Abstract

We propose a method to simulate a two-photon interference fringe by using a density matrix in order to describe various types of imperfections in the input state. Using this method, we numerically discuss the influence of various imperfections in an input state, such as dephasing and misalignment, on the quality (visibility and period) of the two-photon interference fringes. Applying this method to experimental data, we succeeded in numerically reproducing a two-photon interference fringe using the experimentally obtained density matrix, in which almost no free fitting parameters are required. From the results, because the main cause of the degradation of an interference fringe was found to be the limited aperture size of a two-photon detector, we can observe a two-photon interference fringe with a visibility of up to 94% in the experiments if an efficient two-photon absorbing material or a two-photon detector with a sufficiently high spatial resolution can be used.

© 2011 Optical Society of America

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  1. A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733–2736 (2000).
    [Crossref] [PubMed]
  2. P. Kok, A. N. Boto, D. S. Abrams, C. P. Williams, S. L. Braunstein, and J. P. Dowling, “Quantum-interferometric optical lithography: towards arbitrary two-dimensional patterns,” Phys. Rev. A 63, 063407 (2001).
    [Crossref]
  3. G. Bjork and L. L. Sanchez-Soto, “Entangled-state lithography: tailoring any pattern with a single state,” Phys. Rev. Lett. 86, 4516–4519 (2001).
    [Crossref] [PubMed]
  4. M. D’Angelo, M. V. Chekhova, and Y. Shih, “Two-photon diffraction and quantum lithography,” Phys. Rev. Lett. 87, 013602 (2001).
    [Crossref] [PubMed]
  5. T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four entangled photons,” Science 316, 726–729 (2007).
    [Crossref] [PubMed]
  6. K. Edamatsu, R. Shimizu, and T. Itoh, “Measurement of the photonic de Broglie wavelength of entangled photon pairs generated by spontaneous parametric down-conversion,” Phys. Rev. Lett. 89, 213601 (2002).
    [Crossref] [PubMed]
  7. M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature 429, 161–164 (2004).
    [Crossref] [PubMed]
  8. P. Walther, J.-W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, “De Broglie wavelength of a non-local four-photon state,” Nature 429, 158–161 (2004).
    [Crossref] [PubMed]
  9. F. W. Sun, B. H. Liu, Y. F. Huang, Z. Y. Ou, and G. C. Guo, “Observation of the four-photon de Broglie wavelength by state-projection measurement,” Phys. Rev. A 74, 033812 (2006).
    [Crossref]
  10. B. H. Liu, F. W. Sun, Y. X. Gong, Y. F. Huang, G. C. Guo, and Z. Y. Ou, “Four-photon interference with asymmetric beam splitters,” Opt. Lett. 32, 1320–1322 (2007).
    [Crossref] [PubMed]
  11. K. J. Resch, K. L. Pregnell, R. Prevedel, A. Gilchrist, G. J. Pryde, J. L. O’Brien, and A. G. White, “Time-reversal and super-resolving phase measurements,” Phys. Rev. Lett. 98, 223601(2007).
    [Crossref] [PubMed]
  12. J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, “Two-photon interference in a Mach–Zehnder interferometer,” Phys. Rev. Lett. 65, 1348–1351 (1990).
    [Crossref] [PubMed]
  13. E. J. S. Fonseca, C. H. Monken, and S. Pauda, “Measurement of the de Broglie wavelength of a multiphoton wave packet,” Phys. Rev. Lett. 82, 2868–2871 (1999).
    [Crossref]
  14. H. S. Eisenberg, J. F. Hodelin, G. Khoury, and D. Bouwmeester, “Multiphoton path entanglement by nonlocal bunching,” Phys. Rev. Lett. 94, 090502 (2005).
    [Crossref] [PubMed]
  15. Y. Kawabe, H. Fujiwara, R. Okamoto, K. Sasaki, and S. Takeuchi, “Quantum interference fringes beating the diffraction limit,” Opt. Exp. 15, 14244–14250 (2007).
    [Crossref]
  16. Y. H. Kim, S. P. Kulik, and Y. Shih, “High-intensity pulsed source of space-time and polarization double-entangled photon pairs,” Phys. Rev. A 62, 011802 (2000).
    [Crossref]
  17. C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
    [Crossref] [PubMed]
  18. C. C. Gerry and R. A. Campos, “Generation of maximally entangled photonic states with a quantum-optical Fredkin gate,” Phys. Rev. A 64, 063814 (2001).
    [Crossref]
  19. J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate,” Nature 426, 264–267 (2003).
    [Crossref] [PubMed]
  20. E. M. Nagasako, S. J. Bentley, R. W. Boyd, and G. S. Agarwal, “Nonclassical two-photon interferometry and lithography with high-gain parametric amplifiers,” Phys. Rev. A 64, 043802(2001).
    [Crossref]
  21. Y.-H. Wang and L.-M. Kuang, “Nonmaximally entangled state quantum photolithography,” J. Opt. B: Quantum Semiclass. Opt. 5, 405–408 (2003).
    [Crossref]
  22. R. F. Werner, “Quantum states with Einstein–Podolsky–Rosen correlations admitting a hidden-variable model,” Phys. Rev. A 40, 4277–4281 (1989).
    [Crossref] [PubMed]
  23. E. Betzig, J. K. Trautman, T. D. Harris, J. S. Weiner, and R. L. Kostelak, “Breaking the diffraction barrier: optical microscopy of a nanometric scale,” Science 251, 1468–1470 (1991).
    [Crossref] [PubMed]
  24. D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
    [Crossref]
  25. M. B. Nasr, A. F. Abouraddy, M. C. Booth, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, M. Kempe, and R. Wolleschensky, “Biphoton focusing for two-photon excitation,” Phys. Rev. A 65, 023816 (2002).
    [Crossref]

2007 (4)

