Abstract

The phase estimation performance of photonic N00N states propagating in an attenuating medium is analyzed. It is shown that the Heisenberg limit is never achieved and that an attenuated separable state of N photons will actually produce a better phase estimate than an equally attenuated N00N state unless the transmittance of the medium is sufficiently high. Thus, for most practical applications with realistic attenuation, N00N-state-based phase estimation actually performs worse than the standard quantum limit. This performance deficit becomes more pronounced as the number of photons in the signal increases.

© 2008 Optical Society of America

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  1. B. C. Sanders and G. J. Milburn, “Optimal quantum measurements for phase estimation,” Phys. Rev. Lett. 75, 2944-2947 (1995).
    [CrossRef] [PubMed]
  2. 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]
  3. K. T. Kapale, L. D. Didomencio, H. Lee, P. Kok, and J. P. Dowling, “Quantum interferometric sensors,” Concepts Phys. 2, 225-240 (2005).
  4. V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006).
    [CrossRef] [PubMed]
  5. 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]
  6. P. Kok, A. N. Boto, D. S. Abrams, C. P. Williams, S. L. Braunstein, and J. P. Dowling, “Quantum-intereferometric optical lithography: towards arbitrary two-dimensional patterns,” Phys. Rev. A 63, 063407 (2001).
    [CrossRef]
  7. 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]
  8. V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced positioning and clock synchronization,” Nature 412, 417-419 (2001).
    [CrossRef] [PubMed]
  9. R. Jozsa, D. S. Abrams, J. P. Dowling, and C. P. Williams, “Quantum clock synchronization based on shared prior entanglement,” Phys. Rev. Lett. 85, 2010-2013 (2000).
    [CrossRef] [PubMed]
  10. L. A. Lugiato, A. Gatti, and E. Brambilla, “Quantum imaging,” J. Opt. B: Quantum Semiclassical Opt. 4, S176-S183 (2002).
    [CrossRef]
  11. D. V. Strekalov and J. P. Dowling, “Two-photon intereferometry for high-resolution imaging,” J. Mod. Opt. 49, 519-527 (2002).
    [CrossRef]
  12. For a comrehesive review see V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306, 1330-1336 (2004).
    [CrossRef] [PubMed]
  13. Z. Zhao, Y.-A. Chen, A.-N. Zhang, T. Yang, H. J. Briegel, and J.-W. Pan, “Experimental demonstration of five-photon entanglement and open-destination teleportation,” Nature 430, 54-58 (2004).
    [CrossRef] [PubMed]
  14. C.-Y. Lu, X.-Q. Zhou, O. Guhne, W.-B. Gao, J. Zhang, Z.-S. Yuan, A. Goebel, T. Yang, and J.-W. Pan, “Experimental entanglement of six photons in graph states,” Nat. Phys. 3, 91-95 (2007).
    [CrossRef]
  15. D. Leibrfried, E. Knill, S. Seidlin, J. Britton, R. B. Blakestad, J. Chiaverini, D. B. Hume, W. M. Itano, J. D. Jost, C. Langer, R. Ozeri, R. Reichle, and D. J. Wineland, “Creation of six-atom 'Schrodinger cat' state,” Nature 438, 639-642 (2005).
    [CrossRef]
  16. H. Haffner, W. Hansel, C. F. Roos, J. Benhelm, D. Chek-al-kar, M. Chwalla, T. Kober, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Guhne, W. Dur, and R. Blatt, “Scalable multiparticle entanglement of trapped ions,” Nature 438, 643-646 (2005).
    [CrossRef] [PubMed]
  17. P. Walther, J.-W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, “De Broglie wavelength of a non-local four-photon state,” Nature 429, 158-161 (2004).
    [CrossRef] [PubMed]
  18. M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature 429, 161-164 (2004).
    [CrossRef] [PubMed]
  19. V. Meyer, M. A. Rowe, D. Kielpinski, C. A. Sackett, W. M. Itano, C. Monroe, and D. J. Wineland, “Experimental demonstration of entanglement-enhanced rotation angle estimation using trapped ions,” Phys. Rev. Lett. 86, 5870-5873 (2001).
    [CrossRef] [PubMed]
  20. D. Leibrfried, M. D. Barrett, T. Schaetz, J. Britton, J. Chiaverini, W. M. Itano, J. D. Jost, C. Langer, and D. J. Wineland, “Toward Heisenberg-limited spectroscopy with multiparticle entangled states,” Science 304, 1476-1478 (2004).
    [CrossRef]
  21. S. F. Huelga, C. Macchiavello, T. Pellizzari, A. K. Ekert, M. B. Plenio, and J. I. Cirac, “Improvement of frequency standards with quantum entanglement,” Phys. Rev. Lett. 79, 3865-3868 (1997).
    [CrossRef]
  22. J. A. Dunningham, K. Burnett, and S. M. Barnett, “Interferometry below the standard quantum limit with Bose-Einstein condensates,” Phys. Rev. Lett. 89, 150401 (2002).
    [CrossRef] [PubMed]
  23. R. C. Pooser and O. Pfister, “Particle-number scaling of the phase sensitivity in realistic Bayesian twin-mode Heisenberg-limited interferometry,” Phys. Rev. A 69, 043616 (2004).
    [CrossRef]
  24. T. Kim, J. Dunningham, and K. Burnett, “Precision measurement scheme using a quantum interferometer,” Phys. Rev. A 72, 055801 (2005).
    [CrossRef]
  25. J. Dunningham and T. Kim, “Using quantum interferometers to make measurements at the Heisenberg limit,” J. Mod. Opt. 53, 557-571 (2006).
    [CrossRef]
  26. A. V. Chizhov, E. Schmidt, L. Knoll, and D. G. Welsch, “Propogation of entangled light pulses through dispersing and absorbing channels,” J. Opt. B: Quantum Semiclassical Opt. 3, 77-83 (2001).
    [CrossRef]
  27. X. Chen and L. Jiang, “The entanglement and phase measurement performance of the damped NOON state,” J. Phys. B 40, 2799-2808 (2007).
    [CrossRef]
  28. M. A. Rubin and S. Kaushik, “Loss-induced limits to phase measurement and precision with maximally entangled states,” Phys. Rev. A 75, 053805 (2007).
    [CrossRef]
  29. H. Lee, P. Kok, and J. P. Dowling, “A quantum Rosetta stone for interferometry,” J. Mod. Opt. 49, 2325-2338 (2002).
    [CrossRef]
  30. In this and subsequent calculations we have approximated the phase measurement uncertainty by Eq. , so that our quantitative results are valid only when the phase error is small. This approximation has been analyzed in .
  31. L. Pezze and A. Smerzi, “Phase sensitivity of a Mach-Zehnder interferometer,” Phys. Rev. A 73, 011801(R) (2006).
    [CrossRef]
  32. G. A. Durkin and J. P. Dowling, “Local and global distinguishability in quantum interferometry,” Phys. Rev. Lett. 99, 070801 (2007).
    [CrossRef] [PubMed]
  33. R. Loudon, The Quantum Theory of Light (Oxford U. Press, 2000).
  34. The field operator function, b(z), has different physical dimensionality than do the field operators ai, with the former related to the latter by a factor of a square root of length.
  35. J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, “Optical frequency measurements with maximally correlated states,” Phys. Rev. A 54, R4649-R4652 (1996).
    [CrossRef] [PubMed]
  36. C. C. Gerry, “Heisenberg-limit interferometry with four-wave mixers operating in a nonlinear regime,” Phys. Rev. A 61, 043811 (2000).
    [CrossRef]
  37. C. C. Gerry and R. A. Campos, “Generation of maximally entangled states of a Bose-Einstein condensate and Heisenberg-limited phase resolution,” Phys. Rev. A 68, 025602 (2003).
    [CrossRef]
  38. The photons in the separable state are injected into the interferometer at different times, labeled by tk, where k=1,...,N, with measurements independently applied to each photon. The index k is used to properly count these separate injection events.

