Abstract

Phase super-sensitivity is obtained when the sensitivity in a phase measurement goes beyond the quantum shot noise limit, whereas super-resolution is obtained when the interference fringes in an interferometer are narrower than half the input wavelength. Here we show experimentally that these two features can be simultaneously achieved using a relatively simple setup based on Gaussian states and homodyne measurement. Using 430 photons shared between a coherent and a squeezed vacuum state, we demonstrate a 22-fold improvement in the phase resolution, while we observe a 1.7-fold improvement in the sensitivity. In contrast to previous demonstrations of super-resolution and super-sensitivity, this approach is fully deterministic.

© 2018 Optical Society of America

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  1. V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306, 1330–1336 (2004).
    [Crossref]
  2. P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
    [Crossref]
  3. A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81, 299–332 (2009).
    [Crossref]
  4. J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, “Two-photon interference in a Mach-Zehnder interferometer,” Phys. Rev. Lett. 65, 1348–1351 (1990).
    [Crossref]
  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]
  6. 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]
  7. 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]
  8. E. Distante, M. Ježek, and U. L. Andersen, “Deterministic superresolution with coherent states at the shot noise limit,” Phys. Rev. Lett. 111, 033603 (2013).
    [Crossref]
  9. Y. Gao, P. M. Anisimov, C. F. Wildfeuer, J. Luine, H. Lee, and J. P. Dowling, “Super-resolution at the shot-noise limit with coherent states and photon-number-resolving detectors,” J. Opt. Soc. Am. B 27, A170–A174 (2010).
    [Crossref]
  10. L. Cohen, D. Istrati, L. Dovrat, and H. S. Eisenberg, “Super-resolved phase measurements at the shot noise limit by parity measurement,” Opt. Express 22, 11945–11953 (2014).
    [Crossref]
  11. C. C. Gerry, “Heisenberg-limit interferometry with four-wave mixers operating in a nonlinear regime,” Phys. Rev. A 61, 043811 (2000).
    [Crossref]
  12. A simple approach for increasing the fringe resolution by classical means is a multi-pass interferometer. As the mentioned techniques can be mapped to multi-pass configurations, we focus on the single-pass configuration.
  13. J. Jacobson, G. Björk, I. Chuang, and Y. Yamamoto, “Photonic de Broglie waves,” Phys. Rev. Lett. 74, 4835–4838 (1995).
    [Crossref]
  14. E. J. S. Fonseca, C. H. Monken, and S. Pádua, “Measurement of the de Broglie wavelength of a multiphoton wave packet,” Phys. Rev. Lett. 82, 2868–2871 (1999).
    [Crossref]
  15. 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]
  16. K. Jiang, H. Lee, C. C. Gerry, and J. P. Dowling, “Super-resolving quantum radar: coherent-state sources with homodyne detection suffice to beat the diffraction limit,” J. Appl. Phys. 114, 193102 (2013).
    [Crossref]
  17. S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. 72, 3439–3443 (1994).
    [Crossref]
  18. A. Monras, “Optimal phase measurements with pure Gaussian states,” Phys. Rev. A 73, 033821 (2006).
    [Crossref]
  19. 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]
  20. C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
    [Crossref]
  21. J. C. F. Matthews, X.-Q. Zhou, H. Cable, P. J. Shadbolt, D. J. Saunders, G. A. Durkin, G. J. Pryde, and J. L. O’Brien, “Towards practical quantum metrology with photon counting,” npj Quantum Inf. 2, 16023 (2016).
    [Crossref]
  22. H. Grote, K. Danzmann, K. L. Dooley, R. Schnabel, J. Slutsky, and H. Vahlbruch, “First long-term application of squeezed states of light in a gravitational-wave observatory,” Phys. Rev. Lett. 110, 181101 (2013).
    [Crossref]
  23. LIGO Scientific Collaboration, “Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light,” Nat. Photonics 7, 613–619 (2013).
    [Crossref]
  24. D. Bouwmeester, “Quantum physics: high NOON for photons,” Nature 429, 139–141 (2004).
    [Crossref]
  25. Y. Israel, S. Rosen, and Y. Silberberg, “Supersensitive polarization microscopy using NOON states of light,” Phys. Rev. Lett. 112, 103604 (2014).
    [Crossref]
  26. 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]
  27. I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science 328, 879–881 (2010).
    [Crossref]

2016 (1)

J. C. F. Matthews, X.-Q. Zhou, H. Cable, P. J. Shadbolt, D. J. Saunders, G. A. Durkin, G. J. Pryde, and J. L. O’Brien, “Towards practical quantum metrology with photon counting,” npj Quantum Inf. 2, 16023 (2016).
[Crossref]

