Abstract

There has been much recent interest in quantum optical interferometry for applications to metrology, subwavelength imaging, and remote sensing such as in quantum laser radar (LADAR). For quantum LADAR, atmospheric absorption rapidly degrades any quantum state of light, so that for high-photon loss the optimal strategy is to transmit coherent states of light, which suffer no worse loss than the Beer law for classical optical attenuation, and which provides sensitivity at the shot-noise limit. We show that coherent light coupled with photon-number-resolving detectors can provide a super-resolution much below the Rayleigh diffraction limit, with sensitivity no worse than shot noise in terms of the detected photon power.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733–2736 (2000).
    [CrossRef] [PubMed]
  2. V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306, 1330–1336 (2004).
    [CrossRef] [PubMed]
  3. P. Kok, S. L. Braunstein, and J. P. Dowling, “Quantum lithography, entanglement and Heisenberg-limited parameter estimation,” J. Opt. B: Quantum Semiclassical Opt. 6, S811–S815 (2004).
    [CrossRef]
  4. T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four-entangled photons,” Science 316, 726–729 (2007).
    [CrossRef] [PubMed]
  5. J. P. Dowling, “Quantum optical metrology—the lowdown on high-N00N states,” Contemp. Phys. 49, 125–143 (2008).
    [CrossRef]
  6. M. A. Rubin and S. Kaushik, “Loss-induced limits to phase measurement precision with maximally entangled states,” Phys. Rev. A 75, 053805 (2007).
    [CrossRef]
  7. G. Gilbert, M. Hamrick, and Y. S. Weinstein, “Use of maximally entangled N-photon states for practical quantum interferometry,” J. Opt. Soc. Am. B 25, 1336–1340 (2008).
    [CrossRef]
  8. S. D. Huver, C. F. Wildfeuer, and J. P. Dowling, “Entangled Fock states for robust quantum optical metrology, imaging, and sensing,” Phys. Rev. A 78, 063828 (2008).
    [CrossRef]
  9. U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
    [CrossRef] [PubMed]
  10. R. Demkowicz-Dobrzanski, U. Dorner, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Quantum phase estimation with lossy interferometers,” Phys. Rev. A 80, 013825 (2009).
    [CrossRef]
  11. C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
    [CrossRef]
  12. K. J. Resch, K. L. Pregnell, R. Prevedel, A. Gilchrist, G. J. Pryde, J. L. O’Brien, and A. G. White, “Time-reversal and super-resolving phase measurements,” Phys. Rev. Lett. 98, 223601 (2007).
    [CrossRef] [PubMed]
  13. V. Giovannetti, S. Lloyd, L. Maccone, and J. H. Shapiro, “Sub-Rayleigh-diffraction-bound quantum imaging,” Phys. Rev. A 79, 013827 (2009).
    [CrossRef]
  14. For example, two-photon microscopy is capable of resolving sub-wavelength features because it demonstrates amplitude super-resolution. There are many other primarily scanning techniques that are based on sharp amplitude dependence in order to produce high-resolution images pixel by pixel. Something similar could be implemented here given our phase super-resolution where one would have to scan a beam of light across a sample and measure phase shifts on a point-by-point basis. The finite size of the scanning laser beam would have to be deconvoluted out of the final image.
  15. R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantum-mechanical lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A 40, 1371–1384 (1989).
    [CrossRef] [PubMed]
  16. C. C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge Univ. Press, 2005).
  17. M. V. Berry and S. Popescu, “Evolution of quantum superoscillations and optical superresolution without evanescent waves,” J. Phys. A 39, 6965–6977 (2006).
    [CrossRef]
  18. C. C. Gerry, “Heisenberg-limit interferometry with four-wave mixers operating in a nonlinear regime,” Phys. Rev. A 61, 043811 (2000).
    [CrossRef]
  19. C. C. Gerry and R. A. Campos, “Generation of maximally entangled photonic states with a quantum-optical Fredkin gate,” Phys. Rev. A 64, 063814 (2001).
    [CrossRef]
  20. D. Rosenberg, A. E. Lita, A. J. Miller, and S. W. Nam, “Noise-free high-efficiency photon-number-resolving detectors,” Phys. Rev. A 71, 061803(R) (2005).
    [CrossRef]
  21. A. E. Lita, A. J. Miller, and S. W. Nam, “Counting near-infrared single-photons with 95% efficiency,” Opt. Express 16, 3032–3040 (2008).
    [CrossRef] [PubMed]
  22. G. Khoury, H. S. Eisenberg, E. J. S. Fonseca, and D. Bouwmeester, “Nonlinear interferometry via Fock-state projection,” Phys. Rev. Lett. 96, 203601 (2006).
    [CrossRef] [PubMed]
  23. C. F. Wildfeuer, A. J. Pearlman, J. Chen, J. Fan, A. Migdall, and J. P. Dowling, “Resolution and sensitivity of a Fabry–Perot interferometer with a photon-number-resolving detector,” Phys. Rev. A 80, 043822 (2009).
    [CrossRef]
  24. R. A. Campos, C. C. Gerry, and A. Benmoussa, “Optical interferometry at the Heisenberg limit with twin Fock states and parity measurements,” Phys. Rev. A 68, 023810 (2003).
    [CrossRef]
  25. Y. Gao and H. Lee, “Sub-shot-noise quantum optical interferometry: a comparison of entangled state performance within a unified measurement scheme,” J. Mod. Opt. 55, 3319–3327 (2008).
    [CrossRef]
  26. A. Chiruvelli and H. Lee, “Parity measurements in quantum optical metrology,” arXiv:0901.4395.
  27. P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, “Quantum metrology with two-mode squeezed vacuum: parity detection beats the Heisenberg limit,” Phys. Rev. Lett. 104, 103602 (2010).
    [CrossRef] [PubMed]
  28. G. A. Durkin and J. P. Dowling, “Local and global distinguishability in quantum interferometry,” Phys. Rev. Lett. 99, 070801 (2007).
    [CrossRef] [PubMed]
  29. Perkin-Elmer SPCM-AQRH-xx APD operating manual.
  30. C. C. Gerry, A. Benmoussa, and R. A. Campos, “Nonlinear interferometer as a resource for maximally entangled photonic states: application to interferometry,” Phys. Rev. A 66, 013804 (2002).
    [CrossRef]
  31. C. C. Gerry, A. Benmoussa, and R. A. Campos, “Quantum nondemolition measurement of parity and generation of parity eigenstates in optical fields,” Phys. Rev. A 72, 053818 (2005).
    [CrossRef]

