The Test of Entanglement of Polarization States of a Semi-Classical Optical Parametric Oscillator

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1 Amricn Journl of Modrn Physics 07; 6(3: doi: 0.68/j.jmp ISSN: (Print; ISSN: (Onlin Th Tst of Entnglmnt of Polriztion Stts of Smi-Clssicl Opticl Prmtric Oscilltor Grld Mugy Lismdi, Bonifc Otino Ndiny, Josph Ayo Omolo Dprtmnt of Physics nd Mtrils Scinc, Msno Univrsity, Msno, Kny Dprtmnt of Physics, Msind Muliro Univrsity of Scinc nd Tchnology, Kmg, Kny Emil ddrss: (G. M. Lismdi, (B. O. Ndiny, (J. A. Omolo To cit this rticl: Grld Mugy Lismdi, Bonifc Otino Ndiny, Josph Ayo Omolo. Th Tst of Entnglmnt of Polriztion Stts of Smi- Clssicl Opticl Prmtric Oscilltor. Amricn Journl of Modrn Physics. Vol. 6, No. 3, 07, pp doi: 0.68/j.jmp Rcivd: Mrch 9, 07; Accptd: April 0, 07; Pulishd: My 8, 07 Astrct: W study th dynmicl continuous vril ntnglmnt in smi-clssicl Opticl prmtric oscilltor (OPO. Th gnrl tim volving photon polriztion stt vctors rising from xct nlyticl solutions of Hisnrg s qutions of th systm r usd to otin th photon polriztion Bll stt vctors. Th rducd dnsity mtrics of photon polriztion Bll stt vctors of th smi-clssicl OPO produc grtr violtion of CHSH Bll s inqulity yond th Cirl son s inqulity. Kywords: Opticl Prmtric Oscilltor, Bll Stt Vctors, Rducd Dnsity Mtrics, Entnglmnt. Introduction Quntum ntnglmnt is physicl phnomnon tht occurs whn pirs or groups of prticls r gnrtd or intrct in such mnnr tht th quntum stt of ch prticl cnnot dscrid indpndntly; instd, quntum stt is givn for th systm s whol. Entnglmnt hs found mny pplictions in th filds of quntum computtion nd quntum informtion procssing such s quntum cryptogrphy, quntum dns coding, ntnglmnt swpping, quntum lithogrphy nd quntum tlporttion []. Quntum tlporttion (trnsfr of quntum stts twn distnt loctions without n intrvning mdium hs n chivd ovr long distncs [, 3, ]. On August 6, 06, Chin lunchd th world s first quntum stllit, communiction systm incpl of ing hcd nd strtching ovr distnc of 000 m [5]. Continuous vril ntnglmnt ws dmonstrtd for th first tim in th Opticl prmtric oscilltor (OPO oprting low thrshold in 990. Entnglmnt in th ov-thrshold OPO rmind n xprimntl chllng until 005, whn it ws first osrvd y Villr S. nd Cssmiro N. (005 [6]. Entnglmnt fturs for full quntum trtmnt of OPO hs n studid whr tim volution qutions r solvd through writing of th dnsity oprtor qution in th Wignr rprsnttion using quivlnt Lngvin qutions to otin nlyticl rsults [7]. Su nd Tn (006 [8] osrvd two-color ntnglmnt msurd to th ov-thrshold OPO only. Th multiprtit ntur of ntnglmnt ws vrifid y vluting th vn Looc-Furusw critrion for prticulr st of ntnglmnt witnsss dducd from physicl considrtions [9]. Johnsson R. (0 [0] invstigtd thorticlly th conditions undr which multi-mod nno-mchnicl rsontor, oprting s purly mchnicl prmtric oscilltor, cn drivn into highly non-clssicl stts. Quntum ntngld stts of th systm violt Bll inqulitis with homodyn qudrtur