De Broglie Wavelength of a Nonlocal Four-Photon

Transcription

1 1 De Broglie Wvelength of Nonlocl Four-Photon Philip Wlther*, Jin-Wei Pn, Mrkus Aspelmeyer *, Rupert Ursin *, Sr Gsproni* & Anton Zeilinger * * Institut für Experimentlphysik, Universität Wien, Boltzmnngsse 5, 1090 Wien, Austri Physiklisches Institut, Universität eidelerg, Philosophenweg 1, D-6910 eidelerg, Germny Institut für Quntenoptik und Qunteninformtion, Österreichische Akdemie der Wissenschften

2 Superposition is one of the most distinct fetures of quntum theory nd hs een demonstrted in numerous reliztions of Young s clssicl doule -slit interference experiment nd its nlogues 1-5. owever, quntum entnglement 6 - significnt coherent superposition in multiprticle systems - yields phenomen tht re much riche r nd more interesting thn nything tht cn e seen in oneprticle system 7,8. Among them, one importnt type of multi-prticle experiments uses pth-entngled numer-sttes, which exhiit pure higher-order interference nd llow novel pplictions in metrology nd imging 9 such s quntum 10-1 interferometry nd spectroscopy with phse sensitivity t the eisenerg limit or quntum lithogrphy eyond the clssicl diffrction limit 1. Up to now, in opticl implementtions of such schemes lower-order interference effects would lwys decrese the overll performnce t higher prticle numers. They hve thus een limited to two photons 1. We overcome this limittion nd demonstrte liner-optics-sed four-photon interferometer. Oservtion of four-prticle mode -entngled stte is confirmed y interference fringes with periodicity of one qurter of the single -photon wvelength. This scheme cn redily e extended to ritrry photon numers nd thus represents n importnt step towrds relizle pplictions with entnglement-enhnced performnce. To see the origin of multiprticle interference more clerly, consider first simple nlogue to Young s doule slit experiment, i.e. Mch-Zehnder (MZ) interferometer (Figure 1). There, single-photon interference occurs due to the sptil seprtion of two modes of propgtion 1 nd 1 for single prticle entering the interferometer t the first emsplitter. rition of the pth length x induces phse shift ϕ nd gives thus rise to detection proilities P 1+ cos ϕ nd P 1 cos ϕ in ech of the two output modes nd ehind the exit emsplitter. Two-photon interference occurs when 1 nd 1 re the modes of propgtion for stte of two indistinguishle 1 i ϕ photons, i.e. iphoton stte Ψ = ( 0 + e 0 ). This is

3 superposition where either two photons re in mode 1 nd none re in mode 1, or, vice vers, no photons re in mode 1 nd two photons propgte within mode 1. This represents pth-entngled two-photon stte, which exhiits pure two-prticle interference t the output emsplitter. Note tht the pth length vrition x cts on oth photons nd gives thus rise to douled phse shift compred to the single photon cse. Becuse of the unvilility of detectors which re le to distinguish etween N nd N + 1 photons, coincidence mesurements of sptilly seprted photons re required to oserve multi-photon sttes. This cn e chieved, though only proilisticlly, y dding emsplitter in ech of the sptil output modes of the interferometer. The proility to find two photons in either mode or then oscilltes with P 1 + cos( ϕ ) nd P 1 cos( ϕ),, single-photon detection proilities P nd P remin constnt., respectively, while the In the generlized cse of n N-prticle interferometer, the N photons will e in superposition of eing in either mode 1 or 1, resulting in in ϕ ( N 0 + e N ) 1 Ψ = 0. (1) In other words, the pths re entngled in photon numer. ere N (or N ) 1 1 indictes the N -prticle Fock stte in sptil mode 1 (or 1), respe ctively, nd N = 0 represents n empty mode. The phse modultion N ϕ increses linerly with the prticle numer N, which is the origin of ll entnglement-enhnced interferometric schemes. In prticulr, the N -photon detection proility in ech of the interferometer outputs would vry s + cos( N ϕ) P N 1. It hs therefore een suggested to ttriute n effective de Broglie wvelength?/n to the quntum stte. This resemles the cse of hevy mssive molecule consisting of N toms; though here the prticles re in no wy ound to ech other 15.

