( k,l)=(3,12) Trong(b), W p h t h u cvobin |ξ| vi |α|=0.5
(3.22) trong2 F 0 kh i uchohmsiubi. bi tc c i m c a h m W i g n e r p h n g t r n h
( 3 2 2 ) , c h n g t i vt h c a h m WH n h 3 1 3 T r o n g , h n h ( a ) l t h c a h m Wp h t h u c v o p h n t h c v p h n o c a t h a m s α Ch ng tacthth yst n t i c c git r c am c a h m Wigner trong m t sv n g c a kh ng gian pha Vvy, SPAPCS lm t t r n g t h i p h i G a u s s T r o n g h n h (b), c cngconglthc a Wp h t h u c v o b i n |ξ|vb t h a m s (k, l), trong k v l c c h n t n g d n n g c o n g n t c h
∞ n,m=0j,i=1 m e n (0,0)tn g n g viPCS,nhngn g khctn g n g viSPAPCS.Rrnggitrc ctiuc acchmWigneri viSPAPCSlunnhh ni vi k a qk
PCS H n n a, c cngcong cho th y r ng khi c c thams k v l t ngl n, s u c a h m Wigner c ngc t ng cn g i u c ngh a l ,khicngnhiuphotonc th mvohaimodecaPCS,tnhphiGausscaSPAPCScngtng.
G a u s s catrngthinypht h u cvosp h o t o n c thmvoch a i mo dec a PCS, sp h o t o n c th m t ng th phi Gauss c ng t ng theo.phiGausscaSPAPCScaohnkhnhiusovitrngthigcPCS.
Cn h i utiuchunn h ln g anricatrngthihaimode. y,chngtich nti uchu nentropytuyntnh[62]khost an r ichoSPAPCS.Theo,h mentropytuyn t nh E linc nh nghal
2 , (3.23) trong E linlu n th a m n0≤E lin ≤1 N u E linc a m t tr ng th i kh c0 tht r n g t h i br i , E linc ng l n than r i c a tr ng th i c ngcao N u E lin =1, tht r n g t h i can r iltn g x c nh E lincho SPAPCS,u t i n c h n g t i d a v o ρ ab p h n g t r n h
( 3 2 1 ) t n h ρ a = Tr b (ρ ab ).C u ic ng,ch ngt ixcn h c E lin = 1 −Trρ 2 =
((n+q+k)!) 2 , (3.24) tronghs A qkl c x cn h ( 1 4 9 ) Sh ng E 0 trong(3.24)c vi ttngminhl
H nh3.14: thh mentropytuy nt nh E lin ph thucv obi n |ξ| v i φ=π,ε=1,q=
3 Trong h nh (a) v(b)ng cong (0, 0) tngng v i PCS, c cng c n l in g v i SPAPCS.hnh(a),ccn g nttn g vitrn g hpcn h l=4 v k t ng.hnh(b),ccn g n tt n g vitrn g hpcn h k=4 v l t ng.
((n+q+k)!) 2 (n+k)! δ k,l , trong δ k,l l h mdeltaKroneckerpht h u cvoccchs k v l Do khi k l t h E 0 =0
Tk t quvh m entropy tuyt t nh(3.24) v(??), ch ng t i kh os t an r i c aSPAPCSb ng cchvt h TrongH nh3.14,chngt i v thv sph thu c ca E linv o|ξ|vi φ=πv ε=1ivim t sg i t r c a c p t h a m s( k, l), trongc c n g n t l i n m u x a n h(k=l=0)tngn g viPCSvc cn g khclS P A P C S Tronghn h
(a) v( b ) , c c ngcong cho th y r ng khi l c nh v k c t ng l n ho c ng c l i khi kc nh v l t ng, th E lins t ng vim i gi tr| ξ|dn g C c ngcong c a SPAPCS lu n cao h nngcong c a PCS.Thmvo, khigit r c a|ξ| rtln,thc cngconghm E linhitvtinvm tn v.
Nhv y tacthk tlu n r ngr i c a S P A P C S l u n c a o h n c a PCS vim i git r| ξ|dn g v n c t n g cng n usphoton th mvoccmodecaPCStng.
