Abstract

We present measurements of the polarization correlation and photon statistics of photon pairs that emerge from a laser-pumped warm rubidium vapor cell. The photon pairs occur at 780 nm and 1367 nm and are polarization entangled. We measure the autocorrelation of each of the generated fields as well as the cross-correlation function, and observe a strong violation of the two-beam Cauchy-Schwartz inequality. We evaluate the performance of the system as source of heralded single photons at a telecommunication wavelength. We measure the heralded autocorrelation and see that coincidences are suppressed by a factor of ≈ 20 from a Poissonian source at a generation rate of 1500 s−1, a heralding efficiency of 10%, and a narrow spectral width.

© 2011 OSA

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  1. 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, 1984), p. 175.
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  2. A. K. Ekert, “Quantum cryptography based on bell’s theorem,” Phys. Rev. Lett. 67, 661–663 (1991).
    [CrossRef] [PubMed]
  3. L. Duan, M. Lukin, J. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414, 413–418 (2001).
    [CrossRef] [PubMed]
  4. A. Kuzmich, W. P. Bowen, A. D. Booze, 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]
  5. C. H. van der Wal, M. D. Eisaman, A. Andre, R. L. Walsworth, D. F. Phillips, A. S. Zibrov, and M. D. Lukin, “Atomic memory for correlated photon states,” Science 301, 196–200 (2003).
    [CrossRef] [PubMed]
  6. V. Balic, D. A. Braje, P. Kolchin, G. Yin, and S. E. Haris, “Generation of paired photons with controllable waveforms,” Phys. Rev. Lett. 94, 183601 (2005).
  7. S. Du, P. Kolchin, C. Bethangady, G. Yin, and S. E. Harris, “Subnatural linewidth biphotons with controllable temporal length,” Phys. Rev. Lett. 100, 183603 (2008).
    [CrossRef] [PubMed]
  8. T. Chanelière, D. N. Matsukevich, S. D. Jenkins, T. A. B. Kennedy, M. S. Chapman, and A. Kuzmich, “Quantum telecommunication based on atomic cascade transitions,” Phys. Rev. Lett. 96, 093604 (2006).
    [CrossRef] [PubMed]
  9. K. F. Rim, J. Nunn, V. O. Lorenz, B. J. Sussman, K. C. Lee, N. K. Langford, D. Jaksch, and I. A. Walmsley, “Towards high-speed optical quantum memories,” Nat. Photonics 4, 218–221 (2010).
    [CrossRef]
  10. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
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  11. E. A. Goldschmidt, M. D. Eisaman, J. Fan, S. V. Polyakov, and A. Migdall, “Spectrally bright and broad fiber-based heralded single-photon source,” Phys. Rev. A 78, 013844 (2008).
    [CrossRef]
  12. A. R. McMillan, J. Fulconis, M. Halder, C. Xiong, J. G. Rarity, and W. J. Wadsworth, “Narrowband high-fidelity all-fibre source of heralded single photons at 1570 nm,” Opt. Express 17, 6156–6165 (2009).
    [CrossRef] [PubMed]
  13. S. Fasel, O. Alibart, S. Tanzilli, P. Baldi, A. Beveratos, N. Gisin, and H. Zbinden, “High-quality asynchronous heralded single-photon source at telecom wavelength,” N. J. Phys. 6, 163 (2004).
    [CrossRef]
  14. O. Alibart, D. B. Ostrowsky, P. Baldi, and S. Tanzilli, “High-performance guided-wave asynchronous heralded single-photon source,” Opt. Lett. 30, 1539–1541 (2005).
    [CrossRef] [PubMed]
  15. A. G. Radnaev, Y. O. Dudin, R. Zhao, H. H. Jen, S. D. Jenkins, A. Kuzmich, and T. A. B. Kennedy, “A quantum memory with telecom-wavelength conversion,” Nat. Phys. 6, 894–899 (2010).
    [CrossRef]
  16. Y. O. Dudin, A. G. Radnaev, R. Zhao, J. Z. Blumoff, T. A. B. Kennedy, and A. Kuzmich, “Entanglement of light-shift compensated atomic spin waves with telecom light,” Phys. Rev. Lett. 105, 260502 (2010).
    [CrossRef]
  17. R. T. Willis, F. E. Becerra, L. A. Orozco, and S. L. Rolston, “Four-wave mixing in the diamond configuration in an atomic vapor,” Phys. Rev. A 79, 033814 (2009).
    [CrossRef]
  18. F. E. Becerra, R. T. Willis, S. L. Rolston, and L. A. Orozco, “Nondegenerate four-wave mixing in rubidium vapor: the diamond configuration,” Phys. Rev. A 78, 013834 (2008).
    [CrossRef]
  19. W. Z. Zhao, J. E. Simsarian, L. A. Orozco, and G. D. Sprouse, “A computer-based digital feedback control of frequency drift of multiple lasers,” Rev. Sci. Instrum. 69, 3737–3740 (1998).
    [CrossRef]
  20. R. T. Willis, F. E. Becerra, L. A. Orozco, and S. L. Rolston, “Correlated photon pairs generated from a warm atomic ensemble,” Phys. Rev. A 82, 053842 (2010).
    [CrossRef]
  21. J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-varible theories,” Phys. Rev. Lett. 23, 880–884 (1969).
    [CrossRef]
  22. H. J. Metcalf and P. Straten, Laser Cooling and Trapping (Springer, 1999).
    [CrossRef]
  23. Q. Zhou, W. Zhang, J.-R. Cheng, Y.-D. Huang, and J.-D. Peng, “Properties of optical fiber based synchronous heralded single photon sources at 1.5 μm,” Phys. Lett. A 375, 2274–2277 (2011).
    [CrossRef]
  24. A. F. Molisch and B. P. Oehry, Radiation Trapping in Atomic Vapours (Oxford University Press, 1998).
  25. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).
  26. M. D. Reid and D. F. Walls, “Violations of classical inequalities in quantum optics,” Phys. Rev. A 34, 1260–1276 (1986).
    [CrossRef] [PubMed]
  27. Q.-F. Chen, B.-S. Shi, M. Feng, Y.-S. Zhang, and G.-C. Guo, “Non-degenerate nonclassical photon pairs in a hot atomic ensemble,” Opt. Express 16, 21708–21713 (2008).
    [CrossRef] [PubMed]