4 In order to est enefit from such entnglement -enhnced interferometric techniques it is desirle to experimentlly chieve high photon numer N for sttes of the form (1). The specil cse of N = ws relized oth in the originl Young s doule-slit geometry y using colliner production of iphoton sttes vi prmetric down conversion 16 nd in Mch-Zehnder configurtion y using two-photon interference to suppress unwnted single -photon contriutions 17,18. It is commonly elieved tht the reliztion for sttes with N > requires the use of non-liner gtes 19 or N dditionl ncill detectors with single -photon resolution 0. Unfortuntely, ech of these schemes is not fesile with current technologies. We demonstrte how to overcome this limittion giving specific exmple of pure four-photon interferometry. Our proposl is sed on seprting photon pirs into different pirs of modes nd utilizing two-prticle-interferometry rther thn distinguishing photon numers or employing nonliner emsplitters. To chieve this gol, we exploit type-ii spontneous prmetric down-conversion (SPDC) 1. An ultr-violet pulse psses through etrium -orte (BBO) crystl, proilisticlly emitting pirs of energy-degenerte polriztion-entngled photons into the sptil modes 1 nd (Figure ). The U pump em is reflected ck t mirror nd cn thus lso emit entngled photon pirs into the sptil modes 1 nd (Figure ). The setup is ligned to generte the following mximlly entngled iphoton stte ( ) + 1 F = () for ech of the pirs emitted into the pirs of modes 1- nd 1-, respectively. ere (or ) indictes horizontl (or verticl) polriztion of the photon. We first consider the cse where only one pir of entngled photons is emitted on doule pss of the U pulse through the crystl. There re two proility mplitudes

5 5 which will contriute to the emerging two-photon stte, i.e. the pir is emitted either into the pir of modes 1- or into the pir of modes 1-. We then coherently comine the two pirs of modes t the two polrizing emsplitters (PBS). Since the PBS trnsmits horizontlly polrized light nd reflects verticlly polrized light, conditionl on detecting one photon in ech of the output ports nd the iphoton stte will e F i ϕ ( + e ) 1 = () where gin ϕ is the phse modultion of single -photon 17,. The phse ϕ is proportionl to the position of the pump mirror PM, where Dx is the U pth-length chnge. Two-photon interference fringes my now e oserved y performing projection mesurement in the modes nd in the liner polriztion sis ± = ( )( ± ) coincidence 1. Specificlly, the proility of detecting two-fold + is proportionl to P 1 cos( ϕ ), lredy signture of non-loclity, of the two-photon stte (Figure ).. These correltions re Let us now explin how our scheme cn e generlized to four photons nd even higher photon numers. Consider the cse where two pirs of photons re emitted on doule pss of the pump em through the crystl. There re two possiilities which will contriute to n overll four-photon stte, i.e. either y doule-pir emission on one or the other side, where two photon pirs re emitted into the sme pir of modes 1- or 1-, respectively, or one pir of photons is emitted simultneously into ech of the modes 1- nd 1-. We first study the doule-pir emission cse where oth pirs propgte within the sme mode pir. A four-fold coincidence fter the two PBS, i.e. detection of single photon in ech of the output ports,, nd, will either result from

6 6 contriution, if the two photon pirs re oth in 1-, or from contriution, if the two photon pirs re oth in 1-. Temporl overlpping of oth pirs of modes t the two PBS results in i e ϕ +, coherent superposition of forwrd nd c kwrd emission t the sme time, where ll the four ckwrd emitted photons re phse shifted y the pump mirror PM. Introducing pth difference of Dx nd further performing polriztion mesurement in the ± sis to chieve indistinguishility results in interference fringes with one qurter of the single-photon wvelength. owever, with the sme pro ility s the doule-pir emission, one photon pir is emitted into ech of the mode pirs 1- nd 1- y one pulse. Tht second cse of one pir eing emitted forwrds nd one ckwrds will lso result in four -fold coincidences, either from or contriution, where ll the photons hve the sme polriztion. This coherent superposition ( ) i e + ϕ hs n overll phse nd thus fixed reltive phse 5 which is independent of the position of the pump mirror. Note tht this overll phse is hlved compred to the doule-pir emission cse. This is due to the fct tht in this cse in ech contriution only two photons re ffected y the pump mirror. Therefore the complete (for simplicity un-normlized) four-photon stte ehind the PBS cn e written s: ( ) i i e e ϕ ϕ (5) For oserving undistured four photon interference the lst two contriutions hve to e ersed. This cn e chieved y performing proper projection mesurement of the four output modes,, nd into the ± ses; then the numer of +

