Abstract

We discuss how to use coincidence detection to generate unusual, nonsinusoidal interference curves by using not a single detector, but several in coincidence. The method works for both strong (classical) and weak (on the few-photon level) light, although in the latter case the detection becomes probabilistic with low efficiency. Using the method, one can tailor the coincidence measurement setup to obtain essentially any interference pattern. We then use the method to experimentally demonstrate phase-difference state interference patterns in the few-photon regime that are highly nonsinusoidal. We also discuss optimal implementation of the method with regard to fluctuations and success probability, and we analyze the origin and magnitude of errors.

© 2013 Optical Society of America

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  1. C. H. Bennett, F. Bessette, G. Brassard, L. Salvail, and J. Smolin, “Experimental quantum cryptography,” J. Cryptol. 5, 3–28 (1992).
    [CrossRef]
  2. C. Monroe, D. M. Meekhof, B. E. King, W. M. Itano, and D. J. Wineland, “Demonstration of a fundamental quantum logic gate,” Phys. Rev. Lett. 75, 4714–4717 (1995).
    [CrossRef]
  3. C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
    [CrossRef]
  4. D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature 390, 575–579 (1997).
    [CrossRef]
  5. J. M. Jacobson, G. Björk, I. Chuang, and Y. Yamamoto, “Photonic de Broglie waves,” Phys. Rev. Lett. 74, 4835–4838 (1995).
    [CrossRef]
  6. 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]
  7. 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]
  8. E. J. S. Fonseca, C. H. Monken, and S. Pádua, “Measurement of the de Broglie wavelength of a multiphoton wave packet,” Phys. Rev. Lett. 82, 2868–2871 (1999).
    [CrossRef]
  9. 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]
  10. 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]
  11. 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]
  12. M. D’Angelo, M. V. Chekhova, and Y. Shih, “Two-photon diffraction and quantum lithography,” Phys. Rev. Lett. 87, 013602 (2001).
    [CrossRef]
  13. 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]
  14. I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science 328, 879–881 (2010).
    [CrossRef]
  15. T. Tsegaye, J. Söderholm, M. Atatüre, A. Trifonov, G. Björk, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Experimental demonstration of three mutually orthogonal polarization states of entangled photons,” Phys. Rev. Lett. 85, 5013–5017 (2000).
    [CrossRef]
  16. R. Thew, A. Acin, H. Zbinden, and N. Gisin, “Experimental realization of entangled qutrits for quantum communication,” Quantum Inform. Comput. 4, 93–101 (2004).
  17. H. Hofmann, “Generation of highly nonclassical n-photon polarization states by superbunching at a photon bottleneck,” Phys. Rev. A 70, 023812 (2004).
    [CrossRef]
  18. F. W. Sun, Z. Y. Ou, and G. C. Guo, “Projection measurement of the maximally entangled N-photon state for a demonstration of the N-photon de Broglie wavelength,” Phys. Rev. A 73, 023808 (2006).
    [CrossRef]
  19. S. J. Bentley and R. W. Boyd, “Nonlinear optical lithography with ultra-high sub-Rayleigh resolution,” Opt. Express 12, 5735–5740 (2004).
    [CrossRef]
  20. K. J. Resch, K. L. Pregnell, R. Prevedel, A. Gilchrist, G. J. Pryde, J. L. OBrien, and A. G. White, “Time-reversal and super-resolving phase measurements,” Phys. Rev. Lett. 98, 223601 (2007).
    [CrossRef]
  21. C. Kothe, G. Björk, and M. Bourennane, “Arbitrarily high super-resolving phase measurements at telecommunication wavelengths,” Phys. Rev. A 81, 063836 (2010).
    [CrossRef]
  22. S. Shabbir, M. Swillo, and G. Björk, “Synthesis of arbitrary, two-mode, high-visibility N-photon interference patterns,” Phys. Rev. A 87, 053821 (2013).
    [CrossRef]
  23. M. V. Berry and S. Popescu, “Evolution of quantum superoscillations and optical superresolution without evanescent waves,” J. Phys. A 39, 6965–6977 (2006).
    [CrossRef]
  24. G. Khoury, H. S. Eisenberg, E. J. S. Fonseca, and D. Bouwmeester, “Nonlinear interferometry via Fock-state projection,” Phys. Rev. Lett. 96, 203601 (2006).
    [CrossRef]
  25. Y. Gao, P. M. Anisimov, C. F. Wildfeuer, J. Luine, H. Lee, and J. P. Dowling, “Super-resolution at the shot-noise limit with coherent states and photon-number-resolving detectors,” J. Opt. Soc. Am. B 27, A170–A174 (2010).
    [CrossRef]
  26. R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).
    [CrossRef]