msurmnts. Chrrti R. nd Jnish J. (05 [] studid th volution of iprtit ntngld qusi-ll stt in strongly coupld quit oscilltor systm in th prsnc of sttic is, nd xtndd it to th ultr-strong coupling rgim. Aditic pproximtion ws usd to otin rducd dnsity mtrix of th quit for th strong coupling domin in closd form involving linr comintions of th Jcoi tht functions. Aprt from mploying th ditic pproximtion no othr simplifiction hs n md for driving th rducd dnsity mtrix lmnts. A tst of Bll inqulity using polriztion ntngld photons from Spontnous

2 38 Grld Mugy Lismdi t l.: Th Tst of Entnglmnt of Polriztion Stts of Smi-Clssicl Opticl Prmtric Oscilltor Prmtric down Convrsion (SPDC hs shown violtion of Bll s inqulity []. In th currnt ppr, w us rducd dnsity mtrics of polriztion stts of smi-clssicl OPO to tst th violtion of CHSH Bll s inqulity, undr ll conditions of intrction. Th ppr is orgnizd s follows. In sction, th Hmiltonin of th systm is dvlopd nd usd to otin th tim volution oprtor of th systm. In sction 3, th tim volving photon polriztion Bll stts r constructd. In sction, th tst of ntnglmnt of polriztion stts of OPO using rducd dnsity mtrics nd Bll inqulitis is prsntd. A conclusion is providd in sction 5.. Th Tim Evolution Oprtor Th OPO is n ltrntiv tool for non-linr gnrtion of ntngld photons whr pump photon is convrtd into two lowr nrgy ms, nown s th signl photon nd th idlr photon occurring du to xcittion of mtril mdium whn struc y n xtrnl lctromotiv forc. Th signl photon is tn to initilly in th horizontl polriztion stt nd th idlr photon is tn to initilly in th vrticl polriztion stt. Th photon is thrfor in p p suprposition of horizontl nd vrticl polriztion. Th horizontl polriztion stt vctor rprsnts th sic unit vctor 0 nd th vrticl polriztion stt vctor rprsnts th sic unit vctor ccording to th dfinition H 0 0, V 0 Th two-mod Hmiltonin of smi-clssicl OPO is dscrid y [3] ( ˆ + ˆ ˆ+ ˆ ˆ + ˆ ˆ+ ˆ ( H ω + ω + + ( whr ˆ ( ˆ + ħ is th signl nnihiltion (crtion oprtors, ω is th signl frquncy, ˆ ˆ ( + (crtion oprtors, is th idlr nnihiltion ω is th idlr frquncy nd ( is tim-dpndnt intrction prmtr which vris hrmoniclly with tim in th form of rotting pump fild of frquncy ( ω p, ccording to iω t iω t ( ( t ; ( t ; ( t const nt Th dynmics of th smi-clssicl OPO is dscrid through Hisnrg s qutions dˆ ˆ ˆ Η iħ ˆ, Η ħ ˆ ( ω ˆ + t (3 dt ˆ+ ħ dˆ ˆ ˆ ˆ Η i, Η ˆ ( ˆ + ˆ+ ħ t ω (3 dt Sˆ Sˆ σ (5c ;, 0 z z 0 ˆ S+ σ + σ S 0 0 σ σ (5d 0, ˆ x i y ( x i y nd σ i; i x, y, z r th Puli spin oprtors. Eqution (5 is solvd to otin th gnrl form of tim volution photon polriztion vctor, xprssd s Equtions (3 nd (3 cn xprssd in mtrix form s ˆ ħ da ω i ħ dt ( t ( t ω Aˆ ( whr  is th two-componnt photon pir polriztion vctor dfind s ˆ Aˆ ˆ 0 + ˆ ˆ + ˆ ˆ H V Eqution ( cn xprssd in ltrntiv form s whr ( daˆ ˆ ˆ ˆ iħ ħ ˆ ˆ ( Ω S0 + ωsz + ( t S+ + ( t S A (5 dt Ω ω + ω ; ω ω ω, (5 Aˆ ( t U ( t A ˆ(0 A whr UA ( t is th tim volution oprtor, Ω ˆ iωpt ˆ ˆ ωpt ˆ S t tsˆ t S + t S+ + t S z 0 p UA( t z (6 ω (6 nd is constnt dtuning prmtr, ω ω (6c W sustitut for spin oprtorsŝ 0, S ˆz, Ŝ + nd Ŝ (5c-d in (6 to otin th finl mtrix s whr i i ( Ω + ωp t ( Ω + ωp t µ t ν t U A ( t i i ( Ω ωp t ( Ω ωp t ν t µ t (7 p