7 7 projections is different from the numer of projections, sy The overll four -photon mplitude originting from one photon in ech mode then i ϕ ecomes e ( ) nd thus vnishes due to the fixed phse reltion. This is the four-photon equivlent to ong-ou-mndel interference 6 of two photons rriving t the emsplitter. Thus the four-photon detection proility in the sptilly seprted output modes,, nd oscilltes like P 1 + cos( ), i.e. this projection llows the oservtion of unpertured,,, ϕ interference of four-photon stte. Figure comp res this pure four -photon interference effect (Figure c) with the well-known two-photon interference (Figure ) nd the single-photon interference (Figure ) s were otined with the sme setup. Fits to the dt revel reduction of the oscilltion wvelength from 8±6 nm for the single-photon cse over 95±16 nm for the two-photon cse nd 19±9 nm for the four-photon cse. The devition is within experimentl error given y the therml long-term stility of our interferometric setup. This demonstrtes tht the phse modultion of the mirror PM is pplied to ll the sptilly seprted four photons simultneously. Consequently, one hs to tret the fourphoton stte (5) s one oject of the form ψ = ( 0 + e 0 ), 1, 1, 1, 1, 1 i ϕ which is similr to so-clled noon -stte 7. Our four-photon stte hs the dditionl interesting property tht it is nonlocl. It is superposition of four photons either in mode 1 nd or 1 nd. The de -Broglie wvelength feture is then relized y joint nonlocl mesurement on the two photons in nd on one side together with the photons nd on the other. In contrst, ny projection mesurement different from the ove will result in n equl contriution of ll four-photon terms to the interference pttern, including the unwnted contriutions of one photon per sptil mode, s hs een oserved efore 8. For exmple, no pure four -fold wvelength reduction cn e ttined y projecting the

8 8 four-photon stte of (5) onto There, the four-photon detection 1 proility oscilltes like P,,, 1+ cos ( ϕ ) = + cos( ϕ) + cos( ϕ), which is confirmed y compring the simultneously mesured two- nd four-photon coincidences of Figure. The employed method llows the genertion of four-photon sttes nd their susequent utiliztion in pure four-prticle interferometry. The result clerly confirms the theoreticl expecttion tht the de-broglie wvelength of four -photon stte is one fourth of single photon, thus leding to the generl rule λ ( N ) = λ(1) N. This overcomes stte -of -the-rt two-prticle interferometry nd opens new possiilities in quntum metrology nd in quntum imging pplictions, which might e potentilly useful tool for nno-technology 7. It is importnt to note tht, in principle, this scheme cn e extended to higher prticle numers if more sptil modes re involved. The ctul limittion due to low count rtes might eventully e overcome with the next genertion of entngled photon sources.

