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VOLU M E 92, N U M BER 23

PH YSICA L R E VI E W L E T T E RS

week end i ng 11 J UNE 20 0 4

Identi f y i ng Enta nglement Usi ng Q ua nt um Ghost Inter ference a nd I mag i ng
Mi lena D'A ngelo ,1,* Yo on - Ho K i m ,
1 2

2,

Sergei P. Kuli k ,1, a nd Ya n hua Sh i h

1

Depa r t ment of Physics, Un iversit y of Ma r yla nd , Ba lt i more Count y, Ba lt i more , Ma r yla nd , 21250, USA Depa r t ment of Physics, Poha ng Un iversit y of Science a nd Te ch nolog y ( P OST ECH ) , Poha ng, 79 0 -784 , Korea ( Re ceive d 20 Aug ust 20 03; publ ishe d 8 June 20 0 4) We rep or t a qua nt u m i nt er ference a nd i mag i ng exp er i ment wh ich a l lows ident i f y i ng t he ent a ngle ment i n moment um a nd posit ion va r iables of a t wo-phot on system. T he mea surement s show i ndee d t hat t he uncer t a i nt ies i n t he sum of moment a a nd i n t he d i fference of p osit ion s of t he ent a ngle d t wo-phot on sat isf y bot h E PR i ne qua l it ies k1 k2 < mink1 ; k2 and x1 Ъ x2 < minx1 ; x2 . T hese t wo i ne qua l it ies, t oget her, represent a noncla ssica l it y cond it ion. Our mea surement s provide a di re ct way t o d ist i ng uish bet ween qua nt um ent a nglement a nd cla ssica l cor relat ion i n cont i nuous va r iables for system s of pa i rs of phot on s.
DOI: 10.1103/ PhysRevL et t.92. 233601 PACS nu mbers: 42. 50. Xa , 03.65.Ud , 42. 50.St , 42.65. L m

T he concept of mult ipa r t icle qua nt u m ent a nglement , one of t he most sur pr isi ng con sequences of qua nt u m me cha n ics, wa s i nt ro duce d i n t he ver y ea rly days of qua nt u m t heor y [1, 2]. Since t he development of spont a neous pa ra met r ic dow n - conversion (SPDC) a s a n efficient sou rce of t wo-phot on ent a ngle d st at es i n t he 1980's [3], ma ny exp er i ment s have been rea l i ze d t o ex h ibit a nd, a f t er wa rds, t o exploit t he ver y sur pr isi ng qua nt u m effe ct s of ent a ngle d st at es for secu re com mun icat ion , i n for ma t ion processi ng, a nd met rolog y appl icat ion s [4]. Some of t he most i nt r ig ui ng effe ct s of t wo-phot on ent a nglement i n SPDC a re qua nt u m ``g host'' i nter ference a nd i mag i ng [5,6]. T hese effe ct s a re of g reat i mp or t a nce i n p otent ia l appl icat ion s such a s qua nt u m met rolog y a nd l it hog raphy [7 ­ 9]. Re cent ly, it ha s been cla i me d t hat t he t wo-phot on g host i mage ca n be ach ieve d usi ng pa i rs of cla ssica l ly k-vector cor relat e d opt ica l pulses [10]. Ref erence [10], t herefore , ra ises i nt erest i ng question s about f unda ment a l issues of qua nt u m t heor y, na mely: (i) To what extent ca n qua nt u m ent a nglement i n cont i nuous va r iables be si mulat e d wit h cla ssica l ly cor relat e d syst em s, a nd (i i) ca n we exper i ment a l ly d ist i ng uish t hem? I n t h is L et t er, we rep or t a n exper i ment wh ich sheds l ig ht on t hese t wo t ig ht ly relat e d quest ion s. Ou r idea is t o exploit qua nt u m i nt er ference - i mag i ng effe ct s t o ver i f y exper i ment a l ly t he Ei n stei n - Podolsk y-Rosen ( E PR)-t y p e i ne qua l it ies, wh ich a l low d ist i ng uish i ng qua nt u m ent a n glement f rom cla ssica l cor relat ion i n moment u m a nd /or p osit ion va r iables, for t wo-photon system s. By a na lyz i ng t he result s of a t wo-phot on i nt er ference a nd i mag i ng exper i ment , we show qua nt it at ively t hat ent a ngle d twophot on pa i rs ex h ibit bot h moment um - moment u m a nd p osit ion -posit ion E PR -t y p e cor relat ion s, wh ich a re st ronger t ha n a ny cla ssica l cor relat ion. I n cont ra st , pa i rs of pa r t icles havi ng a p er fe ct cla ssica l cor relat ion in moment u m (or posit ion) ca n not ex h ibit a ny cor relat ion i n p osition (or moment u m) , due t o t he uncer t a i nt y pr i n ciple. Ou r exper i ment , t herefore , shows t hat ent a nglement i n moment u m a nd p osition va r iables for t wo-phot on sys233601-1 0031-9 0 0 7= 04 =92(23)=233601(4)$22. 50