Tnhcht phi ci nca trng thi kt hp cp chngcht
3.4.1 Tnhchtnntnghaimode kh o s t t nh ch t n n t ng hai mode c a SPAPSPCS,c c h o phngtrnh(2.11),chngtikhaitrinphn g sai(∆ V ϕ ) 2 i vitrngthi nyrithayvophn g trnh(1.80).Ktqut h u c
Cct h a m s S 1 ,S 2 ,S 3 t r o n g p h n g t r nh( 3 2 5 ) c xcnhtrong trn g hp q≥l v k l l Σ 2 ε 2 n!(n+b 1 )!(2n+b 2 ) n=0 ( n+b 2 )!( n+a 1
Chrng,trongtrn g hp k=l=0,hmn ntngcaSPAPSPCS
Hn h 3 1 5 : C cthc ah m S(ϕ) ph t h u cvobin |ξ| vc pthams ( k,l) vi ε= 1 , ϕ= 0 v q = 9 Hnh(a)tn g n g vitrn g hpc k v l u tng.Hnh(b)tn g ngvitrn g hp k t ngnhng l c n h cquyvd ngcaPCSnhs a u
Phn g trnh(3.29)chr a rngPCSkhngct nhchtnntnghaimode.
3 1 5 nhgin n c a SPAPSPCS pht h u c nht h n ovosl n g photonc thmvomode av b ti m o d e b H nh 3.15 (a) lt h c a h m n n t n g h a i m o d e S(ϕ), h m n y phthucvobin| ξ| vc pthams( k,l).Cct h b a o gmn g congntlinmuxanhlam(5,1) ;ng congntchmch mg chm ut m(7,2);n g c o n g n t c h m g c h m u x a n h l ( 1 0 ,4);n g c o n g n t g c h m u( 1 4 ,7).
C cn g c o n g c h o t h y r n g k h i c k v l t n g n g thi,git r c ahmnntng S(ϕ)cngtngtrntonmingit r c a
|ξ| C ng dd ng nh n th y r ng gitrc a h m n n S(ϕ)pht h u c r t nhiuvothams k i unyc thh i nrt r o n g Hnh3.15(b),bi
2 ngc o n g n tl i nm ux a n h d n g ( 5 ,1) ;n g c o n g n tc h mc h mgchm ut m( 1 0 ,1) ;n g c o n g n tc h mg chm ux a n hl(20,1)v ngcongntgchmu(50,1) C cngco n g tngn g v i
∞ 2 trn g hpthams l c cn h v k t ngln,chngchothyrngkhi kt ngnhanh,gitrc a S (ϕ)sg i mmnhvg i t r −1.
Tv i ckhosttnhchtn ntnghaimodecaSPAPSPCS,chngt i th y r ng vi c t ng slng photonc th m v o mode a vslngphotonbtim o d e bc thlmtngn ncaSPAPSPCS.Trongkhi ,tr ngthigcPCShontonkhngct nhchtnntnghaimode.B n c nh, n n c a S P A P S P C S c t n g c n g m n h k h i s l n g photonthmvomode at ngln.
Ti uki n x cnht nh ch t ph n ktch m hai mode b c cao c atr ng th ia m o d e , c c h o p h n g t r n h ( 1 9 5 ) , c h n g t i t n h t o n chiti tiviSPAPSPCSvt h u c ktqu
! trong u 1 =v 2 =u;u 2 =v 1 =v Khi k=l ,hm R ab (u,v)cdngnh sau
Vitrn g hp k ̸=l ,hm R ab (u,v)cxcnhl
Khi k= l= 0,hm R ab (u,v)q u yvd ngtn g n g viPCSnhs a u
Tk t q u p h n g t r n h ( 3 3 0 ) n ( 3 3 3 ) , c h n g t i n g h i n c u t nh ch t ph n k t chm c a SPAPSPCS b ng c ch kh o s t h m ph n k tch m R ab (u, v) H m n y kh ng chp h t h u c v o b c uv v m c n p h thu c vo s l ngphotonc th m vo k vtr i l C ctht r o n g Hnh3.16(a)v3 1 6 (b)lc cn g congcahmlinkt R ab (u
,v),phthucv ob i nk th p| ξ| vs t h a y i c ac cc pt h a m s( u,v)hoc(k,l)trongkhiccthamskhcccn hl ε =1,φ=0,q=9.
Trong H nh 3.16 (a), bang cong(10,4),(10,3)v ( 1 0 ,2)n g v i trn g h p c n h u nh ng gi m v , ch ng cho th y git r c a R ab gi m khithams v gi mtrntonmingit r c a|ξ|.i unychothymc ph n ktch m c a SPAPSPCS st ng l n n u tham sb c v gi m.Tuyn h i n,c cn g c o n g c nl ib a o g m( 1 2 ,1) ,
( 1 4 ,1)v ( 1 6 ,1)n g vitrn g h p c n h v nh ng t ng u , ch ng cho th y r ng git r c a R ab (u, v)gi m khi tham s u t ng tr n to n mi n gitrc a |ξ|.i u n ycho th y m c ph n ktch m c a SPAPSPCS t ng l n n u tham s u t ng
Hn h 3 1 6 : t h c ahm R ab (u,v) p h t h u cvobin |ξ| viccthamsccnhl ε = 1 , φ = 0 v q = 9 h n h ( a ) , c p t h a m s ( u, v) t h a y i t r o n g k h i k = l=
TrongH n h 3 1 6 ( b ) , n g c o n g( k, l)= ( 0 ,0)tn g n g v i t r n g thibanu caPCS.Ccn g congcnlibaogmn g cong(1 ,1) ,
(3,3)v (5,5)tn g n g v iSPAPSPCS.Rr nglgi trc a R ab (u, v)gi m khi c c tham sk v l t ng ng th i tr n to n mi n git r c a | ξ|.Nc h o thyrngmcp h nktch mcaSPAPSPCStnglnnusln g p h o t o n c t h m v o m o d e a vb t i m o d e b c ng t ng l n.Ngo i ra, ch ng t i c ng th y gitrc a R ab (u, v)iviSPAPSPCS lu nth p h n git r c a P C S t r n t o n m i n g i t r c a |ξ|, dom c ph nktchmc aSPAPSPCScaohnsoviPCS.
