A scheme to realize time-bin entanglement between two photons that never interacted

Science in China Series G: Physics, Mechanics & Astronomy

© 2008 SCIENCE IN CHINA PRESS

Springer-Verlag

Received June 18, 2007; accepted October 19, 2007 doi: 10.1007/s11433-008-0008-7 †Corresponding author (email: [email protected])

www.scichina.com www.springerlink.com Sci China Ser G-Phys Mech Astron | Jan. 2008 | vol. 51 | no. 1 | 38-44

A scheme to realize time-bin entanglement between two photons that never interacted

LU Shan, BAO XiaoHui & YANG Tao† Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, Uni-versity of Science and Technology of China, Hefei 230026, China

We propose a scheme of entangling two photons from two separated sources. Our proposal which is inspired by the time-bin entanglement developed recently, pro-vides a novel alternative for revealing contradiction between quantum nonlocality and local realism based on two independent single photon sources.

quantum nonlocality, time-bin entanglement, linear optics

1 Introduction

Entanglement lies at the heart of quantum information technology. Since the gendenken experi-ment of Einstein-Podolsky-Rosen (EPR) paradox, entanglement has been employed to reveal the contradiction between nonlocality of quantum mechanics and local reality of classic theory. Bell showed that the conflict can be formulated by an inequality, raising the hope of experimental in-vestigation of this problem. Since Bell’s work, numerous experiments have been performed. Normally, the entanglement is realized by having two particles emerging from one common source[1], or generated by interaction between the two particles[2]. Another way to entangle two particles is by entanglement swapping[3]. That is to entangle two particles from the two pairs by projecting the other two particles to one of the four Bell states probabilistically by starting from two entangled pairs. In this case, the generated two particles have never interacted with each other, and even have never met each other. However, it still requires two interacted particles as an interme-diate.

Postselection is another way to entangle two particles. In the experiment of Shih and Alley[4], two photons prepared in two orthogonal polarization states are overlapped (spatially and tempo-rally) on one 50/50 beam splitter (BS). Entanglement is realized by post-selecting the events that two photons left the beam splitter from the two output ports, respectively. The two photons can, in principle, be from two independent sources. However, the two photons have to meet at this beam splitter, thus a loophole exists that the two photons may interact when overlapping at the beam splitter. Yurke and Stoler’s work[5] avoids this potential loophole by designing an optical con-

LU Shan et al. Sci China Ser G-Phys Mech Astron | Jan. 2008 | vol. 51 | no. 1 | 38-44 39

figuration consisting of 4 beam splitters, which can be used to demonstrate EPR effect in the spatial degree of freedom. Two photons from two independent single photon sources impinge on two beam splitters, respectively. Then, the transmitted and reflected components of the two beam splitters are delivered to interference at another two beam splitters, on Alice’s and Bob’s sides, respectively. By post-selecting the events that each of Alice and Bob has one photon, spatial en-tanglement is generated and can be used to test quantum nonlocality. Up to now, several experi-ments have been demonstrated by using this configuration[6,7].

2 Method

In this article, we present a new scheme to generate entanglement between two photons that are never interacted. By using this method, conflict between quantum nonlocality and local realism can be demonstrated. Our work is inspired by the recent development of time-bin entanglement[8－11], which was proposed as a solution to long-distance entanglement distribution in quantum commu-nication. The concept of time-bin qubit has been employed to quantum key distribution based on differential phase shifts[12－14]. Essentially, our scheme is an analog to the post-selection method of Shih[4] and Yurke[5]. The difference is that the two independent photons are entangled in the temporal degree of freedom, instead of polarization or spatial degree of freedom. To understand our scheme, it is useful to start with the conception of individual time-bin photonic qubit, in which the physical basis states | 0⟩ and |1⟩ correspond to early ( | early⟩ ) and late ( | late⟩ ) single-photon wave packets, respectively. The two single photon packets propagate subsequently in one spatial mode, with only temporal difference. For instance, the | early⟩ state implies that the single photon is in the early wave packet, with the late wave packet empty, while the | late⟩ state implies that the single photon is in the late wave packet, with the early wave packet empty. For short, in the fol-lowing, we use | e⟩ to denote | early⟩ , and | l⟩ to denote | late⟩ . Like normal qubit, the two basis of the time-bin qubit can be superposed coherently.

