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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 17 — Aug. 18, 2008
  • pp: 12505–12510
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Observation of quantum interference between a single-photon state and a thermal state generated in optical fibers

Xiaoying Li, Lei Yang, Liang Cui, Zhe Yu Ou, and Daoyin Yu  »View Author Affiliations


Optics Express, Vol. 16, Issue 17, pp. 12505-12510 (2008)
http://dx.doi.org/10.1364/OE.16.012505


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Abstract

We experimentally demonstrate a Hong-Ou-Mandel type of two-photon interference effect with a heralded single-photon state and a thermal state. The light sources in the 1550 nm telecom band are generated from two independent dispersion-shifted fibers via four-wave mixing process. The observed visibility is (82±11)%. This type of interference between independent sources is crucial in quantum information process with independent qubits.

© 2008 Optical Society of America

Quantum information process usually involves quantum bits (qubits) that are independent of each other. Quantum operations are performed on them to make them entangled to each other. If photons are used as qubits, nonlinear optical process is usually needed to make them entangled. However, recent protocol with linear optical elements makes linear quantum optical computation possible [1

1. E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46–52 (2001). [CrossRef] [PubMed]

]. Crucial in this protocol is the Hong-Ou-Mandel (HOM) interference effect [2

2. C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2047 (1987). [CrossRef] [PubMed]

] that allows two independent photons to interfere and generate entanglement between them after some projections such as coincidence measurement. As a matter of fact, creation of entanglement between independent photons was already realized in the schemes such as quantum state teleportation [3

3. D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature 390, 575–579 (1997). [CrossRef]

] and entanglement swapping [4

4. J. W. Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger, “Experimental entanglement swapping: entangling photons that never interacted,” Phys. Rev. Lett. 80, 3891–3894 (1998). [CrossRef]

].

So far, qubits are produced in the form of heralded single-photon state [5

5. C. K. Hong and L. Mandel, “Experimental realization of a localized one-photon state,” Phys. Rev. Lett. 56, 58–60 (1986). [CrossRef] [PubMed]

] from parametric process such as parametric down-conversion (PDC) and four-wave mixing (FWM) [3

3. D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature 390, 575–579 (1997). [CrossRef]

, 4

4. J. W. Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger, “Experimental entanglement swapping: entangling photons that never interacted,” Phys. Rev. Lett. 80, 3891–3894 (1998). [CrossRef]

, 6

6. X. Li, J. Chen, P. L. Voss, J. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications: Improved generation of correlated photons,” Opt. Express 12, 3737–3744 (2004). [CrossRef] [PubMed]

, 7

7. J. Fan, A. Dogariu, and L. J Wang, “Generation of correlated photon pairs in a microstructure fiber,” Opt. Lett. 30, 1530–1532 (2005). [CrossRef] [PubMed]

, 8

8. A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L. M. Duan, and H. J. Kimble, “Generation of Nonclassical Photon Pairs for Scalable Quantum Communication with Atomic Ensembles,” Nature 423, 731–734 (2003). [CrossRef] [PubMed]

, 9

9. X. Li, C. Liang, K. F. Lee, J. Chen, P. L. Voss, and P. Kumar, “Integrable optical-fiber source of polarization-entangled photon pairs in the telecom band,” Phys. Rev. A 73, 052,301 (2006).

, 10

10. H. Takesue, “1.5-um band Hong-Ou-Mandel experiment using photon pairs generated in two independent dispersion shifted fibers,” Appl. Phys. Lett. 90, 204,101 (2007). [CrossRef]

, 11

11. J. Fulconis, O. Alibart, J. L. O’brien, W. J. Wadsworth, and J. G. Rarity, “Nonclassical Interference and Entanglement Generation Using a Photonic Crystal Fiber Pair Photon Source,” Phys. Rev. Lett. 99, 120,501 (2007). [CrossRef]

]. These processes generate nonclassical two-photon states entangled in a variety of degrees of freedom. Recently, interest arises in obtaining a single-photon state from a weak classical source of light such as a weak coherent state. Two-photon interference was demonstrated with a heralded single-photon state and a weak coherent state which provides another source of independent photons [12

12. J. G. Rarity, P. R. Tapster, and R. Loudon, “Non-classical interference between independent sources,” arXiv quant-ph, 9702,032 (1997).