2013

S. Shabbir, M. Swillo, and G. Björk, “Synthesis of arbitrary, two-mode, high-visibility N-photon interference patterns,” Phys. Rev. A 87, 053821 (2013).
[CrossRef]

2010

Y. Gao, P. M. Anisimov, C. F. Wildfeuer, J. Luine, H. Lee, and J. P. Dowling, “Super-resolution at the shot-noise limit with coherent states and photon-number-resolving detectors,” J. Opt. Soc. Am. B 27, A170–A174 (2010).
[CrossRef]

C. Kothe, G. Björk, and M. Bourennane, “Arbitrarily high super-resolving phase measurements at telecommunication wavelengths,” Phys. Rev. A 81, 063836 (2010).
[CrossRef]

I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science 328, 879–881 (2010).
[CrossRef]

2007

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]

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

2006

F. W. Sun, Z. Y. Ou, and G. C. Guo, “Projection measurement of the maximally entangled N-photon state for a demonstration of the N-photon de Broglie wavelength,” Phys. Rev. A 73, 023808 (2006).
[CrossRef]

M. V. Berry and S. Popescu, “Evolution of quantum superoscillations and optical superresolution without evanescent waves,” J. Phys. A 39, 6965–6977 (2006).
[CrossRef]

G. Khoury, H. S. Eisenberg, E. J. S. Fonseca, and D. Bouwmeester, “Nonlinear interferometry via Fock-state projection,” Phys. Rev. Lett. 96, 203601 (2006).
[CrossRef]

2004

S. J. Bentley and R. W. Boyd, “Nonlinear optical lithography with ultra-high sub-Rayleigh resolution,” Opt. Express 12, 5735–5740 (2004).
[CrossRef]

R. Thew, A. Acin, H. Zbinden, and N. Gisin, “Experimental realization of entangled qutrits for quantum communication,” Quantum Inform. Comput. 4, 93–101 (2004).

H. Hofmann, “Generation of highly nonclassical n-photon polarization states by superbunching at a photon bottleneck,” Phys. Rev. A 70, 023812 (2004).
[CrossRef]

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]

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]

2002

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]

2001

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

2000

T. Tsegaye, J. Söderholm, M. Atatüre, A. Trifonov, G. Björk, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Experimental demonstration of three mutually orthogonal polarization states of entangled photons,” Phys. Rev. Lett. 85, 5013–5017 (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]

1999

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

1997

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

1995

J. M. Jacobson, G. Björk, I. Chuang, and Y. Yamamoto, “Photonic de Broglie waves,” Phys. Rev. Lett. 74, 4835–4838 (1995).
[CrossRef]

C. Monroe, D. M. Meekhof, B. E. King, W. M. Itano, and D. J. Wineland, “Demonstration of a fundamental quantum logic gate,” Phys. Rev. Lett. 75, 4714–4717 (1995).
[CrossRef]

1993

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
[CrossRef]

1992

C. H. Bennett, F. Bessette, G. Brassard, L. Salvail, and J. Smolin, “Experimental quantum cryptography,” J. Cryptol. 5, 3–28 (1992).
[CrossRef]

1990

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]

1963

R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).
[CrossRef]

Abrams, D. S.

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]

Acin, A.

R. Thew, A. Acin, H. Zbinden, and N. Gisin, “Experimental realization of entangled qutrits for quantum communication,” Quantum Inform. Comput. 4, 93–101 (2004).

Afek, I.

I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science 328, 879–881 (2010).
[CrossRef]

Ambar, O.

I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science 328, 879–881 (2010).
[CrossRef]

Anisimov, P. M.

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]

Atatüre, M.

T. Tsegaye, J. Söderholm, M. Atatüre, A. Trifonov, G. Björk, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Experimental demonstration of three mutually orthogonal polarization states of entangled photons,” Phys. Rev. Lett. 85, 5013–5017 (2000).
[CrossRef]

Bennett, C. H.