2011

Q. Zhou, W. Zhang, J.-R. Cheng, Y.-D. Huang, and J.-D. Peng, “Properties of optical fiber based synchronous heralded single photon sources at 1.5 μm,” Phys. Lett. A 375, 2274–2277 (2011).
[CrossRef]

2010

K. F. Rim, J. Nunn, V. O. Lorenz, B. J. Sussman, K. C. Lee, N. K. Langford, D. Jaksch, and I. A. Walmsley, “Towards high-speed optical quantum memories,” Nat. Photonics 4, 218–221 (2010).
[CrossRef]

A. G. Radnaev, Y. O. Dudin, R. Zhao, H. H. Jen, S. D. Jenkins, A. Kuzmich, and T. A. B. Kennedy, “A quantum memory with telecom-wavelength conversion,” Nat. Phys. 6, 894–899 (2010).
[CrossRef]

Y. O. Dudin, A. G. Radnaev, R. Zhao, J. Z. Blumoff, T. A. B. Kennedy, and A. Kuzmich, “Entanglement of light-shift compensated atomic spin waves with telecom light,” Phys. Rev. Lett. 105, 260502 (2010).
[CrossRef]

R. T. Willis, F. E. Becerra, L. A. Orozco, and S. L. Rolston, “Correlated photon pairs generated from a warm atomic ensemble,” Phys. Rev. A 82, 053842 (2010).
[CrossRef]

2009

R. T. Willis, F. E. Becerra, L. A. Orozco, and S. L. Rolston, “Four-wave mixing in the diamond configuration in an atomic vapor,” Phys. Rev. A 79, 033814 (2009).
[CrossRef]

A. R. McMillan, J. Fulconis, M. Halder, C. Xiong, J. G. Rarity, and W. J. Wadsworth, “Narrowband high-fidelity all-fibre source of heralded single photons at 1570 nm,” Opt. Express 17, 6156–6165 (2009).
[CrossRef] [PubMed]

2008

Q.-F. Chen, B.-S. Shi, M. Feng, Y.-S. Zhang, and G.-C. Guo, “Non-degenerate nonclassical photon pairs in a hot atomic ensemble,” Opt. Express 16, 21708–21713 (2008).
[CrossRef] [PubMed]

F. E. Becerra, R. T. Willis, S. L. Rolston, and L. A. Orozco, “Nondegenerate four-wave mixing in rubidium vapor: the diamond configuration,” Phys. Rev. A 78, 013834 (2008).
[CrossRef]

E. A. Goldschmidt, M. D. Eisaman, J. Fan, S. V. Polyakov, and A. Migdall, “Spectrally bright and broad fiber-based heralded single-photon source,” Phys. Rev. A 78, 013844 (2008).
[CrossRef]

S. Du, P. Kolchin, C. Bethangady, G. Yin, and S. E. Harris, “Subnatural linewidth biphotons with controllable temporal length,” Phys. Rev. Lett. 100, 183603 (2008).
[CrossRef] [PubMed]

2006

T. Chanelière, D. N. Matsukevich, S. D. Jenkins, T. A. B. Kennedy, M. S. Chapman, and A. Kuzmich, “Quantum telecommunication based on atomic cascade transitions,” Phys. Rev. Lett. 96, 093604 (2006).
[CrossRef] [PubMed]

2005

2004

S. Fasel, O. Alibart, S. Tanzilli, P. Baldi, A. Beveratos, N. Gisin, and H. Zbinden, “High-quality asynchronous heralded single-photon source at telecom wavelength,” N. J. Phys. 6, 163 (2004).
[CrossRef]

2003

A. Kuzmich, W. P. Bowen, A. D. Booze, 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]

C. H. van der Wal, M. D. Eisaman, A. Andre, R. L. Walsworth, D. F. Phillips, A. S. Zibrov, and M. D. Lukin, “Atomic memory for correlated photon states,” Science 301, 196–200 (2003).
[CrossRef] [PubMed]

2002

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[CrossRef]

2001

L. Duan, M. Lukin, J. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414, 413–418 (2001).
[CrossRef] [PubMed]

1999

H. J. Metcalf and P. Straten, Laser Cooling and Trapping (Springer, 1999).
[CrossRef]

1998

A. F. Molisch and B. P. Oehry, Radiation Trapping in Atomic Vapours (Oxford University Press, 1998).

W. Z. Zhao, J. E. Simsarian, L. A. Orozco, and G. D. Sprouse, “A computer-based digital feedback control of frequency drift of multiple lasers,” Rev. Sci. Instrum. 69, 3737–3740 (1998).
[CrossRef]

1995

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

1991

A. K. Ekert, “Quantum cryptography based on bell’s theorem,” Phys. Rev. Lett. 67, 661–663 (1991).
[CrossRef] [PubMed]

1986

M. D. Reid and D. F. Walls, “Violations of classical inequalities in quantum optics,” Phys. Rev. A 34, 1260–1276 (1986).
[CrossRef] [PubMed]

1984

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, 1984), p. 175.
[PubMed]

1969

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

1836

V. Balic, D. A. Braje, P. Kolchin, G. Yin, and S. E. Haris, “Generation of paired photons with controllable waveforms,” Phys. Rev. Lett. 94, 183601 (2005).