T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four entangled photons,” Science 316, 726–729 (2007).
[Crossref] [PubMed]

B. H. Liu, F. W. Sun, Y. X. Gong, Y. F. Huang, G. C. Guo, and Z. Y. Ou, “Four-photon interference with asymmetric beam splitters,” Opt. Lett. 32, 1320–1322 (2007).
[Crossref] [PubMed]

K. J. Resch, K. L. Pregnell, R. Prevedel, A. Gilchrist, G. J. Pryde, J. L. O’Brien, and A. G. White, “Time-reversal and super-resolving phase measurements,” Phys. Rev. Lett. 98, 223601(2007).
[Crossref] [PubMed]

Y. Kawabe, H. Fujiwara, R. Okamoto, K. Sasaki, and S. Takeuchi, “Quantum interference fringes beating the diffraction limit,” Opt. Exp. 15, 14244–14250 (2007).
[Crossref]

2006 (1)

F. W. Sun, B. H. Liu, Y. F. Huang, Z. Y. Ou, and G. C. Guo, “Observation of the four-photon de Broglie wavelength by state-projection measurement,” Phys. Rev. A 74, 033812 (2006).
[Crossref]

2005 (1)

H. S. Eisenberg, J. F. Hodelin, G. Khoury, and D. Bouwmeester, “Multiphoton path entanglement by nonlocal bunching,” Phys. Rev. Lett. 94, 090502 (2005).
[Crossref] [PubMed]

2004 (2)

M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature 429, 161–164 (2004).
[Crossref] [PubMed]

P. Walther, J.-W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, “De Broglie wavelength of a non-local four-photon state,” Nature 429, 158–161 (2004).
[Crossref] [PubMed]

2003 (2)

J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate,” Nature 426, 264–267 (2003).
[Crossref] [PubMed]

Y.-H. Wang and L.-M. Kuang, “Nonmaximally entangled state quantum photolithography,” J. Opt. B: Quantum Semiclass. Opt. 5, 405–408 (2003).
[Crossref]

2002 (2)

M. B. Nasr, A. F. Abouraddy, M. C. Booth, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, M. Kempe, and R. Wolleschensky, “Biphoton focusing for two-photon excitation,” Phys. Rev. A 65, 023816 (2002).
[Crossref]

K. Edamatsu, R. Shimizu, and T. Itoh, “Measurement of the photonic de Broglie wavelength of entangled photon pairs generated by spontaneous parametric down-conversion,” Phys. Rev. Lett. 89, 213601 (2002).
[Crossref] [PubMed]

2001 (6)

P. Kok, A. N. Boto, D. S. Abrams, C. P. Williams, S. L. Braunstein, and J. P. Dowling, “Quantum-interferometric optical lithography: towards arbitrary two-dimensional patterns,” Phys. Rev. A 63, 063407 (2001).
[Crossref]

G. Bjork and L. L. Sanchez-Soto, “Entangled-state lithography: tailoring any pattern with a single state,” Phys. Rev. Lett. 86, 4516–4519 (2001).
[Crossref] [PubMed]

M. D’Angelo, M. V. Chekhova, and Y. Shih, “Two-photon diffraction and quantum lithography,” Phys. Rev. Lett. 87, 013602 (2001).
[Crossref] [PubMed]

E. M. Nagasako, S. J. Bentley, R. W. Boyd, and G. S. Agarwal, “Nonclassical two-photon interferometry and lithography with high-gain parametric amplifiers,” Phys. Rev. A 64, 043802(2001).
[Crossref]

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]

C. C. Gerry and R. A. Campos, “Generation of maximally entangled photonic states with a quantum-optical Fredkin gate,” Phys. Rev. A 64, 063814 (2001).
[Crossref]

2000 (2)

Y. H. Kim, S. P. Kulik, and Y. Shih, “High-intensity pulsed source of space-time and polarization double-entangled photon pairs,” Phys. Rev. A 62, 011802 (2000).
[Crossref]

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733–2736 (2000).
[Crossref] [PubMed]

1999 (1)

E. J. S. Fonseca, C. H. Monken, and S. Pauda, “Measurement of the de Broglie wavelength of a multiphoton wave packet,” Phys. Rev. Lett. 82, 2868–2871 (1999).
[Crossref]

1991 (1)

E. Betzig, J. K. Trautman, T. D. Harris, J. S. Weiner, and R. L. Kostelak, “Breaking the diffraction barrier: optical microscopy of a nanometric scale,” Science 251, 1468–1470 (1991).
[Crossref] [PubMed]

1990 (1)

J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, “Two-photon interference in a Mach–Zehnder interferometer,” Phys. Rev. Lett. 65, 1348–1351 (1990).
[Crossref] [PubMed]

1989 (1)

R. F. Werner, “Quantum states with Einstein–Podolsky–Rosen correlations admitting a hidden-variable model,” Phys. Rev. A 40, 4277–4281 (1989).
[Crossref] [PubMed]

1987 (1)

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
[Crossref] [PubMed]

Abouraddy, A. F.

M. B. Nasr, A. F. Abouraddy, M. C. Booth, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, M. Kempe, and R. Wolleschensky, “Biphoton focusing for two-photon excitation,” Phys. Rev. A 65, 023816 (2002).
[Crossref]

Abrams, D. S.

P. Kok, A. N. Boto, D. S. Abrams, C. P. Williams, S. L. Braunstein, and J. P. Dowling, “Quantum-interferometric optical lithography: towards arbitrary two-dimensional patterns,” Phys. Rev. A 63, 063407 (2001).
[Crossref]

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733–2736 (2000).
[Crossref] [PubMed]

Agarwal, G. S.

E. M. Nagasako, S. J. Bentley, R. W. Boyd, and G. S. Agarwal, “Nonclassical two-photon interferometry and lithography with high-gain parametric amplifiers,” Phys. Rev. A 64, 043802(2001).
[Crossref]

Aspelmeyer, M.