2007 (4)

C.-Y. Lu, X.-Q. Zhou, O. Guhne, W.-B. Gao, J. Zhang, Z.-S. Yuan, A. Goebel, T. Yang, and J.-W. Pan, “Experimental entanglement of six photons in graph states,” Nat. Phys. 3, 91-95 (2007).
[CrossRef]

X. Chen and L. Jiang, “The entanglement and phase measurement performance of the damped NOON state,” J. Phys. B 40, 2799-2808 (2007).
[CrossRef]

M. A. Rubin and S. Kaushik, “Loss-induced limits to phase measurement and precision with maximally entangled states,” Phys. Rev. A 75, 053805 (2007).
[CrossRef]

G. A. Durkin and J. P. Dowling, “Local and global distinguishability in quantum interferometry,” Phys. Rev. Lett. 99, 070801 (2007).
[CrossRef] [PubMed]

2006 (3)

L. Pezze and A. Smerzi, “Phase sensitivity of a Mach-Zehnder interferometer,” Phys. Rev. A 73, 011801(R) (2006).
[CrossRef]

J. Dunningham and T. Kim, “Using quantum interferometers to make measurements at the Heisenberg limit,” J. Mod. Opt. 53, 557-571 (2006).
[CrossRef]

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006).
[CrossRef] [PubMed]

2005 (4)

D. Leibrfried, E. Knill, S. Seidlin, J. Britton, R. B. Blakestad, J. Chiaverini, D. B. Hume, W. M. Itano, J. D. Jost, C. Langer, R. Ozeri, R. Reichle, and D. J. Wineland, “Creation of six-atom 'Schrodinger cat' state,” Nature 438, 639-642 (2005).
[CrossRef]

H. Haffner, W. Hansel, C. F. Roos, J. Benhelm, D. Chek-al-kar, M. Chwalla, T. Kober, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Guhne, W. Dur, and R. Blatt, “Scalable multiparticle entanglement of trapped ions,” Nature 438, 643-646 (2005).
[CrossRef] [PubMed]

T. Kim, J. Dunningham, and K. Burnett, “Precision measurement scheme using a quantum interferometer,” Phys. Rev. A 72, 055801 (2005).
[CrossRef]

K. T. Kapale, L. D. Didomencio, H. Lee, P. Kok, and J. P. Dowling, “Quantum interferometric sensors,” Concepts Phys. 2, 225-240 (2005).

2004 (6)

D. Leibrfried, M. D. Barrett, T. Schaetz, J. Britton, J. Chiaverini, W. M. Itano, J. D. Jost, C. Langer, and D. J. Wineland, “Toward Heisenberg-limited spectroscopy with multiparticle entangled states,” Science 304, 1476-1478 (2004).
[CrossRef]

R. C. Pooser and O. Pfister, “Particle-number scaling of the phase sensitivity in realistic Bayesian twin-mode Heisenberg-limited interferometry,” Phys. Rev. A 69, 043616 (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] [PubMed]

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

For a comrehesive review see V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306, 1330-1336 (2004).
[CrossRef] [PubMed]

Z. Zhao, Y.-A. Chen, A.-N. Zhang, T. Yang, H. J. Briegel, and J.-W. Pan, “Experimental demonstration of five-photon entanglement and open-destination teleportation,” Nature 430, 54-58 (2004).
[CrossRef] [PubMed]

2003 (1)

C. C. Gerry and R. A. Campos, “Generation of maximally entangled states of a Bose-Einstein condensate and Heisenberg-limited phase resolution,” Phys. Rev. A 68, 025602 (2003).
[CrossRef]

2002 (4)

H. Lee, P. Kok, and J. P. Dowling, “A quantum Rosetta stone for interferometry,” J. Mod. Opt. 49, 2325-2338 (2002).
[CrossRef]

J. A. Dunningham, K. Burnett, and S. M. Barnett, “Interferometry below the standard quantum limit with Bose-Einstein condensates,” Phys. Rev. Lett. 89, 150401 (2002).
[CrossRef] [PubMed]

L. A. Lugiato, A. Gatti, and E. Brambilla, “Quantum imaging,” J. Opt. B: Quantum Semiclassical Opt. 4, S176-S183 (2002).
[CrossRef]

D. V. Strekalov and J. P. Dowling, “Two-photon intereferometry for high-resolution imaging,” J. Mod. Opt. 49, 519-527 (2002).
[CrossRef]

2001 (5)

V. Meyer, M. A. Rowe, D. Kielpinski, C. A. Sackett, W. M. Itano, C. Monroe, and D. J. Wineland, “Experimental demonstration of entanglement-enhanced rotation angle estimation using trapped ions,” Phys. Rev. Lett. 86, 5870-5873 (2001).
[CrossRef] [PubMed]

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

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]

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced positioning and clock synchronization,” Nature 412, 417-419 (2001).
[CrossRef] [PubMed]

A. V. Chizhov, E. Schmidt, L. Knoll, and D. G. Welsch, “Propogation of entangled light pulses through dispersing and absorbing channels,” J. Opt. B: Quantum Semiclassical Opt. 3, 77-83 (2001).
[CrossRef]

2000 (3)

C. C. Gerry, “Heisenberg-limit interferometry with four-wave mixers operating in a nonlinear regime,” Phys. Rev. A 61, 043811 (2000).
[CrossRef]

R. Jozsa, D. S. Abrams, J. P. Dowling, and C. P. Williams, “Quantum clock synchronization based on shared prior entanglement,” Phys. Rev. Lett. 85, 2010-2013 (2000).
[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]

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]

1997 (1)

S. F. Huelga, C. Macchiavello, T. Pellizzari, A. K. Ekert, M. B. Plenio, and J. I. Cirac, “Improvement of frequency standards with quantum entanglement,” Phys. Rev. Lett. 79, 3865-3868 (1997).
[CrossRef]

1996 (1)

J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, “Optical frequency measurements with maximally correlated states,” Phys. Rev. A 54, R4649-R4652 (1996).
[CrossRef] [PubMed]

1995 (1)

B. C. Sanders and G. J. Milburn, “Optimal quantum measurements for phase estimation,” Phys. Rev. Lett. 75, 2944-2947 (1995).
[CrossRef] [PubMed]

Abrams, D. S.

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

R. Jozsa, D. S. Abrams, J. P. Dowling, and C. P. Williams, “Quantum clock synchronization based on shared prior entanglement,” Phys. Rev. Lett. 85, 2010-2013 (2000).
[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]

Aspelmeyer, M.

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

Barnett, S. M.

J. A. Dunningham, K. Burnett, and S. M. Barnett, “Interferometry below the standard quantum limit with Bose-Einstein condensates,” Phys. Rev. Lett. 89, 150401 (2002).
[CrossRef] [PubMed]

Barrett, M. D.