Alibart, O.

O. Alibart, D. B. Ostrowsky, P. Baldi, and S. Tanzilli, “High-performance guided-wave asynchronous heralded single-photon source,” Opt. Lett. 30, 1539–1541 (2005).
[CrossRef] [PubMed]

S. Fasel, O. Alibart, S. Tanzilli, P. Baldi, A. Beveratos, N. Gisin, and H. Zbinden, “High-quality asynchronous heralded single-photon source at telecom wavelength,” N. J. Phys. 6, 163 (2004).
[CrossRef]

Andre, A.

C. H. van der Wal, M. D. Eisaman, A. Andre, R. L. Walsworth, D. F. Phillips, A. S. Zibrov, and M. D. Lukin, “Atomic memory for correlated photon states,” Science 301, 196–200 (2003).
[CrossRef] [PubMed]

Baldi, P.

O. Alibart, D. B. Ostrowsky, P. Baldi, and S. Tanzilli, “High-performance guided-wave asynchronous heralded single-photon source,” Opt. Lett. 30, 1539–1541 (2005).
[CrossRef] [PubMed]

S. Fasel, O. Alibart, S. Tanzilli, P. Baldi, A. Beveratos, N. Gisin, and H. Zbinden, “High-quality asynchronous heralded single-photon source at telecom wavelength,” N. J. Phys. 6, 163 (2004).
[CrossRef]

Balic, V.

V. Balic, D. A. Braje, P. Kolchin, G. Yin, and S. E. Haris, “Generation of paired photons with controllable waveforms,” Phys. Rev. Lett. 94, 183601 (2005).

Becerra, F. E.

R. T. Willis, F. E. Becerra, L. A. Orozco, and S. L. Rolston, “Correlated photon pairs generated from a warm atomic ensemble,” Phys. Rev. A 82, 053842 (2010).
[CrossRef]

R. T. Willis, F. E. Becerra, L. A. Orozco, and S. L. Rolston, “Four-wave mixing in the diamond configuration in an atomic vapor,” Phys. Rev. A 79, 033814 (2009).
[CrossRef]

F. E. Becerra, R. T. Willis, S. L. Rolston, and L. A. Orozco, “Nondegenerate four-wave mixing in rubidium vapor: the diamond configuration,” Phys. Rev. A 78, 013834 (2008).
[CrossRef]

Bennett, C. H.

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, 1984), p. 175.
[PubMed]

Bethangady, C.

S. Du, P. Kolchin, C. Bethangady, G. Yin, and S. E. Harris, “Subnatural linewidth biphotons with controllable temporal length,” Phys. Rev. Lett. 100, 183603 (2008).
[CrossRef] [PubMed]

Beveratos, A.

S. Fasel, O. Alibart, S. Tanzilli, P. Baldi, A. Beveratos, N. Gisin, and H. Zbinden, “High-quality asynchronous heralded single-photon source at telecom wavelength,” N. J. Phys. 6, 163 (2004).
[CrossRef]

Blumoff, J. Z.

Y. O. Dudin, A. G. Radnaev, R. Zhao, J. Z. Blumoff, T. A. B. Kennedy, and A. Kuzmich, “Entanglement of light-shift compensated atomic spin waves with telecom light,” Phys. Rev. Lett. 105, 260502 (2010).
[CrossRef]

Boca, A.

A. Kuzmich, W. P. Bowen, A. D. Booze, 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]

Booze, A. D.

A. Kuzmich, W. P. Bowen, A. D. Booze, 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]

Bowen, W. P.

A. Kuzmich, W. P. Bowen, A. D. Booze, 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]

Braje, D. A.

V. Balic, D. A. Braje, P. Kolchin, G. Yin, and S. E. Haris, “Generation of paired photons with controllable waveforms,” Phys. Rev. Lett. 94, 183601 (2005).

Brassard, G.

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, 1984), p. 175.
[PubMed]

Chanelière, T.

T. Chanelière, D. N. Matsukevich, S. D. Jenkins, T. A. B. Kennedy, M. S. Chapman, and A. Kuzmich, “Quantum telecommunication based on atomic cascade transitions,” Phys. Rev. Lett. 96, 093604 (2006).
[CrossRef] [PubMed]

Chapman, M. S.

T. Chanelière, D. N. Matsukevich, S. D. Jenkins, T. A. B. Kennedy, M. S. Chapman, and A. Kuzmich, “Quantum telecommunication based on atomic cascade transitions,” Phys. Rev. Lett. 96, 093604 (2006).
[CrossRef] [PubMed]

Chen, Q.-F.

Cheng, J.-R.

Q. Zhou, W. Zhang, J.-R. Cheng, Y.-D. Huang, and J.-D. Peng, “Properties of optical fiber based synchronous heralded single photon sources at 1.5 μm,” Phys. Lett. A 375, 2274–2277 (2011).
[CrossRef]

Chou, C. W.

A. Kuzmich, W. P. Bowen, A. D. Booze, 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]

Cirac, J.

L. Duan, M. Lukin, J. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414, 413–418 (2001).
[CrossRef] [PubMed]

Clauser, J. F.

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

Du, S.

S. Du, P. Kolchin, C. Bethangady, G. Yin, and S. E. Harris, “Subnatural linewidth biphotons with controllable temporal length,” Phys. Rev. Lett. 100, 183603 (2008).
[CrossRef] [PubMed]

Duan, L.