P. Walther, J.-W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, “De Broglie wavelength of a non-local four-photon state,” Nature 429, 158–161 (2004).
[Crossref] [PubMed]

Bentley, S. J.

E. M. Nagasako, S. J. Bentley, R. W. Boyd, and G. S. Agarwal, “Nonclassical two-photon interferometry and lithography with high-gain parametric amplifiers,” Phys. Rev. A 64, 043802(2001).
[Crossref]

Betzig, E.

E. Betzig, J. K. Trautman, T. D. Harris, J. S. Weiner, and R. L. Kostelak, “Breaking the diffraction barrier: optical microscopy of a nanometric scale,” Science 251, 1468–1470 (1991).
[Crossref] [PubMed]

Bjork, G.

G. Bjork and L. L. Sanchez-Soto, “Entangled-state lithography: tailoring any pattern with a single state,” Phys. Rev. Lett. 86, 4516–4519 (2001).
[Crossref] [PubMed]

Booth, M. C.

M. B. Nasr, A. F. Abouraddy, M. C. Booth, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, M. Kempe, and R. Wolleschensky, “Biphoton focusing for two-photon excitation,” Phys. Rev. A 65, 023816 (2002).
[Crossref]

Boto, A. N.

P. Kok, A. N. Boto, D. S. Abrams, C. P. Williams, S. L. Braunstein, and J. P. Dowling, “Quantum-interferometric optical lithography: towards arbitrary two-dimensional patterns,” Phys. Rev. A 63, 063407 (2001).
[Crossref]

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733–2736 (2000).
[Crossref] [PubMed]

Bouwmeester, D.

H. S. Eisenberg, J. F. Hodelin, G. Khoury, and D. Bouwmeester, “Multiphoton path entanglement by nonlocal bunching,” Phys. Rev. Lett. 94, 090502 (2005).
[Crossref] [PubMed]

Boyd, R. W.

E. M. Nagasako, S. J. Bentley, R. W. Boyd, and G. S. Agarwal, “Nonclassical two-photon interferometry and lithography with high-gain parametric amplifiers,” Phys. Rev. A 64, 043802(2001).
[Crossref]

Branning, D.

J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate,” Nature 426, 264–267 (2003).
[Crossref] [PubMed]

Braunstein, S. L.

P. Kok, A. N. Boto, D. S. Abrams, C. P. Williams, S. L. Braunstein, and J. P. Dowling, “Quantum-interferometric optical lithography: towards arbitrary two-dimensional patterns,” Phys. Rev. A 63, 063407 (2001).
[Crossref]

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733–2736 (2000).
[Crossref] [PubMed]

Campos, R. A.

C. C. Gerry and R. A. Campos, “Generation of maximally entangled photonic states with a quantum-optical Fredkin gate,” Phys. Rev. A 64, 063814 (2001).
[Crossref]

J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, “Two-photon interference in a Mach–Zehnder interferometer,” Phys. Rev. Lett. 65, 1348–1351 (1990).
[Crossref] [PubMed]

Chekhova, M. V.

M. D’Angelo, M. V. Chekhova, and Y. Shih, “Two-photon diffraction and quantum lithography,” Phys. Rev. Lett. 87, 013602 (2001).
[Crossref] [PubMed]

D’Angelo, M.

M. D’Angelo, M. V. Chekhova, and Y. Shih, “Two-photon diffraction and quantum lithography,” Phys. Rev. Lett. 87, 013602 (2001).
[Crossref] [PubMed]

Dowling, J. P.

P. Kok, A. N. Boto, D. S. Abrams, C. P. Williams, S. L. Braunstein, and J. P. Dowling, “Quantum-interferometric optical lithography: towards arbitrary two-dimensional patterns,” Phys. Rev. A 63, 063407 (2001).
[Crossref]

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733–2736 (2000).
[Crossref] [PubMed]

Edamatsu, K.

K. Edamatsu, R. Shimizu, and T. Itoh, “Measurement of the photonic de Broglie wavelength of entangled photon pairs generated by spontaneous parametric down-conversion,” Phys. Rev. Lett. 89, 213601 (2002).
[Crossref] [PubMed]

Eisenberg, H. S.

H. S. Eisenberg, J. F. Hodelin, G. Khoury, and D. Bouwmeester, “Multiphoton path entanglement by nonlocal bunching,” Phys. Rev. Lett. 94, 090502 (2005).
[Crossref] [PubMed]

Fonseca, E. J. S.

E. J. S. Fonseca, C. H. Monken, and S. Pauda, “Measurement of the de Broglie wavelength of a multiphoton wave packet,” Phys. Rev. Lett. 82, 2868–2871 (1999).
[Crossref]

Fujiwara, H.

Y. Kawabe, H. Fujiwara, R. Okamoto, K. Sasaki, and S. Takeuchi, “Quantum interference fringes beating the diffraction limit,” Opt. Exp. 15, 14244–14250 (2007).
[Crossref]

Gasparoni, S.

P. Walther, J.-W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, “De Broglie wavelength of a non-local four-photon state,” Nature 429, 158–161 (2004).
[Crossref] [PubMed]

Gerry, C. C.

C. C. Gerry and R. A. Campos, “Generation of maximally entangled photonic states with a quantum-optical Fredkin gate,” Phys. Rev. A 64, 063814 (2001).
[Crossref]

Gilchrist, A.

K. J. Resch, K. L. Pregnell, R. Prevedel, A. Gilchrist, G. J. Pryde, J. L. O’Brien, and A. G. White, “Time-reversal and super-resolving phase measurements,” Phys. Rev. Lett. 98, 223601(2007).
[Crossref] [PubMed]

Gong, Y. X.

Guo, G. C.