D. Leibrfried, M. D. Barrett, T. Schaetz, J. Britton, J. Chiaverini, W. M. Itano, J. D. Jost, C. Langer, and D. J. Wineland, “Toward Heisenberg-limited spectroscopy with multiparticle entangled states,” Science 304, 1476-1478 (2004).
[CrossRef]

Becher, C.

H. Haffner, W. Hansel, C. F. Roos, J. Benhelm, D. Chek-al-kar, M. Chwalla, T. Kober, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Guhne, W. Dur, and R. Blatt, “Scalable multiparticle entanglement of trapped ions,” Nature 438, 643-646 (2005).
[CrossRef] [PubMed]

Benhelm, J.

H. Haffner, W. Hansel, C. F. Roos, J. Benhelm, D. Chek-al-kar, M. Chwalla, T. Kober, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Guhne, W. Dur, and R. Blatt, “Scalable multiparticle entanglement of trapped ions,” Nature 438, 643-646 (2005).
[CrossRef] [PubMed]

Bjork, G.

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]

Blakestad, R. B.

D. Leibrfried, E. Knill, S. Seidlin, J. Britton, R. B. Blakestad, J. Chiaverini, D. B. Hume, W. M. Itano, J. D. Jost, C. Langer, R. Ozeri, R. Reichle, and D. J. Wineland, “Creation of six-atom 'Schrodinger cat' state,” Nature 438, 639-642 (2005).
[CrossRef]

Blatt, R.

H. Haffner, W. Hansel, C. F. Roos, J. Benhelm, D. Chek-al-kar, M. Chwalla, T. Kober, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Guhne, W. Dur, and R. Blatt, “Scalable multiparticle entanglement of trapped ions,” Nature 438, 643-646 (2005).
[CrossRef] [PubMed]

Bollinger, J. J.

J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, “Optical frequency measurements with maximally correlated states,” Phys. Rev. A 54, R4649-R4652 (1996).
[CrossRef] [PubMed]

Boto, A. N.

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

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

Brambilla, E.

L. A. Lugiato, A. Gatti, and E. Brambilla, “Quantum imaging,” J. Opt. B: Quantum Semiclassical Opt. 4, S176-S183 (2002).
[CrossRef]

Braunstein, S. L.

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

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

Briegel, H. J.

Z. Zhao, Y.-A. Chen, A.-N. Zhang, T. Yang, H. J. Briegel, and J.-W. Pan, “Experimental demonstration of five-photon entanglement and open-destination teleportation,” Nature 430, 54-58 (2004).
[CrossRef] [PubMed]

Britton, J.

D. Leibrfried, E. Knill, S. Seidlin, J. Britton, R. B. Blakestad, J. Chiaverini, D. B. Hume, W. M. Itano, J. D. Jost, C. Langer, R. Ozeri, R. Reichle, and D. J. Wineland, “Creation of six-atom 'Schrodinger cat' state,” Nature 438, 639-642 (2005).
[CrossRef]

D. Leibrfried, M. D. Barrett, T. Schaetz, J. Britton, J. Chiaverini, W. M. Itano, J. D. Jost, C. Langer, and D. J. Wineland, “Toward Heisenberg-limited spectroscopy with multiparticle entangled states,” Science 304, 1476-1478 (2004).
[CrossRef]

Burnett, K.

T. Kim, J. Dunningham, and K. Burnett, “Precision measurement scheme using a quantum interferometer,” Phys. Rev. A 72, 055801 (2005).
[CrossRef]

J. A. Dunningham, K. Burnett, and S. M. Barnett, “Interferometry below the standard quantum limit with Bose-Einstein condensates,” Phys. Rev. Lett. 89, 150401 (2002).
[CrossRef] [PubMed]

Campos, R. A.

C. C. Gerry and R. A. Campos, “Generation of maximally entangled states of a Bose-Einstein condensate and Heisenberg-limited phase resolution,” Phys. Rev. A 68, 025602 (2003).
[CrossRef]

Chek-al-kar, D.

H. Haffner, W. Hansel, C. F. Roos, J. Benhelm, D. Chek-al-kar, M. Chwalla, T. Kober, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Guhne, W. Dur, and R. Blatt, “Scalable multiparticle entanglement of trapped ions,” Nature 438, 643-646 (2005).
[CrossRef] [PubMed]

Chen, X.

X. Chen and L. Jiang, “The entanglement and phase measurement performance of the damped NOON state,” J. Phys. B 40, 2799-2808 (2007).
[CrossRef]

Chen, Y.-A.

Z. Zhao, Y.-A. Chen, A.-N. Zhang, T. Yang, H. J. Briegel, and J.-W. Pan, “Experimental demonstration of five-photon entanglement and open-destination teleportation,” Nature 430, 54-58 (2004).
[CrossRef] [PubMed]

Chiaverini, J.

D. Leibrfried, E. Knill, S. Seidlin, J. Britton, R. B. Blakestad, J. Chiaverini, D. B. Hume, W. M. Itano, J. D. Jost, C. Langer, R. Ozeri, R. Reichle, and D. J. Wineland, “Creation of six-atom 'Schrodinger cat' state,” Nature 438, 639-642 (2005).
[CrossRef]

D. Leibrfried, M. D. Barrett, T. Schaetz, J. Britton, J. Chiaverini, W. M. Itano, J. D. Jost, C. Langer, and D. J. Wineland, “Toward Heisenberg-limited spectroscopy with multiparticle entangled states,” Science 304, 1476-1478 (2004).
[CrossRef]

Chizhov, A. V.

A. V. Chizhov, E. Schmidt, L. Knoll, and D. G. Welsch, “Propogation of entangled light pulses through dispersing and absorbing channels,” J. Opt. B: Quantum Semiclassical Opt. 3, 77-83 (2001).
[CrossRef]

Chwalla, M.

H. Haffner, W. Hansel, C. F. Roos, J. Benhelm, D. Chek-al-kar, M. Chwalla, T. Kober, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Guhne, W. Dur, and R. Blatt, “Scalable multiparticle entanglement of trapped ions,” Nature 438, 643-646 (2005).
[CrossRef] [PubMed]

Cirac, J. I.

S. F. Huelga, C. Macchiavello, T. Pellizzari, A. K. Ekert, M. B. Plenio, and J. I. Cirac, “Improvement of frequency standards with quantum entanglement,” Phys. Rev. Lett. 79, 3865-3868 (1997).
[CrossRef]

Didomencio, L. D.

K. T. Kapale, L. D. Didomencio, H. Lee, P. Kok, and J. P. Dowling, “Quantum interferometric sensors,” Concepts Phys. 2, 225-240 (2005).

Dowling, J. P.

G. A. Durkin and J. P. Dowling, “Local and global distinguishability in quantum interferometry,” Phys. Rev. Lett. 99, 070801 (2007).
[CrossRef] [PubMed]

K. T. Kapale, L. D. Didomencio, H. Lee, P. Kok, and J. P. Dowling, “Quantum interferometric sensors,” Concepts Phys. 2, 225-240 (2005).

D. V. Strekalov and J. P. Dowling, “Two-photon intereferometry for high-resolution imaging,” J. Mod. Opt. 49, 519-527 (2002).
[CrossRef]

H. Lee, P. Kok, and J. P. Dowling, “A quantum Rosetta stone for interferometry,” J. Mod. Opt. 49, 2325-2338 (2002).
[CrossRef]

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

R. Jozsa, D. S. Abrams, J. P. Dowling, and C. P. Williams, “Quantum clock synchronization based on shared prior entanglement,” Phys. Rev. Lett. 85, 2010-2013 (2000).
[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]

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]

Dunningham, J.

