Irrelevance of photon events distinguishability in a class of Bell experiments
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Irrelevance of photon events distinguishability in a class of Bell experiments Marek Z ˙ ukowski, 1 Dagomir Kaszlikowski, 1 and Emilio Santos 2 1 Instytut Fizyki Teoretycznej i Astrofizyki, Uniwersytet Gdan ´ski, PL-80-952 Gdan ´sk, Poland 2 Departamento de Fisica Moderna, Universidad de Cantabria, Santander, Spain ~Received 12 March 1999! We show that the possibility of distinguishing between single- and two-photon detection events, usually not met in the actual experiments, is not a necessary requirement for proof that the experiments of Alley and Shih @Phys. Rev. Lett. 61, 2921 ~1988!# and Ou and Mandel @Phys. Rev. Lett. 61, 50 ~1988!# are modulo a fair sampling assumption, valid tests of local realism. We also give the critical parameters for the experiments to be unconditional tests of local realism, and show that some other interesting phenomena ~involving bosonic- type particle indistinguishability! can be observed during such tests. @S1050-2947~99!50709-9# PACS number~s!: 3.65.Bz, 42.50.Dv, 89.70.1c The first Bell-type experiments that employed the para- metric down-conversion process as the source of entangled photons were those reported in Refs. @1# and @2#. However, the specific traits of those experiments have led to a pro- tracted dispute on their validity as tests of local realism. In this case the issue was not the standard problem of detection efficiency ~which up till now permits a local realistic inter- pretation of all performed experiments!. The trait that distin- guishes the experiments is that, even in the perfect gedanken situation ~which assumes perfect detection!, only in 50% of the detection events does each observer receive a photon; in the other 50% of events one observer receives both photons of a pair while the other observer receives none. The early ‘‘pragmatic’’ approach was to discuss only the events of the first type ~as only such ones lead to spatially separated coin- cidences!. Only those were used as the data input to the Bell inequalities in @1# and @2#. This procedure was soon chal- lenged ~see, e.g., @3,4#, and especially the theoretical analysis of Ref. @5#!, as it raises justified doubts as to whether such experiments could ever be genuine tests of local realism ~as the effective overall collection efficiency of the photon pairs, 50% in the gedanken case, is much below what is usually required for tests of local realism!. Ten years after the first experiments of this type were made, the dispute was finally resolved @6#. It was proposed, that those ‘‘unfavorable’’ cases also be taken into account and that the entire pattern of events be analyzed. In this way one can indeed show that the experiments are a true test of local realism @namely, that the CHSH inequalities are violated by quantum predictions for the idealized case#. The idea was based upon a specific value assignment for the ‘‘wrong events’’ ~see further or @6# itself!. However, the scheme presented by Popescu et al. @6# has one drawback. The authors assumed in their analysis that the detecting scheme employed in the experiment should be able to distinguish between two- and one-photon detections. This was not the case in the actual experiments. The aim of this work is to show that even this is unnecessary; all one needs is use of the specific value assignment procedure of @6#. What is perhaps even more important, problems similar to those sketched above are also shared by the new, potentially highly promising class of electron paramagnetic resonance– Bell-type experiments, which involves the entanglement swapping procedure @7#. Also in this case the first performed experiment did not employ detectors that were able to distin- guish between firings caused by two photons and a single photon @8#. The entanglement swapping experiments thus far have not violated the visibility threshold for local realism (71%); however, in the future the problem of their relation to the Bell theorem will be of fundamental importance ~as entanglement swapping may find application in future quan- tum communication schemes @9#!. The analysis presented in @6# can be adapted to describe such experiments, clearly in- dicating the violation of local realism. Finally, we shall also give a prediction of all effects oc- curring in the experiment. It is quite often overlooked that a kind of Hong-Ou-Mandel dip phenomenon @10# can be ob- served in the experiment. In the class of experiments we consider ~Fig. 1!