Abstract

In this paper, the energy-time entangled photon-pairs at 1.5 μm are generated by the spontaneous four wave mixing (SFWM) in optical fibers under continuous wave (CW) pumping. The energy-time entanglement property is demonstrated experimentally through an experiment of Franson-type interference. Although the generation rates of the noise photons are one order of magnitude higher than that of the photon-pairs under CW pumping, the impact of noise photons can be highly suppressed in the measurement by a narrow time domain filter supported by superconducting nanowire single photon detectors with low timing jitters and time correlated single photon counting (TCSPC) module with high time resolution. The experiment results show that the SFWM in optical fibers under CW pumping provides a simple and practical way to generate energy-time entanglement at 1.5 μm, which has great potential for long-distance quantum information applications over optical fibers.

© 2014 Optical Society of America

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    [CrossRef]

2013

B. Fang, O. Cohen, J. B. Moreno, V. O. Lorenz, “State engineering of photon pairs produced through dual-pump spontaneous four-wave mixing,” Opt. Express 21, 2707–2717 (2013).
[CrossRef] [PubMed]

L. You, X. Yang, Y. He, W. Zhang, D. Liu, W. Zhang, L. Zhang, L. Zhang, X. Liu, S. Chen, Z. Wang, X. Xie, “Jitter analysis of a superconducting nanowire single photon detector,” AIP Advances 3, 072135 (2013).
[CrossRef]

S. Chen, D. Liu, W. Zhang, L. You, Y. He, W. Zhang, X. Yang, G. Wu, M. Ren, H. Zeng, Z. Wang, X. Xie, M. Jiang, “Time-of-flight laser ranging and imaging at 1550 nm using low-jitter superconducting nanowire single-photon detection system,” Applied optics 52, 3241–3245 (2013).
[CrossRef] [PubMed]

2012

J. Pan, Z. Chen, C. Lu, H. Weinfurter, A. Zeilinger, M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84, 777 (2012).
[CrossRef]

2010

2009

2008

2007

2005

2004

K. Inoue, K. Shimizu, “Generation of quantum-correlated photon pairs in optical fiber: influence of spontaneous raman scattering,” Jpn. J. Appl. Phys. 43, 8048 (2004).
[CrossRef]

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

H. de Riedmatten, I. Marcikic, V. Scarani, W. Tittel, H. Zbinden, N. Gisin, “Tailoring photonic entanglement in high-dimensional hilbert spaces,” Phys. Rev. A 69, 050304 (2004).
[CrossRef]

2003

I. Marcikic, H. De Riedmatten, W. Tittel, H. Zbinden, N. Gisin, “Long-distance teleportation of qubits at telecommunication wavelengths,” Nature (London) 421, 509–513 (2003).
[CrossRef]

2002

V. Giovannetti, S. Lloyd, L. Maccone, “Positioning and clock synchronization through entanglement,” Phys. Rev. A 65, 022309 (2002).
[CrossRef]

M. Fiorentino, P. L. Voss, J. E. Sharping, P. Kumar, “All-fiber photon-pair source for quantum communication,” IEEE Photon. Technol. Lett. 27, 491C493 (2002)

2001

L. Wang, C. Hong, S. Friberg, “Generation of correlated photons via four-wave mixing in optical fibres,” J. Opt. B: Quantum and Semiclass. Opt. 3, 346 (2001).
[CrossRef]

J. E. Sharping, M. Fiorentino, P. Kumar, “Observation of twin-beam-type quantum correlation in optical fiber,” Opt. Lett. 26, 367–369 (2001).
[CrossRef]

2000

W. Tittel, J. Brendel, H. Zbinden, N. Gisin, “Quantum cryptography using entangled photons in energy-time bell states,” Phys. Rev. Lett. 84, 4737 (2000).
[CrossRef] [PubMed]

1999

W. Tittel, J. Brendel, N. Gisin, H. Zbinden, “Long-distance bell-type tests using energy-time entangled photons,” Phys. Rev. A 59, 4150 (1999).
[CrossRef]

1989

J. Franson, “Bell inequality for position and time,” Phys. Rev. Lett. 62, 2205–2208 (1989).
[CrossRef] [PubMed]

1969

J. F. Clauser, M. A. Horne, A. Shimony, R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880–884 (1969).
[CrossRef]

1964

J. S. Bell, “On the einstein-podolsky-rosen paradox,” Physics 1, 195–200 (1964).

Ali-Khan, I.