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
[CrossRef]

C. H. Bennett, F. Bessette, G. Brassard, L. Salvail, and J. Smolin, “Experimental quantum cryptography,” J. Cryptol. 5, 3–28 (1992).
[CrossRef]

Bentley, S. J.

Berry, M. V.

M. V. Berry and S. Popescu, “Evolution of quantum superoscillations and optical superresolution without evanescent waves,” J. Phys. A 39, 6965–6977 (2006).
[CrossRef]

Bessette, F.

C. H. Bennett, F. Bessette, G. Brassard, L. Salvail, and J. Smolin, “Experimental quantum cryptography,” J. Cryptol. 5, 3–28 (1992).
[CrossRef]

Björk, G.

S. Shabbir, M. Swillo, and G. Björk, “Synthesis of arbitrary, two-mode, high-visibility N-photon interference patterns,” Phys. Rev. A 87, 053821 (2013).
[CrossRef]

C. Kothe, G. Björk, and M. Bourennane, “Arbitrarily high super-resolving phase measurements at telecommunication wavelengths,” Phys. Rev. A 81, 063836 (2010).
[CrossRef]

T. Tsegaye, J. Söderholm, M. Atatüre, A. Trifonov, G. Björk, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Experimental demonstration of three mutually orthogonal polarization states of entangled photons,” Phys. Rev. Lett. 85, 5013–5017 (2000).
[CrossRef]

J. M. Jacobson, G. Björk, I. Chuang, and Y. Yamamoto, “Photonic de Broglie waves,” Phys. Rev. Lett. 74, 4835–4838 (1995).
[CrossRef]

Boto, A. N.

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]

Bourennane, M.

C. Kothe, G. Björk, and M. Bourennane, “Arbitrarily high super-resolving phase measurements at telecommunication wavelengths,” Phys. Rev. A 81, 063836 (2010).
[CrossRef]

Bouwmeester, D.

G. Khoury, H. S. Eisenberg, E. J. S. Fonseca, and D. Bouwmeester, “Nonlinear interferometry via Fock-state projection,” Phys. Rev. Lett. 96, 203601 (2006).
[CrossRef]

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

Boyd, R. W.

Brassard, G.

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
[CrossRef]

C. H. Bennett, F. Bessette, G. Brassard, L. Salvail, and J. Smolin, “Experimental quantum cryptography,” J. Cryptol. 5, 3–28 (1992).
[CrossRef]

Braunstein, S. L.

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]

Campos, R. A.

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]

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]

Chuang, I.

J. M. Jacobson, G. Björk, I. Chuang, and Y. Yamamoto, “Photonic de Broglie waves,” Phys. Rev. Lett. 74, 4835–4838 (1995).
[CrossRef]

Crépeau, C.

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
[CrossRef]

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]

Dowling, J. P.

Y. Gao, P. M. Anisimov, C. F. Wildfeuer, J. Luine, H. Lee, and J. P. Dowling, “Super-resolution at the shot-noise limit with coherent states and photon-number-resolving detectors,” J. Opt. Soc. Am. B 27, A170–A174 (2010).
[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]

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]

Eibl, M.

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

Eisenberg, H. S.

G. Khoury, H. S. Eisenberg, E. J. S. Fonseca, and D. Bouwmeester, “Nonlinear interferometry via Fock-state projection,” Phys. Rev. Lett. 96, 203601 (2006).
[CrossRef]

Fonseca, E. J. S.

G. Khoury, H. S. Eisenberg, E. J. S. Fonseca, and D. Bouwmeester, “Nonlinear interferometry via Fock-state projection,” Phys. Rev. Lett. 96, 203601 (2006).
[CrossRef]

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

Gao, Y.

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]

Gilchrist, A.

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

Gisin, N.

R. Thew, A. Acin, H. Zbinden, and N. Gisin, “Experimental realization of entangled qutrits for quantum communication,” Quantum Inform. Comput. 4, 93–101 (2004).

Glauber, R. J.

R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).
[CrossRef]

Guo, G. C.

F. W. Sun, Z. Y. Ou, and G. C. Guo, “Projection measurement of the maximally entangled N-photon state for a demonstration of the N-photon de Broglie wavelength,” Phys. Rev. A 73, 023808 (2006).
[CrossRef]

Hofmann, H.