B. H. Liu, F. W. Sun, Y. X. Gong, Y. F. Huang, G. C. Guo, and Z. Y. Ou, “Four-photon interference with asymmetric beam splitters,” Opt. Lett. 32, 1320–1322 (2007).
[Crossref] [PubMed]

F. W. Sun, B. H. Liu, Y. F. Huang, Z. Y. Ou, and G. C. Guo, “Observation of the four-photon de Broglie wavelength by state-projection measurement,” Phys. Rev. A 74, 033812 (2006).
[Crossref]

Harris, T. D.

E. Betzig, J. K. Trautman, T. D. Harris, J. S. Weiner, and R. L. Kostelak, “Breaking the diffraction barrier: optical microscopy of a nanometric scale,” Science 251, 1468–1470 (1991).
[Crossref] [PubMed]

Hodelin, J. F.

H. S. Eisenberg, J. F. Hodelin, G. Khoury, and D. Bouwmeester, “Multiphoton path entanglement by nonlocal bunching,” Phys. Rev. Lett. 94, 090502 (2005).
[Crossref] [PubMed]

Hong, C. K.

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
[Crossref] [PubMed]

Huang, Y. F.

B. H. Liu, F. W. Sun, Y. X. Gong, Y. F. Huang, G. C. Guo, and Z. Y. Ou, “Four-photon interference with asymmetric beam splitters,” Opt. Lett. 32, 1320–1322 (2007).
[Crossref] [PubMed]

F. W. Sun, B. H. Liu, Y. F. Huang, Z. Y. Ou, and G. C. Guo, “Observation of the four-photon de Broglie wavelength by state-projection measurement,” Phys. Rev. A 74, 033812 (2006).
[Crossref]

Itoh, T.

K. Edamatsu, R. Shimizu, and T. Itoh, “Measurement of the photonic de Broglie wavelength of entangled photon pairs generated by spontaneous parametric down-conversion,” Phys. Rev. Lett. 89, 213601 (2002).
[Crossref] [PubMed]

Jakeman, E.

J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, “Two-photon interference in a Mach–Zehnder interferometer,” Phys. Rev. Lett. 65, 1348–1351 (1990).
[Crossref] [PubMed]

James, D. F. V.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]

Kawabe, Y.

Y. Kawabe, H. Fujiwara, R. Okamoto, K. Sasaki, and S. Takeuchi, “Quantum interference fringes beating the diffraction limit,” Opt. Exp. 15, 14244–14250 (2007).
[Crossref]

Kempe, M.

M. B. Nasr, A. F. Abouraddy, M. C. Booth, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, M. Kempe, and R. Wolleschensky, “Biphoton focusing for two-photon excitation,” Phys. Rev. A 65, 023816 (2002).
[Crossref]

Khoury, G.

H. S. Eisenberg, J. F. Hodelin, G. Khoury, and D. Bouwmeester, “Multiphoton path entanglement by nonlocal bunching,” Phys. Rev. Lett. 94, 090502 (2005).
[Crossref] [PubMed]

Kim, Y. H.

Y. H. Kim, S. P. Kulik, and Y. Shih, “High-intensity pulsed source of space-time and polarization double-entangled photon pairs,” Phys. Rev. A 62, 011802 (2000).
[Crossref]

Kok, P.

P. Kok, A. N. Boto, D. S. Abrams, C. P. Williams, S. L. Braunstein, and J. P. Dowling, “Quantum-interferometric optical lithography: towards arbitrary two-dimensional patterns,” Phys. Rev. A 63, 063407 (2001).
[Crossref]

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733–2736 (2000).
[Crossref] [PubMed]

Kostelak, R. L.

E. Betzig, J. K. Trautman, T. D. Harris, J. S. Weiner, and R. L. Kostelak, “Breaking the diffraction barrier: optical microscopy of a nanometric scale,” Science 251, 1468–1470 (1991).
[Crossref] [PubMed]

Kuang, L.-M.

Y.-H. Wang and L.-M. Kuang, “Nonmaximally entangled state quantum photolithography,” J. Opt. B: Quantum Semiclass. Opt. 5, 405–408 (2003).
[Crossref]

Kulik, S. P.

Y. H. Kim, S. P. Kulik, and Y. Shih, “High-intensity pulsed source of space-time and polarization double-entangled photon pairs,” Phys. Rev. A 62, 011802 (2000).
[Crossref]

Kwiat, P. G.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]

Larchuk, T.

J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, “Two-photon interference in a Mach–Zehnder interferometer,” Phys. Rev. Lett. 65, 1348–1351 (1990).
[Crossref] [PubMed]

Liu, B. H.

B. H. Liu, F. W. Sun, Y. X. Gong, Y. F. Huang, G. C. Guo, and Z. Y. Ou, “Four-photon interference with asymmetric beam splitters,” Opt. Lett. 32, 1320–1322 (2007).
[Crossref] [PubMed]

F. W. Sun, B. H. Liu, Y. F. Huang, Z. Y. Ou, and G. C. Guo, “Observation of the four-photon de Broglie wavelength by state-projection measurement,” Phys. Rev. A 74, 033812 (2006).
[Crossref]

Lundeen, J. S.

M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature 429, 161–164 (2004).
[Crossref] [PubMed]

Mandel, L.

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
[Crossref] [PubMed]

Mitchell, M. W.

M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature 429, 161–164 (2004).
[Crossref] [PubMed]

Monken, C. H.

E. J. S. Fonseca, C. H. Monken, and S. Pauda, “Measurement of the de Broglie wavelength of a multiphoton wave packet,” Phys. Rev. Lett. 82, 2868–2871 (1999).
[Crossref]

Munro, W. J.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]

Nagasako, E. M.

E. M. Nagasako, S. J. Bentley, R. W. Boyd, and G. S. Agarwal, “Nonclassical two-photon interferometry and lithography with high-gain parametric amplifiers,” Phys. Rev. A 64, 043802(2001).
[Crossref]

Nagata, T.