In our scheme (see Figure 1), two identical photons 1 and 2 are emitted from two independent sources (PS1 and PS2, respectively) at time t0, and then they are delivered to two input ports of a 50/50 beam splitter (BS0). Before entering BS0, photon 1 takes the short path S1, while the photon 2 takes the long path L2. The difference of the two paths leads to a time delay of δ t between the arrival time of the two photons, which should be larger than the pulse time duration τ. This re-quirement is to assure that the transmitted and reflected components in each outports of BS0 are in temporally separated discrete time bins. Moreover, we assume that the path difference is much larger than the coherence length of the single photons. After leaving the BS0, the two photons will propagate either to Alice’s or to Bob’s side. As illustrated in Figure 2, there are four cases when the two photons pass through the BS0: (a) both transmit; (b) both reflect; (c) photon 1 transmits while photon 2 reflects; (d) photon 1 reflects while photon 2 transmits. Among the four cases, the cases (a) and (b) correspond to the events that both Alice and Bob get one photon; while the cases (c) and (d) correspond the events that both two photons go to the same side, either both Alice’s side or both on Bob’s side. Since we only consider the coincident detection events of Alice and Bob, the last two cases will be filtered out, and only the first two cases will be post-selected. In the following, we will explain the generation of entanglement by post-selection in detail.

As described above, photon 1 takes short path S1 before entering BS0, thus it will reach BS0 early.

40 LU Shan et al. Sci China Ser G-Phys Mech Astron | Jan. 2008 | vol. 51 | no. 1 | 38-44

Figure 1 Schematic of the setup of post-selected time-bin entanglement. PS1 and PS2 are both single photon sources; BS0, BSa, BSa′, BSb and BSb′ are all 50/50 beam splitters; α, β and Φ are phase shifters.

Figure 2 Four possible cases that the two photons pass through the BS0.

So photon 1 is in the early state, which can be written as 1| .e⟩ Photon 2 is in the late state 2| ,l⟩ since it takes the long path before entering BS0 and will arrive late. Hence the whole state of the two photon system reads: 1 2| | .e l⟩ ⊗ ⟩ (1) Note that the relative phase Φ of the two photons is of no physical significance in this context since it can be extracted out as a global phase. The transformation of BS0 can be expressed as

1 a b1| (| i | ),2

e e e⟩ → ⟩ + ⟩ 2 b a1| (| i | ).2

l l l⟩ → ⟩ + ⟩

Now the input two-photon state can be rewritten as

1 2 a b b a

a b a b a a b b

1 1| | (| i | ) (| i | )2 2

1 [(| | | | ) i(| | | | )].2

e l e e l l

e l l e e l e l

⟩ ⊗ ⟩ = ⟩ + ⟩ ⊗ ⟩ + ⟩

= ⟩ ⟩ − ⟩ ⟩ + ⟩ ⟩ + ⟩ ⟩

(2)

From the last line of eq. (2), the last two terms correspond to the events that both photons go to


Alice’s side or both to Bob’s side, while the first two terms correspond to the events that one photon goes to Alice’s side, one to Bob’s side. Thus Alice and Bob can post-select the first two terms by coincident detection. The post-selected state can be written as

a b a b1| (| | | | ),2

e l l eψ − ⟩ = ⟩ ⟩ − ⟩ ⟩ (3)

which is maximal time-bin entanglement state. Note that the post-selected state has been renor-malized here.

The entanglement can be verified with the sophisticated measurement device for time-bin entanglement[9]. It is an unbalanced Mach-Zehnder interferometer, consisting of two arms (one is short and the other is long), the path difference of which should be exactly (within the coherence time of the single photons) equal to the delay δ t. That means

a a b b 2 1.L S L S L S− = − = − Under this condition, each detector may find photon in discrete time slots. Alice, for instance, will find that the photon’s arrival time will be limited in three time slots. It is illustrated in the inset in top right corner of Figure 1. The three peaks represent the time slots corresponding to the different path combinations that the photon has traveled. The amplitude of the peak implies the probability of detecting a single photon. Note that we only consider the cases in which both Alice and Bob detect a photon on their sides. The more complicated cases that both two photons go to the same side are not considered here, since they do not contribute to the following coincidence detection. Here, ta − t0 is the photon arrival time, with respect to the emitting time t0. The left satellite peak represents the earliest time slot, which corresponds to the process that the photon is from PS1 and travels via two shortest paths (S1 and Sa). The right satellite peak means that the photon is from PS2 and takes two longest paths of L2 and La. The process leading to the central peak is more compli-cated, as there are two possible cases that may lead to the detection of a photon in this time slot. One case is that the photon is emitted from PS1 and travels via paths of S1 and La, which is denoted by (S1, La) in the figure. The other case is that the photon is emitted from PS2 and travels via paths of L2 and Sa, which is denoted by (L2, Sa). Similarly, Bob will find that the photon on his side may be detected in three discrete time slots. And the corresponding processes are marked near the time slots, respectively.