H. Hofmann, “Generation of highly nonclassical n-photon polarization states by superbunching at a photon bottleneck,” Phys. Rev. A 70, 023812 (2004).
[CrossRef]

Itano, W. M.

C. Monroe, D. M. Meekhof, B. E. King, W. M. Itano, and D. J. Wineland, “Demonstration of a fundamental quantum logic gate,” Phys. Rev. Lett. 75, 4714–4717 (1995).
[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]

Jacobson, J. M.

J. M. Jacobson, G. Björk, I. Chuang, and Y. Yamamoto, “Photonic de Broglie waves,” Phys. Rev. Lett. 74, 4835–4838 (1995).
[CrossRef]

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]

Jozsa, R.

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
[CrossRef]

Khoury, G.

G. Khoury, H. S. Eisenberg, E. J. S. Fonseca, and D. Bouwmeester, “Nonlinear interferometry via Fock-state projection,” Phys. Rev. Lett. 96, 203601 (2006).
[CrossRef]

King, B. E.

C. Monroe, D. M. Meekhof, B. E. King, W. M. Itano, and D. J. Wineland, “Demonstration of a fundamental quantum logic gate,” Phys. Rev. Lett. 75, 4714–4717 (1995).
[CrossRef]

Kok, P.

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]

Kothe, C.

C. Kothe, G. Björk, and M. Bourennane, “Arbitrarily high super-resolving phase measurements at telecommunication wavelengths,” Phys. Rev. A 81, 063836 (2010).
[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]

Lee, H.

Luine, J.

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]

Mattle, K.

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

Meekhof, D. M.

C. Monroe, D. M. Meekhof, B. E. King, W. M. Itano, and D. J. Wineland, “Demonstration of a fundamental quantum logic gate,” Phys. Rev. Lett. 75, 4714–4717 (1995).
[CrossRef]

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]

Monken, C. H.

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

Monroe, C.

C. Monroe, D. M. Meekhof, B. E. King, W. M. Itano, and D. J. Wineland, “Demonstration of a fundamental quantum logic gate,” Phys. Rev. Lett. 75, 4714–4717 (1995).
[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]

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]

OBrien, J. L.

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

Okamoto, R.

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]

Ou, Z. Y.

F. W. Sun, Z. Y. Ou, and G. C. Guo, “Projection measurement of the maximally entangled N-photon state for a demonstration of the N-photon de Broglie wavelength,” Phys. Rev. A 73, 023808 (2006).
[CrossRef]

Pádua, S.

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

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]

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

Peres, A.

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
[CrossRef]

Popescu, S.

M. V. Berry and S. Popescu, “Evolution of quantum superoscillations and optical superresolution without evanescent waves,” J. Phys. A 39, 6965–6977 (2006).
[CrossRef]

Pregnell, K. L.

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

Prevedel, R.

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

Pryde, G. J.

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

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]

Resch, K. J.

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

Saleh, B. E. A.

T. Tsegaye, J. Söderholm, M. Atatüre, A. Trifonov, G. Björk, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Experimental demonstration of three mutually orthogonal polarization states of entangled photons,” Phys. Rev. Lett. 85, 5013–5017 (2000).
[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]

Salvail, L.

C. H. Bennett, F. Bessette, G. Brassard, L. Salvail, and J. Smolin, “Experimental quantum cryptography,” J. Cryptol. 5, 3–28 (1992).
[CrossRef]

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]

Sergienko, A. V.

T. Tsegaye, J. Söderholm, M. Atatüre, A. Trifonov, G. Björk, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Experimental demonstration of three mutually orthogonal polarization states of entangled photons,” Phys. Rev. Lett. 85, 5013–5017 (2000).
[CrossRef]

Shabbir, S.

S. Shabbir, M. Swillo, and G. Björk, “Synthesis of arbitrary, two-mode, high-visibility N-photon interference patterns,” Phys. Rev. A 87, 053821 (2013).
[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]

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]

Silberberg, Y.

I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science 328, 879–881 (2010).
[CrossRef]

Smolin, J.

C. H. Bennett, F. Bessette, G. Brassard, L. Salvail, and J. Smolin, “Experimental quantum cryptography,” J. Cryptol. 5, 3–28 (1992).
[CrossRef]

Söderholm, J.