2014 (2)

Y. Israel, S. Rosen, and Y. Silberberg, “Supersensitive polarization microscopy using NOON states of light,” Phys. Rev. Lett. 112, 103604 (2014).
[Crossref]

L. Cohen, D. Istrati, L. Dovrat, and H. S. Eisenberg, “Super-resolved phase measurements at the shot noise limit by parity measurement,” Opt. Express 22, 11945–11953 (2014).
[Crossref]

2013 (4)

E. Distante, M. Ježek, and U. L. Andersen, “Deterministic superresolution with coherent states at the shot noise limit,” Phys. Rev. Lett. 111, 033603 (2013).
[Crossref]

H. Grote, K. Danzmann, K. L. Dooley, R. Schnabel, J. Slutsky, and H. Vahlbruch, “First long-term application of squeezed states of light in a gravitational-wave observatory,” Phys. Rev. Lett. 110, 181101 (2013).
[Crossref]

LIGO Scientific Collaboration, “Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light,” Nat. Photonics 7, 613–619 (2013).
[Crossref]

K. Jiang, H. Lee, C. C. Gerry, and J. P. Dowling, “Super-resolving quantum radar: coherent-state sources with homodyne detection suffice to beat the diffraction limit,” J. Appl. Phys. 114, 193102 (2013).
[Crossref]

2010 (3)

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]

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

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

2009 (1)

A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81, 299–332 (2009).
[Crossref]

2007 (3)

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[Crossref]

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]

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]

2006 (1)

A. Monras, “Optimal phase measurements with pure Gaussian states,” Phys. Rev. A 73, 033821 (2006).
[Crossref]

2004 (3)

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]

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

D. Bouwmeester, “Quantum physics: high NOON for photons,” Nature 429, 139–141 (2004).
[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]

1999 (1)

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

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]

1995 (1)

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

1994 (1)

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

1990 (1)

J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, “Two-photon interference in a Mach-Zehnder interferometer,” Phys. Rev. Lett. 65, 1348–1351 (1990).
[Crossref]

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]

Afek, I.

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

Ambar, O.

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

Andersen, U. L.

E. Distante, M. Ježek, and U. L. Andersen, “Deterministic superresolution with coherent states at the shot noise limit,” Phys. Rev. Lett. 111, 033603 (2013).
[Crossref]

Anisimov, P. M.

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

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]

Björk, G.

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

Boto, A. N.

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

Bouwmeester, D.

D. Bouwmeester, “Quantum physics: high NOON for photons,” Nature 429, 139–141 (2004).
[Crossref]

Braunstein, S. L.

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

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

Cable, H.

J. C. F. Matthews, X.-Q. Zhou, H. Cable, P. J. Shadbolt, D. J. Saunders, G. A. Durkin, G. J. Pryde, and J. L. O’Brien, “Towards practical quantum metrology with photon counting,” npj Quantum Inf. 2, 16023 (2016).
[Crossref]

Campos, R. A.

J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, “Two-photon interference in a Mach-Zehnder interferometer,” Phys. Rev. Lett. 65, 1348–1351 (1990).
[Crossref]

Caves, C. M.

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

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

Chuang, I.

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

Cohen, L.

Danzmann, K.

H. Grote, K. Danzmann, K. L. Dooley, R. Schnabel, J. Slutsky, and H. Vahlbruch, “First long-term application of squeezed states of light in a gravitational-wave observatory,” Phys. Rev. Lett. 110, 181101 (2013).
[Crossref]

Distante, E.

E. Distante, M. Ježek, and U. L. Andersen, “Deterministic superresolution with coherent states at the shot noise limit,” Phys. Rev. Lett. 111, 033603 (2013).
[Crossref]

Dooley, K. L.

H. Grote, K. Danzmann, K. L. Dooley, R. Schnabel, J. Slutsky, and H. Vahlbruch, “First long-term application of squeezed states of light in a gravitational-wave observatory,” Phys. Rev. Lett. 110, 181101 (2013).
[Crossref]

Dovrat, L.

Dowling, J. P.

K. Jiang, H. Lee, C. C. Gerry, and J. P. Dowling, “Super-resolving quantum radar: coherent-state sources with homodyne detection suffice to beat the diffraction limit,” J. Appl. Phys. 114, 193102 (2013).
[Crossref]

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]

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

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[Crossref]

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

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]

Durkin, G. A.