L. Duan, M. Lukin, J. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414, 413–418 (2001).
[CrossRef] [PubMed]

Duan, L.-M.

A. Kuzmich, W. P. Bowen, A. D. Booze, 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]

Dudin, Y. O.

Y. O. Dudin, A. G. Radnaev, R. Zhao, J. Z. Blumoff, T. A. B. Kennedy, and A. Kuzmich, “Entanglement of light-shift compensated atomic spin waves with telecom light,” Phys. Rev. Lett. 105, 260502 (2010).
[CrossRef]

A. G. Radnaev, Y. O. Dudin, R. Zhao, H. H. Jen, S. D. Jenkins, A. Kuzmich, and T. A. B. Kennedy, “A quantum memory with telecom-wavelength conversion,” Nat. Phys. 6, 894–899 (2010).
[CrossRef]

Eisaman, M. D.

E. A. Goldschmidt, M. D. Eisaman, J. Fan, S. V. Polyakov, and A. Migdall, “Spectrally bright and broad fiber-based heralded single-photon source,” Phys. Rev. A 78, 013844 (2008).
[CrossRef]

C. H. van der Wal, M. D. Eisaman, A. Andre, R. L. Walsworth, D. F. Phillips, A. S. Zibrov, and M. D. Lukin, “Atomic memory for correlated photon states,” Science 301, 196–200 (2003).
[CrossRef] [PubMed]

Ekert, A. K.

A. K. Ekert, “Quantum cryptography based on bell’s theorem,” Phys. Rev. Lett. 67, 661–663 (1991).
[CrossRef] [PubMed]

Fan, J.

E. A. Goldschmidt, M. D. Eisaman, J. Fan, S. V. Polyakov, and A. Migdall, “Spectrally bright and broad fiber-based heralded single-photon source,” Phys. Rev. A 78, 013844 (2008).
[CrossRef]

Fasel, S.

S. Fasel, O. Alibart, S. Tanzilli, P. Baldi, A. Beveratos, N. Gisin, and H. Zbinden, “High-quality asynchronous heralded single-photon source at telecom wavelength,” N. J. Phys. 6, 163 (2004).
[CrossRef]

Feng, M.

Fulconis, J.

Gisin, N.

S. Fasel, O. Alibart, S. Tanzilli, P. Baldi, A. Beveratos, N. Gisin, and H. Zbinden, “High-quality asynchronous heralded single-photon source at telecom wavelength,” N. J. Phys. 6, 163 (2004).
[CrossRef]

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[CrossRef]

Goldschmidt, E. A.

E. A. Goldschmidt, M. D. Eisaman, J. Fan, S. V. Polyakov, and A. Migdall, “Spectrally bright and broad fiber-based heralded single-photon source,” Phys. Rev. A 78, 013844 (2008).
[CrossRef]

Guo, G.-C.

Halder, M.

Haris, S. E.

V. Balic, D. A. Braje, P. Kolchin, G. Yin, and S. E. Haris, “Generation of paired photons with controllable waveforms,” Phys. Rev. Lett. 94, 183601 (2005).

Harris, S. E.

S. Du, P. Kolchin, C. Bethangady, G. Yin, and S. E. Harris, “Subnatural linewidth biphotons with controllable temporal length,” Phys. Rev. Lett. 100, 183603 (2008).
[CrossRef] [PubMed]

Holt, R. A.

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

Horne, M. A.

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

Huang, Y.-D.

Q. Zhou, W. Zhang, J.-R. Cheng, Y.-D. Huang, and J.-D. Peng, “Properties of optical fiber based synchronous heralded single photon sources at 1.5 μm,” Phys. Lett. A 375, 2274–2277 (2011).
[CrossRef]

Jaksch, D.

K. F. Rim, J. Nunn, V. O. Lorenz, B. J. Sussman, K. C. Lee, N. K. Langford, D. Jaksch, and I. A. Walmsley, “Towards high-speed optical quantum memories,” Nat. Photonics 4, 218–221 (2010).
[CrossRef]

Jen, H. H.

A. G. Radnaev, Y. O. Dudin, R. Zhao, H. H. Jen, S. D. Jenkins, A. Kuzmich, and T. A. B. Kennedy, “A quantum memory with telecom-wavelength conversion,” Nat. Phys. 6, 894–899 (2010).
[CrossRef]

Jenkins, S. D.

A. G. Radnaev, Y. O. Dudin, R. Zhao, H. H. Jen, S. D. Jenkins, A. Kuzmich, and T. A. B. Kennedy, “A quantum memory with telecom-wavelength conversion,” Nat. Phys. 6, 894–899 (2010).
[CrossRef]

T. Chanelière, D. N. Matsukevich, S. D. Jenkins, T. A. B. Kennedy, M. S. Chapman, and A. Kuzmich, “Quantum telecommunication based on atomic cascade transitions,” Phys. Rev. Lett. 96, 093604 (2006).
[CrossRef] [PubMed]

Kennedy, T. A. B.

A. G. Radnaev, Y. O. Dudin, R. Zhao, H. H. Jen, S. D. Jenkins, A. Kuzmich, and T. A. B. Kennedy, “A quantum memory with telecom-wavelength conversion,” Nat. Phys. 6, 894–899 (2010).
[CrossRef]

Y. O. Dudin, A. G. Radnaev, R. Zhao, J. Z. Blumoff, T. A. B. Kennedy, and A. Kuzmich, “Entanglement of light-shift compensated atomic spin waves with telecom light,” Phys. Rev. Lett. 105, 260502 (2010).
[CrossRef]

T. Chanelière, D. N. Matsukevich, S. D. Jenkins, T. A. B. Kennedy, M. S. Chapman, and A. Kuzmich, “Quantum telecommunication based on atomic cascade transitions,” Phys. Rev. Lett. 96, 093604 (2006).
[CrossRef] [PubMed]

Kimble, H. J.