I. Ali-Khan, C. J. Broadbent, J. C. Howell, “Large-alphabet quantum key distribution using energy-time entangled bipartite states,” Phys. Rev. Lett. 98, 060503 (2007).
[CrossRef] [PubMed]

Baek, B.

Bell, J. S.

J. S. Bell, “On the einstein-podolsky-rosen paradox,” Physics 1, 195–200 (1964).

Brainis, E.

E. Brainis, “Four-photon scattering in birefringent fibers,” Phys. Rev. A 79, 023840 (2009).
[CrossRef]

Brendel, J.

W. Tittel, J. Brendel, H. Zbinden, N. Gisin, “Quantum cryptography using entangled photons in energy-time bell states,” Phys. Rev. Lett. 84, 4737 (2000).
[CrossRef] [PubMed]

W. Tittel, J. Brendel, N. Gisin, H. Zbinden, “Long-distance bell-type tests using energy-time entangled photons,” Phys. Rev. A 59, 4150 (1999).
[CrossRef]

Broadbent, C. J.

I. Ali-Khan, C. J. Broadbent, J. C. Howell, “Large-alphabet quantum key distribution using energy-time entangled bipartite states,” Phys. Rev. Lett. 98, 060503 (2007).
[CrossRef] [PubMed]

Cemlyn, B.

Chen, J.

Chen, S.

L. You, X. Yang, Y. He, W. Zhang, D. Liu, W. Zhang, L. Zhang, L. Zhang, X. Liu, S. Chen, Z. Wang, X. Xie, “Jitter analysis of a superconducting nanowire single photon detector,” AIP Advances 3, 072135 (2013).
[CrossRef]

S. Chen, D. Liu, W. Zhang, L. You, Y. He, W. Zhang, X. Yang, G. Wu, M. Ren, H. Zeng, Z. Wang, X. Xie, M. Jiang, “Time-of-flight laser ranging and imaging at 1550 nm using low-jitter superconducting nanowire single-photon detection system,” Applied optics 52, 3241–3245 (2013).
[CrossRef] [PubMed]

Chen, Z.

J. Pan, Z. Chen, C. Lu, H. Weinfurter, A. Zeilinger, M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84, 777 (2012).
[CrossRef]

Cheng, J.

Clark, A.

Clauser, J. F.

J. F. Clauser, M. A. Horne, A. Shimony, R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880–884 (1969).
[CrossRef]

Cohen, O.

de Riedmatten, H.

H. de Riedmatten, I. Marcikic, V. Scarani, W. Tittel, H. Zbinden, N. Gisin, “Tailoring photonic entanglement in high-dimensional hilbert spaces,” Phys. Rev. A 69, 050304 (2004).
[CrossRef]

I. Marcikic, H. De Riedmatten, W. Tittel, H. Zbinden, N. Gisin, “Long-distance teleportation of qubits at telecommunication wavelengths,” Nature (London) 421, 509–513 (2003).
[CrossRef]

Fang, B.

Fejer, M.

Fejer, M. M.

Fiorentino, M.

M. Fiorentino, P. L. Voss, J. E. Sharping, P. Kumar, “All-fiber photon-pair source for quantum communication,” IEEE Photon. Technol. Lett. 27, 491C493 (2002)

J. E. Sharping, M. Fiorentino, P. Kumar, “Observation of twin-beam-type quantum correlation in optical fiber,” Opt. Lett. 26, 367–369 (2001).
[CrossRef]

Franson, J.

J. Franson, “Bell inequality for position and time,” Phys. Rev. Lett. 62, 2205–2208 (1989).
[CrossRef] [PubMed]

Friberg, S.

L. Wang, C. Hong, S. Friberg, “Generation of correlated photons via four-wave mixing in optical fibres,” J. Opt. B: Quantum and Semiclass. Opt. 3, 346 (2001).
[CrossRef]

Fulconis, J.

Giovannetti, V.