T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four entangled photons,” Science 316, 726–729 (2007).
[Crossref] [PubMed]

Nasr, M. B.

M. B. Nasr, A. F. Abouraddy, M. C. Booth, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, M. Kempe, and R. Wolleschensky, “Biphoton focusing for two-photon excitation,” Phys. Rev. A 65, 023816 (2002).
[Crossref]

O’Brien, J. L.

T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four entangled photons,” Science 316, 726–729 (2007).
[Crossref] [PubMed]

K. J. Resch, K. L. Pregnell, R. Prevedel, A. Gilchrist, G. J. Pryde, J. L. O’Brien, and A. G. White, “Time-reversal and super-resolving phase measurements,” Phys. Rev. Lett. 98, 223601(2007).
[Crossref] [PubMed]

J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate,” Nature 426, 264–267 (2003).
[Crossref] [PubMed]

Okamoto, R.

Y. Kawabe, H. Fujiwara, R. Okamoto, K. Sasaki, and S. Takeuchi, “Quantum interference fringes beating the diffraction limit,” Opt. Exp. 15, 14244–14250 (2007).
[Crossref]

T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four entangled photons,” Science 316, 726–729 (2007).
[Crossref] [PubMed]

Ou, Z. Y.

B. H. Liu, F. W. Sun, Y. X. Gong, Y. F. Huang, G. C. Guo, and Z. Y. Ou, “Four-photon interference with asymmetric beam splitters,” Opt. Lett. 32, 1320–1322 (2007).
[Crossref] [PubMed]

F. W. Sun, B. H. Liu, Y. F. Huang, Z. Y. Ou, and G. C. Guo, “Observation of the four-photon de Broglie wavelength by state-projection measurement,” Phys. Rev. A 74, 033812 (2006).
[Crossref]

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
[Crossref] [PubMed]

Pan, J.-W.

P. Walther, J.-W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, “De Broglie wavelength of a non-local four-photon state,” Nature 429, 158–161 (2004).
[Crossref] [PubMed]

Pauda, S.

E. J. S. Fonseca, C. H. Monken, and S. Pauda, “Measurement of the de Broglie wavelength of a multiphoton wave packet,” Phys. Rev. Lett. 82, 2868–2871 (1999).
[Crossref]

Pregnell, K. L.

K. J. Resch, K. L. Pregnell, R. Prevedel, A. Gilchrist, G. J. Pryde, J. L. O’Brien, and A. G. White, “Time-reversal and super-resolving phase measurements,” Phys. Rev. Lett. 98, 223601(2007).
[Crossref] [PubMed]

Prevedel, R.

K. J. Resch, K. L. Pregnell, R. Prevedel, A. Gilchrist, G. J. Pryde, J. L. O’Brien, and A. G. White, “Time-reversal and super-resolving phase measurements,” Phys. Rev. Lett. 98, 223601(2007).
[Crossref] [PubMed]

Pryde, G. J.

K. J. Resch, K. L. Pregnell, R. Prevedel, A. Gilchrist, G. J. Pryde, J. L. O’Brien, and A. G. White, “Time-reversal and super-resolving phase measurements,” Phys. Rev. Lett. 98, 223601(2007).
[Crossref] [PubMed]

J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate,” Nature 426, 264–267 (2003).
[Crossref] [PubMed]

Ralph, T. C.

J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate,” Nature 426, 264–267 (2003).
[Crossref] [PubMed]

Rarity, J. G.

J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, “Two-photon interference in a Mach–Zehnder interferometer,” Phys. Rev. Lett. 65, 1348–1351 (1990).
[Crossref] [PubMed]

Resch, K. J.

K. J. Resch, K. L. Pregnell, R. Prevedel, A. Gilchrist, G. J. Pryde, J. L. O’Brien, and A. G. White, “Time-reversal and super-resolving phase measurements,” Phys. Rev. Lett. 98, 223601(2007).
[Crossref] [PubMed]

Saleh, B. E. A.

M. B. Nasr, A. F. Abouraddy, M. C. Booth, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, M. Kempe, and R. Wolleschensky, “Biphoton focusing for two-photon excitation,” Phys. Rev. A 65, 023816 (2002).
[Crossref]

J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, “Two-photon interference in a Mach–Zehnder interferometer,” Phys. Rev. Lett. 65, 1348–1351 (1990).
[Crossref] [PubMed]

Sanchez-Soto, L. L.

G. Bjork and L. L. Sanchez-Soto, “Entangled-state lithography: tailoring any pattern with a single state,” Phys. Rev. Lett. 86, 4516–4519 (2001).
[Crossref] [PubMed]

Sasaki, K.

T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four entangled photons,” Science 316, 726–729 (2007).
[Crossref] [PubMed]

Y. Kawabe, H. Fujiwara, R. Okamoto, K. Sasaki, and S. Takeuchi, “Quantum interference fringes beating the diffraction limit,” Opt. Exp. 15, 14244–14250 (2007).
[Crossref]

Sergienko, A. V.

M. B. Nasr, A. F. Abouraddy, M. C. Booth, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, M. Kempe, and R. Wolleschensky, “Biphoton focusing for two-photon excitation,” Phys. Rev. A 65, 023816 (2002).
[Crossref]

Shih, Y.

M. D’Angelo, M. V. Chekhova, and Y. Shih, “Two-photon diffraction and quantum lithography,” Phys. Rev. Lett. 87, 013602 (2001).
[Crossref] [PubMed]

Y. H. Kim, S. P. Kulik, and Y. Shih, “High-intensity pulsed source of space-time and polarization double-entangled photon pairs,” Phys. Rev. A 62, 011802 (2000).
[Crossref]

Shimizu, R.

