Abstract

We show how the entangled photons produced in parametric down conversion can be used to improve the sensitivity of a Sagnac interferometer. Two-photon and four-photon coincidences increases the sensitivity by a factor of two and four respectively. Our results apply to sources with arbitrary pumping and squeezing parameters.

© 2007 Optical Society of America

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  1. G. Sagnac, "L’ether lumineux demontre par l’effect du vent relatif d’ether dans un interferometre en rotation uniforme," C. R. Acad. Sci. 157, 708-710 (1913).
  2. G. Bertocchi, O. Alibart, D. B. Ostrowsky, S. Tanzilli, and P. Baldi, "Single-photon Sagnac interferometer," J. Phys. B 39, 1011-1016 (2006).
    [CrossRef]
  3. P. Grangier, G. Roger, and A. Aspect, "Experimental evidence for a photon anticorrelation effect on a beam splitter: a new light on single-photon interferences," Europhys. Lett. 1, 173 (1986).
    [CrossRef]
  4. P. Hariharan, N. Brown, and B. C. Sanders, "Interference of independent laser-beams at the single-photon level," J. Mod. Opt. 40, 113-122 (1993).
    [CrossRef]
  5. A. Zeilinger, "Experiment and the foundations of quantum physics," Rev. Mod. Phys. 71, S288-S297 (1999).
    [CrossRef]
  6. M. J. Holland and K. Burnett, "Interferometric detection of optical-phase shifts at the heisenberg limit," Phys. Rev. Lett. 71, 1355-1358 (1993).
    [CrossRef] [PubMed]
  7. J. P. Dowling, "Correlated input-port, matter-wave interferometer: Quantum-noise limits to the atom-laser gyroscope," Phys. Rev. A 57, 4736-4746 (1998).
    [CrossRef]
  8. A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, " Role of entanglement in two-photon imaging," Phys. Rev. Lett. 87, 123602 (2001).
    [CrossRef] [PubMed]
  9. T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, "Optical imaging by means of 2-photon quantum entanglement," Phys. Rev. A 52, R3429-R3432 (1995).
    [CrossRef] [PubMed]
  10. 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]
  11. E. Yablonovitch and R. B. Vrijen, "Optical projection lithography at half the Rayleigh resolution limit by twophoton exposure," Opt. Eng. 38, 334-338 (1999).
    [CrossRef]
  12. D. V. Korobkin and E. Yablonovitch, "Two-fold spatial resolution enhancement by two-photon exposure of photographic film," Opt. Eng. 41, 1729-1732 (2002).
    [CrossRef]
  13. G. S. Agarwal, R. W. Boyd, E. M. Nagasako, and S. J. Bentley, "Comment on "Quantum interferometric optical lithography: Exploiting entanglement to beat the diffraction limit," Phys. Rev. Lett. 86, 1389-1389 (2001).
    [CrossRef] [PubMed]
  14. G. Bjork, L. L. Sanchez-Soto, and J. Soderholm, "Entangled-state lithography: Tailoring any pattern with a single state," Phys. Rev. Lett. 86, 4516-4519 (2001).
    [CrossRef] [PubMed]
  15. M. D’Angelo, M. V. Chekhova, and Y. Shih, "Two-photon diffraction and quantum lithography," Phys. Rev. Lett. 87, 013602 (2001).
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  18. J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, "2-photon interference in a mach-zehnder interferometer," Phys. Rev. Lett. 65, 1348-1351 (1990).
    [CrossRef] [PubMed]
  19. Z. Y. Ou, X. Y.  Zou, L. J. Wang, and L. Mandel, "Experiment on nonclassical 4th-order interference," Phys. Rev. A 42, 2957-2965 (1990).
    [CrossRef] [PubMed]
  20. 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]
  21. M. Eibl, S. Gaertner,M. Bourennane, C. Kurtsiefer,M. Zukowski, and H. Weinfurter, "Experimental observation of four-photon entanglement from parametric down-conversion," Phys. Rev. Lett. 90, 200403 (2003).
    [CrossRef] [PubMed]
  22. 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 (London) 429, 158-161 (2004).
    [CrossRef]
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    [CrossRef]
  27. F. Jacobs and R. Zamoni, "Laser ring gyro of arbitrary shape and rotation axis," Am. J. Phys. 50, 659-660 (1982).
    [CrossRef]
  28. X. Li, P. L. Voss, J. E. Sharping, and P. Kumar, "Optical-fiber source of polarization-entangled photons in the 1550 nm telecom band," Phys. Rev. Lett. 94, 053601 (2005).
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    [CrossRef] [PubMed]
  31. M. Caminati, F. De Martini, R. Perris, F. Sciarrino, and V. Secondi, "Nonseparable Werner states in spontaneous parametric down-conversion," Phys. Rev. A 73, 032312 (2006).
    [CrossRef]
  32. A. Ourjoumtsev, R. Tualle-Brouri, and P. Grangier,"Quantum homodyne tomography of a two-photon fock state," Phys. Rev. Lett. 96, 213601 (2006).
    [CrossRef] [PubMed]
  33. G. S. Agarwal, K. W. Chan, R. W. Boyd, H. Cable, and J. P. Dowling "Quantum states of light produced by a high-gain optical parametric amplifier for use in quantum lithography," J. Opt. Soc. Am. B 24, 270 (2007).
    [CrossRef]
  34. H. J. Chang, H. Shin, M. N. O’Sullivan-Hale, and R. W. Boyd, "Implementation of subRayleigh lithography using an N-photon absorber," J. Mod. Opt. 53, 2271-2277 (2006).
    [CrossRef]
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2007 (1)