@6# a type-I parametric down-conversion source @11# is used to generate pairs of photons that are degenerated in frequency and polarization ~say x ˆ ) but propagate in two different direc- tions. One of the photons passes through a wave plate ~WP! that rotates its polarization by 90°. The two photons are then directed onto the two input ports of a ~nonpolarizing! ‘‘50- 50’’ beam splitter ~BS!. The observation stations are located in the exit beams of the beam splitter. Each local observer is equipped with a polarizing beam splitter @12# orientated along an arbitrary axis ~which, in principle, can be randomly chosen, in the delayed-choice manner, just before the pho- tons are supposed to arrive!. Behind each polarizing beam splitter are two detectors, D 1 1 , D 1 2 and D 2 1 , D 2 2 , respec- tively, where the lower index indicates the corresponding FIG. 1. Schematic of the setup. For explanations see the main text. RAPID COMMUNICATIONS PHYSICAL REVIEW A OCTOBER 1999 VOLUME 60, NUMBER 4 PRA 60 1050-2947/99/60~4!/2614~4!/$15.00 R2614 ©1999 The American Physical Society
Transcript of Irrelevance of photon events distinguishability in a class of Bell experiments
RAPID COMMUNICATIONS
PHYSICAL REVIEW A OCTOBER 1999VOLUME 60, NUMBER 4
Irrelevance of photon events distinguishability in a class of Bell experiments
Marek Zukowski,1 Dagomir Kaszlikowski,1 and Emilio Santos21Instytut Fizyki Teoretycznej i Astrofizyki, Uniwersytet Gdan´ski, PL-80-952 Gdan´sk, Poland
2Departamento de Fisica Moderna, Universidad de Cantabria, Santander, Spain~Received 12 March 1999!
We show that the possibility of distinguishing between single- and two-photon detection events, usually notmet in the actual experiments, is not a necessary requirement for proof that the experiments of Alley and Shih@Phys. Rev. Lett.61, 2921 ~1988!# and Ou and Mandel@Phys. Rev. Lett.61, 50 ~1988!# are modulo a fairsampling assumption, valid tests of local realism. We also give the critical parameters for the experiments tobe unconditional tests of local realism, and show that some other interesting phenomena~involving bosonic-type particle indistinguishability! can be observed during such tests.@S1050-2947~99!50709-9#
PACS number~s!: 3.65.Bz, 42.50.Dv, 89.70.1c
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The first Bell-type experiments that employed the pametric down-conversion process as the source of entanphotons were those reported in Refs.@1# and @2#. However,the specific traits of those experiments have led to a ptracted dispute on their validity as tests of local realism.this case the issue was not the standard problem of deteefficiency ~which up till now permits a local realistic interpretation of all performed experiments!. The trait that distin-guishes the experiments is that, even in the perfectgedankensituation~which assumes perfect detection!, only in 50% ofthe detection events does each observer receive a photothe other 50% of events one observer receives both phoof a pair while the other observer receives none. The e‘‘pragmatic’’ approach was to discuss only the events offirst type~as only such ones lead to spatially separated ccidences!. Only those were used as the data input to the Binequalities in@1# and @2#. This procedure was soon chalenged~see, e.g.,@3,4#, and especially the theoretical analysof Ref. @5#!, as it raises justified doubts as to whether suexperiments could ever be genuine tests of local realism~asthe effective overall collection efficiency of the photon pai50% in the gedanken case, is much below what is usurequired for tests of local realism!. Ten years after the firsexperiments of this type were made, the dispute was finresolved @6#. It was proposed, that those ‘‘unfavorablecases also be taken into account and that the entire patteevents be analyzed. In this way one can indeed show thaexperiments are a true test of local realism@namely, that theCHSH inequalities are violated by quantum predictionsthe idealized case#. The idea was based upon a specific vaassignment for the ‘‘wrong events’’~see further or@6# itself!.However, the scheme presented by Popescuet al. @6# has onedrawback. The authors assumed in their analysis thatdetecting scheme employed in the experiment should beto distinguish between two- and one-photon detections. Twas not the case in the actual experiments. The aim ofwork is to show that even this is unnecessary; all one neis use of the specific value assignment procedure of@6#.