t em s ca n be ver ifie d exper i ment a l ly a nd suggests t hat t he deg ree of ent a nglement ca n be qua nt ifie d t h rough t he E PR i ne qua l it ies.. T h is result is of pa r t icula r i nt erest , since it represent a `` Bel l's i ne qua l it y'' for cont i nuous va r iables, for t wo-photon system s. As p oi nt e d out by E PR [2], t he most pe cul ia r cha ract erist ic of ent a nglement is it s i ndep endency on t he sele ct e d ba sis: Ent a nglement i n moment u m aut omat ica l ly i mpl ies ent a nglement i n p osition. I ndee d , i n E PR not at ion [2], t he qua nt u m st at e of a n ent a ngle d pa i r of pa r t icles ca n be w r it t en a s x1 ;x2 Z up x1
p

x2 dp

Z

vx x1 x x2 dx;

where x1 (x2 ) is t he va r iable use d t o descr ibe pa r t icle 1 ( pa r t icle 2) , up x1 [ p x2 ] is t he moment u m eigen f unct ion for pa r t icle 1 ( pa r t icle 2), a nd vx x1 [x x2 ] is t he cor resp ond i ng posit ion eigen f unct ion obt a i ne d by Fou r ier t ra n sfor m of up x1 [ p x2 ]. As suggeste d by E PR , a n i mp or t a nt con sequence of ent a nglement app ea rs expl icit ly by con sider i ng t he ca se i n wh ich up x1 [ p x2 ] is a pla ne wave. I ndee d , i n t h is ca se, t he E PR - ent a ngle d st at e a ssumes a n i nterest i ng for m : 1 2h x1 1 p1 ;p2 2h p1 x1 ;x2 Z p1 p2 eip1
x1 =h eip2 x2 =h

dp1 dp

2

Ъ x2 ; Z x1 Ъ x2 eЪ p2 :

ip1 x1 =h eЪip2 x2 =h

dx1 dx2 (1)

A n E PR - ent a ngle d pa i r of pa r t icles is t er i ze d by bot h uncer t a i nt ies p1 p2 x2 0. I n a non ma x i ma l ly ent a ngle d cer t a i nt ies may be d i fferent f rom zero , may st i l l sat isf y bot h i ne qua l it ies:

t herefore cha rac 0 and x1 Ъ system , t hese un never t heless t hey

200 4 T he A mer ica n Physica l Societ y

233601-1


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PH YSICA L R E VI E W L E T T E RS
D1

week end i ng 11 J UNE 20 0 4

p1 p2 < minp1 ; p2 ; x1 Ъ x2 < minx1 ; x2 :

(2)