2006 (4)

M. Caminati, F. De Martini, R. Perris, F. Sciarrino, and V. Secondi, "Nonseparable Werner states in spontaneous parametric down-conversion," Phys. Rev. A 73, 032312 (2006).
[CrossRef]

A. Ourjoumtsev, R. Tualle-Brouri, and P. Grangier,"Quantum homodyne tomography of a two-photon fock state," Phys. Rev. Lett. 96, 213601 (2006).
[CrossRef] [PubMed]

H. J. Chang, H. Shin, M. N. O’Sullivan-Hale, and R. W. Boyd, "Implementation of subRayleigh lithography using an N-photon absorber," J. Mod. Opt. 53, 2271-2277 (2006).
[CrossRef]

G. Bertocchi, O. Alibart, D. B. Ostrowsky, S. Tanzilli, and P. Baldi, "Single-photon Sagnac interferometer," J. Phys. B 39, 1011-1016 (2006).
[CrossRef]

2005 (1)

X. Li, P. L. Voss, J. E. Sharping, and P. Kumar, "Optical-fiber source of polarization-entangled photons in the 1550 nm telecom band," Phys. Rev. Lett. 94, 053601 (2005).
[CrossRef] [PubMed]

2004 (1)

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 (London) 429, 158-161 (2004).
[CrossRef]

2003 (2)

M. Eibl, S. Gaertner,M. Bourennane, C. Kurtsiefer,M. Zukowski, and H. Weinfurter, "Experimental observation of four-photon entanglement from parametric down-conversion," Phys. Rev. Lett. 90, 200403 (2003).
[CrossRef] [PubMed]

G. S. Agarwal and M. O. Scully, "Magneto-optical spectroscopy with entangled photons," Opt. Lett. 28, 462-464 (2003).
[CrossRef] [PubMed]

2002 (3)

O. Steuernagel, "de Broglie wavelength reduction for multiphoton wave packet," Phys. Rev. A 65, 033820 (2002).
[CrossRef]

D. V. Korobkin and E. Yablonovitch, "Two-fold spatial resolution enhancement by two-photon exposure of photographic film," Opt. Eng. 41, 1729-1732 (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 (4)

G. S. Agarwal, R. W. Boyd, E. M. Nagasako, and S. J. Bentley, "Comment on "Quantum interferometric optical lithography: Exploiting entanglement to beat the diffraction limit," Phys. Rev. Lett. 86, 1389-1389 (2001).
[CrossRef] [PubMed]

G. Bjork, L. L. Sanchez-Soto, and J. Soderholm, "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. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, " Role of entanglement in two-photon imaging," Phys. Rev. Lett. 87, 123602 (2001).
[CrossRef] [PubMed]

2000 (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]

1999 (2)

E. Yablonovitch and R. B. Vrijen, "Optical projection lithography at half the Rayleigh resolution limit by twophoton exposure," Opt. Eng. 38, 334-338 (1999).
[CrossRef]

A. Zeilinger, "Experiment and the foundations of quantum physics," Rev. Mod. Phys. 71, S288-S297 (1999).
[CrossRef]

1998 (1)

J. P. Dowling, "Correlated input-port, matter-wave interferometer: Quantum-noise limits to the atom-laser gyroscope," Phys. Rev. A 57, 4736-4746 (1998).
[CrossRef]

1996 (1)

C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wooters, "Mixed-state entanglement and quantum error correction," Phys. Rev. A 54, 3824-3851 (1996).
[CrossRef] [PubMed]

1995 (2)

J. Jacobson, G. Bjork, I. Chuang, and Y. Yamomoto, "Photonic de broglie waves," Phys. Rev. Lett. 74, 4835- 4838 (1995).
[CrossRef] [PubMed]

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, "Optical imaging by means of 2-photon quantum entanglement," Phys. Rev. A 52, R3429-R3432 (1995).
[CrossRef] [PubMed]

1993 (2)

M. J. Holland and K. Burnett, "Interferometric detection of optical-phase shifts at the heisenberg limit," Phys. Rev. Lett. 71, 1355-1358 (1993).
[CrossRef] [PubMed]

P. Hariharan, N. Brown, and B. C. Sanders, "Interference of independent laser-beams at the single-photon level," J. Mod. Opt. 40, 113-122 (1993).
[CrossRef]

1990 (3)

Z. Y. Ou, L. J. Wang, X. Y. Zou, and L. Mandel, "Evidence for phase memory in 2-photon down conversion through entanglement with the vacuum," Phys. Rev. A 41, 566-568 (1990).
[CrossRef] [PubMed]

J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, "2-photon interference in a mach-zehnder interferometer," Phys. Rev. Lett. 65, 1348-1351 (1990).
[CrossRef] [PubMed]

Z. Y. Ou, X. Y.  Zou, L. J. Wang, and L. Mandel, "Experiment on nonclassical 4th-order interference," Phys. Rev. A 42, 2957-2965 (1990).
[CrossRef] [PubMed]

1986 (1)

P. Grangier, G. Roger, and A. Aspect, "Experimental evidence for a photon anticorrelation effect on a beam splitter: a new light on single-photon interferences," Europhys. Lett. 1, 173 (1986).
[CrossRef]

1982 (1)

F. Jacobs and R. Zamoni, "Laser ring gyro of arbitrary shape and rotation axis," Am. J. Phys. 50, 659-660 (1982).
[CrossRef]

1967 (1)

E. J. Post, "Sagnac effect," Rev. Mod. Phys. 39, 475 (1967).
[CrossRef]

1913 (1)

G. Sagnac, "L’ether lumineux demontre par l’effect du vent relatif d’ether dans un interferometre en rotation uniforme," C. R. Acad. Sci. 157, 708-710 (1913).

