Abstract

Here we describe a laboratory procedure by which we have increased the resolution of a measurement of the position of an optical component by a factor of 16. The factor of 16 arises from a four-fold quantum enhancement through the use of an N = 4 N00N state and a four-fold classical enhancement from a quadruple pass through a prism pair. The possibility of achieving supersensitivity using this method is discussed.

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  1. V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics5(4), 222–229 (2011).
    [CrossRef]
  2. 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(13), 2733–2736 (2000).
    [CrossRef] [PubMed]
  3. M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature429(6988), 161–164 (2004).
    [CrossRef] [PubMed]
  4. P. Walther, J. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, “de Broglie wavelength of a nonlocal four-photon state,” Nature429(6988), 158–161 (2004).
    [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,” Science316(5825), 726–729 (2007).
    [CrossRef] [PubMed]
  6. 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]
  7. N. Thomas-Peter, B. Smith, A. Datta, L. Zhang, U. Dorner, and I. Walmsley, “Real-world quantum sensors: evaluating resources for precision measurement,” Phys. Rev. Lett.107(11), 0113603 (2011).
    [CrossRef]
  8. G. M. Gehring, H. Shin, R. Boyd, C. M. Kim, and B. S. Ham, “Tunable optical time delay of quantum signals using a prism pair,” Opt. Express18(18), 19156–19162 (2010).
    [CrossRef] [PubMed]
  9. B. C. Barish and R. Weiss, “LIGO and the detection of gravitational waves,” Phys. Today52(10), 44–50 (1999).
    [CrossRef]
  10. F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica9(7), 686–698 (1942).
    [CrossRef]
  11. F. Marquardt and S. Girvin, “Optomechanics,” Physics2, 40 (2009).
    [CrossRef]
  12. B. L. Higgins, D. W. Berry, S. D. Bartlett, H. M. Wiseman, and G. J. Pryde, “Entanglement-free Heisenberg-limited phase estimation,” Nature450(7168), 393–396 (2007).
    [CrossRef] [PubMed]
  13. S. Braunstein and C. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett.72(22), 3439–3443 (1994).
    [CrossRef] [PubMed]
  14. R. A. Fisher, “Theory of statistical estimation,” Mathematical Proceedings of the Cambridge Philosophical Society22(05), 700–725 (1925).
    [CrossRef]
  15. 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(22), 0223601 (2007).
    [CrossRef]
  16. R. Campos, B. Saleh, and M. Teich, “Quantum-mechanical lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A40(3), 1371–1384 (1989).
    [CrossRef] [PubMed]
  17. H. Shin, K. W. C. Chan, H. J. Chang, and R. W. Boyd, “Quantum spatial superresolution by optical centroid measurements,” Phys. Rev. Lett.107, 083603 (2011).
    [CrossRef] [PubMed]
  18. O. Steuernagel, “de Broglie wavelength reduction for a multiphoton wave packet,” Phys. Rev. A65, 033820 (2002).
    [CrossRef]
  19. J. U. White, “Long optical paths of large aperture,” J. Opt. Soc. Am.32(5), 285–288 (1942).
    [CrossRef]

2011 (3)

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics5(4), 222–229 (2011).
[CrossRef]

N. Thomas-Peter, B. Smith, A. Datta, L. Zhang, U. Dorner, and I. Walmsley, “Real-world quantum sensors: evaluating resources for precision measurement,” Phys. Rev. Lett.107(11), 0113603 (2011).
[CrossRef]

H. Shin, K. W. C. Chan, H. J. Chang, and R. W. Boyd, “Quantum spatial superresolution by optical centroid measurements,” Phys. Rev. Lett.107, 083603 (2011).
[CrossRef] [PubMed]

2010 (1)

2009 (1)

F. Marquardt and S. Girvin, “Optomechanics,” Physics2, 40 (2009).
[CrossRef]

2007 (3)

B. L. Higgins, D. W. Berry, S. D. Bartlett, H. M. Wiseman, and G. J. Pryde, “Entanglement-free Heisenberg-limited phase estimation,” Nature450(7168), 393–396 (2007).
[CrossRef] [PubMed]

T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four-entangled photons,” Science316(5825), 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(22), 0223601 (2007).
[CrossRef]

2004 (2)

M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature429(6988), 161–164 (2004).
[CrossRef] [PubMed]

P. Walther, J. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, “de Broglie wavelength of a nonlocal four-photon state,” Nature429(6988), 158–161 (2004).
[CrossRef] [PubMed]

2002 (2)

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]

O. Steuernagel, “de Broglie wavelength reduction for a multiphoton wave packet,” Phys. Rev. A65, 033820 (2002).
[CrossRef]

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(13), 2733–2736 (2000).
[CrossRef] [PubMed]