A. Kuzmich, W. P. Bowen, A. D. Booze, 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]

Kolchin, P.

S. Du, P. Kolchin, C. Bethangady, G. Yin, and S. E. Harris, “Subnatural linewidth biphotons with controllable temporal length,” Phys. Rev. Lett. 100, 183603 (2008).
[CrossRef] [PubMed]

V. Balic, D. A. Braje, P. Kolchin, G. Yin, and S. E. Haris, “Generation of paired photons with controllable waveforms,” Phys. Rev. Lett. 94, 183601 (2005).

Kuzmich, A.

Y. O. Dudin, A. G. Radnaev, R. Zhao, J. Z. Blumoff, T. A. B. Kennedy, and A. Kuzmich, “Entanglement of light-shift compensated atomic spin waves with telecom light,” Phys. Rev. Lett. 105, 260502 (2010).
[CrossRef]

A. G. Radnaev, Y. O. Dudin, R. Zhao, H. H. Jen, S. D. Jenkins, A. Kuzmich, and T. A. B. Kennedy, “A quantum memory with telecom-wavelength conversion,” Nat. Phys. 6, 894–899 (2010).
[CrossRef]

T. Chanelière, D. N. Matsukevich, S. D. Jenkins, T. A. B. Kennedy, M. S. Chapman, and A. Kuzmich, “Quantum telecommunication based on atomic cascade transitions,” Phys. Rev. Lett. 96, 093604 (2006).
[CrossRef] [PubMed]

A. Kuzmich, W. P. Bowen, A. D. Booze, 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]

Langford, N. K.

K. F. Rim, J. Nunn, V. O. Lorenz, B. J. Sussman, K. C. Lee, N. K. Langford, D. Jaksch, and I. A. Walmsley, “Towards high-speed optical quantum memories,” Nat. Photonics 4, 218–221 (2010).
[CrossRef]

Lee, K. C.

K. F. Rim, J. Nunn, V. O. Lorenz, B. J. Sussman, K. C. Lee, N. K. Langford, D. Jaksch, and I. A. Walmsley, “Towards high-speed optical quantum memories,” Nat. Photonics 4, 218–221 (2010).
[CrossRef]

Lorenz, V. O.

K. F. Rim, J. Nunn, V. O. Lorenz, B. J. Sussman, K. C. Lee, N. K. Langford, D. Jaksch, and I. A. Walmsley, “Towards high-speed optical quantum memories,” Nat. Photonics 4, 218–221 (2010).
[CrossRef]

Lukin, M.

L. Duan, M. Lukin, J. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414, 413–418 (2001).
[CrossRef] [PubMed]

Lukin, M. D.

C. H. van der Wal, M. D. Eisaman, A. Andre, R. L. Walsworth, D. F. Phillips, A. S. Zibrov, and M. D. Lukin, “Atomic memory for correlated photon states,” Science 301, 196–200 (2003).
[CrossRef] [PubMed]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

Matsukevich, D. N.

T. Chanelière, D. N. Matsukevich, S. D. Jenkins, T. A. B. Kennedy, M. S. Chapman, and A. Kuzmich, “Quantum telecommunication based on atomic cascade transitions,” Phys. Rev. Lett. 96, 093604 (2006).
[CrossRef] [PubMed]

McMillan, A. R.

Metcalf, H. J.

H. J. Metcalf and P. Straten, Laser Cooling and Trapping (Springer, 1999).
[CrossRef]

Migdall, A.

E. A. Goldschmidt, M. D. Eisaman, J. Fan, S. V. Polyakov, and A. Migdall, “Spectrally bright and broad fiber-based heralded single-photon source,” Phys. Rev. A 78, 013844 (2008).
[CrossRef]

Molisch, A. F.

A. F. Molisch and B. P. Oehry, Radiation Trapping in Atomic Vapours (Oxford University Press, 1998).

Nunn, J.

K. F. Rim, J. Nunn, V. O. Lorenz, B. J. Sussman, K. C. Lee, N. K. Langford, D. Jaksch, and I. A. Walmsley, “Towards high-speed optical quantum memories,” Nat. Photonics 4, 218–221 (2010).
[CrossRef]

Oehry, B. P.

A. F. Molisch and B. P. Oehry, Radiation Trapping in Atomic Vapours (Oxford University Press, 1998).

Orozco, L. A.

R. T. Willis, F. E. Becerra, L. A. Orozco, and S. L. Rolston, “Correlated photon pairs generated from a warm atomic ensemble,” Phys. Rev. A 82, 053842 (2010).
[CrossRef]

R. T. Willis, F. E. Becerra, L. A. Orozco, and S. L. Rolston, “Four-wave mixing in the diamond configuration in an atomic vapor,” Phys. Rev. A 79, 033814 (2009).
[CrossRef]

F. E. Becerra, R. T. Willis, S. L. Rolston, and L. A. Orozco, “Nondegenerate four-wave mixing in rubidium vapor: the diamond configuration,” Phys. Rev. A 78, 013834 (2008).
[CrossRef]

W. Z. Zhao, J. E. Simsarian, L. A. Orozco, and G. D. Sprouse, “A computer-based digital feedback control of frequency drift of multiple lasers,” Rev. Sci. Instrum. 69, 3737–3740 (1998).
[CrossRef]

Ostrowsky, D. B.