Abouraddy, A. F.

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, " Role of entanglement in two-photon imaging," Phys. Rev. Lett. 87, 123602 (2001).
[CrossRef] [PubMed]

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] [PubMed]

Agarwal, G. S.

Alibart, O.

G. Bertocchi, O. Alibart, D. B. Ostrowsky, S. Tanzilli, and P. Baldi, "Single-photon Sagnac interferometer," J. Phys. B 39, 1011-1016 (2006).
[CrossRef]

Aspect, A.

P. Grangier, G. Roger, and A. Aspect, "Experimental evidence for a photon anticorrelation effect on a beam splitter: a new light on single-photon interferences," Europhys. Lett. 1, 173 (1986).
[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 (London) 429, 158-161 (2004).
[CrossRef]

Baldi, P.

G. Bertocchi, O. Alibart, D. B. Ostrowsky, S. Tanzilli, and P. Baldi, "Single-photon Sagnac interferometer," J. Phys. B 39, 1011-1016 (2006).
[CrossRef]

Bennett, C. H.

C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wooters, "Mixed-state entanglement and quantum error correction," Phys. Rev. A 54, 3824-3851 (1996).
[CrossRef] [PubMed]

Bentley, S. J.

G. S. Agarwal, R. W. Boyd, E. M. Nagasako, and S. J. Bentley, "Comment on "Quantum interferometric optical lithography: Exploiting entanglement to beat the diffraction limit," Phys. Rev. Lett. 86, 1389-1389 (2001).
[CrossRef] [PubMed]

Bertocchi, G.

G. Bertocchi, O. Alibart, D. B. Ostrowsky, S. Tanzilli, and P. Baldi, "Single-photon Sagnac interferometer," J. Phys. B 39, 1011-1016 (2006).
[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] [PubMed]

Bourennane, M.

M. Eibl, S. Gaertner,M. Bourennane, C. Kurtsiefer,M. Zukowski, and H. Weinfurter, "Experimental observation of four-photon entanglement from parametric down-conversion," Phys. Rev. Lett. 90, 200403 (2003).
[CrossRef] [PubMed]

Boyd, R. W.

G. S. Agarwal, K. W. Chan, R. W. Boyd, H. Cable, and J. P. Dowling "Quantum states of light produced by a high-gain optical parametric amplifier for use in quantum lithography," J. Opt. Soc. Am. B 24, 270 (2007).
[CrossRef]

H. J. Chang, H. Shin, M. N. O’Sullivan-Hale, and R. W. Boyd, "Implementation of subRayleigh lithography using an N-photon absorber," J. Mod. Opt. 53, 2271-2277 (2006).
[CrossRef]

G. S. Agarwal, R. W. Boyd, E. M. Nagasako, and S. J. Bentley, "Comment on "Quantum interferometric optical lithography: Exploiting entanglement to beat the diffraction limit," Phys. Rev. Lett. 86, 1389-1389 (2001).
[CrossRef] [PubMed]

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] [PubMed]

Brown, N.

P. Hariharan, N. Brown, and B. C. Sanders, "Interference of independent laser-beams at the single-photon level," J. Mod. Opt. 40, 113-122 (1993).
[CrossRef]

Burnett, K.

M. J. Holland and K. Burnett, "Interferometric detection of optical-phase shifts at the heisenberg limit," Phys. Rev. Lett. 71, 1355-1358 (1993).
[CrossRef] [PubMed]

Cable, H.

Caminati, M.

M. Caminati, F. De Martini, R. Perris, F. Sciarrino, and V. Secondi, "Nonseparable Werner states in spontaneous parametric down-conversion," Phys. Rev. A 73, 032312 (2006).
[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, "2-photon interference in a mach-zehnder interferometer," Phys. Rev. Lett. 65, 1348-1351 (1990).
[CrossRef] [PubMed]

Chan, K. W.

Chang, H. J.

H. J. Chang, H. Shin, M. N. O’Sullivan-Hale, and R. W. Boyd, "Implementation of subRayleigh lithography using an N-photon absorber," J. Mod. Opt. 53, 2271-2277 (2006).
[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] [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]

De Martini, F.

M. Caminati, F. De Martini, R. Perris, F. Sciarrino, and V. Secondi, "Nonseparable Werner states in spontaneous parametric down-conversion," Phys. Rev. A 73, 032312 (2006).
[CrossRef]

DiVincenzo, D. P.

C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wooters, "Mixed-state entanglement and quantum error correction," Phys. Rev. A 54, 3824-3851 (1996).
[CrossRef] [PubMed]

Dowling, J. P.