2010 (1)

P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, “Quantum metrology with two-mode squeezed vacuum: parity detection beats the Heisenberg limit,” Phys. Rev. Lett. 104, 103602 (2010).
[CrossRef] [PubMed]

2009 (4)

C. F. Wildfeuer, A. J. Pearlman, J. Chen, J. Fan, A. Migdall, and J. P. Dowling, “Resolution and sensitivity of a Fabry–Perot interferometer with a photon-number-resolving detector,” Phys. Rev. A 80, 043822 (2009).
[CrossRef]

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
[CrossRef] [PubMed]

R. Demkowicz-Dobrzanski, U. Dorner, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Quantum phase estimation with lossy interferometers,” Phys. Rev. A 80, 013825 (2009).
[CrossRef]

V. Giovannetti, S. Lloyd, L. Maccone, and J. H. Shapiro, “Sub-Rayleigh-diffraction-bound quantum imaging,” Phys. Rev. A 79, 013827 (2009).
[CrossRef]

2008 (5)

J. P. Dowling, “Quantum optical metrology—the lowdown on high-N00N states,” Contemp. Phys. 49, 125–143 (2008).
[CrossRef]

G. Gilbert, M. Hamrick, and Y. S. Weinstein, “Use of maximally entangled N-photon states for practical quantum interferometry,” J. Opt. Soc. Am. B 25, 1336–1340 (2008).
[CrossRef]

S. D. Huver, C. F. Wildfeuer, and J. P. Dowling, “Entangled Fock states for robust quantum optical metrology, imaging, and sensing,” Phys. Rev. A 78, 063828 (2008).
[CrossRef]

A. E. Lita, A. J. Miller, and S. W. Nam, “Counting near-infrared single-photons with 95% efficiency,” Opt. Express 16, 3032–3040 (2008).
[CrossRef] [PubMed]

Y. Gao and H. Lee, “Sub-shot-noise quantum optical interferometry: a comparison of entangled state performance within a unified measurement scheme,” J. Mod. Opt. 55, 3319–3327 (2008).
[CrossRef]

2007 (4)

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

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

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

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

2006 (2)

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

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

2005 (2)