J. Dunningham and T. Kim, “Using quantum interferometers to make measurements at the Heisenberg limit,” J. Mod. Opt. 53, 557-571 (2006).
[CrossRef]

T. Kim, J. Dunningham, and K. Burnett, “Precision measurement scheme using a quantum interferometer,” Phys. Rev. A 72, 055801 (2005).
[CrossRef]

Dunningham, J. A.

J. A. Dunningham, K. Burnett, and S. M. Barnett, “Interferometry below the standard quantum limit with Bose-Einstein condensates,” Phys. Rev. Lett. 89, 150401 (2002).
[CrossRef] [PubMed]

Dur, W.

H. Haffner, W. Hansel, C. F. Roos, J. Benhelm, D. Chek-al-kar, M. Chwalla, T. Kober, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Guhne, W. Dur, and R. Blatt, “Scalable multiparticle entanglement of trapped ions,” Nature 438, 643-646 (2005).
[CrossRef] [PubMed]

Durkin, G. A.

G. A. Durkin and J. P. Dowling, “Local and global distinguishability in quantum interferometry,” Phys. Rev. Lett. 99, 070801 (2007).
[CrossRef] [PubMed]

Ekert, A. K.

S. F. Huelga, C. Macchiavello, T. Pellizzari, A. K. Ekert, M. B. Plenio, and J. I. Cirac, “Improvement of frequency standards with quantum entanglement,” Phys. Rev. Lett. 79, 3865-3868 (1997).
[CrossRef]

Gao, W.-B.

C.-Y. Lu, X.-Q. Zhou, O. Guhne, W.-B. Gao, J. Zhang, Z.-S. Yuan, A. Goebel, T. Yang, and J.-W. Pan, “Experimental entanglement of six photons in graph states,” Nat. Phys. 3, 91-95 (2007).
[CrossRef]

Gasparoni, S.

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

Gatti, A.

L. A. Lugiato, A. Gatti, and E. Brambilla, “Quantum imaging,” J. Opt. B: Quantum Semiclassical Opt. 4, S176-S183 (2002).
[CrossRef]

Gerry, C. C.

C. C. Gerry and R. A. Campos, “Generation of maximally entangled states of a Bose-Einstein condensate and Heisenberg-limited phase resolution,” Phys. Rev. A 68, 025602 (2003).
[CrossRef]

C. C. Gerry, “Heisenberg-limit interferometry with four-wave mixers operating in a nonlinear regime,” Phys. Rev. A 61, 043811 (2000).
[CrossRef]

Giovannetti, V.

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006).
[CrossRef] [PubMed]

For a comrehesive review see V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306, 1330-1336 (2004).
[CrossRef] [PubMed]

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced positioning and clock synchronization,” Nature 412, 417-419 (2001).
[CrossRef] [PubMed]

Goebel, A.

C.-Y. Lu, X.-Q. Zhou, O. Guhne, W.-B. Gao, J. Zhang, Z.-S. Yuan, A. Goebel, T. Yang, and J.-W. Pan, “Experimental entanglement of six photons in graph states,” Nat. Phys. 3, 91-95 (2007).
[CrossRef]

Guhne, O.

C.-Y. Lu, X.-Q. Zhou, O. Guhne, W.-B. Gao, J. Zhang, Z.-S. Yuan, A. Goebel, T. Yang, and J.-W. Pan, “Experimental entanglement of six photons in graph states,” Nat. Phys. 3, 91-95 (2007).
[CrossRef]

H. Haffner, W. Hansel, C. F. Roos, J. Benhelm, D. Chek-al-kar, M. Chwalla, T. Kober, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Guhne, W. Dur, and R. Blatt, “Scalable multiparticle entanglement of trapped ions,” Nature 438, 643-646 (2005).
[CrossRef] [PubMed]

Haffner, H.

H. Haffner, W. Hansel, C. F. Roos, J. Benhelm, D. Chek-al-kar, M. Chwalla, T. Kober, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Guhne, W. Dur, and R. Blatt, “Scalable multiparticle entanglement of trapped ions,” Nature 438, 643-646 (2005).
[CrossRef] [PubMed]

Hansel, W.

H. Haffner, W. Hansel, C. F. Roos, J. Benhelm, D. Chek-al-kar, M. Chwalla, T. Kober, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Guhne, W. Dur, and R. Blatt, “Scalable multiparticle entanglement of trapped ions,” Nature 438, 643-646 (2005).
[CrossRef] [PubMed]

Heinzen, D. J.

J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, “Optical frequency measurements with maximally correlated states,” Phys. Rev. A 54, R4649-R4652 (1996).
[CrossRef] [PubMed]

Huelga, S. F.

S. F. Huelga, C. Macchiavello, T. Pellizzari, A. K. Ekert, M. B. Plenio, and J. I. Cirac, “Improvement of frequency standards with quantum entanglement,” Phys. Rev. Lett. 79, 3865-3868 (1997).
[CrossRef]

Hume, D. B.

D. Leibrfried, E. Knill, S. Seidlin, J. Britton, R. B. Blakestad, J. Chiaverini, D. B. Hume, W. M. Itano, J. D. Jost, C. Langer, R. Ozeri, R. Reichle, and D. J. Wineland, “Creation of six-atom 'Schrodinger cat' state,” Nature 438, 639-642 (2005).
[CrossRef]

Itano, W. M.

D. Leibrfried, E. Knill, S. Seidlin, J. Britton, R. B. Blakestad, J. Chiaverini, D. B. Hume, W. M. Itano, J. D. Jost, C. Langer, R. Ozeri, R. Reichle, and D. J. Wineland, “Creation of six-atom 'Schrodinger cat' state,” Nature 438, 639-642 (2005).
[CrossRef]

D. Leibrfried, M. D. Barrett, T. Schaetz, J. Britton, J. Chiaverini, W. M. Itano, J. D. Jost, C. Langer, and D. J. Wineland, “Toward Heisenberg-limited spectroscopy with multiparticle entangled states,” Science 304, 1476-1478 (2004).
[CrossRef]

V. Meyer, M. A. Rowe, D. Kielpinski, C. A. Sackett, W. M. Itano, C. Monroe, and D. J. Wineland, “Experimental demonstration of entanglement-enhanced rotation angle estimation using trapped ions,” Phys. Rev. Lett. 86, 5870-5873 (2001).
[CrossRef] [PubMed]

J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, “Optical frequency measurements with maximally correlated states,” Phys. Rev. A 54, R4649-R4652 (1996).
[CrossRef] [PubMed]

Jiang, L.

X. Chen and L. Jiang, “The entanglement and phase measurement performance of the damped NOON state,” J. Phys. B 40, 2799-2808 (2007).
[CrossRef]

Jost, J. D.

D. Leibrfried, E. Knill, S. Seidlin, J. Britton, R. B. Blakestad, J. Chiaverini, D. B. Hume, W. M. Itano, J. D. Jost, C. Langer, R. Ozeri, R. Reichle, and D. J. Wineland, “Creation of six-atom 'Schrodinger cat' state,” Nature 438, 639-642 (2005).
[CrossRef]

D. Leibrfried, M. D. Barrett, T. Schaetz, J. Britton, J. Chiaverini, W. M. Itano, J. D. Jost, C. Langer, and D. J. Wineland, “Toward Heisenberg-limited spectroscopy with multiparticle entangled states,” Science 304, 1476-1478 (2004).
[CrossRef]

Jozsa, R.