We now analyze the coincidence detections between the detectors of Alice and Bob. Note that here the coincidence means ta = tb, and the coincidence window tw is determined by electronics unit. To discriminate the three time slots, tw is required to be smaller than the separation time, δ t, be-tween the neighbouring time slots. That is tw <δ t. Based on the analysis, one can conclude that, in fact, the coincidence events can only happen in the central time slots. Neither the events in the left nor that in the right satellite time slots of Alice and Bob can contribute to the coincidence detec-tions, although they seem to satisfy the time condition of coincidence ta = tb. In other words, if Alice’s detector detects a photon in her left satellite time slot, Bob’s detector cannot find a photon in the left satellite time slot, and vice versa. The detection of a photon in the left satellite time slot corresponds to the process (S1, Sa) for Alice, and the process (S1, Sb) for Bob. Both detections correspond to photon 1 emitted by the single photon source of PS1, thus cannot happen at the same time. Similarly, detections of a photon in the right satellite time slots of Alice and Bob correspond to photon 2 emitted by PS2, and thus cannot happen at the same time.

By excluding the possibility of coincidence detection in satellite slots, one can conclude that the


coincidence must happen in central time slots and the processes must be (S1, La) with (L2, Sb), or (L2, Sa) with (S1, Lb). After passing through the BSa′ and BSb′, the two possibilities are indistin-guishable and coherently superposed, leading to two-photon interference. Both Alice and Bob have two detectors (Da+, Da− and Db+, Db−). Hence, there are four combinations of their detectors for joint measurement. The probability for detection by a given combination of detectors depends on the phases α and β of the two interferometers of the two observers. The quantum mechanical treatment leads to

,1 [1 cos( )],4m nP mn α β= − − (4)

where m, n = ±1 denote the detector labels of Alice’s and Bob’s, respectively. This formula for probability Pm,n can be derived as the following.

Assume that the photon pair has been prepared in the time-bin entanglement state of |ψ − ⟩ , as given by eq. (3). Then the probability for detection by the combination of detectors (m, n) is , ,| | ,m n m nP Mψ ψ− −= ⟨ ⟩ (5)

here, the Mm,n is the correlation measurement operator between detector m on Alice’s side, and detector n on Bob’s side, which is defined as , | , , | | | | |,m nM m n m n m m n nα β α β α α β β= ⟩⟨ = ⟩⟨ ⊗ ⟩⟨ (6)

where | mα⟩ and | nβ ⟩ denote the state detected by detectors m and n, with the phase shifters being α and β. They can be expressed as

ia b

ia b

1| (| e | ),2

1| (| e | ).2

m

n

m e l

n e l

α

β

α

β

⟩ = ⟩ + ⟩

⟩ = ⟩ + ⟩ (7)

From eq. (3), (5)－(7), the formula of eq. (4) can be obtained. That is

a b a b1( , ) ( , ) [1 cos( )]4

P D D P D D α β+ + − −= = − − (8)

and

a b a b1( , ) ( , ) [1 cos( )].4

P D D P D D α β+ − − += = + − (9)

One thus obtains cosinusoidal interference fringe, with the visibility of 100%, exceeding 12

≈

71%, which is the upper limit for separable states, therefore demonstrates that the two photons are entangled.

3 Discussion

The method proposed here to realize time-bin entanglement is based on post-selection, which is, essentially, like previous experiment of polarization[4] or spatial[5] entanglement. Nevertheless, it has its own unique features and advantages compared with polarization and spatial entanglement. First, although the two photons from two separate sources are directed to the same beam splitter BS0 with intersected paths, the two photons have never met each other in the sense that they arrived at BS0 not at the same time due to a lag between them introduced by the delay in the path of photon


2. That is, when photon 2 arrived at BS0, it will find no photon there, since photon 1 left δ t before. Hence, the time-bin entanglement scheme proposed here can be regarded as a realization of en-tanglement between two photons that have never interacted, avoiding the loophole that may exist in its polarization counterpart. Second, to close another loophole in the experimental test of nonlo-cality, the two observers Alice and Bob need to be spacelike separated. It is a formidable challenge for keeping the phase stable in the interference setup of spatial entanglement with Alice and Bob separated by large distance. The advantage of the scheme proposed here is that two photon time-bin entanglement can be preserved over a large distance[15].