double-slit signal BBO a1 UV laser a
idler
2

A n ent a ngle d pa i r of pa r t icles is i ndee d descr ibe d i n such a way t hat bot h t he sum of t he moment a a nd d i fference i n t he p osit ion s a re k now n w it h a h ig h deg ree of accu racy even i f bot h t he moment u m a nd p osit ion of each pa r t icle a re complet ely undefine d [2]. A n ent a ngle d pa i r of pa rt icles ca n t hen be descr ibe d on ly a s a whole (i. e. , a s a t wopa r t icle [11]). T h is result is a n i m me d iat e con sequence of t he coherent super p osit ion of t wo -pa r t icle a mpli t udes, wh ich ca n not be ach ieved by a ny system of cla ssica l ly cor relat e d pa i rs of pa r t icles, a s d iscussed i n more det a i l lat er. We prop ose E qs. (2) a s a noncla ssica l it y cond it ion for ident i f y i ng ent a nglement i n moment u m a nd p osit ion va r iables, for m icroscopic bipa r t it e system s. We rep or t a n exper i ment a l ver ificat ion of E q. (2) wh ich exploit s qua nt u m g host i nter ference a nd i mag i ng effe ct s of en t a ngle d t wo-phot on pa i rs. We mea sure k1 k2 f rom a qua nt u m i nt er ference exper i ment a nd x1 Ъ x2 f rom a qua nt u m i mag i ng exper i ment , bot h rea l i ze d using t he sa me SPDC source. To t he best of our k nowle dge , a d i re ct qua nt it at ive ver ificat ion of E qs. (2) for t he t wo-phot on system , i. e. , for rea l moment u m a nd p osition va r iables, ha s not been rep or t e d i n l it erat ure [12]. L et us first exa m i ne whet her SPDC t wo-photon pa i rs would rea l ly ex h ibit E PR -t yp e ent a nglement. Under t he a ssumpt ion t hat t he pump bea m is a pla ne wave a nd t he t ra n sverse d i men sions of t he pump bea m a nd t he down conversion cr yst a l a re much bigger t ha n t he waveleng t h s of t he phot ons, t he qua nt u m stat e of t he SPDC t wophot on pa i rs ca n be w r it t en a s [11,13] X ji !s !i Ъ !p ks ki Ъ kp ay ay j0i; si
s;i

b BS scan scan D3

CC

CC

Imaging Lens

f D2

F IG. 1 (color on l i ne). Schemat ic of t he exper i ment a l set up. Ghost i nt er ference is obser ved i n t he foca l pla ne of t he len s ( Four ier t ra n sfor m pla ne , or moment u m space) t o mea sure t he moment um - moment um cor relat ion , wh i le t he g host i mage is obser ved i n t he t wo-phot on i mag i ng pla ne t o mea sure t he posit ion -posit ion cor relat ion. T he double slit ha s widt h a 0:165 mm a nd slit d ist a nce d 0:4 mm. T he i mag i ng len s ha s foca l leng t h f 510 mm. Releva nt d ist a nces i n t h is exper i ment a re a1 32:5, a2 46:5, a nd b 142 cm.

(3) where !j and kj (wit h j s; i; p) a re t he f re quency a nd wave ve ct or of t he sig na l (s), id ler (i), a nd pump (p), resp e ct ively, a nd ay (ay ) is t he creat ion operat or for t he s i sig na l (id ler) phot on. Si nce i n t h is L et t er we a re i nt erest e d on ly i n t he t ra n sverse cor relat ion of t he ent a ngle d t wo-phot on pa i rs [13], t he qua nt u m st at e used i n our exper i ment is i ndee d ver y close t o t he one of t he or ig i na l E PR - ent a ngle d pa i rs. Ver ificat ion of t he E PR i ne qua l it ies of E q. (2) should t hen be p ossible t h rough ade quat e exp er i ment s rea l i ze d wit h t h is source. A schemat ic of t he exper i ment a l set up ca n be seen i n Fig. 1. T he 351.1 n m l i ne of a n a rgon ion la ser is use d t o pu mp a bet a ba r ium borat e cr ysta l cut for t y p e - I I col l i nea r degenerat e SPDC. Pai rs of or t hogona l ly p ola r i ze d sig na l a nd id ler phot on s at cent ra l waveleng t h i s 702:2 nm, wh ich a re ent a ngle d i n moment u m [ E q. (3)], emerge f rom t he cr ysta l a l most col l i nea rly wit h t he pump la ser. A f t er t he cr ysta l , t he pump la ser bea m is sepa rat e d f rom t he SPDC bea m by a qua r t z d isp ersion pr ism. A p ola r i zat ion bea m spl it t i ng T hompson pr ism sepa rat es 233601-2