, 13

13. T. B. Pittman and J. D. Franson, “Violation of Bell’s Inequality with Photons from Independent Sources,” Phys. Rev. Lett. 90, 240,401 (2003). [CrossRef]

, 14

14. B. Hessmo, P. Usachev, H. Heydari, and G. Bjork, “Experimental Demonstration of Single Photon Nonlocality,” Phys. Rev. Lett. 92, 180,401 (2004). [CrossRef]

].

Investigation of quantum interference of two photons between quantum state and classical state started as early as in 1988 [15

15. Z. Y. Ou, “Quantum theory of fourth-order interference,” Phys. Rev. A 37, 1607–1619 (1988). [CrossRef] [PubMed]

], right after the famous HOM two-photon interference experiment [2

2. C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2047 (1987). [CrossRef] [PubMed]

]. It was predicted [15

15. Z. Y. Ou, “Quantum theory of fourth-order interference,” Phys. Rev. A 37, 1607–1619 (1988). [CrossRef] [PubMed]

] that when the classical field is much weaker than the single-photon state, the visibility of the two-photon interference can approach 100%. The secret behind this is the visibility of two-photon interference involving single-photon state is independent of the ratio between the intensities of the two fields, unlike interference between two classical fields [16

16. Z. Y. Ou and L. Mandel, “Further evidence of nonclassical behavior in optical interference,” Phys. Rev. Lett. 62, 2941–2944 (1989). [CrossRef] [PubMed]

]. In 1991, Tan et al [17

17. S. M. Tan, D. F Walls, and M. J. Collett, “Nonlocality of a single photon,” Phys. Rev. Lett. 66, 252–255 (1991). [CrossRef] [PubMed]

] made the first proposal of using two-photon interference between a single-photon state and a classical coherent state to demonstrate nonlocality in phase correlation. Later, Rarity and Tapster [18

18. J. G. Rarity and P. R. Tapster, “Three-particle entanglement from entangled photon pairs and a weak coherent state,” Phys. Rev. A 59, R35–R38 (1999). [CrossRef]

] proposed a scheme for creating entangled three-photon GHZ state with a coherent state and two-photon state from PDC. Recently, Pittman and Franson [13

13. T. B. Pittman and J. D. Franson, “Violation of Bell’s Inequality with Photons from Independent Sources,” Phys. Rev. Lett. 90, 240,401 (2003). [CrossRef]

] first realized the scheme proposed in Ref.[17

17. S. M. Tan, D. F Walls, and M. J. Collett, “Nonlocality of a single photon,” Phys. Rev. Lett. 66, 252–255 (1991). [CrossRef] [PubMed]

] and demonstrated two-photon interference between a heralded single-photon state and an independent coherent state.

In this paper, we report an experiment of two-photon HOM type interference between a heralded single-photon state and a classical source, which are generated from two independent dispersion-shifted fibers (DSF) via FWM. This classical source is of thermal nature and yet we observed a visibility of (82±11)%, which is obviously larger than the classical limit of 50% in two-photon interference.

Our experiment setup is shown in figure 1. The correlated signal and idler photon pairs are generated by FWM process in a DSF. Two independent sources of photon pairs are obtained by passing the pulsed pump through a 60/40 coupler and then pumping DSF1 and DSF2, respectively. Pump pulses of about 4 ps duration and central wavelength λp=1538.9 nm arrive at 40 MHz rate. DSF1 and DSF2, both with a length of about 300 m and zero dispersion wavelength λ 0=1538±2 nm, are loosely wrapped into coils and merged in liquid nitrogen (77 K) to suppress the Raman scattering. To adjust the pump power of each fiber source independently, an attenuator is placed between the output of the 60/40 coupler and DSF1.

Fig. 1. The experimental setup. FPC, fiber polarization controller; PBS, polarization beam splitter; F, filter; Att, attenuator; SPD, single photon detector.