R. Jozsa, D. S. Abrams, J. P. Dowling, and C. P. Williams, “Quantum clock synchronization based on shared prior entanglement,” Phys. Rev. Lett. 85, 2010-2013 (2000).
[CrossRef] [PubMed]

Kapale, K. T.

K. T. Kapale, L. D. Didomencio, H. Lee, P. Kok, and J. P. Dowling, “Quantum interferometric sensors,” Concepts Phys. 2, 225-240 (2005).

Kaushik, S.

M. A. Rubin and S. Kaushik, “Loss-induced limits to phase measurement and precision with maximally entangled states,” Phys. Rev. A 75, 053805 (2007).
[CrossRef]

Kielpinski, D.

V. Meyer, M. A. Rowe, D. Kielpinski, C. A. Sackett, W. M. Itano, C. Monroe, and D. J. Wineland, “Experimental demonstration of entanglement-enhanced rotation angle estimation using trapped ions,” Phys. Rev. Lett. 86, 5870-5873 (2001).
[CrossRef] [PubMed]

Kim, T.

J. Dunningham and T. Kim, “Using quantum interferometers to make measurements at the Heisenberg limit,” J. Mod. Opt. 53, 557-571 (2006).
[CrossRef]

T. Kim, J. Dunningham, and K. Burnett, “Precision measurement scheme using a quantum interferometer,” Phys. Rev. A 72, 055801 (2005).
[CrossRef]

Knill, E.

D. Leibrfried, E. Knill, S. Seidlin, J. Britton, R. B. Blakestad, J. Chiaverini, D. B. Hume, W. M. Itano, J. D. Jost, C. Langer, R. Ozeri, R. Reichle, and D. J. Wineland, “Creation of six-atom 'Schrodinger cat' state,” Nature 438, 639-642 (2005).
[CrossRef]

Knoll, L.

A. V. Chizhov, E. Schmidt, L. Knoll, and D. G. Welsch, “Propogation of entangled light pulses through dispersing and absorbing channels,” J. Opt. B: Quantum Semiclassical Opt. 3, 77-83 (2001).
[CrossRef]

Kober, T.

H. Haffner, W. Hansel, C. F. Roos, J. Benhelm, D. Chek-al-kar, M. Chwalla, T. Kober, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Guhne, W. Dur, and R. Blatt, “Scalable multiparticle entanglement of trapped ions,” Nature 438, 643-646 (2005).
[CrossRef] [PubMed]

Kok, P.

K. T. Kapale, L. D. Didomencio, H. Lee, P. Kok, and J. P. Dowling, “Quantum interferometric sensors,” Concepts Phys. 2, 225-240 (2005).

H. Lee, P. Kok, and J. P. Dowling, “A quantum Rosetta stone for interferometry,” J. Mod. Opt. 49, 2325-2338 (2002).
[CrossRef]

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

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

Langer, C.

D. Leibrfried, E. Knill, S. Seidlin, J. Britton, R. B. Blakestad, J. Chiaverini, D. B. Hume, W. M. Itano, J. D. Jost, C. Langer, R. Ozeri, R. Reichle, and D. J. Wineland, “Creation of six-atom 'Schrodinger cat' state,” Nature 438, 639-642 (2005).
[CrossRef]

D. Leibrfried, M. D. Barrett, T. Schaetz, J. Britton, J. Chiaverini, W. M. Itano, J. D. Jost, C. Langer, and D. J. Wineland, “Toward Heisenberg-limited spectroscopy with multiparticle entangled states,” Science 304, 1476-1478 (2004).
[CrossRef]

Lee, H.

K. T. Kapale, L. D. Didomencio, H. Lee, P. Kok, and J. P. Dowling, “Quantum interferometric sensors,” Concepts Phys. 2, 225-240 (2005).

H. Lee, P. Kok, and J. P. Dowling, “A quantum Rosetta stone for interferometry,” J. Mod. Opt. 49, 2325-2338 (2002).
[CrossRef]

Leibrfried, D.

D. Leibrfried, E. Knill, S. Seidlin, J. Britton, R. B. Blakestad, J. Chiaverini, D. B. Hume, W. M. Itano, J. D. Jost, C. Langer, R. Ozeri, R. Reichle, and D. J. Wineland, “Creation of six-atom 'Schrodinger cat' state,” Nature 438, 639-642 (2005).
[CrossRef]

D. Leibrfried, M. D. Barrett, T. Schaetz, J. Britton, J. Chiaverini, W. M. Itano, J. D. Jost, C. Langer, and D. J. Wineland, “Toward Heisenberg-limited spectroscopy with multiparticle entangled states,” Science 304, 1476-1478 (2004).
[CrossRef]

Lloyd, S.

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006).
[CrossRef] [PubMed]

For a comrehesive review see V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306, 1330-1336 (2004).
[CrossRef] [PubMed]

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced positioning and clock synchronization,” Nature 412, 417-419 (2001).
[CrossRef] [PubMed]

Loudon, R.

R. Loudon, The Quantum Theory of Light (Oxford U. Press, 2000).

Lu, C.-Y.

C.-Y. Lu, X.-Q. Zhou, O. Guhne, W.-B. Gao, J. Zhang, Z.-S. Yuan, A. Goebel, T. Yang, and J.-W. Pan, “Experimental entanglement of six photons in graph states,” Nat. Phys. 3, 91-95 (2007).
[CrossRef]

Lugiato, L. A.

L. A. Lugiato, A. Gatti, and E. Brambilla, “Quantum imaging,” J. Opt. B: Quantum Semiclassical Opt. 4, S176-S183 (2002).
[CrossRef]

Lundeen, J. S.

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

Macchiavello, C.

S. F. Huelga, C. Macchiavello, T. Pellizzari, A. K. Ekert, M. B. Plenio, and J. I. Cirac, “Improvement of frequency standards with quantum entanglement,” Phys. Rev. Lett. 79, 3865-3868 (1997).
[CrossRef]

Maccone, L.

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006).
[CrossRef] [PubMed]

For a comrehesive review see V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306, 1330-1336 (2004).
[CrossRef] [PubMed]

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced positioning and clock synchronization,” Nature 412, 417-419 (2001).
[CrossRef] [PubMed]

Meyer, V.

V. Meyer, M. A. Rowe, D. Kielpinski, C. A. Sackett, W. M. Itano, C. Monroe, and D. J. Wineland, “Experimental demonstration of entanglement-enhanced rotation angle estimation using trapped ions,” Phys. Rev. Lett. 86, 5870-5873 (2001).
[CrossRef] [PubMed]

Milburn, G. J.

B. C. Sanders and G. J. Milburn, “Optimal quantum measurements for phase estimation,” Phys. Rev. Lett. 75, 2944-2947 (1995).
[CrossRef] [PubMed]

Mitchell, M. W.

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

Monroe, C.

V. Meyer, M. A. Rowe, D. Kielpinski, C. A. Sackett, W. M. Itano, C. Monroe, and D. J. Wineland, “Experimental demonstration of entanglement-enhanced rotation angle estimation using trapped ions,” Phys. Rev. Lett. 86, 5870-5873 (2001).
[CrossRef] [PubMed]

Ozeri, R.