V. Giovannetti, S. Lloyd, L. Maccone, “Positioning and clock synchronization through entanglement,” Phys. Rev. A 65, 022309 (2002).
[CrossRef]

Gisin, N.

H. de Riedmatten, I. Marcikic, V. Scarani, W. Tittel, H. Zbinden, N. Gisin, “Tailoring photonic entanglement in high-dimensional hilbert spaces,” Phys. Rev. A 69, 050304 (2004).
[CrossRef]

I. Marcikic, H. De Riedmatten, W. Tittel, H. Zbinden, N. Gisin, “Long-distance teleportation of qubits at telecommunication wavelengths,” Nature (London) 421, 509–513 (2003).
[CrossRef]

W. Tittel, J. Brendel, H. Zbinden, N. Gisin, “Quantum cryptography using entangled photons in energy-time bell states,” Phys. Rev. Lett. 84, 4737 (2000).
[CrossRef] [PubMed]

W. Tittel, J. Brendel, N. Gisin, H. Zbinden, “Long-distance bell-type tests using energy-time entangled photons,” Phys. Rev. A 59, 4150 (1999).
[CrossRef]

Halder, M.

He, Y.

L. You, X. Yang, Y. He, W. Zhang, D. Liu, W. Zhang, L. Zhang, L. Zhang, X. Liu, S. Chen, Z. Wang, X. Xie, “Jitter analysis of a superconducting nanowire single photon detector,” AIP Advances 3, 072135 (2013).
[CrossRef]

S. Chen, D. Liu, W. Zhang, L. You, Y. He, W. Zhang, X. Yang, G. Wu, M. Ren, H. Zeng, Z. Wang, X. Xie, M. Jiang, “Time-of-flight laser ranging and imaging at 1550 nm using low-jitter superconducting nanowire single-photon detection system,” Applied optics 52, 3241–3245 (2013).
[CrossRef] [PubMed]

Holt, R. A.

J. F. Clauser, M. A. Horne, A. Shimony, R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880–884 (1969).
[CrossRef]

Hong, C.

L. Wang, C. Hong, S. Friberg, “Generation of correlated photons via four-wave mixing in optical fibres,” J. Opt. B: Quantum and Semiclass. Opt. 3, 346 (2001).
[CrossRef]

Horne, M. A.

J. F. Clauser, M. A. Horne, A. Shimony, R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880–884 (1969).
[CrossRef]

Howell, J. C.

I. Ali-Khan, C. J. Broadbent, J. C. Howell, “Large-alphabet quantum key distribution using energy-time entangled bipartite states,” Phys. Rev. Lett. 98, 060503 (2007).
[CrossRef] [PubMed]

Huang, Y.

Inoue, K.

H. Takesue, K. Inoue, “1.5-μm band quantum-correlated photon pair generation in dispersion-shifted fiber: suppression of noise photons by cooling fiber,” Opt. Express 13, 7832–7839 (2005).
[CrossRef] [PubMed]

K. Inoue, K. Shimizu, “Generation of quantum-correlated photon pairs in optical fiber: influence of spontaneous raman scattering,” Jpn. J. Appl. Phys. 43, 8048 (2004).
[CrossRef]

Jiang, M.

S. Chen, D. Liu, W. Zhang, L. You, Y. He, W. Zhang, X. Yang, G. Wu, M. Ren, H. Zeng, Z. Wang, X. Xie, M. Jiang, “Time-of-flight laser ranging and imaging at 1550 nm using low-jitter superconducting nanowire single-photon detection system,” Applied optics 52, 3241–3245 (2013).
[CrossRef] [PubMed]

Kumar, P.

Langrock, C.

Li, X.

Liu, D.

S. Chen, D. Liu, W. Zhang, L. You, Y. He, W. Zhang, X. Yang, G. Wu, M. Ren, H. Zeng, Z. Wang, X. Xie, M. Jiang, “Time-of-flight laser ranging and imaging at 1550 nm using low-jitter superconducting nanowire single-photon detection system,” Applied optics 52, 3241–3245 (2013).
[CrossRef] [PubMed]

L. You, X. Yang, Y. He, W. Zhang, D. Liu, W. Zhang, L. Zhang, L. Zhang, X. Liu, S. Chen, Z. Wang, X. Xie, “Jitter analysis of a superconducting nanowire single photon detector,” AIP Advances 3, 072135 (2013).
[CrossRef]

Liu, X.