t he copropagat i ng sig na l a nd id ler i nt o t wo sepa rat e spat ia l mo des. T he sig na l phot on propagat es t h roug h a double slit t owa rd a det e ct or package (D1 ) con sisti ng of a col le ct ion len s (50 0 m m fo ca l leng t h) a nd a single phot on det e ct or place d i n it s focus. T he id ler phot on propagat es f reely before bei ng col le ct e d by t he i mag i ng len s (f 510 mm). A 50- 50 bea m spl it t er ( BS) is i n ser t e d a f t er t he len s. T he refle ct e d a nd t ra n sm it t e d phot ons a re t hen det e ct e d by single phot on det e ct ors D2 and D3 , resp e ct ively. Each of t hem is mount e d on a n enco der d r iver t o sca n it s own t ra n sverse pla ne. A spe ct ra l filt er cent ere d at 702.2 n m w it h 3 n m ba ndwidt h pre ce des each det e ct or. T he out put pulses of t he det e ct ors a re sent t o a coi nci dence ci rcuit (CC). Coi ncidences a re mea sure d bet ween D1 and D2 a nd bet ween D1 and D3 . T h is set up t herefore a l lows one t o mea sure bot h g host i nt er ference di ff ract ion a nd g host i mage pat t er ns of t he double sl it [5,6]. I ndee d , t he coherent super p osition of biphot on a mpl it udes a l lows exploit i ng t he moment um moment u m cor relat ion t o obt a i n a n i mage ( p osition p osition cor relat ion) by si mply cha ngi ng t he obser vat ion pla ne (D3 , i n st ead of D2 ) [13 ­15]. T he result s a re shown i n Fig. 2. T he single count s on bot h D2 and D3 , wh ich a re sca n ne d i n t he t ra n sverse d i re ct ion , a re fa i rly con st a nt. T he si ngle count i ng rat e of D1 , when t he det e ct or sca n s t he foca l pla ne of t he col le ct ion len s, d id not show a ny i nt er ference f r i nges a s wel l: On ly a wide bel l -shape d pat t er n wa s obser ve d. T h is result is due t o t he fact t hat biphot ons a re generat e d wit h a l l p ossible moment a ki and ks such t hat ks ki kp is sat isfied. I n our exper i ment , t he d ivergence of t he SPDC bea m , wh ich t a kes i nto account t he filt er ba ndwidt h , t he d isp ersion i n t he cr yst a l , a nd t he pha se mat ch i ng condit ion , is such t hat 2:6 mrad =d, where d 0:4 mm is t he d ist a nce be t ween t he slit s a nd 702:2 nm is t he cent ra l wave leng t h of t he SPDC phot ons. Under t h is cond it ion , t he first order i nt er ference - d i ff ract ion pat t er n on D1 is si mply washe d out. 233601-2


VOLU M E 92, N U M BER 23
(a)

PH YSICA L R E VI E W L E T T E RS
70

week end i ng 11 J UNE 20 0 4

D1-D2 Coincidences

160 120

60 50 40

D2 Counts (x103)