D. Rosenberg, A. E. Lita, A. J. Miller, and S. W. Nam, “Noise-free high-efficiency photon-number-resolving detectors,” Phys. Rev. A 71, 061803(R) (2005).
[CrossRef]

C. C. Gerry, A. Benmoussa, and R. A. Campos, “Quantum nondemolition measurement of parity and generation of parity eigenstates in optical fields,” Phys. Rev. A 72, 053818 (2005).
[CrossRef]

2004 (2)

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

P. Kok, S. L. Braunstein, and J. P. Dowling, “Quantum lithography, entanglement and Heisenberg-limited parameter estimation,” J. Opt. B: Quantum Semiclassical Opt. 6, S811–S815 (2004).
[CrossRef]

2003 (1)

R. A. Campos, C. C. Gerry, and A. Benmoussa, “Optical interferometry at the Heisenberg limit with twin Fock states and parity measurements,” Phys. Rev. A 68, 023810 (2003).
[CrossRef]

2002 (1)

C. C. Gerry, A. Benmoussa, and R. A. Campos, “Nonlinear interferometer as a resource for maximally entangled photonic states: application to interferometry,” Phys. Rev. A 66, 013804 (2002).
[CrossRef]

2001 (1)

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

2000 (2)

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

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

1989 (1)

R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantum-mechanical lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A 40, 1371–1384 (1989).
[CrossRef] [PubMed]

1981 (1)

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
[CrossRef]

Abrams, D. S.

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

Anisimov, P. M.

P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, “Quantum metrology with two-mode squeezed vacuum: parity detection beats the Heisenberg limit,” Phys. Rev. Lett. 104, 103602 (2010).
[CrossRef] [PubMed]

Banaszek, K.

R. Demkowicz-Dobrzanski, U. Dorner, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Quantum phase estimation with lossy interferometers,” Phys. Rev. A 80, 013825 (2009).
[CrossRef]

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
[CrossRef] [PubMed]

Benmoussa, A.

C. C. Gerry, A. Benmoussa, and R. A. Campos, “Quantum nondemolition measurement of parity and generation of parity eigenstates in optical fields,” Phys. Rev. A 72, 053818 (2005).
[CrossRef]

R. A. Campos, C. C. Gerry, and A. Benmoussa, “Optical interferometry at the Heisenberg limit with twin Fock states and parity measurements,” Phys. Rev. A 68, 023810 (2003).
[CrossRef]

C. C. Gerry, A. Benmoussa, and R. A. Campos, “Nonlinear interferometer as a resource for maximally entangled photonic states: application to interferometry,” Phys. Rev. A 66, 013804 (2002).
[CrossRef]

Berry, M. V.

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

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]

Bouwmeester, D.

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

Braunstein, S. L.

P. Kok, S. L. Braunstein, and J. P. Dowling, “Quantum lithography, entanglement and Heisenberg-limited parameter estimation,” J. Opt. B: Quantum Semiclassical Opt. 6, S811–S815 (2004).
[CrossRef]

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

Campos, R. A.

C. C. Gerry, A. Benmoussa, and R. A. Campos, “Quantum nondemolition measurement of parity and generation of parity eigenstates in optical fields,” Phys. Rev. A 72, 053818 (2005).
[CrossRef]

R. A. Campos, C. C. Gerry, and A. Benmoussa, “Optical interferometry at the Heisenberg limit with twin Fock states and parity measurements,” Phys. Rev. A 68, 023810 (2003).
[CrossRef]

C. C. Gerry, A. Benmoussa, and R. A. Campos, “Nonlinear interferometer as a resource for maximally entangled photonic states: application to interferometry,” Phys. Rev. A 66, 013804 (2002).
[CrossRef]

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

R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantum-mechanical lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A 40, 1371–1384 (1989).
[CrossRef] [PubMed]

Caves, C. M.

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
[CrossRef]

Chen, J.

C. F. Wildfeuer, A. J. Pearlman, J. Chen, J. Fan, A. Migdall, and J. P. Dowling, “Resolution and sensitivity of a Fabry–Perot interferometer with a photon-number-resolving detector,” Phys. Rev. A 80, 043822 (2009).
[CrossRef]

Chiruvelli, A.

P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, “Quantum metrology with two-mode squeezed vacuum: parity detection beats the Heisenberg limit,” Phys. Rev. Lett. 104, 103602 (2010).
[CrossRef] [PubMed]

A. Chiruvelli and H. Lee, “Parity measurements in quantum optical metrology,” arXiv:0901.4395.

Demkowicz-Dobrzanski, R.