Nhvy,SPAPSPCSltrngthictnhchtphnktch mcaoh n PCS T nh ch t n yc t ng cn g t r o n g t r n g h p t h a y ib cph n ktch m sao cho u t ng ho c v gi m v trn g h p t n g n g t h i sphotonthmvomode av s photonbtmode bc a PCS.
Q − ( a−k)lh mPochhammer. khi nnh n gitrm [13],[29],[71],[89],[90],[94].i v i tr ng th i haimode av b ,hm Wignerc xcn h p h n g trnh(1.101)c vitlididngs au
W= π 4 e 2(γ a ∗ α a +γ b ∗ α b −γ a α a ∗ −γ b α b ∗ ) × b a ⟨ − γ b ,−γ a |ρˆ ab |γ a ,γ b ⟩ ab dγ a dγ 2 b , 2 (3.34) trong α a = |α a |e iφ a ,α b = |α b |e iφ b lc csph ctrongkhnggianpha,
|γ a ⟩ a v|γ b ⟩ b l c ctr ngth ikthpv ρˆ ab l to ntmatr nm t.iviSPAPSPCSto ntmatr nm t ρˆ ab c x cnh(2.12).Thayto ntmatr nm t ρˆ ab t ph ngtr nh(2.12)voph ngtr nh(3.34)vt h c h i n c c b c t n h t o n t a t h u c h m W i g n e r c a
,h m Wigner W phthu c theo ph n th c vph no c a α a trongkhi c c tham sk h c c c h n c n h b a o g m ε=1, ξ=8,|α b |
Hnh3.17(b)lc ct h h mWigner Wt h e o bin| ξ|,n g vicpg i tr( k,l)khcn h a u , c ct h a m s cnl ic c h nc nh l
0 H nh( a ) l t h h m W t r o n g k h ngg i a n p h a p h t h u cv op h nt h cv p h no c a α a v i ξ=8 , |α b |=0.4 v k= l=9 H nh (b)lccthh m W p h t h u c v o bin |ξ| vbt h a m s (k,l) trong khi |α a |=0.5 v |α b |=0.4 q= 12; ε= 1 ;|α a |= 0 5;| α b |
= 0.4;φ a = φ a = φ= 0.ng(0,0) n gvitrngthigcPCS,ccn g cnli ngviSPAPSPCS.c bitl ng(3,5)n g v i t r n g h p c h n g c h t t h m 3 v b t 5 p h o t o n ; n g (6,7)n g vitrn g h pchngchtthm6vb t7photon;n g (9,9) ng vitr ng h p ch ng ch t th m 9 vb t9 photon C ct h n y cho th y, khi ch n c c tham sphh p, vi c th m vb t photon v o c cmodecaPCStoraSPAPSPCScm hmWignercaohn.c bitkhi k=l ,git r c c t i u c a h m Wg i m , t c m h m Wigner t ng,dotnhchtphiGaussc tngcng.
K t quk h o s t c h o t h y m h m Wigner c a SPAPSPCS r tcao, git r h m n y d d n g t−0.3, th m chc n cthn h h n i u nykhngdd ngtmthycctrngthikhc,thmchk h ngthcc c tr ng th ic k h o s t D o v y , S P A P S P C S ltr ng th i phiGaussmnhsoviPCS,PAASTMPCSvS P A P C S p h i Gausscth c t ng cng b ng c ch t ng sphoton th m vb t v o c c m o d e c a PCSkthp vivicchnphhpccthamskhc. k a b
3.4.4.1.nhl n g a n ribngtiuchunentropytuyntnh nhl n g a n ricamttrngthihaimodechngtacths d ngnhiuti uchunkhcnhau[62],[64].Mitiuchuncn h ng unhc i mkhcnhauvp h hpvimthc c trngthinhtnh. ychn g tisd ngtiuchunentropytuyntnh[62]n h ln g anrichoSPAPSPCS.Hmentropytuyntnhc n h ngha
(3.36)Git r c a E linivimttrngthinmtrongon[0;1].Nu E lin =0 th tr ng th i ho n to n kh ng b r i Khi E lin >0th tr ng th i b r i E linc ng l n, r i c ng cao, khi E lin =1tr ng th itr i c c ihayltng.