above , we eva luat e kxj 23 mmЪ1 , for j i; s. So t he g host i nt er ference/d i ff ract ion exper i ment demon st rat es t hat kxs kxi minkxi ; kxs ; for our SPDC source. I n a si m i la r way, we obt a i n xs Ъ xi by st udyi ng t he g host i mage obt a i ne d by mea sur i ng coi ncidences bet ween D1 and D3 [see Fig. 2 ( b)]. To obser ve a g host i mage , t he t wo-photon Gaussia n t h i n len s e quat ion , 1=si 1=so 1=f, where si b and so a1 a2 , must be sat isfied [6]. I n t he idea l sit uat ion , t he det e ct or package D1 is a p er fe ct bucket det e ct or, wh ich det e ct s a ny sig na l phot on t hat ha s not been st oppe d by t he double sl it. T he role of t he double slit is t hen t o mea sure t he lo ca l i zat ion of t he sig na l phot on wit h a n uncer t a i nt y xs , e qua l t o a d. I n t he idea l ca se, count i ng coi ncidences bet ween D1 and D3 ,we would obt a i n t wo re ct a ngles of widt h a0 ma, a nd cent er-t o - cent er d ist a nce d0 md, where m si =so is t he mag n ificat ion. I n our ca se m 1:8, a 0:165 mm, d 0:4 mm, a nd t he cor resp ondi ng idea l result (a0 0:297 mm, d0 0:72 mm) is plot t e d a s a da shed l i ne i n Fig. 2 ( b). To t a ke i nt o account a more rea l ist ic sit uat ion , we fit t he dat a w it h t he convolution of t he double sl it wit h a Gaussia n f unct ion t hat t a kes i nt o account t he fin it e size of D2 . T he compa r ison of t he result i ng fit t i ng cu r ve wit h t he t heoret ica l result , da she d l i ne i n Fig. 2 ( b) , a l lows one t o eva luat e xs Ъ xi a s t he d i fference bet ween t he va ria nces of t he t wo cu r ves: xs Ъ xi 0:11 0:02 mm xs : Note t hat t he cent er-t o - cent er d ist a nce bet ween t he bel l shape d fit t i ng cu r ve a nd t he t wo re ct a ngles is exact ly t he sa me. T he i mp er fe ct cor relat ion i n p osition is ev ident ly sma l ler t ha n t he d ist a nce bet ween t he t wo slit s. Bot h E PR i ne qua l it ies i nt roduce d i n E qs. (2) a re t hen sat isfied by t he moment u m a nd p osit ion va r iables mea sure d on t he ent a ngle d t wo-phot on pa i rs em it t e d by SPDC. As we ment ione d ea rl ier, t h is result is a d i re ct con sequence , i f not t he defin it ion , of qua nt u m cor relat ion : Pa r t icles t hat a re ent a ngle d i n moment u m a re aut omat ica l ly ent a ngle d i n p osition. On ly en t a ngle d t wo-pa r t icle ­ t wo-photon pa i rs ca n sat isf y bot h i ne qua l it ies. A n i nt erest i ng way of underst a ndi ng t h is result is t he fol lowi ng. A n ent a ngle d stat e such a s t he one g iven i n E q. (1) [a nd E q. (3)], ca n be fact ore d [11] by i nt ro duci ng t he va r iables p1 p2 and p1 Ъ p2 =2. T he cor resp ondi ng Fou r ier- conjugat e va r iables a re x1 x2 =2 and x1 Ъ x2 , resp e ct ively. Since p1 p2 and x1 Ъ x2 a re not Fou r ier- conjugat e va r iables, E qs. (2) ca n defin it ely be t r ue si mult a neously. For t h is sa me rea son , a lso t he prod uct of t he t wo uncer t a i nt ies ca n be sma l ler t ha n 1. L et us now con sider a stat ist ica l m i xt u re of pa i rs of qua nt a cla ssica l ly cor relat e d i n moment u m. A n exa mple of such a source is given by a pa i r of bounde d ident ica l 233601-3

80 40 0 -2 -1

30 20 10

(b)
100

x2

-0 (mm)

1

2

120 100 80 60 40

D1-D3 Coincidences

60 40 20

20 0 -1.0 -0.5

x

0.0 3 (mm)

0.5

1.0

F IG. 2 (color on l i ne). Exp er i ment a l dat a. (a) Ghost i nt er ference - d i ff ract ion pat t er n. ( b) Ghost i mage pat t er n. k1 k2 and r1 Ъ r2 a re eva luat e d f rom each of t he fitt ing cu r ves (sol id l i nes). T he squa res a re single count s; t he dot s a re coi ncidence count s.

It is, however, possible t o obser ve a g host i nt er ference d i ff ract ion pat t er n when count i ng coi ncidences bet ween D1 and D2 (D1 is fixe d a nd D2 is sca n ne d i n t he fo ca l pla ne of t he i mag i ng len s) a nd t o obser ve a g host i mage pat t er n i n coi ncidences bet ween D1 and D3 (D1 is, aga i n , fixe d a nd D3 is sca n ne d i n t he i mage pla ne). For g host i nt er ference - d i ff ract ion mea surement , t he det e ct or package D1 plays t he role of a p oi nt l i ke det e ct or. A s shown i n Ref. [5], we exp e ct t he coi ncidence count i ng rat e t o be Rc x2 / sinc2 x2 a=fcos2 x2 d=f; (4)

where x2 is t he t ra n sverse p osition of det e ct or D2 in t he fo ca l pla ne of L2 . Fig u re 2 (a) shows t he g host i nter ference mea surement. T he cont i nuous l i ne i n Fig. 2 (a) is a fit t i ng of t he exper i ment a l dat a , wh ich t a kes i nt o account t he fin it e si ze of t he det e ct ors, t he divergence of t he pump , a nd t he less-t ha n -per fe ct cor relat ion bet ween signa l a nd id ler. It is wel l k nown t hat t he v isibi l it y of t he i nt er ference pat t er n f/ 1 V cos2x2 d=f2 g is relat e d t o t he deg ree of t ra n sverse coherence of t he source. We exploit t h is effe ct t o eva luat e t he less-t ha n -per fe ct t ra n sverse cor relat ion bet ween sig na l a nd id ler phot ons: kxs kxi 2:5 0:6 mmЪ1 : From t he d ivergence of t he SPDC bea m ment ione d 233601-3