3 Amricn Journl of Modrn Physics 07; 6(3: µ ( t cos( t sin( t i p ( t t ω ν t sin( t + ; (7 (7c (7d Similrly, if w considr th signl photon rprsntd y mod to initilly in th vrticl polriztion stt V nd th idlr photon rprsntd y mod to initilly in th horizontl polriztion stt H, nd follow stps givn y qutions ( to (7d, w otin th ltrntiv finl form of tim volution oprtor s i i ( Ω ωp t ( Ω ωp t µ t ν t U B ( t i i ( Ω + ωp t ( Ω + ωp t ν t µ t 3. Bll Stt Vctors For two-prticl systm whr th signl photon is dnotd y mod is tn to initilly in th horizontl polriztion stt nd th idlr photon is dnotd y mod is tn to initilly in th vrticl polriztion stt or vic vrs. Thn th mximum ntngld Bll stt vctor of th photon is th suprposition of horizontl nd vrticl polriztion prsntd in simpl form s [] ( 0 0 ( H H V V φ ± ± ± (9 ( 0 0 ( H V V H ψ ± ± ± (9 Using qutions (, (7 nd (8 th four possil ign stts of th two-prticl systm in (9 nd (9 ts th form Ω + ωp t 0 Ω ωp t A µ ν (0 H U t H t t (8 Ω ωp t 0 Ω + ωp t B µ + ν (0 H U t H t t Ω ωp t 0 Ω + ωp t B ν + µ (0c V U t V t t Ω + ωp t 0 Ω ωp t ν + B µ (0d V U t V t t Using th gnrl tim volving photon polriztion stt vctor qutions (0-d in (9 nd (9, th four tim volution photon polriztion Bll stt vctor rducs to ψ + ψ φ φ + Ωt Ω t cos + t sin ( t i cos( ω sin sinsin pt t + ωpt ( t i cos( ω sin + sinsin pt t ωpt ( t cos + t sin ( t cos t sin ( t sin( ω sin cossin pt t ω pt t sin( ω sin + cossin pt t ω pt t cos + t t sin ( t i cos( ω sin + sinsin pt t ω pt t cos + t sin ( t Ω t cos + t sin ( t i cos( ω sin + sinsin pt t ω pt t sin( ω sin cossin pt t ω pt t cos t sin ( t Ω t cos + t t sin ( t sin( ω sin + cossin pt t ω pt t. Rducd Dnsity of Stt ( ( (c (d Th Clusr Horn Shimony-Holt (CHSH Bll inqulity dfind s S + ( is commonly usd to tst th ntur of ntnglmnt of photon polriztion stts[5]. In ( S rprsnts th rducd dnsity mtrix for Bll stt vctors [6]. Entnglmnt is xhiitd y violtions of th Bll inqulity. Th lrgr th violtion of th Bll inqulity th mor th ntnglmnt prsnt in th systm. W dtrmin rducd dnsity mtrics y otining th trc of dnsity mtrics undr th following conditions of intrction:

4 0 Grld Mugy Lismdi t l.: Th Tst of Entnglmnt of Polriztion Stts of Smi-Clssicl Opticl Prmtric Oscilltor.. Rsonnc At rsonnc, nd (7d rducs to >> (3 ; 0 ( p Tr ˆ ρ ( t Tr φ φ sin ( ω tsin ( t ( p Tr ˆ ρ ( t Tr φ φ + sin ( ω tsin ( t ( Tr ˆ ρ ( t Tr ψ ψ sin ( ω tsin ( t (c p p Tr ˆ ρ ( t Tr ψ ψ + sin ( ω tsin ( t (d W us th condition in (3 nd (3 in polriztion Bll stt vctors (-d to otin th trc of th dnsity mtrics undr rsonnc condition s Th mximum nd lst trcs in (-d t tim t nd t r tultd in tl. Tl. Mximum nd lst trcs of dnsity mtrics for Bll stts (-d undr rsonnc t tim t nd t whr n0,, nd ωp. Tim Tr ( φ+ φ+ Tr nπ t ωp π t + n ωp 0 0 According to