J. C. F. Matthews, X.-Q. Zhou, H. Cable, P. J. Shadbolt, D. J. Saunders, G. A. Durkin, G. J. Pryde, and J. L. O’Brien, “Towards practical quantum metrology with photon counting,” npj Quantum Inf. 2, 16023 (2016).
[Crossref]

Eisenberg, H. S.

Fonseca, E. J. S.

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

Gao, Y.

Gerry, C. C.

K. Jiang, H. Lee, C. C. Gerry, and J. P. Dowling, “Super-resolving quantum radar: coherent-state sources with homodyne detection suffice to beat the diffraction limit,” J. Appl. Phys. 114, 193102 (2013).
[Crossref]

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

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]

Giovannetti, V.

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

Grote, H.

H. Grote, K. Danzmann, K. L. Dooley, R. Schnabel, J. Slutsky, and H. Vahlbruch, “First long-term application of squeezed states of light in a gravitational-wave observatory,” Phys. Rev. Lett. 110, 181101 (2013).
[Crossref]

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]

Israel, Y.

Y. Israel, S. Rosen, and Y. Silberberg, “Supersensitive polarization microscopy using NOON states of light,” Phys. Rev. Lett. 112, 103604 (2014).
[Crossref]

Istrati, D.

Jacobson, J.

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

Jakeman, E.

J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, “Two-photon interference in a Mach-Zehnder interferometer,” Phys. Rev. Lett. 65, 1348–1351 (1990).
[Crossref]

Ježek, M.

E. Distante, M. Ježek, and U. L. Andersen, “Deterministic superresolution with coherent states at the shot noise limit,” Phys. Rev. Lett. 111, 033603 (2013).
[Crossref]

Jiang, K.

K. Jiang, H. Lee, C. C. Gerry, and J. P. Dowling, “Super-resolving quantum radar: coherent-state sources with homodyne detection suffice to beat the diffraction limit,” J. Appl. Phys. 114, 193102 (2013).
[Crossref]

Kok, P.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[Crossref]

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

Larchuk, T.

J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, “Two-photon interference in a Mach-Zehnder interferometer,” Phys. Rev. Lett. 65, 1348–1351 (1990).
[Crossref]

Lee, H.

K. Jiang, H. Lee, C. C. Gerry, and J. P. Dowling, “Super-resolving quantum radar: coherent-state sources with homodyne detection suffice to beat the diffraction limit,” J. Appl. Phys. 114, 193102 (2013).
[Crossref]

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

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]

Lloyd, S.

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

Luine, J.

Lundeen, J. S.

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

Lvovsky, A. I.

A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81, 299–332 (2009).
[Crossref]

Maccone, L.

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

Matthews, J. C. F.

J. C. F. Matthews, X.-Q. Zhou, H. Cable, P. J. Shadbolt, D. J. Saunders, G. A. Durkin, G. J. Pryde, and J. L. O’Brien, “Towards practical quantum metrology with photon counting,” npj Quantum Inf. 2, 16023 (2016).
[Crossref]

Milburn, G. J.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[Crossref]

Mitchell, M. W.

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

Monken, C. H.

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

Monras, A.

A. Monras, “Optimal phase measurements with pure Gaussian states,” Phys. Rev. A 73, 033821 (2006).
[Crossref]

Munro, W. J.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[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]

Nemoto, K.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[Crossref]

O’Brien, J. L.

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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).
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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).
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[Crossref]

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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]

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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]

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J. C. F. Matthews, X.-Q. Zhou, H. Cable, P. J. Shadbolt, D. J. Saunders, G. A. Durkin, G. J. Pryde, and J. L. O’Brien, “Towards practical quantum metrology with photon counting,” npj Quantum Inf. 2, 16023 (2016).
[Crossref]

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]

Ralph, T. C.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
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J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, “Two-photon interference in a Mach-Zehnder interferometer,” Phys. Rev. Lett. 65, 1348–1351 (1990).
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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]

Raymer, M. G.

A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81, 299–332 (2009).
[Crossref]

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]

Rosen, S.

Y. Israel, S. Rosen, and Y. Silberberg, “Supersensitive polarization microscopy using NOON states of light,” Phys. Rev. Lett. 112, 103604 (2014).
[Crossref]

Saleh, B. E. A.

J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, “Two-photon interference in a Mach-Zehnder interferometer,” Phys. Rev. Lett. 65, 1348–1351 (1990).
[Crossref]

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]

Saunders, D. J.