Tto ntmatr nm t ρˆ ab i viSPAPSPCS,cx cnh phngtrnh(2.12),cctont ρˆ 2 v ρ ˆ 2 cxcn h cd ngsau ρˆ 2 = Σ Σ C s,n C j,n+a−a+b−b C ∗
Tphngtrnh(3.39),chngtivitl i E lindidngkhaitrintheoccchs r , s,ivjnh sau
Thayccthamsc xcn h trongphn g trnh(2.10)vophn g trnh(3.40)chngtathuc hm E linn hs a u
Tk t q u b i u t h c ( 3 4 2 ) , c h n g t i t i n h n h v t h k h o s ta n ritheotiuchunentropytuy ntnhcaSPAPSPCSH nh
3.18 C cn g cong lths ph thu c c a E linv o bi n|ξ|vic pthams( k, l)t h a yi t r o n g k h i c c t h a m s k h c c c h n c n h l ε=1;φ=π;q=9.Trongcchnh,n g ntlin( 0 ,0) ngvitrngthigcP CS,ccn g nttcnlin g viSPAPSPCS.Ch a i Hnh
H nh3.18: thcah m E lin ph thucv obi n |ξ| vcpthams (k,l) v i ε=1,φ=π v q = 9 n g n t l i n (0,0) c h a i h n h ( a ) v ( b ) n g v i P C S , c c n g n t t l c a SPAPSPCS h nh (a), cc ngn tttn g n g v i t r n g h p l=q= 9 v k tng.h nh(b),ccngntt tn g ngvitrn g hp k=q=9 trongkhi l tng.
3.18 (a)v3 18 (b) uchothya n r ica SPAPSPCS lun ln hnPCS vi m i|ξ|>0,i un y th yrnh ng vt r g i t r c a |ξ|k h n g ln.a n ricaSPAPSPCSct h t n trn90%ngayck h i git r
|ξ|nh v k ho c l kh ng qu l n V dtrong H nh 3.18 (a), vi k=l=9, E lin ≈0.91t i vt r c | ξ|=5.0 Khi so s nh c cng(3, 9) v (9, 9)H nh 3.18 (a), ch ng ta th y r ng n u k t ng tha n r i c a
S P A P S P C S c ngt ng theo Tuy nhi n, khi k t ngnkhi k=l=q thngcong lcao nh t hay an r i t t nh t,i u n y t h y r k h i s o s n h h a i ng(9,9)v (15 ,9) Cngtngtn h vy , H nh3.18(b),khi lt ngth
(b) anricaSPAPSPCScngtng,khi lt ngn git r m l =k=qth ngcongcaonhtcngngnghavianrittnht. nhgic h i titcctrn g hpcbitnhk h i k= l≤qh o c k= l=q , chn g t i v t h H n h 3 1 9 H n h ( a ) c h o t r n g h p k= l≤q ,ccn g congchothyrngkhi k= lt ngln,mcanr i c ng t ng m i git r c a |ξ| Chonkhi k=l=q , ng cong vtrc a o n h t , t n g n g v itrn g h p S P A P S P C S c a n r i l n n h t trntonminca| ξ|.c bitlk h i ccthams k , l,qt ng,n g thi
H nh3.19: thc ah m E lin ph thucv obi n ξ vbthams (k,l,q) v i ε=1 v φ= π Hnh(a)lt r n g hp k= l t ng,trongkhi q= 8 Hnh(b)lt r n g hp k=l=q t ng. thamn k=l=q tha n r i E lincngtng ln.Dd ngnhn thy i u n y tc c n g c o n g t r o n g H n h 3 1 9 ( b ) , n h n g v t r c g i t r ca|ξ|khln.