For each DSF-based source, which is described in more detail in Ref.[20

20. X. Li, L. Yang, L. Cui, Z. Y. Ou, and D. Yu, “Fiber-based source of photon pairs at telecom band with high temporal coherence and brightness for quantum information processing,” Opt. Lett. 33, 593–595 (2008). [CrossRef] [PubMed]

], signal and idler photon-pairs at wavelengths of 1546.9nm and 1530.9 nm, respectively, are produced by pumping DSFs with the pulsed pump. Signal1 (signal2) and idler1 (idler2) photons co-polarized with the pump are selected by properly adjusting the fiber polarization controller FPC1 (FPC2) placed in front of the polarization beam splitter PBS1 (PBS2). To reliably detect the signal and idler photons, an isolation between the pump and signal/idler photons in excess of 100 dB is required, because of the low efficiency of spontaneous FWM in DSF. We achieve this by passing the output of DSF1 (DSF2) through a dual-band filter F composed by cascading a double grating filter with array-waveguide gratings or a tunable filter [20

20. X. Li, L. Yang, L. Cui, Z. Y. Ou, and D. Yu, “Fiber-based source of photon pairs at telecom band with high temporal coherence and brightness for quantum information processing,” Opt. Lett. 33, 593–595 (2008). [CrossRef] [PubMed]

]. The bandwidth of the pump, signal and idler photons are about 1 nm, 0.33 nm, and 0.7 nm respectively.

The generated signal and idler photons are counted by single photon detectors (SPDs, id200 and PLI-AGD-SC-Rx) operated in the gated Geiger mode. The rate of the 2.5 ns gate is about 600 KHz, which is 1/64 of the repetition rate of the pump pulses, and the dead time of the gate is set to be 10 µs. The quantum efficiency of SPD1, SPD2, and SPD3 is 7%, 10%, and 20%, respectively. The total detection efficiency for signal1 (signal2) and idler2 is 0.6% (0.7%) and 3%, respectively, when the efficiencies of the other transmission components, such as the DSF, PBS, and filter F, are included.

In the experiment, the heralded single-photon state is generated from the signal field of DSF2 by gating on the detection of the idler2 photons with SPD3. The thermal state is produced by DSF1 from only one of the correlated photons, i.e. signal1. This is so because it has been shown that the spontaneousFWM process, which is equivalent to a parametric amplifier, places each field of the two correlated photons in a thermal state [21

21. B. Yurke and M. Potasek, “Obtainment of thermal noise from a pure quantum state,” Phys. Rev. A 36, 3464–3466 (1987). [CrossRef] [PubMed]

]. To confirm this, we measure the normalized intensity correlation function g (2) for the individual signal photons and obtain g (2)=2.00±0.03 [20

20. X. Li, L. Yang, L. Cui, Z. Y. Ou, and D. Yu, “Fiber-based source of photon pairs at telecom band with high temporal coherence and brightness for quantum information processing,” Opt. Lett. 33, 593–595 (2008). [CrossRef] [PubMed]

]. This result shows that the signal photons produced by each of the fiber sources are of thermal nature and have high spatial and temporal coherence. The single photon state (heralded signal2 field) and the thermal state (signal1 field), both having a central wavelength of 1546.9nm and a FWHM of 0.33 nm, are carefully path matched and simultaneously fed into a fiber coupler acting as a 50/50 beam splitter from two input ports, respectively. Before coupling in the 50/50 beam splitter, thermal state in signal1 field is delayed by the reflector mirrors mounted on a translation stage. To ensure the two input fields of the 50/50 coupler have the identical polarization, the polarization of the thermal field is properly adjusted by FPC3. The two output ports of the fiber coupler are detected by SPD1 and SPD2, respectively.

Fig. 2. Three-fold coincidence rate versus the position of the translation stage when the average photon number per pulse of (signal1) thermal field is about (a) 0.26 and (b) 0.38. Three-fold background coincidences of 47 and 40 counts/40min are subtracted from the data in (a) and (b), respectively.