L. You, X. Yang, Y. He, W. Zhang, D. Liu, W. Zhang, L. Zhang, L. Zhang, X. Liu, S. Chen, Z. Wang, X. Xie, “Jitter analysis of a superconducting nanowire single photon detector,” AIP Advances 3, 072135 (2013).
[CrossRef]

Lloyd, S.

V. Giovannetti, S. Lloyd, L. Maccone, “Positioning and clock synchronization through entanglement,” Phys. Rev. A 65, 022309 (2002).
[CrossRef]

Lorenz, V. O.

Lu, C.

J. Pan, Z. Chen, C. Lu, H. Weinfurter, A. Zeilinger, M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84, 777 (2012).
[CrossRef]

Maccone, L.

V. Giovannetti, S. Lloyd, L. Maccone, “Positioning and clock synchronization through entanglement,” Phys. Rev. A 65, 022309 (2002).
[CrossRef]

Marcikic, I.

H. de Riedmatten, I. Marcikic, V. Scarani, W. Tittel, H. Zbinden, N. Gisin, “Tailoring photonic entanglement in high-dimensional hilbert spaces,” Phys. Rev. A 69, 050304 (2004).
[CrossRef]

I. Marcikic, H. De Riedmatten, W. Tittel, H. Zbinden, N. Gisin, “Long-distance teleportation of qubits at telecommunication wavelengths,” Nature (London) 421, 509–513 (2003).
[CrossRef]

Moreno, J. B.

Nam, S. W.

Pan, J.

J. Pan, Z. Chen, C. Lu, H. Weinfurter, A. Zeilinger, M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84, 777 (2012).
[CrossRef]

Peng, J.

Rarity, J. G.

Ren, M.

S. Chen, D. Liu, W. Zhang, L. You, Y. He, W. Zhang, X. Yang, G. Wu, M. Ren, H. Zeng, Z. Wang, X. Xie, M. Jiang, “Time-of-flight laser ranging and imaging at 1550 nm using low-jitter superconducting nanowire single-photon detection system,” Applied optics 52, 3241–3245 (2013).
[CrossRef] [PubMed]

Scarani, V.

H. de Riedmatten, I. Marcikic, V. Scarani, W. Tittel, H. Zbinden, N. Gisin, “Tailoring photonic entanglement in high-dimensional hilbert spaces,” Phys. Rev. A 69, 050304 (2004).
[CrossRef]

Sharping, J.

Sharping, J. E.

M. Fiorentino, P. L. Voss, J. E. Sharping, P. Kumar, “All-fiber photon-pair source for quantum communication,” IEEE Photon. Technol. Lett. 27, 491C493 (2002)

J. E. Sharping, M. Fiorentino, P. Kumar, “Observation of twin-beam-type quantum correlation in optical fiber,” Opt. Lett. 26, 367–369 (2001).
[CrossRef]

Shimizu, K.

K. Inoue, K. Shimizu, “Generation of quantum-correlated photon pairs in optical fiber: influence of spontaneous raman scattering,” Jpn. J. Appl. Phys. 43, 8048 (2004).
[CrossRef]

Shimony, A.

J. F. Clauser, M. A. Horne, A. Shimony, R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880–884 (1969).
[CrossRef]

Takesue, H.

Tittel, W.

H. de Riedmatten, I. Marcikic, V. Scarani, W. Tittel, H. Zbinden, N. Gisin, “Tailoring photonic entanglement in high-dimensional hilbert spaces,” Phys. Rev. A 69, 050304 (2004).
[CrossRef]

I. Marcikic, H. De Riedmatten, W. Tittel, H. Zbinden, N. Gisin, “Long-distance teleportation of qubits at telecommunication wavelengths,” Nature (London) 421, 509–513 (2003).
[CrossRef]

W. Tittel, J. Brendel, H. Zbinden, N. Gisin, “Quantum cryptography using entangled photons in energy-time bell states,” Phys. Rev. Lett. 84, 4737 (2000).
[CrossRef] [PubMed]

W. Tittel, J. Brendel, N. Gisin, H. Zbinden, “Long-distance bell-type tests using energy-time entangled photons,” Phys. Rev. A 59, 4150 (1999).
[CrossRef]

Voss, P.