1999 (1)

B. C. Barish and R. Weiss, “LIGO and the detection of gravitational waves,” Phys. Today52(10), 44–50 (1999).
[CrossRef]

1994 (1)

S. Braunstein and C. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett.72(22), 3439–3443 (1994).
[CrossRef] [PubMed]

1989 (1)

R. Campos, B. Saleh, and M. Teich, “Quantum-mechanical lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A40(3), 1371–1384 (1989).
[CrossRef] [PubMed]

1942 (2)

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica9(7), 686–698 (1942).
[CrossRef]

J. U. White, “Long optical paths of large aperture,” J. Opt. Soc. Am.32(5), 285–288 (1942).
[CrossRef]

1925 (1)

R. A. Fisher, “Theory of statistical estimation,” Mathematical Proceedings of the Cambridge Philosophical Society22(05), 700–725 (1925).
[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(13), 2733–2736 (2000).
[CrossRef] [PubMed]

Aspelmeyer, M.

P. Walther, J. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, “de Broglie wavelength of a nonlocal four-photon state,” Nature429(6988), 158–161 (2004).
[CrossRef] [PubMed]

Barish, B. C.

B. C. Barish and R. Weiss, “LIGO and the detection of gravitational waves,” Phys. Today52(10), 44–50 (1999).
[CrossRef]

Bartlett, S. D.

B. L. Higgins, D. W. Berry, S. D. Bartlett, H. M. Wiseman, and G. J. Pryde, “Entanglement-free Heisenberg-limited phase estimation,” Nature450(7168), 393–396 (2007).
[CrossRef] [PubMed]

Berry, D. W.

B. L. Higgins, D. W. Berry, S. D. Bartlett, H. M. Wiseman, and G. J. Pryde, “Entanglement-free Heisenberg-limited phase estimation,” Nature450(7168), 393–396 (2007).
[CrossRef] [PubMed]

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(13), 2733–2736 (2000).
[CrossRef] [PubMed]

Boyd, R.

Boyd, R. W.

H. Shin, K. W. C. Chan, H. J. Chang, and R. W. Boyd, “Quantum spatial superresolution by optical centroid measurements,” Phys. Rev. Lett.107, 083603 (2011).
[CrossRef] [PubMed]

Braunstein, S.

S. Braunstein and C. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett.72(22), 3439–3443 (1994).
[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(13), 2733–2736 (2000).
[CrossRef] [PubMed]

Campos, R.

R. Campos, B. Saleh, and M. Teich, “Quantum-mechanical lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A40(3), 1371–1384 (1989).
[CrossRef] [PubMed]

Caves, C.

S. Braunstein and C. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett.72(22), 3439–3443 (1994).
[CrossRef] [PubMed]

Chan, K. W. C.

H. Shin, K. W. C. Chan, H. J. Chang, and R. W. Boyd, “Quantum spatial superresolution by optical centroid measurements,” Phys. Rev. Lett.107, 083603 (2011).
[CrossRef] [PubMed]

Chang, H. J.

H. Shin, K. W. C. Chan, H. J. Chang, and R. W. Boyd, “Quantum spatial superresolution by optical centroid measurements,” Phys. Rev. Lett.107, 083603 (2011).
[CrossRef] [PubMed]

Datta, A.

N. Thomas-Peter, B. Smith, A. Datta, L. Zhang, U. Dorner, and I. Walmsley, “Real-world quantum sensors: evaluating resources for precision measurement,” Phys. Rev. Lett.107(11), 0113603 (2011).
[CrossRef]

Dorner, U.

N. Thomas-Peter, B. Smith, A. Datta, L. Zhang, U. Dorner, and I. Walmsley, “Real-world quantum sensors: evaluating resources for precision measurement,” Phys. Rev. Lett.107(11), 0113603 (2011).
[CrossRef]

Dowling, J. 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(13), 2733–2736 (2000).
[CrossRef] [PubMed]

Fisher, R. A.

R. A. Fisher, “Theory of statistical estimation,” Mathematical Proceedings of the Cambridge Philosophical Society22(05), 700–725 (1925).
[CrossRef]

Gasparoni, S.

P. Walther, J. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, “de Broglie wavelength of a nonlocal four-photon state,” Nature429(6988), 158–161 (2004).
[CrossRef] [PubMed]

Gehring, G. M.

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(22), 0223601 (2007).
[CrossRef]

Giovannetti, V.

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics5(4), 222–229 (2011).
[CrossRef]

Girvin, S.

F. Marquardt and S. Girvin, “Optomechanics,” Physics2, 40 (2009).
[CrossRef]

Ham, B. S.

Higgins, B. L.

