Abstract

We propose and implement a procedure for enhancing the sensitivity with which one can determine the phase shift experienced by a thermal light beam possessing on average fewer than four photons in passing through an interferometer. Our procedure entails subtracting exactly one (which can be generalized to m) photon from the light field exiting an interferometer containing a phase-shifting element in one of its arms. As a consequence of the process of photon subtraction, the mean photon number and signal-to-noise ratio (SNR) of the resulting light field are increased, leading to an enhancement of the SNR of the interferometric signal for that fraction of the incoming data that leads to photon subtraction.

© 2017 Optical Society of America

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  1. B. P. Abbott, et al., “Observation of gravitational waves from a binary black hole merger,” Phys. Rev. Lett. 116, 061102 (2016).
    [Crossref]
  2. C. M. Caves, “Quantum limits on noise in linear-amplifiers,” Phys. Rev. D 26, 1817–1839 (1982).
    [Crossref]
  3. C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
    [Crossref]
  4. P. Grangier, R. E. Slusher, B. Yurke, and A. LaPorta, “Squeezed-light-enhanced polarization interferometer,” Phys. Rev. Lett. 59, 2153–2156 (1987).
    [Crossref]
  5. V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306, 1330–1336 (2004).
  6. V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006).
    [Crossref]
  7. C. C. Gerry and J. Mimih, “The parity operator in quantum optical metrology,” Contemp. Phys. 51, 497–511 (2010).
    [Crossref]
  8. V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
    [Crossref]
  9. M. D’Angelo, M. V. Chekhova, and Y. Shih, “Two-photon diffraction and quantum lithography,” Phys. Rev. Lett. 87, 013602 (2001).
    [Crossref]
  10. 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]
  11. 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]
  12. 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]
  13. 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]
  14. M. Kacprowicz, R. Demkowicz-Dobrzański, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Experimental quantum-enhanced estimation of a lossy phase shift,” Nat. Photonics 4, 357–360 (2010).
    [Crossref]
  15. M. A. Usuga, C. R. Müller, C. Wittmann, P. Marek, R. Filip, C. Marquardt, G. Leuchs, and U. L. Andersen, “Noise-powered probabilistic concentration of phase information,” Nat. Phys. 6, 767–771 (2010).
    [Crossref]
  16. F. Ferreyrol, M. Barbieri, R. Blandino, S. Fossier, R. Tualle-Brouri, and P. Grangier, “Implementation of a nondeterministic optical noiseless amplifier,” Phys. Rev. Lett. 104, 123603 (2010).
    [Crossref]
  17. G. Y. Xiang, T. C. Ralph, A. P. Lund, N. Walk, and G. J. Pryde, “Heralded noiseless linear amplification and distillation of entanglement,” Nat. Photonics 4, 316–319 (2010).
    [Crossref]
  18. A. Zavatta, J. Fiurášek, and M. Bellini, “A high-fidelity noiseless amplifier for quantum light states,” Nat. Photonics 5, 52–60 (2011).
    [Crossref]
  19. J. Park, J. Joo, A. Zavatta, M. Bellini, and H. Jeong, “Efficient noiseless linear amplification for light fields with larger amplitudes,” Opt. Express 24, 1331–1346 (2016).
    [Crossref]
  20. V. Parigi, A. Zavatta, M. Kim, and M. Bellini, “Probing quantum commutation rules by addition and subtraction of single photons to/from a light field,” Science 317, 1890–1893 (2007).