Voss, P. L.

M. Fiorentino, P. L. Voss, J. E. Sharping, P. Kumar, “All-fiber photon-pair source for quantum communication,” IEEE Photon. Technol. Lett. 27, 491C493 (2002)

Wadsworth, W. J.

Wang, L.

L. Wang, C. Hong, S. Friberg, “Generation of correlated photons via four-wave mixing in optical fibres,” J. Opt. B: Quantum and Semiclass. Opt. 3, 346 (2001).
[CrossRef]

Wang, Z.

S. Chen, D. Liu, W. Zhang, L. You, Y. He, W. Zhang, X. Yang, G. Wu, M. Ren, H. Zeng, Z. Wang, X. Xie, M. Jiang, “Time-of-flight laser ranging and imaging at 1550 nm using low-jitter superconducting nanowire single-photon detection system,” Applied optics 52, 3241–3245 (2013).
[CrossRef] [PubMed]

L. You, X. Yang, Y. He, W. Zhang, D. Liu, W. Zhang, L. Zhang, L. Zhang, X. Liu, S. Chen, Z. Wang, X. Xie, “Jitter analysis of a superconducting nanowire single photon detector,” AIP Advances 3, 072135 (2013).
[CrossRef]

Weinfurter, H.

J. Pan, Z. Chen, C. Lu, H. Weinfurter, A. Zeilinger, M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84, 777 (2012).
[CrossRef]

Wu, G.

S. Chen, D. Liu, W. Zhang, L. You, Y. He, W. Zhang, X. Yang, G. Wu, M. Ren, H. Zeng, Z. Wang, X. Xie, M. Jiang, “Time-of-flight laser ranging and imaging at 1550 nm using low-jitter superconducting nanowire single-photon detection system,” Applied optics 52, 3241–3245 (2013).
[CrossRef] [PubMed]

Xie, X.

L. You, X. Yang, Y. He, W. Zhang, D. Liu, W. Zhang, L. Zhang, L. Zhang, X. Liu, S. Chen, Z. Wang, X. Xie, “Jitter analysis of a superconducting nanowire single photon detector,” AIP Advances 3, 072135 (2013).
[CrossRef]

S. Chen, D. Liu, W. Zhang, L. You, Y. He, W. Zhang, X. Yang, G. Wu, M. Ren, H. Zeng, Z. Wang, X. Xie, M. Jiang, “Time-of-flight laser ranging and imaging at 1550 nm using low-jitter superconducting nanowire single-photon detection system,” Applied optics 52, 3241–3245 (2013).
[CrossRef] [PubMed]

Q. Zhang, H. Takesue, S. W. Nam, C. Langrock, X. Xie, B. Baek, M. Fejer, Y. Yamamoto, “Distribution of time-energy entanglement over 100 km fiber using superconducting singlephoton detectors,” Opt. Express 16, 5776–5781 (2008).
[CrossRef] [PubMed]

Q. Zhang, X. Xie, H. Takesue, S. W. Nam, C. Langrock, M. M. Fejer, Y. Yamamoto, ”Correlated photon-pair generation in reverse-proton-exchange PPLN waveguides with integrated mode demultiplexer at 10 GHz clock,” Opt. Express 15, 10288–10293 (2007)
[CrossRef] [PubMed]

Xiong, C.

Yamamoto, Y.

Yang, X.

L. You, X. Yang, Y. He, W. Zhang, D. Liu, W. Zhang, L. Zhang, L. Zhang, X. Liu, S. Chen, Z. Wang, X. Xie, “Jitter analysis of a superconducting nanowire single photon detector,” AIP Advances 3, 072135 (2013).
[CrossRef]

S. Chen, D. Liu, W. Zhang, L. You, Y. He, W. Zhang, X. Yang, G. Wu, M. Ren, H. Zeng, Z. Wang, X. Xie, M. Jiang, “Time-of-flight laser ranging and imaging at 1550 nm using low-jitter superconducting nanowire single-photon detection system,” Applied optics 52, 3241–3245 (2013).
[CrossRef] [PubMed]

You, L.