B. L. Higgins, D. W. Berry, S. D. Bartlett, H. M. Wiseman, and G. J. Pryde, “Entanglement-free Heisenberg-limited phase estimation,” Nature450(7168), 393–396 (2007).
[CrossRef] [PubMed]

Kim, C. M.

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(13), 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]

Lloyd, S.

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics5(4), 222–229 (2011).
[CrossRef]

Lundeen, J. S.

M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature429(6988), 161–164 (2004).
[CrossRef] [PubMed]

Maccone, L.

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics5(4), 222–229 (2011).
[CrossRef]

Marquardt, F.

F. Marquardt and S. Girvin, “Optomechanics,” Physics2, 40 (2009).
[CrossRef]

Mitchell, M. W.

M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature429(6988), 161–164 (2004).
[CrossRef] [PubMed]

Nagata, T.

T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four-entangled photons,” Science316(5825), 726–729 (2007).
[CrossRef] [PubMed]

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,” Science316(5825), 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(22), 0223601 (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,” Science316(5825), 726–729 (2007).
[CrossRef] [PubMed]

Pan, J.

P. Walther, J. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, “de Broglie wavelength of a nonlocal four-photon state,” Nature429(6988), 158–161 (2004).
[CrossRef] [PubMed]

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(22), 0223601 (2007).
[CrossRef]

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(22), 0223601 (2007).
[CrossRef]

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(22), 0223601 (2007).
[CrossRef]

B. L. Higgins, D. W. Berry, S. D. Bartlett, H. M. Wiseman, and G. J. Pryde, “Entanglement-free Heisenberg-limited phase estimation,” Nature450(7168), 393–396 (2007).
[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(22), 0223601 (2007).
[CrossRef]

Saleh, B.

R. Campos, B. Saleh, and M. Teich, “Quantum-mechanical lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A40(3), 1371–1384 (1989).
[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,” Science316(5825), 726–729 (2007).
[CrossRef] [PubMed]

Shin, H.

H. Shin, K. W. C. Chan, H. J. Chang, and R. W. Boyd, “Quantum spatial superresolution by optical centroid measurements,” Phys. Rev. Lett.107, 083603 (2011).
[CrossRef] [PubMed]

G. M. Gehring, H. Shin, R. Boyd, C. M. Kim, and B. S. Ham, “Tunable optical time delay of quantum signals using a prism pair,” Opt. Express18(18), 19156–19162 (2010).
[CrossRef] [PubMed]

Smith, B.

N. Thomas-Peter, B. Smith, A. Datta, L. Zhang, U. Dorner, and I. Walmsley, “Real-world quantum sensors: evaluating resources for precision measurement,” Phys. Rev. Lett.107(11), 0113603 (2011).
[CrossRef]

Steinberg, A. M.

M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature429(6988), 161–164 (2004).
[CrossRef] [PubMed]

Steuernagel, O.

O. Steuernagel, “de Broglie wavelength reduction for a multiphoton wave packet,” Phys. Rev. A65, 033820 (2002).
[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,” Science316(5825), 726–729 (2007).
[CrossRef] [PubMed]

Teich, M.

R. Campos, B. Saleh, and M. Teich, “Quantum-mechanical lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A40(3), 1371–1384 (1989).
[CrossRef] [PubMed]

Thomas-Peter, N.

N. Thomas-Peter, B. Smith, A. Datta, L. Zhang, U. Dorner, and I. Walmsley, “Real-world quantum sensors: evaluating resources for precision measurement,” Phys. Rev. Lett.107(11), 0113603 (2011).
[CrossRef]

Ursin, R.

P. Walther, J. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, “de Broglie wavelength of a nonlocal four-photon state,” Nature429(6988), 158–161 (2004).
[CrossRef] [PubMed]

Walmsley, I.

N. Thomas-Peter, B. Smith, A. Datta, L. Zhang, U. Dorner, and I. Walmsley, “Real-world quantum sensors: evaluating resources for precision measurement,” Phys. Rev. Lett.107(11), 0113603 (2011).
[CrossRef]

Walther, P.

P. Walther, J. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, “de Broglie wavelength of a nonlocal four-photon state,” Nature429(6988), 158–161 (2004).
[CrossRef] [PubMed]

Weiss, R.

B. C. Barish and R. Weiss, “LIGO and the detection of gravitational waves,” Phys. Today52(10), 44–50 (1999).
[CrossRef]

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(22), 0223601 (2007).
[CrossRef]

White, J. U.

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(13), 2733–2736 (2000).
[CrossRef] [PubMed]

Wiseman, H. M.

B. L. Higgins, D. W. Berry, S. D. Bartlett, H. M. Wiseman, and G. J. Pryde, “Entanglement-free Heisenberg-limited phase estimation,” Nature450(7168), 393–396 (2007).
[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]

Zeilinger, A.