    [Crossref]
  21. Y. Zhai, F. E. Becerra, B. L. Glebov, J. Wen, A. E. Lita, B. Calkins, T. Gerrits, J. Fan, S. W. Nam, and A. Migdall, “Photon-number-resolved detection of photon-subtracted thermal light,” Opt. Lett. 38, 2171–2173 (2013).
    [Crossref]
  22. J. Wenger, R. Tualle-Brouri, and P. Grangier, “Non-Gaussian statistics from individual pulses of squeezed light,” Phys. Rev. Lett. 92, 153601 (2004).
    [Crossref]
  23. A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312, 83–86 (2006).
    [Crossref]
  24. J. S. Neergaard-Nielsen, B. M. Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik, “Generation of a superposition of odd photon number states for quantum information networks,” Phys. Rev. Lett. 97, 083604 (2006).
    [Crossref]
  25. A. Ourjoumtsev, F. Ferreyrol, R. Tualle-Brouri, and P. Grangier, “Preparation of non-local superpositions of quasi-classical light states,” Nat. Phys. 5, 189–192 (2009).
    [Crossref]
  26. H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Entanglement distillation from Gaussian input states,” Nat. Photonics 4, 178–181 (2010).
  27. S. Rosenblum, O. Bechler, I. Shomroni, Y. Lovsky, G. Guendelman, and B. Dayan, “Extraction of a single photon from an optical pulse,” Nat. Photonics 10, 19–22 (2016).
    [Crossref]
  28. R. Carranza and C. C. Gerry, “Photon-subtracted two-mode squeezed vacuum states and applications to quantum optical interferometry,” J. Opt. Soc. Am. B 29, 2581–2587 (2012).
    [Crossref]
  29. R. Birrittella and C. C. Gerry, “Quantum optical interferometry via the mixing of coherent and photon-subtracted squeezed vacuum states of light,” J. Opt. Soc. Am. B 31, 586–588 (2014).
    [Crossref]
  30. D. Braun, P. Jian, O. Pinel, and N. Treps, “Precision measurements with photon-subtracted or photon-added Gaussian states,” Phys. Rev. A 90, 013821 (2014).
    [Crossref]
  31. M. D. Vidrighin, O. Dahlsten, M. Barbieri, M. S. Kim, V. Vedral, and I. A. Walmsley, “Photonic Maxwell’s demon,” Phys. Rev. Lett. 116, 050401 (2016).
    [Crossref]
  32. e.g., Eq. (2) can be derived using a procedure described in Chap. 6 of C. C. Gerry and P. Knight, Introductory Quantum Optics (Cambridge University, 2004).
  33. A. Zavatta, V. Parigi, M. S. Kim, H. Jeong, and M. Bellini, “Experimental demonstration of the bosonic commutation relation via superpositions of quantum operations on thermal light fields,” Phys. Rev. Lett. 103, 140406 (2009).
    [Crossref]
  34. C. G. Parazzoli, B. E. Koltenbah, D. R. Gerwe, P. S. Idell, B. T. Gard, R. Birrittella, S. M. Hashemi Rafsanjani, M. Mirhosseini, O. S. Magana-Loaiza, J. P. Dowling, C. C. Gerry, R. W. Boyd, and B. C. Capron, “Enhanced thermal object imaging by photon addition and subtraction,” arXiv:1609:02780 (2016).
  35. F. T. Arecchi, “Measurement of the statistical distribution of Gaussian and laser sources,” Phys. Rev. Lett. 15, 912–916 (1965).
    [Crossref]
  36. J. Honer, R. Löw, H. Weimer, T. Pfau, and H. P. Büchler, “Artificial atoms can do more than atoms: deterministic single photon subtraction from arbitrary light fields,” Phys. Rev. Lett. 107, 093601 (2011).
    [Crossref]
  37. C. I. Osorio, N. Bruno, N. Sangouard, H. Zbinden, N. Gisin, and R. T. Thew, “Heralded photon amplification for quantum communication,” Phys. Rev. A 86, 023815 (2012).
    [Crossref]