D. Leibrfried, E. Knill, S. Seidlin, J. Britton, R. B. Blakestad, J. Chiaverini, D. B. Hume, W. M. Itano, J. D. Jost, C. Langer, R. Ozeri, R. Reichle, and D. J. Wineland, “Creation of six-atom 'Schrodinger cat' state,” Nature 438, 639-642 (2005).
[CrossRef]

Pan, J.-W.

C.-Y. Lu, X.-Q. Zhou, O. Guhne, W.-B. Gao, J. Zhang, Z.-S. Yuan, A. Goebel, T. Yang, and J.-W. Pan, “Experimental entanglement of six photons in graph states,” Nat. Phys. 3, 91-95 (2007).
[CrossRef]

Z. Zhao, Y.-A. Chen, A.-N. Zhang, T. Yang, H. J. Briegel, and J.-W. Pan, “Experimental demonstration of five-photon entanglement and open-destination teleportation,” Nature 430, 54-58 (2004).
[CrossRef] [PubMed]

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

Pellizzari, T.

S. F. Huelga, C. Macchiavello, T. Pellizzari, A. K. Ekert, M. B. Plenio, and J. I. Cirac, “Improvement of frequency standards with quantum entanglement,” Phys. Rev. Lett. 79, 3865-3868 (1997).
[CrossRef]

Pezze, L.

L. Pezze and A. Smerzi, “Phase sensitivity of a Mach-Zehnder interferometer,” Phys. Rev. A 73, 011801(R) (2006).
[CrossRef]

Pfister, O.

R. C. Pooser and O. Pfister, “Particle-number scaling of the phase sensitivity in realistic Bayesian twin-mode Heisenberg-limited interferometry,” Phys. Rev. A 69, 043616 (2004).
[CrossRef]

Plenio, M. B.

S. F. Huelga, C. Macchiavello, T. Pellizzari, A. K. Ekert, M. B. Plenio, and J. I. Cirac, “Improvement of frequency standards with quantum entanglement,” Phys. Rev. Lett. 79, 3865-3868 (1997).
[CrossRef]

Pooser, R. C.

R. C. Pooser and O. Pfister, “Particle-number scaling of the phase sensitivity in realistic Bayesian twin-mode Heisenberg-limited interferometry,” Phys. Rev. A 69, 043616 (2004).
[CrossRef]

Rapol, U. D.

H. Haffner, W. Hansel, C. F. Roos, J. Benhelm, D. Chek-al-kar, M. Chwalla, T. Kober, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Guhne, W. Dur, and R. Blatt, “Scalable multiparticle entanglement of trapped ions,” Nature 438, 643-646 (2005).
[CrossRef] [PubMed]

Reichle, R.

D. Leibrfried, E. Knill, S. Seidlin, J. Britton, R. B. Blakestad, J. Chiaverini, D. B. Hume, W. M. Itano, J. D. Jost, C. Langer, R. Ozeri, R. Reichle, and D. J. Wineland, “Creation of six-atom 'Schrodinger cat' state,” Nature 438, 639-642 (2005).
[CrossRef]

Riebe, M.

H. Haffner, W. Hansel, C. F. Roos, J. Benhelm, D. Chek-al-kar, M. Chwalla, T. Kober, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Guhne, W. Dur, and R. Blatt, “Scalable multiparticle entanglement of trapped ions,” Nature 438, 643-646 (2005).
[CrossRef] [PubMed]

Roos, C. F.

H. Haffner, W. Hansel, C. F. Roos, J. Benhelm, D. Chek-al-kar, M. Chwalla, T. Kober, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Guhne, W. Dur, and R. Blatt, “Scalable multiparticle entanglement of trapped ions,” Nature 438, 643-646 (2005).
[CrossRef] [PubMed]

Rowe, M. A.

V. Meyer, M. A. Rowe, D. Kielpinski, C. A. Sackett, W. M. Itano, C. Monroe, and D. J. Wineland, “Experimental demonstration of entanglement-enhanced rotation angle estimation using trapped ions,” Phys. Rev. Lett. 86, 5870-5873 (2001).
[CrossRef] [PubMed]

Rubin, M. A.

M. A. Rubin and S. Kaushik, “Loss-induced limits to phase measurement and precision with maximally entangled states,” Phys. Rev. A 75, 053805 (2007).
[CrossRef]

Sackett, C. A.

V. Meyer, M. A. Rowe, D. Kielpinski, C. A. Sackett, W. M. Itano, C. Monroe, and D. J. Wineland, “Experimental demonstration of entanglement-enhanced rotation angle estimation using trapped ions,” Phys. Rev. Lett. 86, 5870-5873 (2001).
[CrossRef] [PubMed]

Sanchez-Soto, L. L.

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]

Sanders, B. C.

B. C. Sanders and G. J. Milburn, “Optimal quantum measurements for phase estimation,” Phys. Rev. Lett. 75, 2944-2947 (1995).
[CrossRef] [PubMed]

Schaetz, T.

D. Leibrfried, M. D. Barrett, T. Schaetz, J. Britton, J. Chiaverini, W. M. Itano, J. D. Jost, C. Langer, and D. J. Wineland, “Toward Heisenberg-limited spectroscopy with multiparticle entangled states,” Science 304, 1476-1478 (2004).
[CrossRef]

Schmidt, E.

A. V. Chizhov, E. Schmidt, L. Knoll, and D. G. Welsch, “Propogation of entangled light pulses through dispersing and absorbing channels,” J. Opt. B: Quantum Semiclassical Opt. 3, 77-83 (2001).
[CrossRef]

Schmidt, P. O.

H. Haffner, W. Hansel, C. F. Roos, J. Benhelm, D. Chek-al-kar, M. Chwalla, T. Kober, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Guhne, W. Dur, and R. Blatt, “Scalable multiparticle entanglement of trapped ions,” Nature 438, 643-646 (2005).
[CrossRef] [PubMed]

Seidlin, S.

D. Leibrfried, E. Knill, S. Seidlin, J. Britton, R. B. Blakestad, J. Chiaverini, D. B. Hume, W. M. Itano, J. D. Jost, C. Langer, R. Ozeri, R. Reichle, and D. J. Wineland, “Creation of six-atom 'Schrodinger cat' state,” Nature 438, 639-642 (2005).
[CrossRef]

Smerzi, A.

L. Pezze and A. Smerzi, “Phase sensitivity of a Mach-Zehnder interferometer,” Phys. Rev. A 73, 011801(R) (2006).
[CrossRef]

Soderholm, J.

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]

Steinberg, A. M.

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

Strekalov, D. V.

D. V. Strekalov and J. P. Dowling, “Two-photon intereferometry for high-resolution imaging,” J. Mod. Opt. 49, 519-527 (2002).
[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] [PubMed]

Walther, P.

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

Welsch, D. G.

A. V. Chizhov, E. Schmidt, L. Knoll, and D. G. Welsch, “Propogation of entangled light pulses through dispersing and absorbing channels,” J. Opt. B: Quantum Semiclassical Opt. 3, 77-83 (2001).
[CrossRef]

Williams, C. P.

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

R. Jozsa, D. S. Abrams, J. P. Dowling, and C. P. Williams, “Quantum clock synchronization based on shared prior entanglement,” Phys. Rev. Lett. 85, 2010-2013 (2000).
[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]

Wineland, D. J.