R. Demkowicz-Dobrzanski, U. Dorner, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Quantum phase estimation with lossy interferometers,” Phys. Rev. A 80, 013825 (2009).
[CrossRef]

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
[CrossRef] [PubMed]

Dorner, U.

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
[CrossRef] [PubMed]

R. Demkowicz-Dobrzanski, U. Dorner, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Quantum phase estimation with lossy interferometers,” Phys. Rev. A 80, 013825 (2009).
[CrossRef]

Dowling, J. P.

P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, “Quantum metrology with two-mode squeezed vacuum: parity detection beats the Heisenberg limit,” Phys. Rev. Lett. 104, 103602 (2010).
[CrossRef] [PubMed]

C. F. Wildfeuer, A. J. Pearlman, J. Chen, J. Fan, A. Migdall, and J. P. Dowling, “Resolution and sensitivity of a Fabry–Perot interferometer with a photon-number-resolving detector,” Phys. Rev. A 80, 043822 (2009).
[CrossRef]

S. D. Huver, C. F. Wildfeuer, and J. P. Dowling, “Entangled Fock states for robust quantum optical metrology, imaging, and sensing,” Phys. Rev. A 78, 063828 (2008).
[CrossRef]

J. P. Dowling, “Quantum optical metrology—the lowdown on high-N00N states,” Contemp. Phys. 49, 125–143 (2008).
[CrossRef]

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

P. Kok, S. L. Braunstein, and J. P. Dowling, “Quantum lithography, entanglement and Heisenberg-limited parameter estimation,” J. Opt. B: Quantum Semiclassical Opt. 6, S811–S815 (2004).
[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]

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]

Eisenberg, H. S.

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

Fan, J.

C. F. Wildfeuer, A. J. Pearlman, J. Chen, J. Fan, A. Migdall, and J. P. Dowling, “Resolution and sensitivity of a Fabry–Perot interferometer with a photon-number-resolving detector,” Phys. Rev. A 80, 043822 (2009).
[CrossRef]

Fonseca, E. J. S.

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

Gao, Y.

Y. Gao and H. Lee, “Sub-shot-noise quantum optical interferometry: a comparison of entangled state performance within a unified measurement scheme,” J. Mod. Opt. 55, 3319–3327 (2008).
[CrossRef]

Gerry, C. C.

C. C. Gerry, A. Benmoussa, and R. A. Campos, “Quantum nondemolition measurement of parity and generation of parity eigenstates in optical fields,” Phys. Rev. A 72, 053818 (2005).
[CrossRef]

R. A. Campos, C. C. Gerry, and A. Benmoussa, “Optical interferometry at the Heisenberg limit with twin Fock states and parity measurements,” Phys. Rev. A 68, 023810 (2003).
[CrossRef]

C. C. Gerry, A. Benmoussa, and R. A. Campos, “Nonlinear interferometer as a resource for maximally entangled photonic states: application to interferometry,” Phys. Rev. A 66, 013804 (2002).
[CrossRef]

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

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 P. L. Knight, Introductory Quantum Optics (Cambridge Univ. Press, 2005).

Gilbert, G.

Gilchrist, A.

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

Giovannetti, V.

V. Giovannetti, S. Lloyd, L. Maccone, and J. H. Shapiro, “Sub-Rayleigh-diffraction-bound quantum imaging,” Phys. Rev. A 79, 013827 (2009).
[CrossRef]

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

Hamrick, M.

Huver, S. D.

P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, “Quantum metrology with two-mode squeezed vacuum: parity detection beats the Heisenberg limit,” Phys. Rev. Lett. 104, 103602 (2010).
[CrossRef] [PubMed]

S. D. Huver, C. F. Wildfeuer, and J. P. Dowling, “Entangled Fock states for robust quantum optical metrology, imaging, and sensing,” Phys. Rev. A 78, 063828 (2008).
[CrossRef]

Kaushik, S.

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

Khoury, G.

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

Knight, P. L.

C. C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge Univ. Press, 2005).

Kok, P.

P. Kok, S. L. Braunstein, and J. P. Dowling, “Quantum lithography, entanglement and Heisenberg-limited parameter estimation,” J. Opt. B: Quantum Semiclassical Opt. 6, S811–S815 (2004).
[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]

Lee, H.