The observation of the interference between the single photon state and the thermal state is achieved by measuring three-fold coincidences versus the position of the translation stage. During the measurement, the single counts, which are the outputs of SPD1, SPD2, and SPD3, and two-fold coincidence between arbitrary two SPDs are also recorded by the coincidence counting system. For an ideal single photon state, there is no need to modify the measured three-fold coincidences if the dark counts of the SPDs are negligible. But for the heralded single photon state, background counts caused by the multiphoton pairs of signal2 and idler2 fields should be subtracted. To reliably remove the background, single counts, two-fold and three-fold coincidence are measured when only the heralded single photon state is presented in the input of the 50/50 beam-splitter. The direct measurement of the background counts, i.e. the three-fold coincidences due to multiphoton pair events, should be consistent with the calculated results obtained by using the measured single counts and two-fold coincidences, and the experimental results agree with the prediction. Figure 2 shows the three-fold coincidences rate as a function of the position of the translation stage or the relative delay in the optical paths between the signal1 and signal2 fields, after subtraction of a background counts of 47/40min and 40/40min in Fig. 2(a) and (b), respectively. The production rate of photon pairs originated from DSF2 is about 0.15 pairs/pulse, and the average photon number of the signal1 (thermal) field produced by DSF1 is about 0.26 and 0.38 photons/pulse for the data in Fig. 2(a) and (b), respectively. The solid curves are the least-square fits to the data, which show a dip of (82±11)%and (67±11)% visibility are obtained in Fig. 2(a) and (b), respectively.

âD1=(â1+â2)2
âD2=(â1â2)2
(1)

where â 1 and â 2 are the field operators for the thermal state and single-photon state, respectively. The two-photon coincidence probability P 2 is proportional to [15

15. Z. Y. Ou, “Quantum theory of fourth-order interference,” Phys. Rev. A 37, 1607–1619 (1988). [CrossRef] [PubMed]

]

aD1(t)aD2(t)aD2(t)aD1(t)=14(a1(t)a1(t)a1(t)a1(t)+a2(t)a2(t)a2(t)a2(t)
+a1(t)a2(t)a2(t)a1(t)+a2(t)a1(t)a1(t)a2(t)
a2(t)a1(t)a2(t)a1(t)a1(t)a2(t)a1(t)a2(t))
=14(2n¯2+2n¯2n¯I12)
(2)

where is the average photon number of the thermal field, and I 12=〈a 2 (t)a 1(t′)a 2(t′)a 1(t)〉/ is the interference term, that leads to two-photon interference and depends on the overlap between signal1 and signal2 fields at the 50/50 beam splitter. When t=t′,I 12=1 corresponding to the bottom of the dip in figure 2 but when t and t′ are very different, I 12=0 corresponding to the wings in figure 2. From Eqs (1)–(2), we have P2=n¯2(n¯+1I12) . So the visibility is simply V=11+n¯ . It is clear that V approaches to 1 or 100% visibility if the average photon number of the thermal state n ̅≪1.

Experimentally, we can estimate from measured efficiencies of detectors and single counts and obtain =0.26±0.035 and =0.38±0.05, which leads to Vtheory=0.79±0.03 and Vtheory=0.72±0.03. These values are consistent with the results in Fig. 2.

Acknowledgment

This work was supported in part by NCET-060238, the NSF of China (No. 60578024, No. 10774111), Foundation for Key Project of Ministry of Education of China(No. 107027), the State Key Development Program for Basic Research of China (No. 2003CB314904), the Specialized Research Fund for the Doctoral Program of Higher Education of China (No.20070056084) and 111 Project B07014.

References and links

1.

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46–52 (2001). [CrossRef] [PubMed]

2.

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2047 (1987). [CrossRef] [PubMed]

3.

D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature 390, 575–579 (1997). [CrossRef]

4.

J. W. Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger, “Experimental entanglement swapping: entangling photons that never interacted,” Phys. Rev. Lett. 80, 3891–3894 (1998). [CrossRef]

5.

C. K. Hong and L. Mandel, “Experimental realization of a localized one-photon state,” Phys. Rev. Lett. 56, 58–60 (1986). [CrossRef] [PubMed]

6.

X. Li, J. Chen, P. L. Voss, J. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications: Improved generation of correlated photons,” Opt. Express 12, 3737–3744 (2004). [CrossRef] [PubMed]

7.

J. Fan, A. Dogariu, and L. J Wang, “Generation of correlated photon pairs in a microstructure fiber,” Opt. Lett. 30, 1530–1532 (2005). [CrossRef] [PubMed]

8.