T. Tsegaye, J. Söderholm, M. Atatüre, A. Trifonov, G. Björk, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Experimental demonstration of three mutually orthogonal polarization states of entangled photons,” Phys. Rev. Lett. 85, 5013–5017 (2000).
[CrossRef]

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]

Sun, F. W.

F. W. Sun, Z. Y. Ou, and G. C. Guo, “Projection measurement of the maximally entangled N-photon state for a demonstration of the N-photon de Broglie wavelength,” Phys. Rev. A 73, 023808 (2006).
[CrossRef]

Swillo, M.

S. Shabbir, M. Swillo, and G. Björk, “Synthesis of arbitrary, two-mode, high-visibility N-photon interference patterns,” Phys. Rev. A 87, 053821 (2013).
[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]

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]

Teich, M. C.

T. Tsegaye, J. Söderholm, M. Atatüre, A. Trifonov, G. Björk, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Experimental demonstration of three mutually orthogonal polarization states of entangled photons,” Phys. Rev. Lett. 85, 5013–5017 (2000).
[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]

Thew, R.

R. Thew, A. Acin, H. Zbinden, and N. Gisin, “Experimental realization of entangled qutrits for quantum communication,” Quantum Inform. Comput. 4, 93–101 (2004).

Trifonov, A.

T. Tsegaye, J. Söderholm, M. Atatüre, A. Trifonov, G. Björk, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Experimental demonstration of three mutually orthogonal polarization states of entangled photons,” Phys. Rev. Lett. 85, 5013–5017 (2000).
[CrossRef]

Tsegaye, T.

T. Tsegaye, J. Söderholm, M. Atatüre, A. Trifonov, G. Björk, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Experimental demonstration of three mutually orthogonal polarization states of entangled photons,” Phys. Rev. Lett. 85, 5013–5017 (2000).
[CrossRef]

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]

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]

Weinfurter, H.

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

White, A. G.

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

Wildfeuer, C. F.

Williams, C. P.

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]

Wineland, D. J.

C. Monroe, D. M. Meekhof, B. E. King, W. M. Itano, and D. J. Wineland, “Demonstration of a fundamental quantum logic gate,” Phys. Rev. Lett. 75, 4714–4717 (1995).
[CrossRef]

Wootters, W. K.

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
[CrossRef]

Yamamoto, Y.

J. M. Jacobson, G. Björk, I. Chuang, and Y. Yamamoto, “Photonic de Broglie waves,” Phys. Rev. Lett. 74, 4835–4838 (1995).
[CrossRef]

Zbinden, H.

R. Thew, A. Acin, H. Zbinden, and N. Gisin, “Experimental realization of entangled qutrits for quantum communication,” Quantum Inform. Comput. 4, 93–101 (2004).

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]

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

J. Cryptol.

C. H. Bennett, F. Bessette, G. Brassard, L. Salvail, and J. Smolin, “Experimental quantum cryptography,” J. Cryptol. 5, 3–28 (1992).
[CrossRef]

J. Opt. Soc. Am. B

J. Phys. A

M. V. Berry and S. Popescu, “Evolution of quantum superoscillations and optical superresolution without evanescent waves,” J. Phys. A 39, 6965–6977 (2006).
[CrossRef]

Nature

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

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]

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]

Opt. Express

Phys. Rev.

R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).
[CrossRef]

Phys. Rev. A

C. Kothe, G. Björk, and M. Bourennane, “Arbitrarily high super-resolving phase measurements at telecommunication wavelengths,” Phys. Rev. A 81, 063836 (2010).
[CrossRef]

S. Shabbir, M. Swillo, and G. Björk, “Synthesis of arbitrary, two-mode, high-visibility N-photon interference patterns,” Phys. Rev. A 87, 053821 (2013).
[CrossRef]

H. Hofmann, “Generation of highly nonclassical n-photon polarization states by superbunching at a photon bottleneck,” Phys. Rev. A 70, 023812 (2004).
[CrossRef]

F. W. Sun, Z. Y. Ou, and G. C. Guo, “Projection measurement of the maximally entangled N-photon state for a demonstration of the N-photon de Broglie wavelength,” Phys. Rev. A 73, 023808 (2006).
[CrossRef]

Phys. Rev. Lett.