What is perhaps even more important, problems similathose sketched above are also shared by the new, potenhighly promising class of electron paramagnetic resonanBell-type experiments, which involves the entanglemswapping procedure@7#. Also in this case the first performe
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experiment did not employ detectors that were able to disguish between firings caused by two photons and a sinphoton@8#. The entanglement swapping experiments thushave not violated the visibility threshold for local realis(71%); however, in the future the problem of their relatioto the Bell theorem will be of fundamental importance~asentanglement swapping may find application in future qutum communication schemes@9#!. The analysis presented i@6# can be adapted to describe such experiments, clearlydicating the violation of local realism.
Finally, we shall also give a prediction of all effects ocurring in the experiment. It is quite often overlooked thakind of Hong-Ou-Mandel dip phenomenon@10# can be ob-served in the experiment.
In the class of experiments we consider~Fig. 1! @6# atype-I parametric down-conversion source@11# is used togenerate pairs of photons that are degenerated in frequ
and polarization~sayx) but propagate in two different directions. One of the photons passes through a wave plate~WP!that rotates its polarization by 90°. The two photons are thdirected onto the two input ports of a~nonpolarizing! ‘‘50-50’’ beam splitter~BS!. The observation stations are locatein the exit beams of the beam splitter. Each local observeequipped with a polarizing beam splitter@12# orientatedalong an arbitrary axis~which, in principle, can be randomlychosen, in the delayed-choice manner, just before the ptons are supposed to arrive!. Behind each polarizing beamsplitter are two detectors,D1
1 , D12 and D2
1 ,D22 , respec-
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FIG. 1. Schematic of the setup. For explanations see the mtext.
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PRA 60 R2615IRRELEVANCE OF PHOTON EVENTS . . .
observer and the upper index the two exit ports of the poized beam splitter~‘‘ 1’’ meaning parallel with the polarization axis of the beam splitter and ‘‘2 ’’ meaning orthogonalto this axis!. All optical paths are assumed to be equal.
Let us calculate the quantum predictions for the expment. We will use the second quantization formalism, whis very convenient here, since the whole phenomenonserved in the experiment rests upon the indistinguishabof photons.
After the action of the wave plate one can approximthe quantum mechanical state describing two photons eming from a nonlinear crystal along the ‘‘signal’’ and th‘‘idler’’ beam by
uC0&5a1xW†
a2yW† u0&, ~1!
wherea1xW† anda2yW
† are creation operators andu0& denotes the
vacuum state. SubscriptsxW ,yW decode the polarization of thphoton~either along thexW or the yW axis!. The beam-splitteraction can be described by
a1xW†
51
A2~ icxW
†1dxW
†!, a2yW
†5
1
A2~cyW
†1 idyW
†!, ~2!
wherecxW† ,dxW
† ,cyW† ,dyW
† are operators describing output modesthe beam splitter (c stands for the first observer andd for thesecond one!. Thus our stateuC0& changes to
uC&51
2~ icxW
†cyW
†2cxW
†dyW
†1cyW
†dxW
†1 idxW
†dyW
†!u0&. ~3!
Next comes the action of the polarizers in both beamwhich can be described as
nxW†5cos~u1!ni
†1sin~u1!n'† , nyW
†5sin~u1!ni
†2cos~u1!n'† ,~4!
wheren†5c† or d†, ni† describes the mode parallel to pola
izer’s axis, andn'† describes the mode perpendicular to p
larizer’s axis;u is the angle between thexW axis and polariz-er’s axis. Thus the final state reaching the detector read
uc f inal&51
2 Fsin~u12u2!uci ,di&1cos~u12u2!ucid'&
2cos~u12u2!uc' ,di&1sin~u12u2!uc' ,d'&
1 i1
A2sin~2u1!u2ci&1 i
1
A2sin~2u1!u2c'&
2 icos~2u1!uc' ,ci&1 i1
A2sin~2u2!u2di&
1 i1
A2sin~2u2!u2d'&2 icos~2u2!udi ,d'&G , ~5!
where, e.g.,uci ,di& denotes one photon in the modeci and
one in di , whereasu2ci&5(1/A2)ci†2
u0& denotes two pho-tons in the modeci .