D3 Counts (x103)

80


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week end i ng 11 J UNE 20 0 4

g uns wh ich em it (qua nt u m) pa r t icles wh i le rot at i ng si mult a neously, i n such a way t hat t he moment a of t he t wo pa r t icles a re a lways e qua l i n mo dulus but wit h opposit e d i re ct ion. Each pa i r of i ndep endent but cor relat e d pa rt icles, fire d at a cer t a i n a ngle at a g iven t i me , may be descr ibe d by jj i12 ay kj ay Ъkj j0i: 1 2 I f each pa i r of pa r t icles ha s (non - negat ive) probabi l it y Pkj of bei ng em it t e d by t he source , t he result i ng i n coherent st at ist ica l m i xt u re is descr ibe d by t he fol lowi ng den sit y mat r i x: X X 12 Pkj jj i12 12 hj j Pkj j
j ; (5) 1 2
k
j

exper i ment s [18]. T h is is a quit e di fferent approach wit h resp e ct t o Bel l's i ne qua l it y a nd may represent a n exten sion of Bel l's i ne qua l it y i n opt ics. T he aut hors t ha n k M. H. Rubi n a nd V. P rot opopescu for helpf ul com ment s a nd d iscussions. T h is work wa s supp or t e d , i n pa r t , by NASA-CASPR prog ra m , NSF, a nd ON R. Y. H. K. a lso ack nowle dges suppor t f rom t he Korea Resea rch Foundat ion ( KR F-2003- 0 70-C00 02 4).

k

j

where jkj i11 hkj j and jЪkj i22 hЪkj j are t he den sit y mat r ices for pa r t icles 1 a nd 2 , resp e ct ively, be long i ng t o t he jth pa i r. It is wel l k now n t hat , for each pa r t icle t o propagat e wit h such a p er fe ct ly wel l - define d moment u m , t he t ra n sverse d i men sion of t he source ha s t o be i nfin it e [16 ,17]. T herefore , pa i rs of pa r t icles wit h a p er fe ct moment u m - moment u m cor relat ion do not ex h ibit a ny p osit ion -posit ion cor relat ion. I n t he more rea l ist ic ca se of fin it e t ra n sverse d i men sion of t he source , t he p osit ion -posit ion cor relat ion i mproves at t he expen se of t he moment u m - moment u m cor relat ion : Each pa r t icle is a lways d i ff ract e d i ndep endent ly. I n genera l , a ny at t empt t o i mprove t he cla ssica l cor relat ion i n one va r iable i nev i t ably worsens t he cor relat ion i n t he ot her. T hus, a system of cla ssica l ly cor relat e d pa r t icles ca n never sat isf y bot h E PR i ne qua l it ies [ E qs. (2)]. I n conclusion , a ny source of cla ssica l ly cor relat e d pa i rs of qua nt a (i) ca n never ach ieve per fe ct cor relat ion i n bot h moment u m a nd p osit ion va r iables, a nd (i i) ca n never sat isf y t he pa i r of E PR i ne qua l it ies [ E qs. (2)]. I n sum ma r y, we have exper i ment a l ly demon st rat e d t hat SPDC t wo-photon pa i rs sat isf y bot h E PR i ne qua l i t ies of E qs. (2). I n doi ng so, we have shown t hat ent a ngle d pa r t icles ex h ibit a l most per fe ct cor relat ion i n bot h mo ment u m a nd p osition va r iables. Cla ssica l ly cor relat e d pa i rs of pa r t icles ca n not ex h ibit such behav ior. T he mea surement descr ibe d i n t h is L et t er t hus prov ides a d i re ct way t o d ist i ng uish bet ween qua nt u m ent a nglement a nd cla ssica l cor relat ion i n moment u m a nd /or p osition va r i ables, for t wo-phot on system s. A n i mp or t a nt pract ica l con sequence is t hat on ly t he non lo ca l cor relat ion i mpl icit i n ent a ngle d system s a l lows one t o ``overcome'' t he usua l d i ff ract ion l i m it a nd t o obt a i n super- resolve d i mages, a s prop ose d a nd demon st rat e d i n Refs. [8,9,17]. Fu r t hermore , ou r exper i ment shows t hat a d ist i nct ion bet ween cla ssica l ly cor relat e d a nd qua nt u m ent a ngle d system s, i n moment u m a nd /or p osition va r iables, ca n be rea l i ze d exper i ment a l ly t h roug h t he st udy of g host i mag i ng-t y p e