D. Leibrfried, E. Knill, S. Seidlin, J. Britton, R. B. Blakestad, J. Chiaverini, D. B. Hume, W. M. Itano, J. D. Jost, C. Langer, R. Ozeri, R. Reichle, and D. J. Wineland, “Creation of six-atom 'Schrodinger cat' state,” Nature 438, 639-642 (2005).
[CrossRef]

D. Leibrfried, M. D. Barrett, T. Schaetz, J. Britton, J. Chiaverini, W. M. Itano, J. D. Jost, C. Langer, and D. J. Wineland, “Toward Heisenberg-limited spectroscopy with multiparticle entangled states,” Science 304, 1476-1478 (2004).
[CrossRef]

V. Meyer, M. A. Rowe, D. Kielpinski, C. A. Sackett, W. M. Itano, C. Monroe, and D. J. Wineland, “Experimental demonstration of entanglement-enhanced rotation angle estimation using trapped ions,” Phys. Rev. Lett. 86, 5870-5873 (2001).
[CrossRef] [PubMed]

J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, “Optical frequency measurements with maximally correlated states,” Phys. Rev. A 54, R4649-R4652 (1996).
[CrossRef] [PubMed]

Yang, T.

C.-Y. Lu, X.-Q. Zhou, O. Guhne, W.-B. Gao, J. Zhang, Z.-S. Yuan, A. Goebel, T. Yang, and J.-W. Pan, “Experimental entanglement of six photons in graph states,” Nat. Phys. 3, 91-95 (2007).
[CrossRef]

Z. Zhao, Y.-A. Chen, A.-N. Zhang, T. Yang, H. J. Briegel, and J.-W. Pan, “Experimental demonstration of five-photon entanglement and open-destination teleportation,” Nature 430, 54-58 (2004).
[CrossRef] [PubMed]

Yuan, Z.-S.

C.-Y. Lu, X.-Q. Zhou, O. Guhne, W.-B. Gao, J. Zhang, Z.-S. Yuan, A. Goebel, T. Yang, and J.-W. Pan, “Experimental entanglement of six photons in graph states,” Nat. Phys. 3, 91-95 (2007).
[CrossRef]

Zeilinger, A.

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

Zhang, A.-N.

Z. Zhao, Y.-A. Chen, A.-N. Zhang, T. Yang, H. J. Briegel, and J.-W. Pan, “Experimental demonstration of five-photon entanglement and open-destination teleportation,” Nature 430, 54-58 (2004).
[CrossRef] [PubMed]

Zhang, J.

C.-Y. Lu, X.-Q. Zhou, O. Guhne, W.-B. Gao, J. Zhang, Z.-S. Yuan, A. Goebel, T. Yang, and J.-W. Pan, “Experimental entanglement of six photons in graph states,” Nat. Phys. 3, 91-95 (2007).
[CrossRef]

Zhao, Z.

Z. Zhao, Y.-A. Chen, A.-N. Zhang, T. Yang, H. J. Briegel, and J.-W. Pan, “Experimental demonstration of five-photon entanglement and open-destination teleportation,” Nature 430, 54-58 (2004).
[CrossRef] [PubMed]

Zhou, X.-Q.

C.-Y. Lu, X.-Q. Zhou, O. Guhne, W.-B. Gao, J. Zhang, Z.-S. Yuan, A. Goebel, T. Yang, and J.-W. Pan, “Experimental entanglement of six photons in graph states,” Nat. Phys. 3, 91-95 (2007).
[CrossRef]

Concepts Phys. (1)

K. T. Kapale, L. D. Didomencio, H. Lee, P. Kok, and J. P. Dowling, “Quantum interferometric sensors,” Concepts Phys. 2, 225-240 (2005).

J. Mod. Opt. (3)

D. V. Strekalov and J. P. Dowling, “Two-photon intereferometry for high-resolution imaging,” J. Mod. Opt. 49, 519-527 (2002).
[CrossRef]

J. Dunningham and T. Kim, “Using quantum interferometers to make measurements at the Heisenberg limit,” J. Mod. Opt. 53, 557-571 (2006).
[CrossRef]

H. Lee, P. Kok, and J. P. Dowling, “A quantum Rosetta stone for interferometry,” J. Mod. Opt. 49, 2325-2338 (2002).
[CrossRef]

J. Opt. B: Quantum Semiclassical Opt. (2)

A. V. Chizhov, E. Schmidt, L. Knoll, and D. G. Welsch, “Propogation of entangled light pulses through dispersing and absorbing channels,” J. Opt. B: Quantum Semiclassical Opt. 3, 77-83 (2001).
[CrossRef]

L. A. Lugiato, A. Gatti, and E. Brambilla, “Quantum imaging,” J. Opt. B: Quantum Semiclassical Opt. 4, S176-S183 (2002).
[CrossRef]

J. Phys. B (1)

X. Chen and L. Jiang, “The entanglement and phase measurement performance of the damped NOON state,” J. Phys. B 40, 2799-2808 (2007).
[CrossRef]

Nat. Phys. (1)

C.-Y. Lu, X.-Q. Zhou, O. Guhne, W.-B. Gao, J. Zhang, Z.-S. Yuan, A. Goebel, T. Yang, and J.-W. Pan, “Experimental entanglement of six photons in graph states,” Nat. Phys. 3, 91-95 (2007).
[CrossRef]

Nature (6)

D. Leibrfried, E. Knill, S. Seidlin, J. Britton, R. B. Blakestad, J. Chiaverini, D. B. Hume, W. M. Itano, J. D. Jost, C. Langer, R. Ozeri, R. Reichle, and D. J. Wineland, “Creation of six-atom 'Schrodinger cat' state,” Nature 438, 639-642 (2005).
[CrossRef]

H. Haffner, W. Hansel, C. F. Roos, J. Benhelm, D. Chek-al-kar, M. Chwalla, T. Kober, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Guhne, W. Dur, and R. Blatt, “Scalable multiparticle entanglement of trapped ions,” Nature 438, 643-646 (2005).
[CrossRef] [PubMed]

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

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

Z. Zhao, Y.-A. Chen, A.-N. Zhang, T. Yang, H. J. Briegel, and J.-W. Pan, “Experimental demonstration of five-photon entanglement and open-destination teleportation,” Nature 430, 54-58 (2004).
[CrossRef] [PubMed]

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced positioning and clock synchronization,” Nature 412, 417-419 (2001).
[CrossRef] [PubMed]

Phys. Rev. A (9)

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

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]

M. A. Rubin and S. Kaushik, “Loss-induced limits to phase measurement and precision with maximally entangled states,” Phys. Rev. A 75, 053805 (2007).
[CrossRef]

R. C. Pooser and O. Pfister, “Particle-number scaling of the phase sensitivity in realistic Bayesian twin-mode Heisenberg-limited interferometry,” Phys. Rev. A 69, 043616 (2004).
[CrossRef]

T. Kim, J. Dunningham, and K. Burnett, “Precision measurement scheme using a quantum interferometer,” Phys. Rev. A 72, 055801 (2005).
[CrossRef]

L. Pezze and A. Smerzi, “Phase sensitivity of a Mach-Zehnder interferometer,” Phys. Rev. A 73, 011801(R) (2006).
[CrossRef]