CHSH Bll inqulity (, th Bll stts ( nd (c undr rsonnc produc w ntnglmnt. Th Bll stts ( nd (d produc strong ntnglmnt ov th Cirl son s inqulity t th tim t... Vry W Intrction At vry w intrction nd (7d rducs to ; >> (5 W us th condition in (5 nd (5 in polriztion Bll stt vctors (-d to otin th trc of th dnsity mtrics undr vry w intrction condition s << (5 ( + + ( ( + + ( Tr φ φ Tr φ φ Tr ψ ψ Tr ψ ψ sin ( t (6 Th mximum nd lst trcs rising from (6 undr vry w intrction r tultd in tl. Tl. Mximum nd lst trcs of dnsity mtrics for Bll stts (6 undr vry w intrction t tim t nd t whr n0,,.. Tim Tr ( φ+ φ+ Tr n π t π t + n According to tl (, ll th Bll stts in (6 undr vry w intrction produc w ntnglmnt. Bll stt vctors (-d to otin th trc of th dnsity mtrics undr w intrction condition s.3. W Intrction At w intrction, nd (7d rducs to < (7 + ; > (7 W us th condition in (7 nd (7 in polriztion 8 ˆ 8 sin ( sin sin ( Trρ t Tr + φ+ φ+ t + + ω t t + ( p + + ( ( + ( + 6 sin ( ω tsin ( t + sin ( t + ( p + ( + Tr ˆ ρ ( t Tr sin sin ( φ φ ω pt t + ( sin ( t + sin ( ω sin ( + pt t ( + ( + (8 (8

5 Amricn Journl of Modrn Physics 07; 6(3: 37-8 ˆ 8 Trρ sin ( sin sin t Tr ( + ψ + ψ + t + + ω t t + + ( p + ( ( + 6 sin ( ω tsin ( t + sin ( t + ( p + ( + (8c Tr ˆ ρ ( t Tr sin sin ( ψ ψ ω + pt t ( + t ω pt t sin ( + sin sin ( + ( + ( + (8d Th mximum nd lst trcs corrsponding to (8-d undr w intrction r givn in tl 3. Tl 3. Mximum nd lst trcs of dnsity mtrics for Bll stts (8-8d undr w intrction t tim t nd t whr n0,,. nd + ω p. Tim Tr ( φ+ φ+ Tr nπ t + π t + n < 3 > < 3 > According to CHSH Bll inqulity, th Bll stts (8 nd (8c undr w intrction produc strong ntnglmnt ov th Cirl son s inqulity t th timt. Th Bll stts (8 nd (8d produc strong ntnglmnt ov th cirl son s inqulity t th tim t. Th trc rducs s th vlu of incrss i.. >... Mdium Strngth Intrction > (9 nd (7d rducs to (7 t <. Th trc of th dnsity mtrics undr mdium strngth intrction condition t th form rprsntd in (8-d. Th mximum nd lst trcs corrsponding to (8-d undr mdium strngth intrction r givn in tl. At mdium strngth intrction, Tl. Mximum nd lst trcs of dnsity mtrics for Bll stts (8-8d undr mdium strngth intrction t tim t nd t whr n0,,. nd + ω p. Tim Tr ( φ+ φ+ Tr t nπ + π t + n According to tl (, th Bll stts in (8 nd (8c undr mdium strngth intrction produc strong ntnglmnt ov th Cirl son s inqulity t th tim t. Th Bll stts (8 nd (8d produc strong ntnglmnt ov th cirl son s inqulity t th tim t. Th trc rducs s th vlu of dcrss i.. <..5. Criticl (Thrshold Intrctions At Criticl (thrshold intrction, (0 nd (7d coms (7 t /. Th rducd dnsity mtrics t th sm formt s thos prsntd in qutions (8-d. Th mximum nd lst trcs corrsponding to (8-d undr criticl strngth intrction r givn in tl 5. Tl 5. Mximum nd lst trcs of dnsity mtrics for Bll stts (8-d undr criticl intrction t tim t nd t whr n0,, nd + ω p. Tim Tr ( φ+ φ+ Tr t nπ + π t + n