J. C. F. Matthews, X.-Q. Zhou, H. Cable, P. J. Shadbolt, D. J. Saunders, G. A. Durkin, G. J. Pryde, and J. L. O’Brien, “Towards practical quantum metrology with photon counting,” npj Quantum Inf. 2, 16023 (2016).
[Crossref]

Schnabel, R.

H. Grote, K. Danzmann, K. L. Dooley, R. Schnabel, J. Slutsky, and H. Vahlbruch, “First long-term application of squeezed states of light in a gravitational-wave observatory,” Phys. Rev. Lett. 110, 181101 (2013).
[Crossref]

Shadbolt, P. J.

J. C. F. Matthews, X.-Q. Zhou, H. Cable, P. J. Shadbolt, D. J. Saunders, G. A. Durkin, G. J. Pryde, and J. L. O’Brien, “Towards practical quantum metrology with photon counting,” npj Quantum Inf. 2, 16023 (2016).
[Crossref]

Silberberg, Y.

Y. Israel, S. Rosen, and Y. Silberberg, “Supersensitive polarization microscopy using NOON states of light,” Phys. Rev. Lett. 112, 103604 (2014).
[Crossref]

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

Slutsky, J.

H. Grote, K. Danzmann, K. L. Dooley, R. Schnabel, J. Slutsky, and H. Vahlbruch, “First long-term application of squeezed states of light in a gravitational-wave observatory,” Phys. Rev. Lett. 110, 181101 (2013).
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M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature 429, 161–164 (2004).
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T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four-entangled photons,” Science 316, 726–729 (2007).
[Crossref]

Tapster, P. R.

J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, “Two-photon interference in a Mach-Zehnder interferometer,” Phys. Rev. Lett. 65, 1348–1351 (1990).
[Crossref]

Teich, M. C.

J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, “Two-photon interference in a Mach-Zehnder interferometer,” Phys. Rev. Lett. 65, 1348–1351 (1990).
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Vahlbruch, H.

H. Grote, K. Danzmann, K. L. Dooley, R. Schnabel, J. Slutsky, and H. Vahlbruch, “First long-term application of squeezed states of light in a gravitational-wave observatory,” Phys. Rev. Lett. 110, 181101 (2013).
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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).
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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).
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J. C. F. Matthews, X.-Q. Zhou, H. Cable, P. J. Shadbolt, D. J. Saunders, G. A. Durkin, G. J. Pryde, and J. L. O’Brien, “Towards practical quantum metrology with photon counting,” npj Quantum Inf. 2, 16023 (2016).
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K. Jiang, H. Lee, C. C. Gerry, and J. P. Dowling, “Super-resolving quantum radar: coherent-state sources with homodyne detection suffice to beat the diffraction limit,” J. Appl. Phys. 114, 193102 (2013).
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J. Opt. Soc. Am. B (1)

Nat. Photonics (1)

LIGO Scientific Collaboration, “Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light,” Nat. Photonics 7, 613–619 (2013).
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Nature (2)

D. Bouwmeester, “Quantum physics: high NOON for photons,” Nature 429, 139–141 (2004).
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M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature 429, 161–164 (2004).
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J. C. F. Matthews, X.-Q. Zhou, H. Cable, P. J. Shadbolt, D. J. Saunders, G. A. Durkin, G. J. Pryde, and J. L. O’Brien, “Towards practical quantum metrology with photon counting,” npj Quantum Inf. 2, 16023 (2016).
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[Crossref]

Y. Israel, S. Rosen, and Y. Silberberg, “Supersensitive polarization microscopy using NOON states of light,” Phys. Rev. Lett. 112, 103604 (2014).
[Crossref]

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).
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[Crossref]

E. Distante, M. Ježek, and U. L. Andersen, “Deterministic superresolution with coherent states at the shot noise limit,” Phys. Rev. Lett. 111, 033603 (2013).
[Crossref]

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

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

J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, “Two-photon interference in a Mach-Zehnder interferometer,” Phys. Rev. Lett. 65, 1348–1351 (1990).
[Crossref]

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]

Rev. Mod. Phys. (2)

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[Crossref]

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Science (3)

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306, 1330–1336 (2004).
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I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science 328, 879–881 (2010).
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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]

Other (1)

A simple approach for increasing the fringe resolution by classical means is a multi-pass interferometer. As the mentioned techniques can be mapped to multi-pass configurations, we focus on the single-pass configuration.