A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L. M. Duan, and H. J. Kimble, “Generation of Nonclassical Photon Pairs for Scalable Quantum Communication with Atomic Ensembles,” Nature 423, 731–734 (2003). [CrossRef] [PubMed]

9.

X. Li, C. Liang, K. F. Lee, J. Chen, P. L. Voss, and P. Kumar, “Integrable optical-fiber source of polarization-entangled photon pairs in the telecom band,” Phys. Rev. A 73, 052,301 (2006).

10.

H. Takesue, “1.5-um band Hong-Ou-Mandel experiment using photon pairs generated in two independent dispersion shifted fibers,” Appl. Phys. Lett. 90, 204,101 (2007). [CrossRef]

11.

J. Fulconis, O. Alibart, J. L. O’brien, W. J. Wadsworth, and J. G. Rarity, “Nonclassical Interference and Entanglement Generation Using a Photonic Crystal Fiber Pair Photon Source,” Phys. Rev. Lett. 99, 120,501 (2007). [CrossRef]

12.

J. G. Rarity, P. R. Tapster, and R. Loudon, “Non-classical interference between independent sources,” arXiv quant-ph, 9702,032 (1997).

13.

T. B. Pittman and J. D. Franson, “Violation of Bell’s Inequality with Photons from Independent Sources,” Phys. Rev. Lett. 90, 240,401 (2003). [CrossRef]

14.

B. Hessmo, P. Usachev, H. Heydari, and G. Bjork, “Experimental Demonstration of Single Photon Nonlocality,” Phys. Rev. Lett. 92, 180,401 (2004). [CrossRef]

15.

Z. Y. Ou, “Quantum theory of fourth-order interference,” Phys. Rev. A 37, 1607–1619 (1988). [CrossRef] [PubMed]

16.

Z. Y. Ou and L. Mandel, “Further evidence of nonclassical behavior in optical interference,” Phys. Rev. Lett. 62, 2941–2944 (1989). [CrossRef] [PubMed]

17.

S. M. Tan, D. F Walls, and M. J. Collett, “Nonlocality of a single photon,” Phys. Rev. Lett. 66, 252–255 (1991). [CrossRef] [PubMed]

18.

J. G. Rarity and P. R. Tapster, “Three-particle entanglement from entangled photon pairs and a weak coherent state,” Phys. Rev. A 59, R35–R38 (1999). [CrossRef]

19.

C. H. Bennett and G. Brassard, “Quantum cryptography: Public key distribution and coin tossing,” in Proceedings of the IEEE International Conference on Computers, Systems and Signal Processing, Bangalore, India, (IEEE, New York) pp. 175–179 (1984). [PubMed]

20.

X. Li, L. Yang, L. Cui, Z. Y. Ou, and D. Yu, “Fiber-based source of photon pairs at telecom band with high temporal coherence and brightness for quantum information processing,” Opt. Lett. 33, 593–595 (2008). [CrossRef] [PubMed]

21.

B. Yurke and M. Potasek, “Obtainment of thermal noise from a pure quantum state,” Phys. Rev. A 36, 3464–3466 (1987). [CrossRef] [PubMed]

OCIS Codes
(190.4370) Nonlinear optics : Nonlinear optics, fibers
(190.4410) Nonlinear optics : Nonlinear optics, parametric processes
(270.0270) Quantum optics : Quantum optics

ToC Category:
Quantum Optics

History
Original Manuscript: June 9, 2008
Revised Manuscript: July 18, 2008
Manuscript Accepted: July 23, 2008
Published: August 4, 2008

Citation
Xiaoying Li, Lei Yang, Liang Cui, Zhe Yu Ou, and Daoyin Yu, "Observation of quantum interference between a single-photon state and a thermal state generated in optical fibers," Opt. Express 16, 12505-12510 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-17-12505