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Let us denote byP( i ,u1 ; j ,u2) the joint probability for theoutcomei to be registered by observer 1 when her polariis oriented along the direction that makes an angleu1 withthexW direction and the outcomej to be registered by observe2 when her polarizer is oriented along the direction thmakes an angleu2 with the xW direction. Herei , j 51 – 6 andhave the following meaning@6#: 15one photon inD2, nophoton in D1; 25one photon inD1, no photon inD2;35no photons; 45one photon inD1 and one photon inD2;55two photons inD1, no photon inD2; 65two photons inD2, no photons inD1.
The quantum predictions for joint probabilities of thoevents are given by
P~1,u1 ;1,u2!5P~2,u1 ;2,u2!5 18 @12cos 2~u12u2!#,
~6!
P~2,u1 ;1,u2!5P~1,u1 ;2,u2!5 18 @11cos 2~u12u2!#,
~7!
P~5,u1 ;3,u2!5P~6,u1 ;3,u2!5 18 sin2~2u1!, ~8!
P~3,u1 ;5,u2!5P~3,u1 ;6,u2!5 18 sin2~2u2!, ~9!
P~4,u1 ;3,u2!5 14 cos2~2u1!, ~10!
P~3,u1 ;4,u2!5 14 cos2~2u2!. ~11!
Following @6# we associate with each outcome registeredthe observers 1 and 2 a corresponding valueai and bj , re-spectively, wherea15b1521 while all the other values areequal to 1. Let us denote byE(u1 ,u2) the expectation valueof their product,
E~u1 ,u2!5(i , j
aibj P~ i ,u1 ; j ,u2!. ~12!
After simple calculations one has
E~c1 ,c2!52 12 cos~c11c2!1 1
2 , ~13!
where we have put 2uk5(21)k21ck .The above formula for the correlation function is valid
one assumes that it is possible to distinguish between sinand double-photon detection. This is usually not the caThus it is convenient to have a parametera that measures thedistinguishability of the double and single detection at odetector (0<a<1, and gives the probability of distinguishing by the employed detecting scheme of the double coun!.The partial distinguishability blurs the distinction betweevents 1 and 6~2 and 5! and thus part of the events of typeis interpreted as being of type 1 and is ascribed by the loobserver a wrong value; e.g., an event of type 6, if bphotons go to the ‘‘2 ’’ exit of the polarizer, can be inter-preted as a firing due to a single photon and is ascribe21 value. Please note that events like 1 or 2 in statioaccompanied by 3~no photon! at station 2 do not contributeto the correlation function because for anyaP(1,u1 ;3,u2)5P(2,u1 ;3,u2).
If the parametera is taken into account, the correlatiofunction acquires the following form:
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R2616 PRA 60ZUKOWSKI, KASZLIKOWSKI, AND SANTOS
E~c1 ,c2 ;a!52 12 cos~c11c2!1 1
2 a
1 14 ~12a!~cos2 c11cos2 c2!. ~14!
In this case after the insertion of the quantum correlatfunction ~14! into the CHSH inequality,
22<E~c1 ,c2 ;a!1E~c18 ,c2 ;a!
1E~c1 ,c28 ;a!2E~c18 ,c28 ;a!<2,
one obtains
22<2 12 @cos~c11c2!1cos~c181c2!
1cos~c11c28!2cos~c181c28!]
1a1 12 ~12a!~cos2 c11cos2 c2!<2. ~15!
The interesting feature of this inequality is that it canviolated for all values ofa. What is perhaps even more important, it can be robustly violated even when one is not ato distinguish between single and double clicks at alla50). The actual value of the CHSH expression can reacthis case 2.337 12~a numerical result!, which is only slightlyless than the maximal value fora51, which is A211'2.414 21. Therefore we conclude that in the experimone can observe violations of local realism even if one isable to distinguish between the double and single countone detector. That is, the essential feature of the method@6# to reveal violations of local realism in the experimentthis type is the specific value assignment scheme and nodouble-single photon counts distinguishability.