2016 (4)

B. P. Abbott, et al., “Observation of gravitational waves from a binary black hole merger,” Phys. Rev. Lett. 116, 061102 (2016).
[Crossref]

S. Rosenblum, O. Bechler, I. Shomroni, Y. Lovsky, G. Guendelman, and B. Dayan, “Extraction of a single photon from an optical pulse,” Nat. Photonics 10, 19–22 (2016).
[Crossref]

M. D. Vidrighin, O. Dahlsten, M. Barbieri, M. S. Kim, V. Vedral, and I. A. Walmsley, “Photonic Maxwell’s demon,” Phys. Rev. Lett. 116, 050401 (2016).
[Crossref]

J. Park, J. Joo, A. Zavatta, M. Bellini, and H. Jeong, “Efficient noiseless linear amplification for light fields with larger amplitudes,” Opt. Express 24, 1331–1346 (2016).
[Crossref]

2014 (2)

R. Birrittella and C. C. Gerry, “Quantum optical interferometry via the mixing of coherent and photon-subtracted squeezed vacuum states of light,” J. Opt. Soc. Am. B 31, 586–588 (2014).
[Crossref]

D. Braun, P. Jian, O. Pinel, and N. Treps, “Precision measurements with photon-subtracted or photon-added Gaussian states,” Phys. Rev. A 90, 013821 (2014).
[Crossref]

2013 (1)

2012 (2)

R. Carranza and C. C. Gerry, “Photon-subtracted two-mode squeezed vacuum states and applications to quantum optical interferometry,” J. Opt. Soc. Am. B 29, 2581–2587 (2012).
[Crossref]

C. I. Osorio, N. Bruno, N. Sangouard, H. Zbinden, N. Gisin, and R. T. Thew, “Heralded photon amplification for quantum communication,” Phys. Rev. A 86, 023815 (2012).
[Crossref]

2011 (3)

J. Honer, R. Löw, H. Weimer, T. Pfau, and H. P. Büchler, “Artificial atoms can do more than atoms: deterministic single photon subtraction from arbitrary light fields,” Phys. Rev. Lett. 107, 093601 (2011).
[Crossref]

A. Zavatta, J. Fiurášek, and M. Bellini, “A high-fidelity noiseless amplifier for quantum light states,” Nat. Photonics 5, 52–60 (2011).
[Crossref]

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
[Crossref]

2010 (6)

C. C. Gerry and J. Mimih, “The parity operator in quantum optical metrology,” Contemp. Phys. 51, 497–511 (2010).
[Crossref]

M. Kacprowicz, R. Demkowicz-Dobrzański, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Experimental quantum-enhanced estimation of a lossy phase shift,” Nat. Photonics 4, 357–360 (2010).
[Crossref]

M. A. Usuga, C. R. Müller, C. Wittmann, P. Marek, R. Filip, C. Marquardt, G. Leuchs, and U. L. Andersen, “Noise-powered probabilistic concentration of phase information,” Nat. Phys. 6, 767–771 (2010).
[Crossref]

F. Ferreyrol, M. Barbieri, R. Blandino, S. Fossier, R. Tualle-Brouri, and P. Grangier, “Implementation of a nondeterministic optical noiseless amplifier,” Phys. Rev. Lett. 104, 123603 (2010).
[Crossref]

G. Y. Xiang, T. C. Ralph, A. P. Lund, N. Walk, and G. J. Pryde, “Heralded noiseless linear amplification and distillation of entanglement,” Nat. Photonics 4, 316–319 (2010).
[Crossref]

H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Entanglement distillation from Gaussian input states,” Nat. Photonics 4, 178–181 (2010).

2009 (2)

A. Ourjoumtsev, F. Ferreyrol, R. Tualle-Brouri, and P. Grangier, “Preparation of non-local superpositions of quasi-classical light states,” Nat. Phys. 5, 189–192 (2009).
[Crossref]

A. Zavatta, V. Parigi, M. S. Kim, H. Jeong, and M. Bellini, “Experimental demonstration of the bosonic commutation relation via superpositions of quantum operations on thermal light fields,” Phys. Rev. Lett. 103, 140406 (2009).
[Crossref]

2007 (2)

V. Parigi, A. Zavatta, M. Kim, and M. Bellini, “Probing quantum commutation rules by addition and subtraction of single photons to/from a light field,” Science 317, 1890–1893 (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 (3)

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

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312, 83–86 (2006).
[Crossref]

J. S. Neergaard-Nielsen, B. M. Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik, “Generation of a superposition of odd photon number states for quantum information networks,” Phys. Rev. Lett. 97, 083604 (2006).
[Crossref]

2004 (4)

J. Wenger, R. Tualle-Brouri, and P. Grangier, “Non-Gaussian statistics from individual pulses of squeezed light,” Phys. Rev. Lett. 92, 153601 (2004).
[Crossref]

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

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]

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]

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]

2001 (1)

M. D’Angelo, M. V. Chekhova, and Y. Shih, “Two-photon diffraction and quantum lithography,” Phys. Rev. Lett. 87, 013602 (2001).
[Crossref]

1987 (1)

P. Grangier, R. E. Slusher, B. Yurke, and A. LaPorta, “Squeezed-light-enhanced polarization interferometer,” Phys. Rev. Lett. 59, 2153–2156 (1987).
[Crossref]

1982 (1)

C. M. Caves, “Quantum limits on noise in linear-amplifiers,” Phys. Rev. D 26, 1817–1839 (1982).
[Crossref]

1981 (1)

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

1965 (1)

F. T. Arecchi, “Measurement of the statistical distribution of Gaussian and laser sources,” Phys. Rev. Lett. 15, 912–916 (1965).
[Crossref]

Abbott, B. P.