G. S. Agarwal, K. W. Chan, R. W. Boyd, H. Cable, and J. P. Dowling "Quantum states of light produced by a high-gain optical parametric amplifier for use in quantum lithography," J. Opt. Soc. Am. B 24, 270 (2007).
[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]

J. P. Dowling, "Correlated input-port, matter-wave interferometer: Quantum-noise limits to the atom-laser gyroscope," Phys. Rev. A 57, 4736-4746 (1998).
[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] [PubMed]

Eibl, M.

M. Eibl, S. Gaertner,M. Bourennane, C. Kurtsiefer,M. Zukowski, and H. Weinfurter, "Experimental observation of four-photon entanglement from parametric down-conversion," Phys. Rev. Lett. 90, 200403 (2003).
[CrossRef] [PubMed]

Gaertner, S.

M. Eibl, S. Gaertner,M. Bourennane, C. Kurtsiefer,M. Zukowski, and H. Weinfurter, "Experimental observation of four-photon entanglement from parametric down-conversion," Phys. Rev. Lett. 90, 200403 (2003).
[CrossRef] [PubMed]

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 (London) 429, 158-161 (2004).
[CrossRef]

Grangier, P.

A. Ourjoumtsev, R. Tualle-Brouri, and P. Grangier,"Quantum homodyne tomography of a two-photon fock state," Phys. Rev. Lett. 96, 213601 (2006).
[CrossRef] [PubMed]

P. Grangier, G. Roger, and A. Aspect, "Experimental evidence for a photon anticorrelation effect on a beam splitter: a new light on single-photon interferences," Europhys. Lett. 1, 173 (1986).
[CrossRef]

Hariharan, P.

P. Hariharan, N. Brown, and B. C. Sanders, "Interference of independent laser-beams at the single-photon level," J. Mod. Opt. 40, 113-122 (1993).
[CrossRef]

Holland, M. J.

M. J. Holland and K. Burnett, "Interferometric detection of optical-phase shifts at the heisenberg limit," Phys. Rev. Lett. 71, 1355-1358 (1993).
[CrossRef] [PubMed]

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]

Jacobs, F.

F. Jacobs and R. Zamoni, "Laser ring gyro of arbitrary shape and rotation axis," Am. J. Phys. 50, 659-660 (1982).
[CrossRef]

Jacobson, J.

J. Jacobson, G. Bjork, I. Chuang, and Y. Yamomoto, "Photonic de broglie waves," Phys. Rev. Lett. 74, 4835- 4838 (1995).
[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, "2-photon interference in a mach-zehnder interferometer," Phys. Rev. Lett. 65, 1348-1351 (1990).
[CrossRef] [PubMed]

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] [PubMed]

Korobkin, D. V.

D. V. Korobkin and E. Yablonovitch, "Two-fold spatial resolution enhancement by two-photon exposure of photographic film," Opt. Eng. 41, 1729-1732 (2002).
[CrossRef]

Kumar, P.

X. Li, P. L. Voss, J. E. Sharping, and P. Kumar, "Optical-fiber source of polarization-entangled photons in the 1550 nm telecom band," Phys. Rev. Lett. 94, 053601 (2005).
[CrossRef] [PubMed]

Kurtsiefer, C.

M. Eibl, S. Gaertner,M. Bourennane, C. Kurtsiefer,M. Zukowski, and H. Weinfurter, "Experimental observation of four-photon entanglement from parametric down-conversion," Phys. Rev. Lett. 90, 200403 (2003).
[CrossRef] [PubMed]

Larchuk, T.

J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, "2-photon interference in a mach-zehnder interferometer," Phys. Rev. Lett. 65, 1348-1351 (1990).
[CrossRef] [PubMed]

Li, X.

X. Li, P. L. Voss, J. E. Sharping, and P. Kumar, "Optical-fiber source of polarization-entangled photons in the 1550 nm telecom band," Phys. Rev. Lett. 94, 053601 (2005).
[CrossRef] [PubMed]

Mandel, L.

Z. Y. Ou, L. J. Wang, X. Y. Zou, and L. Mandel, "Evidence for phase memory in 2-photon down conversion through entanglement with the vacuum," Phys. Rev. A 41, 566-568 (1990).
[CrossRef] [PubMed]

Z. Y. Ou, X. Y.  Zou, L. J. Wang, and L. Mandel, "Experiment on nonclassical 4th-order interference," Phys. Rev. A 42, 2957-2965 (1990).
[CrossRef] [PubMed]

Nagasako, E. M.

G. S. Agarwal, R. W. Boyd, E. M. Nagasako, and S. J. Bentley, "Comment on "Quantum interferometric optical lithography: Exploiting entanglement to beat the diffraction limit," Phys. Rev. Lett. 86, 1389-1389 (2001).
[CrossRef] [PubMed]

O’Sullivan-Hale, M. N.

H. J. Chang, H. Shin, M. N. O’Sullivan-Hale, and R. W. Boyd, "Implementation of subRayleigh lithography using an N-photon absorber," J. Mod. Opt. 53, 2271-2277 (2006).
[CrossRef]

Ostrowsky, D. B.

G. Bertocchi, O. Alibart, D. B. Ostrowsky, S. Tanzilli, and P. Baldi, "Single-photon Sagnac interferometer," J. Phys. B 39, 1011-1016 (2006).
[CrossRef]

Ou, Z. Y.