The specific angles at which the maximum violationthe CHSH inequality is achieved fora50 differ very muchfrom those fora51 ~for which the standard result is reproduced!, and they read ~in radians! c152.937 98, c1854.255 13,c2520.202 41, andc2851.117 08.
Let us notice that with the setup of Fig. 1 one is ableobserve effects of similar nature to the famous Hong-OMandel dip@10#. These are revealed by the probabilities ptaining to the wrong events~8!–~11!. Simply, for certainorientations of the polarizers, if the two photons emergeone side of the experiment only, then they must exitpolarizing beam splitter via a single output port~this effect isdue to the bosonic-type indistinguishability of photons; s@10#!.
Finally let us discuss what the critical efficiency of thdetection of experiments of this type is. To this end, in ocalculations we will use a very simple model of imperfedetections: we insert a beam splitter with reflectivityA12h,in front of an ideal detector, which observes only the tramitted light. This results in the system behaving just likedetector of efficiencyh. If we assume that the incoming lighis described by a creation operatora†, then the transmittedmode is denoted asta
† whereas reflected mode is denotedr a
† and one has
a†5A12hr a††
1Ahta†† . ~16!
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For instance, if one takes the following part of the state vtor ~5!
cuu†duu
†u0&, ~17!
the beam-splitter model of an imperfect detector transforthis term into
@~12h!r cuu
† r duu
† 1Ah~12h!r cuu
† tduu
†
1Ah~12h!tcuu
† r duu
† 1htcuu
† tduu
† #u0&. ~18!
The probabilities now read
P~3,u1 ;2,u2!5P~2,u1 ;3,u2!,~19!
P~1,u1 ;3,u2!5P~3,u1 ;1,u2!5h~12h!,
P~1,u1 ;1,u2!5P~2,u1 ;2,u2!5 14 h2@sin~u12u2!#2,
~20!
P~2,u1 ;1,u2!5P~1,u;2,u2!5 14 h2@cos~u12u2!#2,
~21!
P~5,u1 ;3,u2!5P~6,u1 ;3,u2!5 18 h2@sin~2u1!#2, ~22!
P~3,u1 ;5,u2!5P~3,u1 ;6,u2!5 18 h2@sin~2u2!#2, ~23!
P~4,u1 ;3,u2!5 14 h2@cos~2u1!#2, ~24!
P~3,u1 ;4,u2!5 14 h2@cos~2u2!#2. ~25!
The correlation function, which includes the inefficiencof the detection, reads
E~c1 ,c2 ;h,a!5h2E~c1 ,c2 ;a!1~12h!2, ~26!
where E(c1 ,c2 ;a) is given by Eq.~14!. We have tacitlyassumed here that the parametersa andh are independent oeach other~this assumption may not hold for specific tecnical arrangements!. Putting this prediction into the CHSHinequality, assuming thata51 ~full distinguishability!, weobtain a minimum quantum efficiency needed for violatiof local realism equal to 0.91, whereas for other values oawe have for a50,h50.926; for a50.5,h50.92; for a50.75,h50.92; and fora50.875,h50.91. One should notehere that the method of value assignment of@6# is in accor-dance with the method given by Garg and Mermin@13# forthe optimal estimation of required detector quantum eciency to violate local realism in a Bell test. Thus the otained efficiencies are indeed the lowest possible, and sthat experiments of this type are not good candidates fo‘‘loophole-free’’ Bell test @14#; nevertheless, due to the facthat the whole observable effect is a consequence of quanprinciple of particle indistinguishability, such tests are veinteresting in themselves—they reveal the entanglementherently associated with this principle.
M.Z. was supported by the University of Gdansk undGrant No. BW/5400-5-0202-8. D.K. was supported by tKBN under Grant No. 2 P03B 096 15.
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PRA 60 R2617IRRELEVANCE OF PHOTON EVENTS . . .
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