Trngthikthpcpchngchtthmphotonvb tphotoncd ng c chophng tr nh (2.11).x c nh m cr i c a PAASTM-PCS theo ti u chu n m i, ch ng t i t m d ng khai tri n c a h m r iℜphn g t r n h ( 2 4 6 ) l m c i u , u ti n, ch ng t
A s,n A r,n+a− a |ξ| 2n ξ ∗a s −a r (n+a s )δ a− b, a − b i t nh c cilng N 1 , N 2 ,E 1 , E 2 vE 3 cnh ngh aphng tr nh (2.47)iviPAASTMPCS.Ktquthucl
Dt h yrng,miquanhg i ahaithams k v l c n h hn g rtlnn hmr i ℜ Cthtacmtstrnghpcbitsau:
2 a b a b k=l±2hoc k=l±4,khihmri ℜ lnlt cd ng
+Khi k̸ =l+j,j ∈ {−4;−2;2;4}th E 1 ,E 2 ,E 3n gth itrittiu nnhmriℜcd ng
+c bitkhi k=l=q ,th N 1 =N 2,khih mriℜcd ngnginsau
Trongphn g trnh(3.51)v( 3 5 2 ) , cci ln g N 1 v N 2 c xcn h nhsa u
N ˆ 2 E −DN ˆE 2 −DN ˆE 2 , (3.54) trongc ctrt r u n g bnhc xcn h c cphn g trnh(2.13)n (2.16).
( 3 4 9 ) v ( 3 5 0 ) , ch ngti v c cthc h o b n t r n g h p k= l+2, l=k+2, k= l+4vl=k+4nhH nh 3.20 Trong,ng(k, l) =(0,0)lthh m r i ℜ c a PCS, c cngc n l ilc a SPAPSPCS vi k v l t ng d n vth a m n b n trn g h p n u t r n uti n ta dd n g t h y r n g t r o n g cb n trn g h p , r i c a SPAPSPCS l n h n PCSngkvngmg i t r c a|ξ| khngln.ivicctrn g hp k=l+2( hnha)v k=l+ 4(h nh c)r i c a
S P A P S P C S t n g n h a n h k h i kv l t n g k h u vccgitr|ξ| nh.Trongkhic ngkhuvc,trnghp l=k+2
Hn h 3 2 0 : t h h m r i ℜ c a S P A P S P C S p h t h u c t h e o b i n |ξ| v b t h a m s ( k, l) ,v i q= 8, ε= 1, φ= 0 c c h n h ( a ) , ( b ) , ( c ) v ( d ) n g c o n g (k, l) = (0,0) n g v i PCS, ccng c n l ilc a SPAPSPCS v i k v l t ng d n H nh (a)n g v i t r n g h p k=l+2 ,h nh (b)n g v i t r n g h p l=k+2 ,h nh (c)n g v i t r n g h p k= l+4 ,hnh(d)ngvitrnghp l=k+4
(hnhb)v l =k+2(hnhd)r icaSPAPSPCStngchmhnkhi k v l tng. ivihmriℜ c cphn g trnh(3.51)v( 3 5 2 ) , chngtiti nhnhvc cthc ahmℜchocctrn g hp k=l̸=q ,v k=l=q
H nh 3.21 (a) v( b ) h n h ( a ) , n g (k, l, q) = (0,0,8)tn g n g v i PCS,ccngc nlilcaSPAPSPCS.Ccthc h o thykhi k= lt ng th r i t ng r t nhanh v ng|ξ|nh , th m chg i t r ℜ ≈1ngayti| ξ|
= 0 vi k = l = q = 8.thyr i uc bitn y,c h ngt iv c c th h nh (b), trong k= l=qt
Hn h 3 2 1 : t h h m r i ℜ c a S P A P S P C S p h t h u c t h e o b i n |ξ| v b t h a m s ( k, l, q) v i ε =1,φ= 0 h nh(a), n g cong ( 0 ,0,8) n g vitrngthiPCS,ccn g cnlilcaSPAPS PCSvi q=8 trongkhi k=l tngdn.Hnh(b)lt r n g hp k=l=q tngdn. ngth ik h i k= l=qt n g , g i t r c aℜc n g t n gl n N h v y c tht h y , r i c a SPAPSPCS g n nht ltn g t r n t o n m i n |ξ|trongtrn g hp k,l,qb n g nhauvc h b ngvin v.Lur ng,khi k=l=q thSPAPSPCSct nhi xng cao nh t vp h n b p h o t o n haimode.th yri uny,chngtaxemlibiudinc aSPAPSPCS phng tr nh (2.8).Rr ng tr ng th i n ylth p c a hai tr ng th ith nh ph n Khi k=l=q th kh ng ch s photon hai mode a v b c am i tr ng th i th nh ph n b ng nhau mt n g s p h o t o n m i m o d e c a chai tr ng th i th nh ph n c ng b ng nhau.i un y h mr ng, vi cth m vb t p h o t o n s a o c h o S P A P S P C S c t n h i x n g c n g c a o v p h n bp h o t o n h a i m o d e t h an r i c a tr ng th i c ng c ng l n, th mchtgnmcltng. so s nh ti u chu nan r i m i v i ti u chu n entropy tuy n t nh,chngtasosnhccthcc ngbt ham s( k,l,q)hnh3.21(b)v
3.19 (b) Rr ngiv i chai ti u chu n, khi k=l=q an r i c aSPAPSPCS utm c r t cao v g n v i tr ng h plt ng ngaytkhigitr|ξ|chtvinv.n g thi,khigitrca k,l,q tngln m r i c a SPAPSPCS c ng t ng i m kh c bi tlv ng m| ξ|cgitrn h , a n r i c a S P A P S P C S t h e o t i u c h u n a n r i m i tgit r tr n 0.95 vt n g c h m n m t n v T r o n g k h i v i t i u c h u n e n t r o p y tuy n t nh, r it t0.5vt n g k h n h a n h n0.95, sau t ng r tchmn mtn v.