P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, “Quantum metrology with two-mode squeezed vacuum: parity detection beats the Heisenberg limit,” Phys. Rev. Lett. 104, 103602 (2010).
[CrossRef] [PubMed]

Y. Gao and H. Lee, “Sub-shot-noise quantum optical interferometry: a comparison of entangled state performance within a unified measurement scheme,” J. Mod. Opt. 55, 3319–3327 (2008).
[CrossRef]

A. Chiruvelli and H. Lee, “Parity measurements in quantum optical metrology,” arXiv:0901.4395.

Lita, A. E.

A. E. Lita, A. J. Miller, and S. W. Nam, “Counting near-infrared single-photons with 95% efficiency,” Opt. Express 16, 3032–3040 (2008).
[CrossRef] [PubMed]

D. Rosenberg, A. E. Lita, A. J. Miller, and S. W. Nam, “Noise-free high-efficiency photon-number-resolving detectors,” Phys. Rev. A 71, 061803(R) (2005).
[CrossRef]

Lloyd, S.

V. Giovannetti, S. Lloyd, L. Maccone, and J. H. Shapiro, “Sub-Rayleigh-diffraction-bound quantum imaging,” Phys. Rev. A 79, 013827 (2009).
[CrossRef]

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

Lundeen, J. S.

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
[CrossRef] [PubMed]

R. Demkowicz-Dobrzanski, U. Dorner, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Quantum phase estimation with lossy interferometers,” Phys. Rev. A 80, 013825 (2009).
[CrossRef]

Maccone, L.

V. Giovannetti, S. Lloyd, L. Maccone, and J. H. Shapiro, “Sub-Rayleigh-diffraction-bound quantum imaging,” Phys. Rev. A 79, 013827 (2009).
[CrossRef]

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

Migdall, A.

C. F. Wildfeuer, A. J. Pearlman, J. Chen, J. Fan, A. Migdall, and J. P. Dowling, “Resolution and sensitivity of a Fabry–Perot interferometer with a photon-number-resolving detector,” Phys. Rev. A 80, 043822 (2009).
[CrossRef]

Miller, A. J.

A. E. Lita, A. J. Miller, and S. W. Nam, “Counting near-infrared single-photons with 95% efficiency,” Opt. Express 16, 3032–3040 (2008).
[CrossRef] [PubMed]

D. Rosenberg, A. E. Lita, A. J. Miller, and S. W. Nam, “Noise-free high-efficiency photon-number-resolving detectors,” Phys. Rev. A 71, 061803(R) (2005).
[CrossRef]

Nagata, T.

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

Nam, S. W.

A. E. Lita, A. J. Miller, and S. W. Nam, “Counting near-infrared single-photons with 95% efficiency,” Opt. Express 16, 3032–3040 (2008).
[CrossRef] [PubMed]

D. Rosenberg, A. E. Lita, A. J. Miller, and S. W. Nam, “Noise-free high-efficiency photon-number-resolving detectors,” Phys. Rev. A 71, 061803(R) (2005).
[CrossRef]

O’Brien, J. L.

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

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

Okamoto, R.

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

Pearlman, A. J.

C. F. Wildfeuer, A. J. Pearlman, J. Chen, J. Fan, A. Migdall, and J. P. Dowling, “Resolution and sensitivity of a Fabry–Perot interferometer with a photon-number-resolving detector,” Phys. Rev. A 80, 043822 (2009).
[CrossRef]

Plick, W. N.

P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, “Quantum metrology with two-mode squeezed vacuum: parity detection beats the Heisenberg limit,” Phys. Rev. Lett. 104, 103602 (2010).
[CrossRef] [PubMed]

Popescu, S.

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

Pregnell, K. L.

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

Prevedel, R.

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

Pryde, G. J.

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

Raterman, G. M.

P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, “Quantum metrology with two-mode squeezed vacuum: parity detection beats the Heisenberg limit,” Phys. Rev. Lett. 104, 103602 (2010).
[CrossRef] [PubMed]

Resch, K. J.

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

Rosenberg, D.

D. Rosenberg, A. E. Lita, A. J. Miller, and S. W. Nam, “Noise-free high-efficiency photon-number-resolving detectors,” Phys. Rev. A 71, 061803(R) (2005).
[CrossRef]

Rubin, M. A.

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

Saleh, B. E. A.

R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantum-mechanical lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A 40, 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,” Science 316, 726–729 (2007).
[CrossRef] [PubMed]

Shapiro, J. H.