Z. Y. Ou, X. Y.  Zou, L. J. Wang, and L. Mandel, "Experiment on nonclassical 4th-order interference," Phys. Rev. A 42, 2957-2965 (1990).
[CrossRef] [PubMed]

Z. Y. Ou, L. J. Wang, X. Y. Zou, and L. Mandel, "Evidence for phase memory in 2-photon down conversion through entanglement with the vacuum," Phys. Rev. A 41, 566-568 (1990).
[CrossRef] [PubMed]

Ourjoumtsev, A.

A. Ourjoumtsev, R. Tualle-Brouri, and P. Grangier,"Quantum homodyne tomography of a two-photon fock state," Phys. Rev. Lett. 96, 213601 (2006).
[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 (London) 429, 158-161 (2004).
[CrossRef]

Perris, R.

M. Caminati, F. De Martini, R. Perris, F. Sciarrino, and V. Secondi, "Nonseparable Werner states in spontaneous parametric down-conversion," Phys. Rev. A 73, 032312 (2006).
[CrossRef]

Pittman, T. B.

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, "Optical imaging by means of 2-photon quantum entanglement," Phys. Rev. A 52, R3429-R3432 (1995).
[CrossRef] [PubMed]

Post, E. J.

E. J. Post, "Sagnac effect," Rev. Mod. Phys. 39, 475 (1967).
[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, "2-photon interference in a mach-zehnder interferometer," Phys. Rev. Lett. 65, 1348-1351 (1990).
[CrossRef] [PubMed]

Roger, G.

P. Grangier, G. Roger, and A. Aspect, "Experimental evidence for a photon anticorrelation effect on a beam splitter: a new light on single-photon interferences," Europhys. Lett. 1, 173 (1986).
[CrossRef]

Sagnac, G.

G. Sagnac, "L’ether lumineux demontre par l’effect du vent relatif d’ether dans un interferometre en rotation uniforme," C. R. Acad. Sci. 157, 708-710 (1913).

Saleh, B. E. A.

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, " Role of entanglement in two-photon imaging," Phys. Rev. Lett. 87, 123602 (2001).
[CrossRef] [PubMed]

J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, "2-photon interference in a mach-zehnder interferometer," Phys. Rev. Lett. 65, 1348-1351 (1990).
[CrossRef] [PubMed]

Sanders, B. C.

P. Hariharan, N. Brown, and B. C. Sanders, "Interference of independent laser-beams at the single-photon level," J. Mod. Opt. 40, 113-122 (1993).
[CrossRef]

Sciarrino, F.

M. Caminati, F. De Martini, R. Perris, F. Sciarrino, and V. Secondi, "Nonseparable Werner states in spontaneous parametric down-conversion," Phys. Rev. A 73, 032312 (2006).
[CrossRef]

Scully, M. O.

Secondi, V.

M. Caminati, F. De Martini, R. Perris, F. Sciarrino, and V. Secondi, "Nonseparable Werner states in spontaneous parametric down-conversion," Phys. Rev. A 73, 032312 (2006).
[CrossRef]

Sergienko, A. V.

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, " Role of entanglement in two-photon imaging," Phys. Rev. Lett. 87, 123602 (2001).
[CrossRef] [PubMed]

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, "Optical imaging by means of 2-photon quantum entanglement," Phys. Rev. A 52, R3429-R3432 (1995).
[CrossRef] [PubMed]

Sharping, J. E.

X. Li, P. L. Voss, J. E. Sharping, and P. Kumar, "Optical-fiber source of polarization-entangled photons in the 1550 nm telecom band," Phys. Rev. Lett. 94, 053601 (2005).
[CrossRef] [PubMed]

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]

Shih, Y. H.

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, "Optical imaging by means of 2-photon quantum entanglement," Phys. Rev. A 52, R3429-R3432 (1995).
[CrossRef] [PubMed]

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]

Shin, H.

H. J. Chang, H. Shin, M. N. O’Sullivan-Hale, and R. W. Boyd, "Implementation of subRayleigh lithography using an N-photon absorber," J. Mod. Opt. 53, 2271-2277 (2006).
[CrossRef]

Smolin, J. A.

C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wooters, "Mixed-state entanglement and quantum error correction," Phys. Rev. A 54, 3824-3851 (1996).
[CrossRef] [PubMed]

Steuernagel, O.

O. Steuernagel, "de Broglie wavelength reduction for multiphoton wave packet," Phys. Rev. A 65, 033820 (2002).
[CrossRef]

Strekalov, D. V.

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, "Optical imaging by means of 2-photon quantum entanglement," Phys. Rev. A 52, R3429-R3432 (1995).
[CrossRef] [PubMed]

Tanzilli, S.

G. Bertocchi, O. Alibart, D. B. Ostrowsky, S. Tanzilli, and P. Baldi, "Single-photon Sagnac interferometer," J. Phys. B 39, 1011-1016 (2006).
[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, "2-photon interference in a mach-zehnder interferometer," Phys. Rev. Lett. 65, 1348-1351 (1990).
[CrossRef] [PubMed]

Teich, M. C.

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, " Role of entanglement in two-photon imaging," Phys. Rev. Lett. 87, 123602 (2001).
[CrossRef] [PubMed]

J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, "2-photon interference in a mach-zehnder interferometer," Phys. Rev. Lett. 65, 1348-1351 (1990).
[CrossRef] [PubMed]

Tualle-Brouri, R.