S. Chen, D. Liu, W. Zhang, L. You, Y. He, W. Zhang, X. Yang, G. Wu, M. Ren, H. Zeng, Z. Wang, X. Xie, M. Jiang, “Time-of-flight laser ranging and imaging at 1550 nm using low-jitter superconducting nanowire single-photon detection system,” Applied optics 52, 3241–3245 (2013).
[CrossRef] [PubMed]

L. You, X. Yang, Y. He, W. Zhang, D. Liu, W. Zhang, L. Zhang, L. Zhang, X. Liu, S. Chen, Z. Wang, X. Xie, “Jitter analysis of a superconducting nanowire single photon detector,” AIP Advances 3, 072135 (2013).
[CrossRef]

Zbinden, H.

H. de Riedmatten, I. Marcikic, V. Scarani, W. Tittel, H. Zbinden, N. Gisin, “Tailoring photonic entanglement in high-dimensional hilbert spaces,” Phys. Rev. A 69, 050304 (2004).
[CrossRef]

I. Marcikic, H. De Riedmatten, W. Tittel, H. Zbinden, N. Gisin, “Long-distance teleportation of qubits at telecommunication wavelengths,” Nature (London) 421, 509–513 (2003).
[CrossRef]

W. Tittel, J. Brendel, H. Zbinden, N. Gisin, “Quantum cryptography using entangled photons in energy-time bell states,” Phys. Rev. Lett. 84, 4737 (2000).
[CrossRef] [PubMed]

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Figures (6)

Fig. 1
Fig. 1

Experiment setup. A CW laser was used as the pump light source; EDFA: erbium doped fiber amplifier; VOA: variable optical attenuator; DWDM: dense wavelength division multiplexing device; FPC: fiber polarization controller; SNSPD: superconducting nanowire single photon detector; TCSPC: time correlated single photon counting module.

Fig. 2
Fig. 2

Single side photon count rates. Squares and circles were results of signal and idler photons, respectively. The curve-fitting results are shown in the figure.

Fig. 3
Fig. 3

Measurement results of coincidence count. (a) Typical experimental result of the coincidence count when the pump power is 2.92 mW. (b) Coincidence count rates and accidental coincidence count rates under different pump levels.

Fig. 4
Fig. 4

Calculated results of R, Rs and Ri under different pump levels. (a) R under different pump levels: squares are calculated results, the red line is the fitting curve of R = 5.38 × 105P2. (b) Rs and Ri under different pump levels. Squares and circles are calculated results, blue solid and red dashed lines are fitting curves of Rs = 1.38 × 107P, Ri = 2.00 × 107P.

Fig. 5
Fig. 5

CARs under different pump levels. The red squares are measured results. The blue line is the calculated result according to Eq. (6) utilizing the fitting curves shown in Fig. 4.

Fig. 6
Fig. 6

Quantum interference of energy-time entanglement generated in the optical fiber under CW pumping. (a) Experimental setup. Photon pairs are generated in optical fibers under CW pumping, two UMZIs are realized utilizing a commercial 10 GHz DQPSK demodulator. α and β are the additional phases in the long arms of the two UMZI, respectively. An additional filter system for the signal and idler photons is used to get rid of the impact of the beam splitter (BS1) combining the two UMZIs in the DQPSK. (b) A typical result of the coincidence measurement and the applied time domain filter. (c) The measured interference fringes of the coincidence counts when changing α under β = 3.41 rad and β = 4.83 rad.

Equations (9)

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C s = η s R + η s R s + D s
C i = η i R + η i R i + D i
C a = C s C i τ
C c = η s η i R + C a
C a η s η i R s R i τ + η s η i R ( R s + R i ) τ
CAR = C c C a R [ R s R i + R ( R s + R i ) ] τ + 1
| Φ = 1 2 [ | short s | short i + exp ( i ( α + β ) ) | long s | long i ]
S = | E ( α 1 , β 1 ) + E ( α 1 , β 2 ) + E ( α 2 , β 1 ) E ( α 2 , β 2 ) | 2
E ( α , β ) = V cos ( α + β )

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