Peng, J.-D.

Q. Zhou, W. Zhang, J.-R. Cheng, Y.-D. Huang, and J.-D. Peng, “Properties of optical fiber based synchronous heralded single photon sources at 1.5 μm,” Phys. Lett. A 375, 2274–2277 (2011).
[CrossRef]

Phillips, D. F.

C. H. van der Wal, M. D. Eisaman, A. Andre, R. L. Walsworth, D. F. Phillips, A. S. Zibrov, and M. D. Lukin, “Atomic memory for correlated photon states,” Science 301, 196–200 (2003).
[CrossRef] [PubMed]

Polyakov, S. V.

E. A. Goldschmidt, M. D. Eisaman, J. Fan, S. V. Polyakov, and A. Migdall, “Spectrally bright and broad fiber-based heralded single-photon source,” Phys. Rev. A 78, 013844 (2008).
[CrossRef]

Radnaev, A. G.

A. G. Radnaev, Y. O. Dudin, R. Zhao, H. H. Jen, S. D. Jenkins, A. Kuzmich, and T. A. B. Kennedy, “A quantum memory with telecom-wavelength conversion,” Nat. Phys. 6, 894–899 (2010).
[CrossRef]

Y. O. Dudin, A. G. Radnaev, R. Zhao, J. Z. Blumoff, T. A. B. Kennedy, and A. Kuzmich, “Entanglement of light-shift compensated atomic spin waves with telecom light,” Phys. Rev. Lett. 105, 260502 (2010).
[CrossRef]

Rarity, J. G.

Reid, M. D.

M. D. Reid and D. F. Walls, “Violations of classical inequalities in quantum optics,” Phys. Rev. A 34, 1260–1276 (1986).
[CrossRef] [PubMed]

Ribordy, G.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[CrossRef]

Rim, K. F.

K. F. Rim, J. Nunn, V. O. Lorenz, B. J. Sussman, K. C. Lee, N. K. Langford, D. Jaksch, and I. A. Walmsley, “Towards high-speed optical quantum memories,” Nat. Photonics 4, 218–221 (2010).
[CrossRef]

Rolston, S. L.

R. T. Willis, F. E. Becerra, L. A. Orozco, and S. L. Rolston, “Correlated photon pairs generated from a warm atomic ensemble,” Phys. Rev. A 82, 053842 (2010).
[CrossRef]

R. T. Willis, F. E. Becerra, L. A. Orozco, and S. L. Rolston, “Four-wave mixing in the diamond configuration in an atomic vapor,” Phys. Rev. A 79, 033814 (2009).
[CrossRef]

F. E. Becerra, R. T. Willis, S. L. Rolston, and L. A. Orozco, “Nondegenerate four-wave mixing in rubidium vapor: the diamond configuration,” Phys. Rev. A 78, 013834 (2008).
[CrossRef]

Shi, B.-S.

Shimony, A.

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

Simsarian, J. E.

W. Z. Zhao, J. E. Simsarian, L. A. Orozco, and G. D. Sprouse, “A computer-based digital feedback control of frequency drift of multiple lasers,” Rev. Sci. Instrum. 69, 3737–3740 (1998).
[CrossRef]

Sprouse, G. D.

W. Z. Zhao, J. E. Simsarian, L. A. Orozco, and G. D. Sprouse, “A computer-based digital feedback control of frequency drift of multiple lasers,” Rev. Sci. Instrum. 69, 3737–3740 (1998).
[CrossRef]

Straten, P.

H. J. Metcalf and P. Straten, Laser Cooling and Trapping (Springer, 1999).
[CrossRef]

Sussman, B. J.

K. F. Rim, J. Nunn, V. O. Lorenz, B. J. Sussman, K. C. Lee, N. K. Langford, D. Jaksch, and I. A. Walmsley, “Towards high-speed optical quantum memories,” Nat. Photonics 4, 218–221 (2010).
[CrossRef]

Tanzilli, S.

O. Alibart, D. B. Ostrowsky, P. Baldi, and S. Tanzilli, “High-performance guided-wave asynchronous heralded single-photon source,” Opt. Lett. 30, 1539–1541 (2005).
[CrossRef] [PubMed]

S. Fasel, O. Alibart, S. Tanzilli, P. Baldi, A. Beveratos, N. Gisin, and H. Zbinden, “High-quality asynchronous heralded single-photon source at telecom wavelength,” N. J. Phys. 6, 163 (2004).
[CrossRef]

Tittel, W.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[CrossRef]

van der Wal, C. H.

C. H. van der Wal, M. D. Eisaman, A. Andre, R. L. Walsworth, D. F. Phillips, A. S. Zibrov, and M. D. Lukin, “Atomic memory for correlated photon states,” Science 301, 196–200 (2003).
[CrossRef] [PubMed]

Wadsworth, W. J.

Walls, D. F.

M. D. Reid and D. F. Walls, “Violations of classical inequalities in quantum optics,” Phys. Rev. A 34, 1260–1276 (1986).
[CrossRef] [PubMed]

Walmsley, I. A.

K. F. Rim, J. Nunn, V. O. Lorenz, B. J. Sussman, K. C. Lee, N. K. Langford, D. Jaksch, and I. A. Walmsley, “Towards high-speed optical quantum memories,” Nat. Photonics 4, 218–221 (2010).
[CrossRef]

Walsworth, R. L.

C. H. van der Wal, M. D. Eisaman, A. Andre, R. L. Walsworth, D. F. Phillips, A. S. Zibrov, and M. D. Lukin, “Atomic memory for correlated photon states,” Science 301, 196–200 (2003).
[CrossRef] [PubMed]

Willis, R. T.