K. Edamatsu, R. Shimizu, and T. Itoh, “Measurement of the photonic de Broglie wavelength of entangled photon pairs generated by spontaneous parametric down-conversion,” Phys. Rev. Lett. 89, 213601 (2002).
[Crossref] [PubMed]

Steinberg, A. M.

M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature 429, 161–164 (2004).
[Crossref] [PubMed]

Sun, F. W.

B. H. Liu, F. W. Sun, Y. X. Gong, Y. F. Huang, G. C. Guo, and Z. Y. Ou, “Four-photon interference with asymmetric beam splitters,” Opt. Lett. 32, 1320–1322 (2007).
[Crossref] [PubMed]

F. W. Sun, B. H. Liu, Y. F. Huang, Z. Y. Ou, and G. C. Guo, “Observation of the four-photon de Broglie wavelength by state-projection measurement,” Phys. Rev. A 74, 033812 (2006).
[Crossref]

Takeuchi, S.

T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four entangled photons,” Science 316, 726–729 (2007).
[Crossref] [PubMed]

Y. Kawabe, H. Fujiwara, R. Okamoto, K. Sasaki, and S. Takeuchi, “Quantum interference fringes beating the diffraction limit,” Opt. Exp. 15, 14244–14250 (2007).
[Crossref]

Tapster, P. R.

J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, “Two-photon interference in a Mach–Zehnder interferometer,” Phys. Rev. Lett. 65, 1348–1351 (1990).
[Crossref] [PubMed]

Teich, M. C.

M. B. Nasr, A. F. Abouraddy, M. C. Booth, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, M. Kempe, and R. Wolleschensky, “Biphoton focusing for two-photon excitation,” Phys. Rev. A 65, 023816 (2002).
[Crossref]

J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, “Two-photon interference in a Mach–Zehnder interferometer,” Phys. Rev. Lett. 65, 1348–1351 (1990).
[Crossref] [PubMed]

Trautman, J. K.

E. Betzig, J. K. Trautman, T. D. Harris, J. S. Weiner, and R. L. Kostelak, “Breaking the diffraction barrier: optical microscopy of a nanometric scale,” Science 251, 1468–1470 (1991).
[Crossref] [PubMed]

Ursin, R.

P. Walther, J.-W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, “De Broglie wavelength of a non-local four-photon state,” Nature 429, 158–161 (2004).
[Crossref] [PubMed]

Walther, P.

P. Walther, J.-W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, “De Broglie wavelength of a non-local four-photon state,” Nature 429, 158–161 (2004).
[Crossref] [PubMed]

Wang, Y.-H.

Y.-H. Wang and L.-M. Kuang, “Nonmaximally entangled state quantum photolithography,” J. Opt. B: Quantum Semiclass. Opt. 5, 405–408 (2003).
[Crossref]

Weiner, J. S.

E. Betzig, J. K. Trautman, T. D. Harris, J. S. Weiner, and R. L. Kostelak, “Breaking the diffraction barrier: optical microscopy of a nanometric scale,” Science 251, 1468–1470 (1991).
[Crossref] [PubMed]

Werner, R. F.

R. F. Werner, “Quantum states with Einstein–Podolsky–Rosen correlations admitting a hidden-variable model,” Phys. Rev. A 40, 4277–4281 (1989).
[Crossref] [PubMed]

White, A. G.

K. J. Resch, K. L. Pregnell, R. Prevedel, A. Gilchrist, G. J. Pryde, J. L. O’Brien, and A. G. White, “Time-reversal and super-resolving phase measurements,” Phys. Rev. Lett. 98, 223601(2007).
[Crossref] [PubMed]

J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate,” Nature 426, 264–267 (2003).
[Crossref] [PubMed]

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]

Williams, C. P.

P. Kok, A. N. Boto, D. S. Abrams, C. P. Williams, S. L. Braunstein, and J. P. Dowling, “Quantum-interferometric optical lithography: towards arbitrary two-dimensional patterns,” Phys. Rev. A 63, 063407 (2001).
[Crossref]

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733–2736 (2000).
[Crossref] [PubMed]

Wolleschensky, R.

M. B. Nasr, A. F. Abouraddy, M. C. Booth, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, M. Kempe, and R. Wolleschensky, “Biphoton focusing for two-photon excitation,” Phys. Rev. A 65, 023816 (2002).
[Crossref]

Zeilinger, A.

P. Walther, J.-W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, “De Broglie wavelength of a non-local four-photon state,” Nature 429, 158–161 (2004).
[Crossref] [PubMed]

J. Opt. B: Quantum Semiclass. Opt. (1)

Y.-H. Wang and L.-M. Kuang, “Nonmaximally entangled state quantum photolithography,” J. Opt. B: Quantum Semiclass. Opt. 5, 405–408 (2003).
[Crossref]

Nature (3)

M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature 429, 161–164 (2004).
[Crossref] [PubMed]

P. Walther, J.-W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, “De Broglie wavelength of a non-local four-photon state,” Nature 429, 158–161 (2004).
[Crossref] [PubMed]

J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate,” Nature 426, 264–267 (2003).
[Crossref] [PubMed]

Opt. Exp. (1)

Y. Kawabe, H. Fujiwara, R. Okamoto, K. Sasaki, and S. Takeuchi, “Quantum interference fringes beating the diffraction limit,” Opt. Exp. 15, 14244–14250 (2007).
[Crossref]

Opt. Lett. (1)

Phys. Rev. A (8)

Y. H. Kim, S. P. Kulik, and Y. Shih, “High-intensity pulsed source of space-time and polarization double-entangled photon pairs,” Phys. Rev. A 62, 011802 (2000).
[Crossref]

E. M. Nagasako, S. J. Bentley, R. W. Boyd, and G. S. Agarwal, “Nonclassical two-photon interferometry and lithography with high-gain parametric amplifiers,” Phys. Rev. A 64, 043802(2001).
[Crossref]