P. Walther, J. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, “de Broglie wavelength of a nonlocal four-photon state,” Nature429(6988), 158–161 (2004).
[CrossRef] [PubMed]

Zernike, F.

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica9(7), 686–698 (1942).
[CrossRef]

Zhang, L.

N. Thomas-Peter, B. Smith, A. Datta, L. Zhang, U. Dorner, and I. Walmsley, “Real-world quantum sensors: evaluating resources for precision measurement,” Phys. Rev. Lett.107(11), 0113603 (2011).
[CrossRef]

J. Opt. Soc. Am. (1)

Mathematical Proceedings of the Cambridge Philosophical Society (1)

R. A. Fisher, “Theory of statistical estimation,” Mathematical Proceedings of the Cambridge Philosophical Society22(05), 700–725 (1925).
[CrossRef]

Nat. Photonics (1)

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics5(4), 222–229 (2011).
[CrossRef]

Nature (3)

M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature429(6988), 161–164 (2004).
[CrossRef] [PubMed]

P. Walther, J. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, “de Broglie wavelength of a nonlocal four-photon state,” Nature429(6988), 158–161 (2004).
[CrossRef] [PubMed]

B. L. Higgins, D. W. Berry, S. D. Bartlett, H. M. Wiseman, and G. J. Pryde, “Entanglement-free Heisenberg-limited phase estimation,” Nature450(7168), 393–396 (2007).
[CrossRef] [PubMed]

Opt. Eng. (1)

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. Express (1)

Phys. Rev. A (2)

R. Campos, B. Saleh, and M. Teich, “Quantum-mechanical lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A40(3), 1371–1384 (1989).
[CrossRef] [PubMed]

O. Steuernagel, “de Broglie wavelength reduction for a multiphoton wave packet,” Phys. Rev. A65, 033820 (2002).
[CrossRef]

Phys. Rev. Lett. (5)

H. Shin, K. W. C. Chan, H. J. Chang, and R. W. Boyd, “Quantum spatial superresolution by optical centroid measurements,” Phys. Rev. Lett.107, 083603 (2011).
[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(22), 0223601 (2007).
[CrossRef]

N. Thomas-Peter, B. Smith, A. Datta, L. Zhang, U. Dorner, and I. Walmsley, “Real-world quantum sensors: evaluating resources for precision measurement,” Phys. Rev. Lett.107(11), 0113603 (2011).
[CrossRef]

S. Braunstein and C. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett.72(22), 3439–3443 (1994).
[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(13), 2733–2736 (2000).
[CrossRef] [PubMed]

Phys. Today (1)

B. C. Barish and R. Weiss, “LIGO and the detection of gravitational waves,” Phys. Today52(10), 44–50 (1999).
[CrossRef]

Physica (1)

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica9(7), 686–698 (1942).
[CrossRef]

Physics (1)

F. Marquardt and S. Girvin, “Optomechanics,” Physics2, 40 (2009).
[CrossRef]

Science (1)

T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four-entangled photons,” Science316(5825), 726–729 (2007).
[CrossRef] [PubMed]

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Figures (2)

Fig. 1
Fig. 1

(a) Schematic of the experimental setup. BS: non-polarizing beam splitter, IF: interference filter, PZT: Piezoelectric actuator, PP: prism pair, and APD: detector. Inset: The detailed prism pair alignment and the beam path for M = 2 through the prism pair. d: the lateral displacement of two prisms, x: the longitudinal movement of a prism, and L: the hypotenuse length. (b) Simplified schematic of the setup, and the four-photon states at each stage I, II, and III. ϕ is the phase difference between two paths, A and B.

Fig. 2
Fig. 2

Experimental data showing interference patterns obtained as a function of piezo voltage. (a) Single-photon count rates for a strongly attenuated coherent state at 800 nm without any passes through the prism pair (M = 1). Two-photon count rates for the N = 2 N00N state with (b) M = 2 and (c) M = 4 multiple passes through the prism pair. (d) Four-photon count rates for the N = 4 N00N state for M = 4 multiple passes through the prism pair, and (e) the zoomed-in plot of the dashed box in (d). The interference patterns in (a)–(d) were performed under identical experimental conditions. The solid curves are theoretical fits to the data. Error bars show the standard deviation of counts ( ± counts ).

Equations (3)

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Δ ϕ ( N , M ) 1 / ( NMV η g η net q )
| Ψ 2 = 1 2 ( e i 2 ϕ | 20 A B + | 02 A B )
| Ψ 4 = 1 2 ( e i 4 ϕ | 40 AB + | 04 AB ) + 1 2 e i 2 ϕ | 22 AB ,

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