Our scheme is within the reach of current technology. In practice, PS1 and PS2 can employ heralded single photon source relying on spontaneous parametric down-conversion (SPDC) process[16]. We mention that, the heralded single photon source are not ideal single photon source since there are cases, in which two (or even more) photon pairs are created in the SPDC process of either source[3]. Nevertheless, this possibility can be ruled out by triggered detection procedures, and thus has no significant implication on the experiment result, where 4-fold coincidence is re-quired (with two detectors as trigger). To make the emitted photons from the two independent sources interference, the SPDC process must be pumped by short (femto-second) pulsed laser, which is required to be well synchronized[17,18]. Contrary to Frason-type energy-time entanglement source[19], the coherence of the pump laser in our scheme is of no importance. Moreover, compared with previous time-bin entanglement source, our scheme holds another advantage. As explained above, the post-selected entanglement state is insensitive to instability of the relative phase Φ between the input long and short paths before BS0. In principle, the experiment can be performed in fibre system by replacing the bulk beam splitters by their fibre counterpart. Especially, the un-balanced Mach-Zehnder interferometers in Alice and Bob’s sides can adopt the corresponding fibre device which is routinely used in previous time-bin experiment[9].

4 Conclusion

In conclusion, we have presented a novel linear-optical scheme for entangling two photons from independent sources. Our scheme is a time-bin counterpart of previous post-selection polarization or spatial entanglement. The two photons have never met each other. Thus, our scheme can be employed to reveal the contradiction between quantum nonlocality and local realism. The advan-tage of the time-bin entanglement scheme has been shown, compared with the previous polariza-tion or spatial entanglement scheme. The feasibility of the scheme is within the reach of current technology. By combining the post-selection method and the time-bin technique developed re-cently, we present an alternative approach to test the fundamental physical issue of quantum nonlocality.

1 Freedman S J, Clauser J F. Experimental test of local hidden-variable theories. Phys Rev Lett, 1972, 28(14): 938－941 2 Hagley E, Maître X, Nogues G, et al. Generation of Einstein-Podolsky-Rosen pairs of atoms. Phys Rev Lett, 1997, 79(1):

1－5 3 Pan J W, Bouwmeester D, Weinfurter H, et al. Experimental entanglement swapping: Entangling photons that never inter-

acted. Phys Rev Lett, 1998, 80(18): 3891－3894 4 Shih Y H, Alley C O. New type of Einstein-Podolsky-Rosen-Bohm experiment using pairs of light quanta produced by

optical parametric down conversion. Phys Rev Lett, 1988, 61(26): 2921－2924 5 Yurke B, Stoler D. Bell’s-inequality experiments using independent-particle sources. Phys Rev A, 1992, 46(5): 2229－2234 6 Lombardi E, Sciarrino F, Popescu S, et al. Teleportation of a vacuum-one-photon qubit. Phys Rev Lett, 2002, 88(7): 070402


7 Rösch J, Jing X M, Yin J, et al. Non-local generation of entanglement of photons which do not meet each other. eprint arXiv: quant-ph 0609135

8 Brendel J, Mohler E, Martienssen W. Time-resolved dual-beam two-photon interferences with high visibility. Phys Rev Lett, 1991, 66(9): 1142－1145

9 Brendel J, Gisin N, Tittel W, et al. Pulsed energy-time entangled twin-photon source for quantum communication. Phys Rev Lett, 1999, 82(12): 2594－2597

10 Tittel W, Brendel J, Zbinden H, et al. Quantum cryptography using entangled photons in Energy-Time Bell States. Phys Rev Lett, 2000, 84(20): 4737－4740

11 Inoue K, Takesue H. Quantum key distribution using entangled-photon trains with no basis selection. Phys Rev A, 2006, 73(3): 032332

12 Inoue K, Waks E, Yamamoto Y. Differential phase shift quantum key distribution. Phys Rev Lett, 2002, 89(3): 037902 13 Inoue K, Waks E, Yamamoto Y. Differential-phase-shift quantum key distribution using coherent light. Phys Rev A, 2003,

68(2): 022317 14 Wang W Y, Wang C, Wen K, et al. Quantum key distribution network based on differential phase shift. Chin Phys Lett, 2007,

24(6): 1463－1466 15 Tittel W, Brendel J, Zbinden H, et al. Violation of Bell Inequalities by photons more than 10 km apart. Phys Rev Lett, 1998,

81(17): 3563－3566 16 Pittman T, Jacobs B, Franson J. Heralding single photons from pulsed parametric down-conversion. Opt Comm, 2005,

246(4-6): 545－550 17 Yang T, Zhang Q, Chen T Y, et al. Experimental synchronization of independent entangled photon sources. Phys Rev Lett,

2006, 96(11): 110501 18 Kaltenbaek R, Blauensteiner B, Zukowski M, et al. Experimental interference of independent photons. Phys Rev Lett, 2006,

96(24): 240502 19 Franson J D. Bell inequality for position and time. Phys Rev Lett, 1989, 62(19): 2205－2208

A scheme to realize time-bin entanglement between two photons that never interacted

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