R. T. Willis, F. E. Becerra, L. A. Orozco, and S. L. Rolston, “Correlated photon pairs generated from a warm atomic ensemble,” Phys. Rev. A 82, 053842 (2010).
[CrossRef]

R. T. Willis, F. E. Becerra, L. A. Orozco, and S. L. Rolston, “Four-wave mixing in the diamond configuration in an atomic vapor,” Phys. Rev. A 79, 033814 (2009).
[CrossRef]

F. E. Becerra, R. T. Willis, S. L. Rolston, and L. A. Orozco, “Nondegenerate four-wave mixing in rubidium vapor: the diamond configuration,” Phys. Rev. A 78, 013834 (2008).
[CrossRef]

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

Xiong, C.

Yin, G.

S. Du, P. Kolchin, C. Bethangady, G. Yin, and S. E. Harris, “Subnatural linewidth biphotons with controllable temporal length,” Phys. Rev. Lett. 100, 183603 (2008).
[CrossRef] [PubMed]

V. Balic, D. A. Braje, P. Kolchin, G. Yin, and S. E. Haris, “Generation of paired photons with controllable waveforms,” Phys. Rev. Lett. 94, 183601 (2005).

Zbinden, H.

S. Fasel, O. Alibart, S. Tanzilli, P. Baldi, A. Beveratos, N. Gisin, and H. Zbinden, “High-quality asynchronous heralded single-photon source at telecom wavelength,” N. J. Phys. 6, 163 (2004).
[CrossRef]

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[CrossRef]

Zhang, W.

Q. Zhou, W. Zhang, J.-R. Cheng, Y.-D. Huang, and J.-D. Peng, “Properties of optical fiber based synchronous heralded single photon sources at 1.5 μm,” Phys. Lett. A 375, 2274–2277 (2011).
[CrossRef]

Zhang, Y.-S.

Zhao, R.

Y. O. Dudin, A. G. Radnaev, R. Zhao, J. Z. Blumoff, T. A. B. Kennedy, and A. Kuzmich, “Entanglement of light-shift compensated atomic spin waves with telecom light,” Phys. Rev. Lett. 105, 260502 (2010).
[CrossRef]

A. G. Radnaev, Y. O. Dudin, R. Zhao, H. H. Jen, S. D. Jenkins, A. Kuzmich, and T. A. B. Kennedy, “A quantum memory with telecom-wavelength conversion,” Nat. Phys. 6, 894–899 (2010).
[CrossRef]

Zhao, W. Z.

W. Z. Zhao, J. E. Simsarian, L. A. Orozco, and G. D. Sprouse, “A computer-based digital feedback control of frequency drift of multiple lasers,” Rev. Sci. Instrum. 69, 3737–3740 (1998).
[CrossRef]

Zhou, Q.

Q. Zhou, W. Zhang, J.-R. Cheng, Y.-D. Huang, and J.-D. Peng, “Properties of optical fiber based synchronous heralded single photon sources at 1.5 μm,” Phys. Lett. A 375, 2274–2277 (2011).
[CrossRef]

Zibrov, A. S.

C. H. van der Wal, M. D. Eisaman, A. Andre, R. L. Walsworth, D. F. Phillips, A. S. Zibrov, and M. D. Lukin, “Atomic memory for correlated photon states,” Science 301, 196–200 (2003).
[CrossRef] [PubMed]

Zoller, P.

L. Duan, M. Lukin, J. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414, 413–418 (2001).
[CrossRef] [PubMed]

N. J. Phys.

S. Fasel, O. Alibart, S. Tanzilli, P. Baldi, A. Beveratos, N. Gisin, and H. Zbinden, “High-quality asynchronous heralded single-photon source at telecom wavelength,” N. J. Phys. 6, 163 (2004).
[CrossRef]

Nat. Photonics

K. F. Rim, J. Nunn, V. O. Lorenz, B. J. Sussman, K. C. Lee, N. K. Langford, D. Jaksch, and I. A. Walmsley, “Towards high-speed optical quantum memories,” Nat. Photonics 4, 218–221 (2010).
[CrossRef]

Nat. Phys.

A. G. Radnaev, Y. O. Dudin, R. Zhao, H. H. Jen, S. D. Jenkins, A. Kuzmich, and T. A. B. Kennedy, “A quantum memory with telecom-wavelength conversion,” Nat. Phys. 6, 894–899 (2010).
[CrossRef]

Nature

L. Duan, M. Lukin, J. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414, 413–418 (2001).
[CrossRef] [PubMed]

A. Kuzmich, W. P. Bowen, A. D. Booze, 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]

Opt. Express

Opt. Lett.

Phys. Lett. A

Q. Zhou, W. Zhang, J.-R. Cheng, Y.-D. Huang, and J.-D. Peng, “Properties of optical fiber based synchronous heralded single photon sources at 1.5 μm,” Phys. Lett. A 375, 2274–2277 (2011).
[CrossRef]

Phys. Rev. A

R. T. Willis, F. E. Becerra, L. A. Orozco, and S. L. Rolston, “Correlated photon pairs generated from a warm atomic ensemble,” Phys. Rev. A 82, 053842 (2010).
[CrossRef]

R. T. Willis, F. E. Becerra, L. A. Orozco, and S. L. Rolston, “Four-wave mixing in the diamond configuration in an atomic vapor,” Phys. Rev. A 79, 033814 (2009).
[CrossRef]

F. E. Becerra, R. T. Willis, S. L. Rolston, and L. A. Orozco, “Nondegenerate four-wave mixing in rubidium vapor: the diamond configuration,” Phys. Rev. A 78, 013834 (2008).
[CrossRef]

E. A. Goldschmidt, M. D. Eisaman, J. Fan, S. V. Polyakov, and A. Migdall, “Spectrally bright and broad fiber-based heralded single-photon source,” Phys. Rev. A 78, 013844 (2008).
[CrossRef]

M. D. Reid and D. F. Walls, “Violations of classical inequalities in quantum optics,” Phys. Rev. A 34, 1260–1276 (1986).
[CrossRef] [PubMed]

Phys. Rev. Lett.