9 9 1. Young, T. Experiments nd clcultions reltive to physicl optics. Phil. Trns. Roy. Soc. of Lond. 9, 1-16 (180).. Mrton, L. Electron interferometer. Phys. Rev. 85, (195).. Ruch,., Treimer W. & Bonse, U. Test of Single Crystl Neutron Interferometer, Phys. Lett. A 7, (197).. Keith, D.W., Ekstrom, C.R., Pritchrd, D.E. An Interferometer for Atoms, Phys. Rev. Lett. 66, (1991). 5. Arndt, M., Nirz, O., os -Andree, J., Keller, C., vn der Zouw, G. & Zeilinger, A. Wve -Prticle Dulity of C60 Molecules. Nture 01, (1999). 6. Schrödinger, E. Die gegenwärtige Sitution in der Quntenmechnik. Nturwissenschften, , 8-88, 8-89 (195). 7. orne M.A. & Zeilinger A. A Bell-Type Experiment Using Liner Moment. Symposium on the Foundtions of Modern Physics, Joensuu, Lthi P. & Mittelsted P. (ed.), 5 (1985). 8. Greenerger, D., orne, M., Zeilinger, A., Multiprticle Interferometry nd the Superposition Principle. Physics Tody 8, -9 (199). 9. Lee,., Kok, P. & Dowling, J. P. Quntum Imging nd Metrology. Proc. Sixth Interntionl Conference on Quntum Communiction, Mesurement nd Computing, Shpiro, J.. & irot, O. (ed.), Rinton Press, (00). 10. Yurke, B. Input Sttes for Enhncement of Fermion Interferometer Sensitivity. Phys. Rev. Lett. 56, (1986). 11. ollnd, M. J. & Burnett, K. Interferometric Detection of Opticl Phse Shifts t the eisenerg Limit. Phys. Rev. Lett. 71, (199).

10 10 1. Bollinger, J. J. Itno, W. M., Winelnd, D. J. & einzen, D. J. Optiml frequency mesurements with mximlly correlted sttes. Phys. Rev. A 5, R69-R65 (1996). 1. Boto, A., Kok, P., Arms, D., Brunstein S., Willims C. & Dowling J. Quntum Interferometric Opticl Lithogrphy: Exploiting Entnglement to Bet the Diffrction Limit. Phys. Rev. Lett. 85, 7-76 (000). 1. Steuerngel, O. de Broglie wvelength reduction for multiphoton wve pcket. Phys. Rev. A. 65, 080 (00). 15. Jcoson, J., Björk, G., Chung, I. & Ymmoto Y. Photonic de Broglie Wves. Phys. Rev. Lett. 7, (1995). 16. Fonsec, E. J. S., Monken, C.. & Pdu, S. Mesurement of the de Broglie Wvelength of Multiphoton Wve Pcket. Phys. Rev. Lett. 8, (1999). 17. Ou, Z. Y., Wng, L. J., Zou, X. Y. & Mndel, L. Evidence for phse memory in two-photon down conversion through entnglement with the vcuum. Phys. Rev. A 1, (1990). 18. Edmtsu, K., Shimizu, R. & Itoh, T. Mesurement of the Photonic de Broglie Wvelength of Entngled Photon Pirs Generted y Spontneous Prmetric Down-Conversion. Phys. Rev. Lett. 89, 1601 (1995). 19. Gerry, C. C. & Cmpos, R. A. Genertion of mximlly entngled photonic sttes with quntum-opticl Fredkin gte. Phys. Rev. A 6, 0681 (001) 0. Kok, P., Lee,. & Dowling, J. Cretion of lrge -photon numer pth entnglement conditioned on photodetection. Phys. Rev. A 65, 0510 (00). 1. Kwit, P. G., et l. New igh-intensity Source of Polriztion-Entngled Photon Pirs. Phys. Rev. Lett. 75, 7-1 (1995).

11 11. Pn, J.-W., Gsproni, S., Ursin, R., Weihs, G. & Zeilinger, A. Experimentl entnglement purifiction of ritrry unknown sttes. Nture, 17- (00).. Shih, Y.. & Alley, C. O. New Type of Einstein-Podolsky-Rosen-Bohm Experiment Using Pirs of Light Qunt Produced y Opticl Prmetric Down Conversion. Phys. Rev. Lett. 61, 91-9 (1988).. Ou, Z. Y. & Mndel, L. ioltion of Bell's Inequlity nd Clssicl Proility in Two-Photon Correltion Experiment. Phys. Rev. Lett. 61, 50-5 (1988). 5. Simon, C. & Pn, J. -W. Polriztion Entnglement Purifiction using Sptil Entnglement. Phys. Rev. Lett. 89, (00). 6. ong, C. K., Ou, Z. Y. & Mndel, L. Mesurement of supicosecond time intervls etween two photons y interference. Phys. Rev. Lett. 59, 0-06 (1987). 7. Kok, P. et l. Quntum interferometric opticl lithogrphy: towrds ritrry twodimensionl ptterns. Phys. Rev. A 6, 0607/1-8 (001). 8. Lms-Linres, A., owell, J. C. & Bouwmeester, D. Stimulted emission of polriztion-entngled photons. Nture 1, (001). Acknowledgements: We thnk C. Brukner nd K. Resch for discussions. This work ws supported y the Austrin Science Foundtion (FWF), project numer SFB 015 P06, y the Europen Commission, contrct no. IST , RAMBOQ, nd y the Alexnder von umoldt-foundtion.