j 1

j 2

[1]

[2] [3] [4] [5] [6] [7] [8]

[9] [10] [11] [12]

[13] [14] [15] [16]

[17] [18]

*Ele ct ron ic add ress: d m i lena1@u mbc. e du Ele ct ron ic add ress: yo on ho@post e ch.ac. k r Per ma nent add ress: Depa r t ment of Physics, Moscow St at e Un iversit y, Moscow, Russia. Ё E. Sch rod i nger, i n Qua nt u m T heor y a nd Mea surement , e d it e d by J. A. W heeler a nd W. H. Z u rek ( P r i ncet on Un iversit y P ress, New York , 1983). A. Ei n stei n , B. Podolsk y, a nd N. Rosen , Phys. Rev. 47, 777 (1935). D. N. K lysh ko , Phot on a nd Non l i nea r Optics (Gordon a nd Breach Science , New York , 1988). J. P. Dowl i ng a nd G. J. Mi lbur n , qua nt -ph /02060 91; A. Migda l l , Phys. To day 52 , No. 1, 41 (199 9). D. V. St reka lov et a l. , Phys. Rev. L et t. 74 , 360 0 (19 95). T. B. P it t ma n et a l. , Phys. Rev. A 52 , R3429 (1995). V. Giova n net t i , S. Lloyd , a nd L. Maccone , Phys. Rev. A 65, 022309 (20 02); Y. Sh i h (unpubl ishe d). A. N. Bot o , P. Kok , D. S. Abra m s, S. L. Braun st ei n , C. P. Wi l l ia m s, a nd J. P. Dowl i ng, Phys. Rev. L et t. 85, 2733 (20 0 0). M. D'A ngelo , M. V. C hek hova , a nd Y. Sh i h , Phys. Rev. Let t. 87, 013602 (20 01). R. S. Ben n i n k , S. J. Bent ley, a nd R. W. Boyd, Phys. Rev. Let t. 89, 113601 (20 02). M. H. Rubi n , D. N. K lysh ko , Y. H. Sh i h , a nd A. V. Sergien ko , Phys. Rev. A 50, 5122 (19 9 4). A si m i la r noncla ssica l it y relat ion ha s been ver ifie d exp er i ment a l ly for a n OPO [Z. Y. O u, S. F. Perei ra , H. J. K i mble , a nd K. C. Peng, Phys. Rev. L et t. 68, 3663 (19 92)]. T he qua nt it ies mea sure d i n t he exp er i ment a re t he spe ct ra l comp onent s of t he quad rat ure -pha se a mpl i t udes ( pha se a nd i nt en sit y) of t he sig na l a nd id ler bea m s. M. H. Rubi n , Phys. Rev. A 54 , 5349 (1996). M. H. Rubi n , qua nt -ph /0303188. A. Gat t i , E. Bra mbi l la , a nd L. A. Lug iat o, Phys. Rev. Let t. 90, 133603 (20 03) D. Be d ford a nd F. Sel ler i , Let t. Nuovo Ci ment o 42 , 325 (1985); M. J. Col let a nd R. L oudon , Nat ure ( L ondon) 326 , 6 71 (1987). Y.-H. K i m a nd Y. H. Sh i h , Found. Phys. 29, 1849 (19 9 9). Ca ncel lat ion of first order effe ct s is a prere quisit e for ma k i ng t h is d ist i nct ion si nce qua nt u m g host i nt er ference - i mag i ng phenomena a re second order effe ct s. A lso , t wo i ndep endent mea surement s a re re qui re d: one i n moment um space ( Fou r ier t ra n sfor m pla ne) a nd a not her one i n posit ion space (i mage pla ne).

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