Y. O. Dudin, A. G. Radnaev, R. Zhao, J. Z. Blumoff, T. A. B. Kennedy, and A. Kuzmich, “Entanglement of light-shift compensated atomic spin waves with telecom light,” Phys. Rev. Lett. 105, 260502 (2010).
[CrossRef]

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

A. K. Ekert, “Quantum cryptography based on bell’s theorem,” Phys. Rev. Lett. 67, 661–663 (1991).
[CrossRef] [PubMed]

V. Balic, D. A. Braje, P. Kolchin, G. Yin, and S. E. Haris, “Generation of paired photons with controllable waveforms,” Phys. Rev. Lett. 94, 183601 (2005).

S. Du, P. Kolchin, C. Bethangady, G. Yin, and S. E. Harris, “Subnatural linewidth biphotons with controllable temporal length,” Phys. Rev. Lett. 100, 183603 (2008).
[CrossRef] [PubMed]

T. Chanelière, D. N. Matsukevich, S. D. Jenkins, T. A. B. Kennedy, M. S. Chapman, and A. Kuzmich, “Quantum telecommunication based on atomic cascade transitions,” Phys. Rev. Lett. 96, 093604 (2006).
[CrossRef] [PubMed]

Rev. Mod. Phys.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[CrossRef]

Rev. Sci. Instrum.

W. Z. Zhao, J. E. Simsarian, L. A. Orozco, and G. D. Sprouse, “A computer-based digital feedback control of frequency drift of multiple lasers,” Rev. Sci. Instrum. 69, 3737–3740 (1998).
[CrossRef]

Science

C. H. van der Wal, M. D. Eisaman, A. Andre, R. L. Walsworth, D. F. Phillips, A. S. Zibrov, and M. D. Lukin, “Atomic memory for correlated photon states,” Science 301, 196–200 (2003).
[CrossRef] [PubMed]

Other

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, 1984), p. 175.
[PubMed]

H. J. Metcalf and P. Straten, Laser Cooling and Trapping (Springer, 1999).
[CrossRef]

A. F. Molisch and B. P. Oehry, Radiation Trapping in Atomic Vapours (Oxford University Press, 1998).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

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Figures (7)

Fig. 1
Fig. 1

A) The geometry of the experiment: the angles are θ 1 = 2°, θ 2 = 0.7°, and θ 3 = 2.7°. An auxiliary 85Rb filter cell can intersect the 780 nm beam as indicated. B) The diamond configuration in 85Rb. C) Schematic of detection set up. For polarization correlation measurements the beam splitters (BS) are replaced with polarizing beamsplitters (PBS) and half-wave plates are inserted into each beam path.

Fig. 2
Fig. 2

(Color online) Polarization correlations: Coincidence counts (filled and empty circles) in 100 s as a function of 780 nm polarizer angle for two angles of the 1367 nm polarizer when the pump 795 nm and 1324 nm have parallel polarizations and using the filter cell at 323 K. The experimental data (circles) fit to the function cos2780 – Θ1367) (solid lines). The filter cell reduces the background coincidences and increase the visibility to 92(2)%.

Fig. 3
Fig. 3

A) Geometry used in the calculation. The pumps are polarized along the z-axis and all fields propagate along the x-axis. B) Simplified atomic level structure along with the coupling fields. C) Full level atomic structure which is taken into account in the calculation. The dashed lines show an example of a path out of the mF = 0 ground state, along with the Clebsh Gordan weights, that contributes to the 4WM and returns to the same level.

Fig. 4
Fig. 4

Autocorrelation function for the 1367 nm light field conditioned on the arrival of a 780 nm photon.

Fig. 5
Fig. 5

The normalized autocorrelation function of the 780-nm light field at two different temperatures.

Fig. 6
Fig. 6

Normalized autocorrelation function of the 1367 nm light field.

Fig. 7
Fig. 7

The unnormalized cross-correlation function for the 780 nm and the 1367 nm fields showing the time correlation between them.

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

V I = h ¯ i , j Ω 1 i j σ ^ b a i j e i Δ 1 i j t + h ¯ i , j Ω 2 i j σ ^ c b i j e i Δ 2 i j t + H . C . + k i , j ( λ ) g k i j a ^ k ( λ ) σ ^ c d i j e i Δ k i j t + q i , j ( λ ) g q i j b ^ q ( λ ) σ ^ d a i j e i Δ q i j t + H . C .
| ψ 4 W M k , q λ , γ j , k , l Ω 1 j α Ω 2 k , j λ g k * l k γ g q * α l 𝒲 α j k l ( t ) a ^ k ( λ ) b ^ q ( γ ) | 0 0 a .
| ψ α = cos ( χ α ) | H H + sin ( χ α ) | V V ,
g ˜ ( 2 ) ( 0 ) = P ˜ I 1 , I 2 ( 0 ) P ˜ I 1 P ˜ I 2 .
R = [ g cross ( 2 ) ( τ ) ] 2 g S ( 2 ) ( 0 ) g I ( 2 ) ( 0 ) < 1.

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