F. W. Sun, B. H. Liu, Y. F. Huang, Z. Y. Ou, and G. C. Guo, “Observation of the four-photon de Broglie wavelength by state-projection measurement,” Phys. Rev. A 74, 033812 (2006).
[Crossref]

P. Kok, A. N. Boto, D. S. Abrams, C. P. Williams, S. L. Braunstein, and J. P. Dowling, “Quantum-interferometric optical lithography: towards arbitrary two-dimensional patterns,” Phys. Rev. A 63, 063407 (2001).
[Crossref]

R. F. Werner, “Quantum states with Einstein–Podolsky–Rosen correlations admitting a hidden-variable model,” Phys. Rev. A 40, 4277–4281 (1989).
[Crossref] [PubMed]

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]

M. B. Nasr, A. F. Abouraddy, M. C. Booth, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, M. Kempe, and R. Wolleschensky, “Biphoton focusing for two-photon excitation,” Phys. Rev. A 65, 023816 (2002).
[Crossref]

C. C. Gerry and R. A. Campos, “Generation of maximally entangled photonic states with a quantum-optical Fredkin gate,” Phys. Rev. A 64, 063814 (2001).
[Crossref]

Phys. Rev. Lett. (9)

G. Bjork and L. L. Sanchez-Soto, “Entangled-state lithography: tailoring any pattern with a single state,” Phys. Rev. Lett. 86, 4516–4519 (2001).
[Crossref] [PubMed]

M. D’Angelo, M. V. Chekhova, and Y. Shih, “Two-photon diffraction and quantum lithography,” Phys. Rev. Lett. 87, 013602 (2001).
[Crossref] [PubMed]

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733–2736 (2000).
[Crossref] [PubMed]

K. Edamatsu, R. Shimizu, and T. Itoh, “Measurement of the photonic de Broglie wavelength of entangled photon pairs generated by spontaneous parametric down-conversion,” Phys. Rev. Lett. 89, 213601 (2002).
[Crossref] [PubMed]

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
[Crossref] [PubMed]

K. J. Resch, K. L. Pregnell, R. Prevedel, A. Gilchrist, G. J. Pryde, J. L. O’Brien, and A. G. White, “Time-reversal and super-resolving phase measurements,” Phys. Rev. Lett. 98, 223601(2007).
[Crossref] [PubMed]

J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, “Two-photon interference in a Mach–Zehnder interferometer,” Phys. Rev. Lett. 65, 1348–1351 (1990).
[Crossref] [PubMed]

E. J. S. Fonseca, C. H. Monken, and S. Pauda, “Measurement of the de Broglie wavelength of a multiphoton wave packet,” Phys. Rev. Lett. 82, 2868–2871 (1999).
[Crossref]

H. S. Eisenberg, J. F. Hodelin, G. Khoury, and D. Bouwmeester, “Multiphoton path entanglement by nonlocal bunching,” Phys. Rev. Lett. 94, 090502 (2005).
[Crossref] [PubMed]

Science (2)

T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four entangled photons,” Science 316, 726–729 (2007).
[Crossref] [PubMed]

E. Betzig, J. K. Trautman, T. D. Harris, J. S. Weiner, and R. L. Kostelak, “Breaking the diffraction barrier: optical microscopy of a nanometric scale,” Science 251, 1468–1470 (1991).
[Crossref] [PubMed]

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Figures (8)

Fig. 1
Fig. 1

Numerical model to simulate two-photon interference pattern.

Fig. 2
Fig. 2

Calculated interference fringes using maximally entangled state by changing the relative phase ξ from 0 (top) to π (bottom) in Eq. (1): (a) ξ = 0 , (b) ξ = π / 4 , (c) ξ = π / 2 , (d) ξ = 3 π / 4 , and (e) ξ = π .

Fig. 3
Fig. 3

Calculated interference fringes and density matrices using Werner state by changing the fraction t from 0 (top) to 1 (bottom); (a) t = 1 , (b) t = 0.75 , (c) t = 0.5 , (d) t = 0.25 , and (e) t = 0 .

Fig. 4
Fig. 4

Calculated interference fringes and density matrices using nonmaximally entangled states with changing γ: γ = (a) 45 ° , (b) 30 ° , (c) 20 ° , (d) 10 ° , and (e) 0 ° .

Fig. 5
Fig. 5

Calculated interference fringes and density matrices using the superposition state of the nonorthogonal linearly polarized two-photon states by changing the tilting angle: x = (a) 60 ° , (b) 10 , (c) 0 ° , (d) 10 ° , and (e) 60 ° .

Fig. 6
Fig. 6

Schematic of experimental setup. Photon pairs entangled in polarization are generated from two beta barium borate crystals (BBOs) and passed through an interference filter (IF) and a single-mode fiber (SMF). Then, the entangled photons are spatially separated into two paths depending on their polarization in a calcite crystal (Cal). The polarization in one path is rotated by 90 ° by using a half-wave plate (HWP), giving a two-photon NOON state. An aspheric lens (L) is used to focus the photons to a small spot and form an interference fringe in the focal plane. An NSOM probe with an elliptical opening (inset) is scanned with a piezoactuator along the focal plane. The SMF output of the probe is divided by a 50% beam splitter (50/50 BS) and detected by two SPCMs.

Fig. 7
Fig. 7

Optical setup for the quantum state tomography measurement of the incident polarization entangled state: HWP, half-wave plate; QWP, quarter-wave plate; DF, dichroic filter; CF, color filter; IF, interference filter; SMF, single-mode fiber; Pol, polarizer; 50/50BS, 50% beam splitter; and D1, 2, SPCM.