V. Giovannetti, S. Lloyd, L. Maccone, and J. H. Shapiro, “Sub-Rayleigh-diffraction-bound quantum imaging,” Phys. Rev. A 79, 013827 (2009).
[CrossRef]

Smith, B. J.

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
[CrossRef] [PubMed]

R. Demkowicz-Dobrzanski, U. Dorner, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Quantum phase estimation with lossy interferometers,” Phys. Rev. A 80, 013825 (2009).
[CrossRef]

Takeuchi, S.

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

Teich, M. C.

R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantum-mechanical lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A 40, 1371–1384 (1989).
[CrossRef] [PubMed]

Walmsley, I. A.

R. Demkowicz-Dobrzanski, U. Dorner, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Quantum phase estimation with lossy interferometers,” Phys. Rev. A 80, 013825 (2009).
[CrossRef]

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
[CrossRef] [PubMed]

Wasilewski, W.

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
[CrossRef] [PubMed]

R. Demkowicz-Dobrzanski, U. Dorner, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Quantum phase estimation with lossy interferometers,” Phys. Rev. A 80, 013825 (2009).
[CrossRef]

Weinstein, Y. S.

White, A. G.

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

Wildfeuer, C. F.

C. F. Wildfeuer, A. J. Pearlman, J. Chen, J. Fan, A. Migdall, and J. P. Dowling, “Resolution and sensitivity of a Fabry–Perot interferometer with a photon-number-resolving detector,” Phys. Rev. A 80, 043822 (2009).
[CrossRef]

S. D. Huver, C. F. Wildfeuer, and J. P. Dowling, “Entangled Fock states for robust quantum optical metrology, imaging, and sensing,” Phys. Rev. A 78, 063828 (2008).
[CrossRef]

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]

Contemp. Phys. (1)

J. P. Dowling, “Quantum optical metrology—the lowdown on high-N00N states,” Contemp. Phys. 49, 125–143 (2008).
[CrossRef]

J. Mod. Opt. (1)

Y. Gao and H. Lee, “Sub-shot-noise quantum optical interferometry: a comparison of entangled state performance within a unified measurement scheme,” J. Mod. Opt. 55, 3319–3327 (2008).
[CrossRef]

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

P. Kok, S. L. Braunstein, and J. P. Dowling, “Quantum lithography, entanglement and Heisenberg-limited parameter estimation,” J. Opt. B: Quantum Semiclassical Opt. 6, S811–S815 (2004).
[CrossRef]

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

J. Phys. A (1)

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

Opt. Express (1)

Phys. Rev. A (12)

R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantum-mechanical lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A 40, 1371–1384 (1989).
[CrossRef] [PubMed]

C. F. Wildfeuer, A. J. Pearlman, J. Chen, J. Fan, A. Migdall, and J. P. Dowling, “Resolution and sensitivity of a Fabry–Perot interferometer with a photon-number-resolving detector,” Phys. Rev. A 80, 043822 (2009).
[CrossRef]

R. A. Campos, C. C. Gerry, and A. Benmoussa, “Optical interferometry at the Heisenberg limit with twin Fock states and parity measurements,” Phys. Rev. A 68, 023810 (2003).
[CrossRef]

C. C. Gerry, A. Benmoussa, and R. A. Campos, “Nonlinear interferometer as a resource for maximally entangled photonic states: application to interferometry,” Phys. Rev. A 66, 013804 (2002).
[CrossRef]

C. C. Gerry, A. Benmoussa, and R. A. Campos, “Quantum nondemolition measurement of parity and generation of parity eigenstates in optical fields,” Phys. Rev. A 72, 053818 (2005).
[CrossRef]

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 photonic states with a quantum-optical Fredkin gate,” Phys. Rev. A 64, 063814 (2001).
[CrossRef]

D. Rosenberg, A. E. Lita, A. J. Miller, and S. W. Nam, “Noise-free high-efficiency photon-number-resolving detectors,” Phys. Rev. A 71, 061803(R) (2005).
[CrossRef]

R. Demkowicz-Dobrzanski, U. Dorner, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Quantum phase estimation with lossy interferometers,” Phys. Rev. A 80, 013825 (2009).
[CrossRef]

V. Giovannetti, S. Lloyd, L. Maccone, and J. H. Shapiro, “Sub-Rayleigh-diffraction-bound quantum imaging,” Phys. Rev. A 79, 013827 (2009).
[CrossRef]