Supplementary Material (1)

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» Supplement 1       Supplemental Document

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Figures (5)

Fig. 1.
Fig. 1. Scheme of the approach. A coherent state |α⟩ and a vacuum squeezed state |ξ⟩ are interfered on the first beam splitter. Insets show Wigner functions of the respective states, simulated for ς=1e and α=10. In one of the resulting modes, a variable phase shifter is placed. At the second beam splitter the modes interfere again, producing the depicted Wigner functions. Eight superimposed distributions illustrate the effect of the phase shift, that is, each distribution is separated by π4. If at Δϕ=0 the squeezed vacuum state leaves the upper arm, the coherent state exits the lower one. Finally, the state is projected onto the quadrature eigenstate ⟨p| and partitioned by Π^.

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Fig. 2.
Fig. 2. Performance of the protocol for α=10 under variation of the bin size and squeezing parameter. (a) Improvement of the FWHM compared to the Rayleigh criterion (2π/3). The achieved FWHM is extracted numerically from the response of a pure state. The improvement is monotonic in the sense that a smaller bin size a always leads to a higher resolution. (b) Maximum sensitivity compared to the SNL. The region with negative values describes the parameter space where the SNL is surpassed. The threshold is marked by the bold line. (c) Unlike case (b), we set the purity P=1/2.

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Fig. 3.
Fig. 3. Experimental implementation. A vacuum squeezed state, created by parametric downconversion, and a coherent state, generated via an electro-optic modulator (EOM), are sent into a polarization-based Mach–Zehnder interferometer (MZI). A quarter-wave plate in combination with a motorized half-wave plate (Δϕ) forms the equivalent phase shift of a MZI where the two modes are spatially separated. The piezo transducers P1 and P2 stabilize the phase between the input states and the local oscillator (LO), respectively. A half-wave plate in front of the last polarizing beam splitter is used to balance the photocurrent in the homodyne detector (HD). All cubes represent polarizing beam splitters.

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Fig. 4.
Fig. 4. Results achieved for an input state with (a) |α|2≈30.7 (N≈33.6) and (b) |α|2≈427 (N≈430). Left: The fringe after applying the dichotomy operator Π^. A dashed line follows the fringe of a standard interferometer. Its FWHM is (a) 5.7 and (b) 22.2 times larger compared to our result. Right: The sensitivity derived from the experimental data. In a range of about ±0.1  rad, the SNL was surpassed by a factor of (a) 1.5 and (b) 1.7. The uncertainty of each data point is well within the “□” symbol. We attribute the symmetric deviations at the wings to a systematic anomaly in the set phase shift controlled by the HWP.

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Fig. 5.
Fig. 5. Summary of experimental results. The solid orange lines are theoretical predictions derived from the measured squeezing parameters and displacement amplitude. Each cross symbolizes a measurement run. The uncertainties are much smaller than the symbol size. (a) The FWHM under variation of the total average photon number of the input state. It always beats the Rayleigh criterion of 2π/3. Comparing the theoretically predicted FWHM proves a stable performance of the setup. (b) A comparison to four sensitivity limits. As for the resolution, the theoretical prediction affirms our experimental results. The SNL was outperformed throughout the experiment at a scaling of N−0.56; the Heisenberg scaling of 1/N is, however, not attainable. Using no windowing (a→∞), a better sensitivity and a scaling of N−0.57 can be achieved; however, it comes at the cost of super-resolution. The ultimate bound of our protocol follows N−3/4, assuming no losses and restrictions on the squeezing degree.

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Equations (8)

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Win(x1,p1,x2,p2)=W|α⟩(x1,p1)W|ξ⟩(x2,p2)=2 exp(−2((x1−α)2+p12))π·2P exp(−2(P2ς2x22+p22ς2))π,
Π^0=∫−aadp|p⟩⟨p|,Π^1=I^−Π^0.
⟨Π^⟩a→0,P=1=2ς2 exp(−2ς2|α|2 sin2 ϕ2ς2(ς2−1)cos ϕ+(ς4−1)cos2 ϕ+(ς2+1)2)(ς4−1)cos2 ϕ+2(ς2−1)ς2 cos ϕ+(ς2+1)2.
⟨Π^⟩=12 erf(2aς)[erf(2c1c−)+erf(2c1c+)],
c±=a±12|α| sin ϕ
c1=P2ς2(ς2(cos ϕ+1)2+2(1−cos ϕ))−cos2 ϕ+14P2ς2.
σ=ΔΠ^/|d/dϕ⟨Π^⟩|,
σ=|c132(2−c2)c2π/2/[exp(−2c−2/c1)(c1′c−+αc1 cos ϕ)+exp(−2c+2/c1)(c1′c+−αc1 cos ϕ)]|,

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