Fig. 8
Fig. 8

Experimentally obtained density matrix [(a) real and (b) imaginary parts] after compensation. (c) Calculated two-photon inter ference fringe from Eq. (9) using density matrices (a) and (b). (d) Experimentally obtained two-photon interference fringe using polarization entangled state. The accumulation time for one data point was 5 s , and the error bars show ± counts assuming Poisson statistics. The solid curve indicates the fitting function of Eq. (12). (e) Calculated two-photon interference fringe from Eq. (11) using the experimentally obtained density matrix.

Equations (33)

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

| ψ = 1 2 { | 2 V s + exp ( i ξ ) | 2 H s } ,
| ψ = 1 2 { | 2 H c | 0 d + exp ( i ξ ) | 0 c | 2 H d } .
ϕ = 2 k r sin θ .
P ( r ) = 1 2 ( 1 + cos 2 ϕ ) = 1 2 { 1 + cos ( 4 k r sin θ ) } ,
ρ s = b 11 | 2 V s 2 V | s + b 12 | 2 V s V H | s + b 13 | 2 V s 2 H | s + b 12 * | V H s 2 V | s + b 22 | V H s V H | s + b 23 | V H s 2 H | s + b 13 * | 2 H s 2 V | s + b 23 * | 2 H s V H | s + ( 1 b 11 b 22 ) | 2 H s 2 H | s ,
ρ m = b 11 | 2 e | 0 f 0 | f 2 | e + b 12 | 2 e | 0 f 1 | f 1 | e + b 13 | 2 e | 0 f 2 | f 0 | e + b 12 * | 1 e | 1 f 0 | f 2 | e + b 22 | 1 e | 1 f 1 | f 1 | e + b 23 | 1 e | 1 f 2 | f 0 | e + b 13 * | 0 e | 2 f 0 | f 2 | e + b 23 * | 0 e | 2 f 1 | f 1 | e + ( 1 b 11 b 22 ) | 0 e | 2 f 2 | f 0 | e .
P ( r ) = T r { ρ m E ^ ( ) ( r ) E ^ ( ) ( r ) E ^ ( + ) ( r ) E ^ ( + ) ( r ) } ,
E ^ ( + ) ( r ) = 1 2 { a ^ e exp ( i k e · r ) + a ^ f exp ( i k f · r ) } , E ^ ( ) ( r ) = { E ^ ( + ) ( r ) } ,
P ( ϕ ) = 1 2 [ 1 + b 22 + ( b 13 + b 13 * ) cos 2 ϕ + i ( b 13 b 13 * ) sin 2 ϕ + 2 ( b 12 + b 23 + b 12 * + b 23 * ) cos ϕ + 2 i ( b 12 + b 23 b 12 * b 23 * ) sin ϕ ] .
P ( r ) = 1 2 [ 1 + b 22 + ( b 13 + b 13 * ) cos ( 4 k r sin θ ) + i ( b 13 b 13 * ) sin ( 4 k r sin θ ) + 2 ( b 12 + b 23 + b 12 * + b 23 * ) cos ( 2 k r sin θ ) + 2 i ( b 12 + b 23 b 12 * b 23 * ) sin ( 2 k r sin θ ) ] ,
P gauss ( r ) = a exp { ( r r 0 ) 2 2 σ 2 } × P ( r ) .
P obs ( r ) = 0 r rect ( x / d ) P gauss ( r x ) d x ,
rect ( x / d ) = { 0 if | x | > d / 2 1 otherwise ,
ρ w = t | ψ ψ | + 1 t 3 I ( 3 ) ,
P ( ϕ ) = 1 2 ( 1 t 3 + 1 + t cos 2 ϕ ) .
| ψ = cos γ | 2 H + sin γ | 2 V ,
P ( ϕ ) = 1 2 ( 1 + sin 2 γ cos 2 ϕ ) ,
| 2 x = 1 2 a ^ x a ^ x | 0 = 1 2 ( a ^ H cos x + a ^ V sin x ) ( a ^ H cos x + a ^ V sin x ) | 0 ,
| ψ = c 2 ( | 2 V + α | 2 x ) = c 2 ( α cos 2 x | 2 H + 2 α cos x sin x | H V + ( 1 + α sin 2 x ) | 2 V ) ,
c = 2 1 + 2 α sin 2 x + α 2 ,
α = 2 | cos ( x π 4 ) | .
P ( ϕ ) = 1 2 { 1 + c 2 α 2 sin 2 2 x 4 + c 2 α sin 2 x ( 1 + α ) cos ϕ + c 2 α cos 2 x ( 1 + α sin 2 x ) cos 2 ϕ } .
ρ 4 × 4 = A 11 | V V V V | + A 12 | V V V H | + A 13 | V V H V | + A 14 | V V H H | + A 12 * | V H V V | + A 22 | V H V H | + A 23 | V H H V | + A 24 | V H H H | + A 13 * | H V V V | + A 23 * | H V V H | + A 33 | H V H V | + A 34 | H V H H | + A 14 * | H H V V | + A 24 * | H H V H | + A 34 | H H H V | + ( 1 A 11 A 22 A 33 ) | H H H H | .
ρ 3 × 3 = B 11 | V V V V | + B 12 | V V V H | + B 13 | V V H H | + B 12 * | V H V V | + B 22 | V H V H | + B 23 | V H H H | + B 13 * | H H V V | + B 23 * | H H V H | + ( 1 B 11 B 22 ) | H H H H | .
a ^ γ a ^ δ = 1 2 1 1 1 1 a ^ α a ^ β ,
A 11 | V V V V | = A 11 | V β | V α V | α V | β .
A 11 | V V V V | = A 11 2 ( | V V γ | V V δ ) ( V V | δ + V V | γ ) .
A 11 | V V V V | = A 11 2 ( | V V γ V V | γ ) .
B 11 = A 11 2 .
B 12 = 1 2 2 A 12 + A 13 2 ,
B 13 = A 14 2 ,
B 22 = 1 4 A 22 + A 23 + A 33 3 ,
B 23 = 1 2 2 A 24 + A 34 2 .

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