12 1 Figure1 A two-mode Mch-Zehnder interferometer. The phse is chnged y vrying the pth length vi the position of mirror. Single-photon interference occurs due to the sptil seprtion of two possile modes of propgtion 1 nd 1 for single prticle entering the interferometer t the first emsplitter (BS). Two-photon interference cn e chieved when 1 nd 1 re the two possile modes of propgtion for iphoton stte.

13 1 Figure In our experiment the required four -photon stte is produced y type-ii spontneous prmetric down-conversion (SPDC). A 00 fs pulse t centrl Uwvelength of 95 nm nd t repetition rte of 76 Mz psses through et -riumorte (BBO) crystl proilisticlly emitting pirs of energy-dege nerte polriztionentngled photons t 790 nm into the sptil modes 1 nd. The U pump em is reflected ck t mirror nd might thus emit second pir into the sptil modes 1 nd. The proility of single -pir cretion is on the order of p (in our setup ), while the proility to crete two pirs is proportionl to p. nm ndwidth filters (F) nd coupling into single-mode fires in front of ech detector enles good temporl nd sptil overlp of the photon-wvepckets t the polrizing emsplitters (PBS). The U-pump is reflected y the pump mirror PM, which is mounted on computer -controlled trnsltion stge. By scnning the position of PM with step size of 1 µm nd performing fine djustment of the position of M, we chieved the temporl overlp of modes 1 nd 1, nd of modes nd. An dditionl piezo trnsltion stge is used to move the pump mirror PM nd to perform chnge of the phse etween four photons emitted into modes 1 nd reltive to the four photons emitted into 1 nd. The detection of the sptilly seprted -photon coincidences ehind 5 polrizer (Pol) while vrying the position of PM leds to the oserved interference fringes.

14 1 Figure Experimentl demonstrtion of pure one-, two- nd four- photon interference. The two- nd four- photon interference is recorded simultneously, while for the onephoton interferometry the pulsed lser hs een switched from mode-locking to continuous-wve (cw) mode. Single photon rte in mode fter performing projection mesurement in the liner polriztion sis ± = ( )( ± ) 1. For this interference pttern, the pump lser is used in the CW mode t 790nm (insted of the mode-locked frequency-douled mode t 95nm). A Mch-Zehnder configurtion for modes 1-1 rises for light scttered from the BBO-crystl when pssing through the crystl. By moving the pump mirror PM interference fringes pper for single photons with centrl wvelength of 790nm which corresponds to the down-converted photons. Note tht, due to the ck reflection of the pump em, the chnge in the opticl pth is twice s lrge s in the position of the pump mirror. The two-photon coincidence rte corresponding to the detection in mode nd fter projecting onto +. c Performing projection onto results in pure four-photon interference due to projection onto the (non-locl) pth-entngled four-photon stte 1 i ϕ ψ = ( 0 + e 0 ). 1, 1, 1, 1,

15 15 Figure Two- nd four-photon interference without proper post-selection. Twophoton interference fter pssing two 5 polrizer, projecting onto + + in the mode nd. Simultneously mesured four-photon coincidences fter projecting onto the stte This leds to dditionl, unwnted two-photon interference terms resulting in the mximlly entngled stte Ψ i ϕ i ϕ 0 + e 0 + e 1, 1, 1, 1, 1, 1, chnge in the opticl pth is twice the movement of the pump mirr or PM.. Agin, the effective

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