J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, “Optical frequency measurements with maximally correlated states,” Phys. Rev. A 54, R4649-R4652 (1996).
[CrossRef] [PubMed]

C. C. Gerry, “Heisenberg-limit interferometry with four-wave mixers operating in a nonlinear regime,” Phys. Rev. A 61, 043811 (2000).
[CrossRef]

C. C. Gerry and R. A. Campos, “Generation of maximally entangled states of a Bose-Einstein condensate and Heisenberg-limited phase resolution,” Phys. Rev. A 68, 025602 (2003).
[CrossRef]

Phys. Rev. Lett. (9)

G. A. Durkin and J. P. Dowling, “Local and global distinguishability in quantum interferometry,” Phys. Rev. Lett. 99, 070801 (2007).
[CrossRef] [PubMed]

S. F. Huelga, C. Macchiavello, T. Pellizzari, A. K. Ekert, M. B. Plenio, and J. I. Cirac, “Improvement of frequency standards with quantum entanglement,” Phys. Rev. Lett. 79, 3865-3868 (1997).
[CrossRef]

J. A. Dunningham, K. Burnett, and S. M. Barnett, “Interferometry below the standard quantum limit with Bose-Einstein condensates,” Phys. Rev. Lett. 89, 150401 (2002).
[CrossRef] [PubMed]

B. C. Sanders and G. J. Milburn, “Optimal quantum measurements for phase estimation,” Phys. Rev. Lett. 75, 2944-2947 (1995).
[CrossRef] [PubMed]

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006).
[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]

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]

R. Jozsa, D. S. Abrams, J. P. Dowling, and C. P. Williams, “Quantum clock synchronization based on shared prior entanglement,” Phys. Rev. Lett. 85, 2010-2013 (2000).
[CrossRef] [PubMed]

V. Meyer, M. A. Rowe, D. Kielpinski, C. A. Sackett, W. M. Itano, C. Monroe, and D. J. Wineland, “Experimental demonstration of entanglement-enhanced rotation angle estimation using trapped ions,” Phys. Rev. Lett. 86, 5870-5873 (2001).
[CrossRef] [PubMed]

Science (2)

D. Leibrfried, M. D. Barrett, T. Schaetz, J. Britton, J. Chiaverini, W. M. Itano, J. D. Jost, C. Langer, and D. J. Wineland, “Toward Heisenberg-limited spectroscopy with multiparticle entangled states,” Science 304, 1476-1478 (2004).
[CrossRef]

For a comrehesive review see V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306, 1330-1336 (2004).
[CrossRef] [PubMed]

Other (4)

R. Loudon, The Quantum Theory of Light (Oxford U. Press, 2000).

The field operator function, b(z), has different physical dimensionality than do the field operators ai, with the former related to the latter by a factor of a square root of length.

The photons in the separable state are injected into the interferometer at different times, labeled by tk, where k=1,...,N, with measurements independently applied to each photon. The index k is used to properly count these separate injection events.

In this and subsequent calculations we have approximated the phase measurement uncertainty by Eq. , so that our quantitative results are valid only when the phase error is small. This approximation has been analyzed in .

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Figures (2)

Fig. 1
Fig. 1

Top, phase error for attenuated N 0 0 N states for N = 2 (dashed curve) and N = 4 (dashed–dotted curve), with α 1 = 0.6 , α 2 = 0.5 . The lower and upper horizontal lines depict, respectively, the standard quantum limits ( 1 N ) for N = 4 and N = 2 . Bottom, phase error in the limit of decreasing attenuation for N 0 0 N states for N = 2 and N = 4 . The upper two curves are for α 1 = 0.8 , α 2 = 0.6 . The lower two curves are for α 1 = 0.999999 , α 2 = 0.99 . In the top figure, the Heisenberg limits ( 1 N ) for N = 2 and N = 4 , respectively, are denoted by the horizontal lines at δ ϕ = 0.5 and 0.25. All curve values are in radians.

Fig. 2
Fig. 2

Left, comparison of (1) minimum phase error for attenuated entangled N 0 0 N states with (2) minimum phase error for attenuated separable N-photon states for N = 2 . The abscissa is the transmittance of the long arm of the interferometer. For each case there are three plotted curves: transmittance in the short arm is α 1 = 1.0 (solid curve), α 1 = 0.6 (long-dashed curve), and α 1 = 0.3 (short-dashed curve). Phase errors are in radians. Center, transmittance curves for which the N 0 0 N state method breaks even with the N-photon separable-state method in terms of the minimum phase error. The N 0 0 N state performance is superior in the region above and to the right of the curves. Right, minimum transmittance necessary for N 0 0 N states to outperform separable states when α 1 = α 2 = α min .

Equations (19)

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ψ N 00 N = 1 2 ( N 0 + 0 N ) = 1 2 ( a 1 N N ! + a 2 N N ! ) v ,
ψ N 00 N = 1 2 ( a 1 N N ! + e i N ϕ a 2 N N ! ) v = 1 2 ( N 0 + e i N ϕ 0 N ) .
A D = N 0 0 N + 0 N N 0 = 1 N ! ( a 1 N v v a 2 N + a 2 N v v a 1 N ) .
Δ A D A D 2 A D 2 = sin N ϕ ,
d A D d ϕ = N sin N ϕ ,
δ ϕ Δ A D d A D d ϕ = 1 N .
a i e ( i η i ω c K i 2 ) L i a i + i K i 0 L i d z e ( i η i ω c K i 2 ) ( L i z ) b ( z ) ,
ψ N 00 N = 1 2 N ! [ e ( i η 1 ω c K 1 2 ) N L 1 a 1 N + e ( i η 2 ω c K 2 2 ) N L 2 e i N ϕ a 2 N + ] v ,
A D = 0 N N 0 + N 0 0 N = 1 N ! ( a 1 N v v a 2 N + a 2 N v v a 1 N ) ,
Δ A D = [ 1 2 ( α 1 N 2 α 1 N α 2 N + α 2 N ) + ( α 1 α 2 ) N sin 2 N ( ϕ ϕ 0 ) ] 1 2 ,
d A D d ϕ = N ( α 1 α 2 ) N 2 sin N ( ϕ ϕ 0 ) ,
δ ϕ = 1 2 ( 1 α 1 N 2 + 1 α 2 N ) + sin 2 N ( ϕ ϕ 0 ) N sin N ( ϕ ϕ 0 ) ,
ψ N = 1 2 N ( 10 + 01 ) N = 1 2 N k = 1 N ( a k , 1 + a k , 2 ) v ,
ψ N = 1 2 N k = 1 N [ e ( i η 1 ω c K 1 2 ) L 1 a k , 1 + e ( i η 2 ω c K 2 2 ) L 2 e i ϕ a k , 2 + ] v ,
A R k = 1 N A R ( k ) ,
A R ( k ) = 01 k k 10 + 10 k k 01 = a k , 1 v v a k , 2 + a k , 2 v v a k , 1 .
Δ A R = [ N 2 ( α 1 2 α 1 α 2 + α 2 ) + N ( α 1 α 2 ) sin 2 ( ϕ ϕ 0 ) ] 1 2 ,
d A D d ϕ = N α 1 α 2 sin ( ϕ ϕ 0 ) .
δ ϕ = 1 2 ( 1 α 1 2 + 1 α 2 ) + sin 2 ( ϕ ϕ 0 ) N sin ( ϕ ϕ 0 ) .

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