A. Ourjoumtsev, R. Tualle-Brouri, and P. Grangier,"Quantum homodyne tomography of a two-photon fock state," Phys. Rev. Lett. 96, 213601 (2006).
[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 (London) 429, 158-161 (2004).
[CrossRef]

Voss, P. L.

X. Li, P. L. Voss, J. E. Sharping, and P. Kumar, "Optical-fiber source of polarization-entangled photons in the 1550 nm telecom band," Phys. Rev. Lett. 94, 053601 (2005).
[CrossRef] [PubMed]

Vrijen, R. B.

E. Yablonovitch and R. B. Vrijen, "Optical projection lithography at half the Rayleigh resolution limit by twophoton exposure," Opt. Eng. 38, 334-338 (1999).
[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 (London) 429, 158-161 (2004).
[CrossRef]

Wang, L. J.

Z. Y. Ou, X. Y.  Zou, L. J. Wang, and L. Mandel, "Experiment on nonclassical 4th-order interference," Phys. Rev. A 42, 2957-2965 (1990).
[CrossRef] [PubMed]

Z. Y. Ou, L. J. Wang, X. Y. Zou, and L. Mandel, "Evidence for phase memory in 2-photon down conversion through entanglement with the vacuum," Phys. Rev. A 41, 566-568 (1990).
[CrossRef] [PubMed]

Weinfurter, H.

M. Eibl, S. Gaertner,M. Bourennane, C. Kurtsiefer,M. Zukowski, and H. Weinfurter, "Experimental observation of four-photon entanglement from parametric down-conversion," Phys. Rev. Lett. 90, 200403 (2003).
[CrossRef] [PubMed]

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] [PubMed]

Wooters, W. K.

C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wooters, "Mixed-state entanglement and quantum error correction," Phys. Rev. A 54, 3824-3851 (1996).
[CrossRef] [PubMed]

Yablonovitch, E.

D. V. Korobkin and E. Yablonovitch, "Two-fold spatial resolution enhancement by two-photon exposure of photographic film," Opt. Eng. 41, 1729-1732 (2002).
[CrossRef]

E. Yablonovitch and R. B. Vrijen, "Optical projection lithography at half the Rayleigh resolution limit by twophoton exposure," Opt. Eng. 38, 334-338 (1999).
[CrossRef]

Zamoni, R.

F. Jacobs and R. Zamoni, "Laser ring gyro of arbitrary shape and rotation axis," Am. J. Phys. 50, 659-660 (1982).
[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 (London) 429, 158-161 (2004).
[CrossRef]

A. Zeilinger, "Experiment and the foundations of quantum physics," Rev. Mod. Phys. 71, S288-S297 (1999).
[CrossRef]

Zou, X. Y.

Z. Y. Ou, L. J. Wang, X. Y. Zou, and L. Mandel, "Evidence for phase memory in 2-photon down conversion through entanglement with the vacuum," Phys. Rev. A 41, 566-568 (1990).
[CrossRef] [PubMed]

Z. Y. Ou, X. Y.  Zou, L. J. Wang, and L. Mandel, "Experiment on nonclassical 4th-order interference," Phys. Rev. A 42, 2957-2965 (1990).
[CrossRef] [PubMed]

Zukowski, M.

M. Eibl, S. Gaertner,M. Bourennane, C. Kurtsiefer,M. Zukowski, and H. Weinfurter, "Experimental observation of four-photon entanglement from parametric down-conversion," Phys. Rev. Lett. 90, 200403 (2003).
[CrossRef] [PubMed]

Am. J. Phys. (1)

F. Jacobs and R. Zamoni, "Laser ring gyro of arbitrary shape and rotation axis," Am. J. Phys. 50, 659-660 (1982).
[CrossRef]

C. R. Acad. Sci. (1)

G. Sagnac, "L’ether lumineux demontre par l’effect du vent relatif d’ether dans un interferometre en rotation uniforme," C. R. Acad. Sci. 157, 708-710 (1913).

Europhys. Lett. (1)

P. Grangier, G. Roger, and A. Aspect, "Experimental evidence for a photon anticorrelation effect on a beam splitter: a new light on single-photon interferences," Europhys. Lett. 1, 173 (1986).
[CrossRef]

J. Mod. Opt. (2)

P. Hariharan, N. Brown, and B. C. Sanders, "Interference of independent laser-beams at the single-photon level," J. Mod. Opt. 40, 113-122 (1993).
[CrossRef]

H. J. Chang, H. Shin, M. N. O’Sullivan-Hale, and R. W. Boyd, "Implementation of subRayleigh lithography using an N-photon absorber," J. Mod. Opt. 53, 2271-2277 (2006).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Phys. B (1)

G. Bertocchi, O. Alibart, D. B. Ostrowsky, S. Tanzilli, and P. Baldi, "Single-photon Sagnac interferometer," J. Phys. B 39, 1011-1016 (2006).
[CrossRef]

Nature (London) (1)

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 (London) 429, 158-161 (2004).
[CrossRef]

Opt. Eng. (2)

E. Yablonovitch and R. B. Vrijen, "Optical projection lithography at half the Rayleigh resolution limit by twophoton exposure," Opt. Eng. 38, 334-338 (1999).
[CrossRef]

D. V. Korobkin and E. Yablonovitch, "Two-fold spatial resolution enhancement by two-photon exposure of photographic film," Opt. Eng. 41, 1729-1732 (2002).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (7)