S. D. Huver, C. F. Wildfeuer, and J. P. Dowling, “Entangled Fock states for robust quantum optical metrology, imaging, and sensing,” Phys. Rev. A 78, 063828 (2008).
[CrossRef]

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

Phys. Rev. D (1)

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
[CrossRef]

Phys. Rev. Lett. (6)

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

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
[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]

P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, “Quantum metrology with two-mode squeezed vacuum: parity detection beats the Heisenberg limit,” Phys. Rev. Lett. 104, 103602 (2010).
[CrossRef] [PubMed]

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

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

Science (2)

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

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

Other (4)

For example, two-photon microscopy is capable of resolving sub-wavelength features because it demonstrates amplitude super-resolution. There are many other primarily scanning techniques that are based on sharp amplitude dependence in order to produce high-resolution images pixel by pixel. Something similar could be implemented here given our phase super-resolution where one would have to scan a beam of light across a sample and measure phase shifts on a point-by-point basis. The finite size of the scanning laser beam would have to be deconvoluted out of the final image.

C. C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge Univ. Press, 2005).

A. Chiruvelli and H. Lee, “Parity measurements in quantum optical metrology,” arXiv:0901.4395.

Perkin-Elmer SPCM-AQRH-xx APD operating manual.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Here we indicate the Mach–Zehnder interferometer used in the calculations. The coherent state is incident in mode A and vacuum in mode B at the left at Line I. After the first beam splitter transformation we have the two-mode coherent state of Eq. (1), as indicated at Line II. After the phase shifter φ this state becomes the two-mode coherent state of Eq. (2). At Line III we also implement the detection schemes corresponding to the operators N ̂ A B , ν ̂ A B , and μ ̂ A B . Finally, after the final beam splitter, we implement the parity operator Π ̂ A detection of Eq. (9) in the upper mode.

Fig. 2
Fig. 2

This plot shows the expectation value μ ̂ A B of Eq. (6) plotted as a function of the phase shift φ (solid curve) for a return power of n ¯ = 100 . For reference we plot the normalized “classical" two-port difference signal (dashed curve). We see that the plot of the μ ̂ A B is super-resolving and is narrower than the classical curve by a factor of δ φ = 1 n ¯ = 1 10 , as given in Eq. (8).

Fig. 3
Fig. 3

In this plot we depict the sensitivity expression Δ φ μ 2 of Eq. (7), again for the return power of n ¯ = 100 (solid curve). The horizontal dashed line indicates the shot-noise limit of Δ φ SNL 2 = 1 n ¯ = 1 100 . We see that the sensitivity of the super-resolving μ ̂ A B detection scheme hits the SNL at φ = 0 , as indicated by expanding Eq. (7) in a power series.

Fig. 4
Fig. 4

Simulated parity detection as a function of interferometer phase shift for perfect detection, TES detection with η = 0.95 , and APD detection with η = 0.72 , compared with the theoretical prediction.

Equations (10)

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

| α 2 , α 2 = e n ¯ 2 n , m = 0 ( α 2 ) n + m n ! m ! | n , m ,
| ψ A B = | α e i φ 2 , α 2 = e n ¯ 2 n , m = 0 ( e i φ n ¯ 2 ) n ( n ¯ 2 ) m n ! m ! | n , m ,
Δ φ N 2 = Δ N ̂ A B 2 | N ̂ A B φ | 2 = 2 N e n ¯ N ! 2 n ¯ N cos 2 N φ 2 n ¯ N sin 2 N φ 1 N 2 ,
Δ φ ν 2 = { e n ¯ + e 3 n ¯ 2 e n ¯ cos φ [ 1 + cos ( n ¯ sin φ ) ] } × e n ¯ cos φ csc 2 ( φ + n ¯ 2 sin φ ) 2 n ¯ 2 ,
μ ̂ A B = M , M = 0 | M , M M , M | ,
μ ̂ A B = ψ | A B μ ̂ A B | ψ A B = e 2 n ¯ sin 2 ( φ 2 ) ,
Δ φ μ 2 = e 4 n ¯ sin 2 ( φ 2 ) 1 n ¯ 2 sin 2 φ ,
| μ ̂ A B | φ 0 e n ¯ φ 2 2 ,
Π ̂ A = ( 1 ) n ̂ A = e i π a ̂ a ̂ ,
ψ | A B μ ̂ A B | ψ A B α , 0 | U ̂ ( Π ̂ A I ̂ B ) U ̂ | α , 0 ,

Metrics