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References

  1. E. Knill, R. Laflamme, and G. J. Milburn, "A scheme for efficient quantum computation with linear optics," Nature 409,46-52 (2001). [CrossRef] [PubMed]
  2. C. K. Hong, Z. Y. Ou, and L. Mandel, "Measurement of subpicosecond time intervals between two photons by interference," Phys. Rev. Lett. 59, 2044-2047 (1987). [CrossRef] [PubMed]
  3. D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, "Experimental quantum teleportation," Nature 390, 575-579 (1997). [CrossRef]
  4. J. W. Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger, "Experimental entanglement swapping: entangling photons that never interacted," Phys. Rev. Lett. 80, 3891-3894 (1998). [CrossRef]
  5. C. K. Hong and L. Mandel, "Experimental realization of a localized one-photon state," Phys. Rev. Lett. 56, 58-60 (1986). [CrossRef] [PubMed]
  6. X. Li, J. Chen, P. L. Voss, J. Sharping, and P. Kumar, "All-fiber photon-pair source for quantum communications: Improved generation of correlated photons," Opt. Express 12, 3737-3744 (2004). [CrossRef] [PubMed]
  7. J. Fan, A. Dogariu, and L. J. Wang, "Generation of correlated photon pairs in a microstructure fiber," Opt. Lett. 30, 1530-1532 (2005). [CrossRef] [PubMed]
  8. A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L. M. Duan, and H. J. Kimble, "Generation of Nonclassical Photon Pairs for Scalable Quantum Communication with Atomic Ensembles," Nature 423, 731-734 (2003). [CrossRef] [PubMed]
  9. X. Li, C. Liang, K. F. Lee, J. Chen, P. L. Voss, and P. Kumar, "Integrable optical-fiber source of polarizationentangled photon pairs in the telecom band," Phys. Rev. A 73, 052,301 (2006).
  10. H. Takesue, "1.5-um band Hong-Ou-Mandel experiment using photon pairs generated in two independent dispersion shifted fibers," Appl. Phys. Lett. 90, 204,101 (2007). [CrossRef]
  11. J. Fulconis, O. Alibart, J. L. O�??brien, W. J. Wadsworth, and J. G. Rarity, "Nonclassical Interference and Entanglement Generation Using a Photonic Crystal Fiber Pair Photon Source," Phys. Rev. Lett. 99, 120,501 (2007). [CrossRef]
  12. J. G. Rarity, P. R. Tapster, and R. Loudon, "Non-classical interference between independent sources," arXiv quant-ph, 9702,032 (1997).
  13. T. B. Pittman and J. D. Franson, "Violation of Bell�??s Inequality with Photons from Independent Sources," Phys. Rev. Lett. 90, 240,401 (2003). [CrossRef]
  14. B. Hessmo, P. Usachev, H. Heydari, and G. Bjork, "Experimental Demonstration of Single Photon Nonlocality," Phys. Rev. Lett. 92, 180,401 (2004). [CrossRef]
  15. Z. Y. Ou, "Quantum theory of fourth-order interference," Phys. Rev. A 37, 1607-1619 (1988). [CrossRef] [PubMed]
  16. Z. Y. Ou and L. Mandel, "Further evidence of nonclassical behavior in optical interference," Phys. Rev. Lett. 62, 2941-2944 (1989). [CrossRef] [PubMed]
  17. S. M. Tan, D. F. Walls, and M. J. Collett, "Nonlocality of a single photon," Phys. Rev. Lett. 66, 252-255 (1991). [CrossRef] [PubMed]
  18. J. G. Rarity and P. R. Tapster, "Three-particle entanglement from entangled photon pairs and a weak coherent state," Phys. Rev. A 59, R35-R38 (1999). [CrossRef]
  19. C. H. Bennett and G. Brassard, "Quantum cryptography: Public key distribution and coin tossing," in Proceedings of the IEEE International Conference on Computers, Systems and Signal Processing, Bangalore, India, (IEEE, New York) pp. 175-179 (1984). [PubMed]
  20. X. Li, L. Yang, L. Cui, Z. Y. Ou, and D. Yu, "Fiber-based source of photon pairs at telecom band with high temporal coherence and brightness for quantum information processing," Opt. Lett. 33, 593-595 (2008). [CrossRef] [PubMed]
  21. B. Yurke and M. Potasek, "Obtainment of thermal noise from a pure quantum state," Phys. Rev. A 36, 3464-3466 (1987). [CrossRef] [PubMed]

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