Z. Y. Ou, L. J. Wang, X. Y. Zou, and L. Mandel, "Evidence for phase memory in 2-photon down conversion through entanglement with the vacuum," Phys. Rev. A 41, 566-568 (1990).
[CrossRef] [PubMed]

Z. Y. Ou, X. Y.  Zou, L. J. Wang, and L. Mandel, "Experiment on nonclassical 4th-order interference," Phys. Rev. A 42, 2957-2965 (1990).
[CrossRef] [PubMed]

J. P. Dowling, "Correlated input-port, matter-wave interferometer: Quantum-noise limits to the atom-laser gyroscope," Phys. Rev. A 57, 4736-4746 (1998).
[CrossRef]

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, "Optical imaging by means of 2-photon quantum entanglement," Phys. Rev. A 52, R3429-R3432 (1995).
[CrossRef] [PubMed]

O. Steuernagel, "de Broglie wavelength reduction for multiphoton wave packet," Phys. Rev. A 65, 033820 (2002).
[CrossRef]

C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wooters, "Mixed-state entanglement and quantum error correction," Phys. Rev. A 54, 3824-3851 (1996).
[CrossRef] [PubMed]

M. Caminati, F. De Martini, R. Perris, F. Sciarrino, and V. Secondi, "Nonseparable Werner states in spontaneous parametric down-conversion," Phys. Rev. A 73, 032312 (2006).
[CrossRef]

Phys. Rev. Lett. (12)

A. Ourjoumtsev, R. Tualle-Brouri, and P. Grangier,"Quantum homodyne tomography of a two-photon fock state," Phys. Rev. Lett. 96, 213601 (2006).
[CrossRef] [PubMed]

J. Jacobson, G. Bjork, I. Chuang, and Y. Yamomoto, "Photonic de broglie waves," Phys. Rev. Lett. 74, 4835- 4838 (1995).
[CrossRef] [PubMed]

X. Li, P. L. Voss, J. E. Sharping, and P. Kumar, "Optical-fiber source of polarization-entangled photons in the 1550 nm telecom band," Phys. Rev. Lett. 94, 053601 (2005).
[CrossRef] [PubMed]

M. J. Holland and K. Burnett, "Interferometric detection of optical-phase shifts at the heisenberg limit," Phys. Rev. Lett. 71, 1355-1358 (1993).
[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]

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, " Role of entanglement in two-photon imaging," Phys. Rev. Lett. 87, 123602 (2001).
[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]

M. Eibl, S. Gaertner,M. Bourennane, C. Kurtsiefer,M. Zukowski, and H. Weinfurter, "Experimental observation of four-photon entanglement from parametric down-conversion," Phys. Rev. Lett. 90, 200403 (2003).
[CrossRef] [PubMed]

J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, "2-photon interference in a mach-zehnder interferometer," Phys. Rev. Lett. 65, 1348-1351 (1990).
[CrossRef] [PubMed]

G. S. Agarwal, R. W. Boyd, E. M. Nagasako, and S. J. Bentley, "Comment on "Quantum interferometric optical lithography: Exploiting entanglement to beat the diffraction limit," Phys. Rev. Lett. 86, 1389-1389 (2001).
[CrossRef] [PubMed]

G. Bjork, L. L. Sanchez-Soto, and J. Soderholm, "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]

Rev. Mod. Phys. (2)

A. Zeilinger, "Experiment and the foundations of quantum physics," Rev. Mod. Phys. 71, S288-S297 (1999).
[CrossRef]

E. J. Post, "Sagnac effect," Rev. Mod. Phys. 39, 475 (1967).
[CrossRef]

Other (3)

M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information, (Cambridge University Press, Cambridge, 2000).

W. Schleich and M. O. Scully, 1984 modern trends in Atomic and Molecular Physics, Proceedings of Les Houches Summer School, Session 38, R. Stora and G. Grynberg, eds., North Holland, Amsterdam
[PubMed]

H. Lefevre, The Fiber-Optic Gyroscope, (Artech House, Boston, 1993).

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Figures (5)

Fig. 1.
Fig. 1.

Schematics of a Sagnac ring interferometer.

Fig. 2.
Fig. 2.

The Sagnac interferometer setup with classical input. The input field Ein is separated by the beam splitter into two counter-propagating waves tEin and rEin . Because of the rotation they end up at the beam splitter at different times (t 1,t 2).

Fig. 3.
Fig. 3.

The equivalent optical network diagram of the Sagnac interferometer for quantized fields. “ϕ” represents the phase shift provided by the rotating loop of the interferometer. The detectors D 3 and D 4 with the extra beam splitters (dashed lines) are to be used for four-photon coincidence counting.

Fig. 4.
Fig. 4.

Normalized four-photon probability in coincidence in 2-by-2 (solid line) and 3-by-1 (dashed line) detection scheme described by Eqs. (32) and (34) respectively.

Fig. 5.
Fig. 5.

The Sagnac interferometer setup for four-photon coincidence detection in 3-by-1 scheme.