Nhv y, sau khi kh o s tr i c a SPAPSPCS b ng ti u chu nentropytuyn t nhvcti uchunan r i mi,chng t i th y r ngSPAPSPCSltr ng th ican r i r t m nh.Chai ti u chu ncktquk h t n g n g khiu chothyr i caSPAPSPCSctngcn g k h i t n g s p h o t o n t h m v o v b t i h a i m o d e c a P C S , n g th ian r it tnh t khi k= l=q , t clkhi tr ng th i n ycsphoton hai mode c n b ng.i m kh c bi txy ra v ng m|ξ|cgitrn h N h n g i m tngngv kh c bi tcng tn g tnhk h i p dngiviPAASTMPCS.
Tnhchtphici nbc caocatrngthikthpbb a chngchtthmphotonbamode
Tc c h m n n t ng ba mode b c caoc x cnhp h n g t r n h (1.85) v( 1 8 6 ) , c h n g t i t n h t o n v t h u c c c b i u t h c t n g minhcachngi viSTMPATCS[95]nhsau
Hnh3 2 2 : thh m S U (m) ph t h u cvobin r v b thams ( h,k,l) v i p=q= 0 v ε=λ=σ=1 h nh(a) m= 1 ,hnh(b) m=2 v bnh(c) m=3 trong B i,j,w , C u v D i,j,w ca r a c c p h n g t r n h ( 1 6 3 ) ,
Bi u th c gi i t ch trong phn g t r n h ( 3 5 5 ) v ( 3 5 6 ) c h o p h p c h n g tinghincutnhchtnntngbamodebccaocaSTMPATCS.TrongH nh 3.22, ch ng t i vt h h m S U (m)pht h c v o b i n r khi p=q= 0vε=λ=σ=1vim t sgitrkh c nhauc a h, k,l v m Th nhn y, ch ng tactht h y r n g m c n n c n ng cao b ng c ch t ng h,k,lc ngnh r v ibtkbc mn o.Tuynhin,mcmca S U (m)sg i m n u m cgit r c a o L ur ng tr ng th i kth p b ba kh ngt n t isn n t ng ba mode Do,sxu t hi n t nh ch t n n t ng bamode bc cao trong STMPATCS ch ng tvai trquan tr ng c a vi c th mphoton.
Chngt it p =q = 0 ,ε=λ = σ = 1v m =2 ,t r o n g t r n g h p h+k+l lh ng s v v ths ph thu c c a S U (m)v o r vi h,k v l cthayi H nh3.23.Mcn ntngbamodebccaoll n nh t khi sl n g p h o t o n c t h m v o h a i m o d e l n h n h t
T r o n g H nh3.24,chngt ic ngvb i u h m S U (m)phthuc λ v σ vi p=q= 0,h=k= l=ε=1,r=4v m = 2 Trongtrn g hpny,mc nntngbamodebccaolcaonhtkhic| λ|v| σ|uthpnht. Σ
Hnh3 2 5 : t h h m S V (m) p h t h u cvobi n r v b t h a m s ( h,k,l) k h i p =q= 0 v ε=λ=σ=1 iv i S V (m),trongH nh3.25,ch ngt ivt h h m n y phthucvobin r khi p=q=0 v m=2 ivimtsgitrca h,k v l Dd ngthyrngSTMPATCSkhngthh i ntnhchtnntngba modeb ccaotrong V ˆvc cgitrc a S V (m)lu nkh ngm.i uny cgiithchlv t r ngthiny t ntitnhchtnnt ngbamodeb ccaotrong U ˆ.
3.5.2.1.Tnhchtphnktchmn modebccao kh o s t t nh ch t ph n k t ch mn mode b c cao cho STM-PATCS, ch ngt it nh h m R x (m)( 1 9 3 ) v nh nc k tqui vimode al
( 3 5 8 ) v (3.59)chophpchngtikho sttnhchtphnktchmn modebccaoch oSTMPATCS.ivimode a ,trongHnh3.26,chngtiv thc a R a (m)p hn g tr nh(3 57) di dngm th mc a r k h i p=q=0ivim tsgitrc a(h, k,l)v m K tquc h o t h y r n g m c ph n ktchm nmode b c caoivimode ac t ng c ngbngcchtngbc m Tuynhin,ns g i mnutngsp h o t o n h , kv lc t hmvo.Ngoira,khi mnh v h,kvll n,hm R a (m)trnn m h n trong v ng gitrcao c a r i ung chn alkhi m l n,STMPATCSluntntitnhchtphnktchmnmodebccao.