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

T. Tsegaye, J. Söderholm, M. Atatüre, A. Trifonov, G. Björk, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Experimental demonstration of three mutually orthogonal polarization states of entangled photons,” Phys. Rev. Lett. 85, 5013–5017 (2000).
[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]

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

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

J. M. Jacobson, G. Björk, I. Chuang, and Y. Yamamoto, “Photonic de Broglie waves,” Phys. Rev. Lett. 74, 4835–4838 (1995).
[CrossRef]

C. Monroe, D. M. Meekhof, B. E. King, W. M. Itano, and D. J. Wineland, “Demonstration of a fundamental quantum logic gate,” Phys. Rev. Lett. 75, 4714–4717 (1995).
[CrossRef]

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
[CrossRef]

G. Khoury, H. S. Eisenberg, E. J. S. Fonseca, and D. Bouwmeester, “Nonlinear interferometry via Fock-state projection,” Phys. Rev. Lett. 96, 203601 (2006).
[CrossRef]

Quantum Inform. Comput.

R. Thew, A. Acin, H. Zbinden, and N. Gisin, “Experimental realization of entangled qutrits for quantum communication,” Quantum Inform. Comput. 4, 93–101 (2004).

Science

I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science 328, 879–881 (2010).
[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]

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Figures (5)

Fig. 1.
Fig. 1.

Schematic of a setup that creates arbitrary interference patterns. Light is sequentially divided between N interferometers using N1 splitters Sn with reflectance Rn. Each interferometer combines the two beams, one phase shifted w.r.t. another by θn, via a beam splitter BSn with reflectivity and transmittivity rn and tn, respectively. Adjusting these parameters and multiplying the signals from all the unprimed (shaded) detectors Dn gives the desired interference curve as a function of ϕ.

Fig. 2.
Fig. 2.

Schematic of a setup needed to project out any two-mode, N-photon state from a linearly (diagonal) polarized input state. A wave plate is used to impart a birefringence ϕ, from 02π, to the input, which is subsequently split into N arms with the use of N1 beam splitters. In each arm n, a fixed birefringence θn is added between the two polarization modes. A half-wave plate HWPn adjusts the relative amplitudes. Coincident detection of single photons in each of the unprimed (shaded) detectors Dn then projects out the required state.

Fig. 3.
Fig. 3.

Phase-difference state projection probability pattern |ξ4(0)|U^(ϕ)|ξ4(0)|2. The solid (black) line represents raw data points connected by straight lines. No background subtraction has been done. The red dashed line represents a two-parameter (height and horizontal position) fit of the theoretically expected result. The inset is a magnification of the boxed portion of the plot. Error bars show ±σ.

Fig. 4.
Fig. 4.

Phase-difference state projection probability patterns |ξ29(πm/15)|U^(ϕ)|ξ29(πm/15)|2 for m=0,,29 are shown. As can be seen, the function is 2π periodic.

Fig. 5.
Fig. 5.

Phase-difference state projection probability pattern nominally |ξ29(0)|U^(ϕ)|ξ29(0)|2. The solid (black) line represents raw data points connected by straight lines. No background subtraction has been done. The red dashed line represents a two-parameter (height and horizontal position) fit of the theoretically expected result. The inset is a magnification of the boxed portion of the plot. Error bars show ±σ.

Equations (41)