Equations (47)

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

t = 2 π R c ,
t 1 = 2 πR c + R Ω ,
t 2 = 2 πR c R Ω .
Δ t = t 2 t 1 = 4 πR 2 Ω c 2 R 2 Ω 2 .
Δ t = 4 πR 2 Ω c 2 .
ϕ = ω Δ t = 8 π λc A Ω ,
ϕ = 4 πLR Ω λc .
E 1 = r rE in e iωt 2 + t 2 E in e t 1 = E in e t 2 ie 2 sin ( ϕ 2 ) ,
E 2 = r tE in e iωt 1 + t r E in e t 2 = E in e t 2 ie 2 cos ( ϕ 2 ) ,
I 1 = E 1 2 = E in 2 sin 2 ( ϕ 2 ) ,
I 2 = E 2 2 = E in 2 cos 2 ( ϕ 2 ) ,
( b 1 b 2 ) = 1 2 ( 1 i i 1 ) BS out ( 1 0 0 e ) SAGNAC 1 2 ( 1 i i 1 ) BS in ( a 1 a 2 ) ,
= ie 2 ( sin ( ϕ 2 ) cos ( ϕ 2 ) cos ( ϕ 2 ) sin ( ϕ 2 ) ) ( a 1 a 2 ) ,
b i = j = 1 2 S ij a j ,
a i b i = j = 1 2 S ij a j U a i U ,
a i b i = j = 1 2 S ij * a j U a i U ,
U 10 = U a 1 00 = U a 1 U U 00 = Ua 1 U 00 .
U a i U = j = 1 2 S ji a j .
U 10 = sin ( ϕ 2 ) 10 + cos ( ϕ 2 ) 01 ,
U 11 = 1 2 sin ( ϕ ) ( 20 + 02 ) + cos ( ϕ ) 11 ,
I 1 b 1 b 1 = sin 2 ( ϕ 2 ) ,
I 2 b 2 b 2 = cos 2 ( ϕ 2 ) ,
I 12 b 1 b 2 b 2 b 1 = cos 2 ( ϕ ) ,
11 20 + 02 2 20 + e i 2 ϕ 02 2 ,
ψ = 1 cosh r n = 0 ( e tanh r ) n n 1 n 2 .
E = cos h 2 r log 2 ( cosh 2 r ) sinh 2 r log 2 ( sinh 2 r ) .
I 1 b 1 b 1 = sinh 2 r = b 2 b 2 I 2 ,
g 12 2 = b 1 b 2 b 2 b 1 b 1 b 1 b 2 b 2 1 = cos 2 ( ϕ ) coth 2 r ,
P 2 = T r [ 11 11 ρ ( ϕ ) ] 11 U ψ 2
= tanh 2 r cosh 2 r cos 2 ( ϕ ) ,
P 4 = 1 D 1 1 D 3 1 D 2 1 D 4 U B U S ψ , 0 v 1 0 v 2 2
= 1 D 1 1 D 3 1 D 2 1 D 4 U B U S ψ , 0 v 1 0 v 2 ψ , 0 v 1 0 v 2 U S U B 1 D 1 1 D 3 1 D 2 1 D 4
U B 1 D 1 1 D 3 1 D 2 1 D 4 = U B D 1 D 3 D 2 D 4 U B U B 0000
= ( t 1 * b 1 + r 1 * v 1 ) ( t 2 * b 2 + r 2 * v 2 ) ( r 1 * b 1 + t 1 * v 1 ) ( r 2 * b 2 + t 2 * v 2 ) 0000
= t 1 * t 2 * r 1 * r 2 * b 1 2 b 2 2 0000 + ,
U S t 1 * t 2 * r 1 * r 2 * b 1 2 b 2 2 0000 = t 1 * t 2 * r 1 * r 2 * U S b 1 2 b 2 2 U S U S 0000
= t 1 * t 2 * r 1 * r 2 * ( sin ( ϕ 2 ) a 1 + cos ( ϕ 2 ) a 2 ) 2 × ( cos ( ϕ 2 ) a 1 + sin ( ϕ 2 ) a 2 ) 2 0000
= ( 1 2 sin ( ϕ ) ( a 1 2 + a 2 2 ) + cos ( ϕ ) a 1 a 2 ) 2 0000
= 1 2 [ 1 + 3 cos ( 2 ϕ ) ] 2200 + ,
P 4 = tanh 4 r cosh 2 r t 1 t 2 r 1 r 2 2 1 4 [ 1 + 3 cos ( 2 ϕ ) ] 2 .
22 = a 1 2 2 a 2 2 2 00 1 2 ( iu + l 2 ) 2 ( u + il 2 ) 2 00
1 8 { ( ib 1 + b 2 2 ) 4 + e i 4 ϕ ( b 1 + ib 2 2 ) 4 + 2 e i 2 ϕ ( ib 1 + b 2 2 ) 2 ( b 1 + ib 2 2 ) 2 } 00
= 1 4 { e i 2 ϕ sin ( 2 ϕ ) b 1 3 b 2 + } 00
e i 2 ϕ 4 sin ( 2 ϕ ) { ( r 1 d 1 + t 1 ( r 2 d 2 + t 2 d 3 ) ) 3 b 2 } 0000
= e i 2 ϕ 4 sin ( 2 ϕ ) { 6 r 1 t 1 2 r 2 t 2 d 1 d 2 d 3 d 2 + } 0000
= 3 2 e i 2 ϕ sin ( 2 ϕ ) r 1 t 1 2 r 2 t 2 1111 .
P 4 ( 3 by 1 ) = tanh 4 r cosh 2 r t 1 2 t 2 r 1 r 2 2 9 4 sin 2 ( 2 ϕ ) .

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