Khi h+k+lc cnh, t clt ng s photon c th m v olm t h ng s , trong H nh 3.27,chng t ivt h h m R a (m)phthucv o bi n rkhip=q=0vm =2vic c git r k h c n h a u c a h, kv l Ch ng
(1, 1, 4) tactht h y r n g m c ph n ktch mnmode b c cao l nnhtkhislngphotonc thmvomode al nh nht.Mtkhc,
Hn h3 2 6 : th h m R a (m) p h t h u cvobin r v bthams ( h,k,l) k h i p=q= 0 v ε =λ=σ= 1 Hnh(a),(b)v( c ) lnlt n g vi m=1 , m=2 v m =3
Hnh3.28,khi p = q= 0 ,h= k= l=λ=1v r= 4 ,m ca R a (m)lcao nh t n u λ=σ=0, t clc c photon ch c th m v o mode a Lur ngccktqun utrncngngchoccmode b v c
Davohmphnktchmhaimodebccaoc chop h ngtrnh(1.96), chngtivitlihmnytheohaimode xvyψ s a u
⟨xˆ †m xˆ m yψˆ † yψˆ⟩+⟨yψˆ †m yψˆ m xˆ † xˆ⟩ −1 (3.61) Mtt r ngt h ia m o d e c t nhp h nk tc h mh a i m o d e b c m k h i
R x,yψ (m)q=l th F avgi m.i u n y cho th y, t i m i git r | ξ|,khi k , q,l t h a yψi saocho k = q = l t h F avtgit r caonhttcl trungthctrungbnhlt tnht.H nh4.5(b),cngtn g tn h vy,ccngc ongntt chothy,khi lt ngdnngit r l =q(ch l ≤q )tht r u n g thctrungbnh F avtng.Bncnhc h a i hnh
H nh4.5: thh m F av ph thucv obi n |ξ| vbbathams (k,q,l) v i ε=1; φ=0 Hnh(a)ngvitrn g hp k t ng,hnh(b)n g vitrn g hp l t ng,hnh(c)n g vicct rnghp k=l=q t ng. bt n m 2007 trong b i b o [30] C cngc n l i trong (c) t ng ngvic cS P A P S P C S c k , l,qn g t h it n g d n v lun t h ai uk i n k=l=q C cngcong n y cho th y,trung th c trung b nh c c ic a SPAPSPCS lu n cao h n trn g h p t tnh t c a PCS, vd trongh nh (b)c F av ≈82%t i |ξ|≈ 10 Th m v o,trung th c trungbnhtgitrcngcaonu k=l=qc g i trcngln.
Nhv y tacthnh nthy r ng, vi cp d ng SPAPSPCS v o vi ntiln g tmttrngthikthptronggiaothco ccthnhphntr c giao lr t t h n h c n g v t r u n g t h c t r u n g b n h k h n g n h n g v t quatr ngh pt tnh t c a PCS (76%)mc n dd n g ttr n 80%nhngvtrcbinkthpthp.
Trong chn g n y , c h n g t i p d ng c c tr ng th i phic i nhaimodemibaogmPAASTMPCS,SPAPCSv SPAPSPCSvoqut rnhvintiln g tmttrngthikthp.Cbatrngthim iu c sdng l m ngu n r i cho qutr nh vi n t i b ng giao th co c cthnhphntrcgiao.RingPAASTMPCSc p dngchocg i a o thc otngsh tvh i uphanhmsosnhhaigiaothccsd ng.
K t qu cho th y,ivigiao th coc c th nh ph n tr c giao,cbatr ng th i m iu lngu n r it tcho qu tr nh vi n t i.Cthtrungth c trung b nhiviPAASTMPCSctht n70%mi ncgitr| ξ| tk h o ngvichcn vt r ln.iviSPAPCSvS P A P S P C S trung th c trung b nh ccicao h ntrung th c c ci iviPCS (76%) vcthttr n80%khi|ξ|cgit r k h o n g v i nvv c c tham sc c h n p h h p t r u n g t h c t r u n g b n h i v i c b a tr ng th i m iuc t ng cn g k h i t n g s p h o t o n c t h m h o c bthocct h mvb tphhpv occmodecaPCS.