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

|ξN(γ)=1N+1n=0Nexp(iγn)|n,Nn,
|ψ=n=0Ncn|n,Nn=n=0Nbn(a^)n(b^)Nn|0,0,
n=0Nbnzn=bN(zz1)(zz2)(zzN).
|ψ=bNΠn=1NNnN1(a^z1b^)NN(a^zNb^)|0,0,
|ξ4(0)=15(|0,4+|1,3+|2,2+|3,1+|4,0)=1230[(b^)4+2(b^)3a^+6(b^)2(a^)2+2b^(a^)3+(a^)4]|0,0.
z4+2z3+6z2+2z+1=(z+ei1.441)(z+ei0.514)(z+ei0.514)(z+ei1.441).
|ξ4(0)=215[(a^+ei1.441b^)2(a^+ei0.514b^)2×(a^+ei0.514b^)2(a^+ei1.441b^)2]|0,0.
U^(ϕ)=exp[iϕ(a^a^b^b^)/2]
|ξ4(0)|U^(ϕ)|ξ4(0)|2=125|1+eiϕ+ei2ϕ+ei3ϕ+ei4ϕ|2.
|n=0Nbneinϕ|2,
EnI0Rn2(rneiϕ+tneiθn),
|n=0Nbnzn|2|Πn=1N(z+zn)|2.
α4exp(|α|2)n=04|n,4nn!(4n)!.
|Ξ4(0)=18[(b^)4+(b^)3a^+(b^)2(a^)2+b^(a^)3+(a^)4]|0,0=18(4!|0,4+3!|1,3++4!|4,0),
|Ξ4(0)|U^(ϕ)|α,α|2=|α|8exp[2|α|2]64|ξ4(0)|U^(ϕ)|ξ4(0)|2.
z4+z3+z2+z+1=(z+ei3π/5)(z+eiπ/5)(z+eiπ/5)(z+ei3π/5).
|Ξ4(0)=215[(a^+ei3π/5b^)2(a^+eiπ/5b^)2×(a^+eiπ/5b^)2(a^+ei3π/5b^)2]|0,0.
[1P0(|β)]NP1(|β)NP1(|β)N=[1P0(|β)]NP1(|β)N1N|β|22.
|α|21.
P(ϕ)=|ξ4(0)|eiϕ(a^a^b^b^)/2|ξ4(0)|2=125|1+eiϕ+ei2ϕ+ei3ϕ+ei4ϕ|2=125|(eiϕ+ei3ϕ/5)(eiϕ+eiϕ/5)(eiϕ+eiϕ/5)(eiϕ+ei3ϕ/5)|2=125|eiϕ/2+ei(ϕ/2+3ϕ/5)|2|eiϕ/2+ei(ϕ/2+ϕ/5)|2|eiϕ/2+ei(ϕ/2ϕ/5)|2|eiϕ/2+ei(ϕ/23ϕ/5)|2.
Pn(ϕ)=|α,α|eiϕ(a^a^b^b^)/212(|0,1+eiθn|1,0)|2=12|α,α|(eiϕ/2|0,1+ei(ϕ/2+θn)|1,0)|2=|α|2e|α|22|eiϕ/2+ei(ϕ/2+θn)|2,
P(ϕ)=1625|α|8e4|α|2Πk=14Pk(ϕ).
PMax=|ψIn|ψProj|2.
|ψIn=n=0Ncn|n,Nn,
|ψInn=0Ncn(a^)n(b^)(Nn)n!(Nn)!|0,0,0,0n=0Ncn((RA^+1Ra^)nn!×(RB^+1Rb^)(Nn)(Nn)!)|0,0,0,0.
(R[1R]N1)1/2n=0Ncn(n|n1,Nn,1,0+Nn|n,Nn1,0,1)
R(1R)N1n=1N(n+Nn)|cn|2=NR(1R)N1.
P1=[(N1)/N]N1.
POBO=(N1N)N1(N2N1)N2··12=N!NN.
Ps=PMaxPOBOη4=3η432.
|NOON4(π)=12(|0,4|4,0)=12·4![(b^)4(a^)4]|0,0,
1z4=(z+1)(z1)(z+i)(zi)
|NOON4(π)=13((a^+b^)2(a^b^)2×(a^+ib^)2(a^ib^)2)|0,0.
N(N1)2R2(1R)N2.
P1+1=2(N1)N(N2N)N2.
P1&1=(N1N)N1(N2N1)N2=N1N(N2N)N2.
PTBT=2N/2N!NN.
In=1/2|cos(ϱn)eiϕ/2+sin(ϱn)ei(ϕ/2+θn)|2η=1/2[1+sin(2ϱn)cos(ϕ+θn)]η,
PnNIn,
(1/2)NnN[1+sin(2ϱn)cos(ϕ+θn)]η.
σdP2=nN[(Pη)2ση2+(Pρn)2σρn2+(Pθn)2σθn2]+(Pϕ)2σϕ2=P2N(σηη)2+nN[(2Pcos(2ρn)cos(ϕ+θn)1+sin(2ρn)cos(ϕ+θn))2σρn2+(Psin(2ρn)sin(ϕ+θn)1+sin(2ρn)cos(ϕ+θn))2σθn2]+[nNPsin(ϕ+θn)sin(2ρn)1+sin(2ρn)cos(ϕ+θn)]2σϕ2.

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