6 Grld Mugy Lismdi t l.: Th Tst of Entnglmnt of Polriztion Stts of Smi-Clssicl Opticl Prmtric Oscilltor According to CHSH Bll inqulity, th Bll stts (8 nd (8c undr criticl intrction produc strong ntnglmnt t th tim t. Th Bll stts (8 nd (8d produc strong ntnglmnt ov th cirl son s inqulity t th tim t. 5. Conclusion Th smi-clssicl OPO is good systm for dmonstrtion of dynmicl volution of ntnglmnt of polriztion stts y us of Bll stts whos ntnglmnt is tstd y us of rducd dnsity mtrics in CHSH Bll inqulity. Th CHSH Bll inqulity is violtd undr w intrction, mdium strngth nd criticl intrction for th Bll stt vctors ( nd (c hnc producing strong ntnglmnt. Th trcs of dnsity mtrics undr vry w intrction for ll th four Bll stt vctors do not violt th CHSH Bll inqulity hnc producing w ntnglmnt which is dynmic in ntur. Th CHSH Bll inqulity is violtd to giv highr trc of S undr rsonnc, w intrction, mdium strngth nd criticl intrction for Bll stt vctors ( nd (d hnc producing dynmiclly strongr ntnglmnt yond th Cirl son s inqulity of S< sttd for quntum thory. This prsnts th OPO s n importnt tool in quntum optics for possil implmnttion of quntum y communiction protocols in quntum mchnics such s quntum tlporttion, quntum y distriution, nd dns coding. Rfrncs [] Bnntt H. nd Brssrd G. (993. Tlporting n unnown quntum stt vi dul clssicl nd Einstin-Podolsy-Rosn chnnls". Phys. Rv. Ltt. 70, 895. [] Ruprt U. (00. "Quntum tlporttion cross th Dnu". ccssd on 6/7/0. [3] Hrst V. nd Schidl F. (0. "Quntum tlporttion ovr 3 ilomtrs using ctiv fd-forwrd". Bicod: Ntur 89.,69M. doi:0.038/ntur 7. [] Td T. (0. "Dtrministic quntum tlporttion of photonic quntum its y hyrid tchniqu". Ntur 500, 35. DOI: /ntur 366. [5] Adms W. (06. Chin s Quntum Cryptogrphy: Tls from (Quntum Crypt. Ntionl lwrviw. ccssd on 5//07. [6] Villr S. nd Cssmiro N. (005. Gnrtion of Bright Two-Color Continuous Vril Entnglmnt," Phys. Rv. Ltt. 95, [7] Villr S. nd Cssmiro N. (006. Entnglmnt in th ov-thrshold opticl prmtric oscilltor. org/pdf/qunt-ph/ ccssd on //05. [8] Su X. nd Tn A. (006. Exprimntl dmonstrtion of quntum ntnglmnt twn frquncy-nondgnrt opticl twin ms," Opt. Ltt. 3, 33. DOI:0. 36/OL. [9] Shhrohshhi R. nd Pfistr O. (0. Lrg-scl multiprtit ntnglmnt in th quntum opticl frquncy com of dpltd-pump opticl prmtric oscilltor. org/pdf/ ccssd on /6/05. [0] Johnsson R. (0. Entngld-stt gnrtion nd Bll inqulity violtions in nnomchnicl rsontors. Phys. Rv. B90, [] Chrrti R. nd Jnish J. (05. Qusi-Bll stts in strongly coupld quit-oscilltor systm nd thir dlocliztion in th phs spc. Phys. Rv. ltt 88,830. [] Erhrd P. H. (999. Violtion of Bll s inqulity. Phys. Rv. A., 60, RR773. [3] Ayo J. O. (008. Rvisiting non-dgnrt prmtric down-convrsion. Prmn-journl of physics Vol. 7, No. 6 pp [] Ling A. (008. Entngld stt prprtion for opticl quntum communiction. Ntionl univrsity of Singpor. [5] Munro W. J., Nmoto K. nd Whit A. G. (00. Th Bll inqulity: msur of ntnglmnt. Journl of Modrn Optics, Vol. 8, No. 7, [6] Prsil J. (998. Quntum informtion nd computtion: Lctur Nots for physics 9. Cliforni. lorntz lidnuniv.ni/quntum computrs/prsill/ph.