KhipdngPAASTMPCSvoqut r nhvintivigiaothco tngsh t vh i upha,t r u n g thctrungbnhcaqut r nhvinti c t ng cn g t r o n g t r n g h p t n g s p h o t o n c t h m v o m o d e aho c b tmode b c a PCS ho cltrn g h p g i m b i n kth p α Khi αl k h n h ,t r u n g thctrungbnhcaqut r nhvintidd ng t nx p x 100% Do PAASTMPCS phh p v igiao th co t n g sh tvh i uphahngiaoth co ccthnhphntr cgiaonubincatrngthikth pcvintilkhnh
Hinnay,cclnhvcnhthngtinln g t,mytnhln g tvvi nt iln g t cnh ngbc ph t tri n m nh m Vi c t m ra c ctr ng th i m icc c t nh ch t phic i ncao,cbi tlt nh ch ta n r i hay vi c t m ra c c phn g ph p t ng cn g m c th nh c ng c ac c nhi mvlngtnhv i n t iln g tv na n g lnh ng v nc n quantmnghincunhiu tl t h u y tnthcnghim.Tn h ngvn c p thi t,ch ngti l a ch n nghi n c u t i lu nn n y v thucmtsktquminhsau:
Thn h t , c h n g t i ara hai tr ng th i phi c i n hai modemibn g phn g phpthm,btphoton.utin,bngcchthmvb t nhxccphotonlnhaimodecaPCSchngtitoratrngthikthpcpthmvb t photonhaimode(PAASTMPCS).Tip, chngti d ng ph ng ph p th m v b t p h o t o n k h n g n h x t o r a t r n g thikthpcpchngchtthmphotonvb tphoton(SPAPSPCS).
Thh a i , c h n g t ixu tc ti u chu na n r i m i c h o h h a i mode th ng qua c c to n th i u s h t v h i u p h a c d n g
H e r m i t e C c ktquso s nh vnhgicho th y ti u chu nan r i m i r t hi u qutrong trong vi cdt m vn h l n g an r i c a c c tr ng th i haimode m i K t qu kh o s tan r i c a c c tr ng th i hai mode theoti u chu na n r i m i k h t n g n g so viti u chu n entropy tuy ntnh.Tuynhina n ritheotiuchunmithn g caohn.
Thba, chng t inghi n c u c c t nh ch t c a hai tr ng th i phic i n m i c xu t chn g h a i v h a i t r n g t h i p h i c i n h a i v ba mode m ic xu t g n y K t quc h o t h y , c c t r n g t h i n y ctnhphicincao,cthhinthngquatnhchtnn,tnhphn ktchm,tnhphiGaussvt nhchta n ri.Tnhphici ncachng ctngcn g bngcchthayi sp h o t o n thm,btvoccmodec a tr ng th i g c vic c tham suv o c ch n m t c ch ph h p.B n c nh, m t st n h c h t p h i c i ntrong tr ng h p b c cao th t thnbcthp,inhnhnhi viSTMPATCS.c bit,khisosnh an r i c a c c tr ng th i PCS, PAASTMPCS, SPAPCS vSPAPSPCS,chngtinhnthyvicthm,btphotont oratrngthimic an r i cao h n tr ng th igc banu Ngo i ra, trng h p th m, btphoton kh ng nhxthn g t o r a c c t r n g t h i m i canrit th nsovitrngh pnhx.
Tht ,ch ng t isd ng c c tr ng th i hai mode m i bao g mPAASTMPCS, SPAPCS vS P A P S P C S l m n g u n a n r i c h o q u t r n h vintiln g tmttrngthikthp.Davot r u n g thctrungb nh,ch ngtith yr ngc cqutr nhvi nti lr tt h n h c n g trungthctrungbnhcaccqut r nhvintic t ngcn g vc a o h nngks o v itrn g h p sd ng PCS khi th m, btsp h o t o n hai modec a PCS.iving n r ilSPAPCS vSPAPSPCS,trungthctrungbnhcaccqut r nhvintit git r c a o trn8 0%khisd ng giao th co c c th nh ph n tr c giao Tuy nhi n, vingu n r i lPAASTMPCS,t r u n g t h c t r u n g b n h c a q u t r n h v i n t i c t h l n tr n 90% khisd ng giao th co t ng sh t vh i u p h a , v i i uki nbincatrngthikthpcvintilnh
Nh vy , m cti u c ntc c at ilu n n c ho nth nh Lu nn cth c ti pt cnghi n c u vp h t t r i n t h e o n h i u h ng kh c nhau Hng thnh tlti pt cnghi n c u c c t nh ch t phicinbccaocacctrngthihaimodemi.Hn g thh a i lxutccmhnhct htoracctrngthihaimodemibngthcnghim.
Hn g t h t l p d ng ti u chu na n r i m i c h o n h i u h tr ng th i t m ra c c tr ng th ia n r i m i c a n r i c a o c n g n h x cnhc nhnghtrngthiphhpvitiuchunminy.
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