B. P. Abbott, et al., “Observation of gravitational waves from a binary black hole merger,” Phys. Rev. Lett. 116, 061102 (2016).
[Crossref]

Andersen, U. L.

M. A. Usuga, C. R. Müller, C. Wittmann, P. Marek, R. Filip, C. Marquardt, G. Leuchs, and U. L. Andersen, “Noise-powered probabilistic concentration of phase information,” Nat. Phys. 6, 767–771 (2010).
[Crossref]

Arecchi, F. T.

F. T. Arecchi, “Measurement of the statistical distribution of Gaussian and laser sources,” Phys. Rev. Lett. 15, 912–916 (1965).
[Crossref]

Aspelmeyer, M.

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

Banaszek, K.

M. Kacprowicz, R. Demkowicz-Dobrzański, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Experimental quantum-enhanced estimation of a lossy phase shift,” Nat. Photonics 4, 357–360 (2010).
[Crossref]

Barbieri, M.

M. D. Vidrighin, O. Dahlsten, M. Barbieri, M. S. Kim, V. Vedral, and I. A. Walmsley, “Photonic Maxwell’s demon,” Phys. Rev. Lett. 116, 050401 (2016).
[Crossref]

F. Ferreyrol, M. Barbieri, R. Blandino, S. Fossier, R. Tualle-Brouri, and P. Grangier, “Implementation of a nondeterministic optical noiseless amplifier,” Phys. Rev. Lett. 104, 123603 (2010).
[Crossref]

Becerra, F. E.

Bechler, O.

S. Rosenblum, O. Bechler, I. Shomroni, Y. Lovsky, G. Guendelman, and B. Dayan, “Extraction of a single photon from an optical pulse,” Nat. Photonics 10, 19–22 (2016).
[Crossref]

Bellini, M.

J. Park, J. Joo, A. Zavatta, M. Bellini, and H. Jeong, “Efficient noiseless linear amplification for light fields with larger amplitudes,” Opt. Express 24, 1331–1346 (2016).
[Crossref]

A. Zavatta, J. Fiurášek, and M. Bellini, “A high-fidelity noiseless amplifier for quantum light states,” Nat. Photonics 5, 52–60 (2011).
[Crossref]

A. Zavatta, V. Parigi, M. S. Kim, H. Jeong, and M. Bellini, “Experimental demonstration of the bosonic commutation relation via superpositions of quantum operations on thermal light fields,” Phys. Rev. Lett. 103, 140406 (2009).
[Crossref]

V. Parigi, A. Zavatta, M. Kim, and M. Bellini, “Probing quantum commutation rules by addition and subtraction of single photons to/from a light field,” Science 317, 1890–1893 (2007).
[Crossref]

Benmoussa, A.

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]

Birrittella, R.

R. Birrittella and C. C. Gerry, “Quantum optical interferometry via the mixing of coherent and photon-subtracted squeezed vacuum states of light,” J. Opt. Soc. Am. B 31, 586–588 (2014).
[Crossref]

C. G. Parazzoli, B. E. Koltenbah, D. R. Gerwe, P. S. Idell, B. T. Gard, R. Birrittella, S. M. Hashemi Rafsanjani, M. Mirhosseini, O. S. Magana-Loaiza, J. P. Dowling, C. C. Gerry, R. W. Boyd, and B. C. Capron, “Enhanced thermal object imaging by photon addition and subtraction,” arXiv:1609:02780 (2016).

Blandino, R.

F. Ferreyrol, M. Barbieri, R. Blandino, S. Fossier, R. Tualle-Brouri, and P. Grangier, “Implementation of a nondeterministic optical noiseless amplifier,” Phys. Rev. Lett. 104, 123603 (2010).
[Crossref]

Boyd, R. W.

C. G. Parazzoli, B. E. Koltenbah, D. R. Gerwe, P. S. Idell, B. T. Gard, R. Birrittella, S. M. Hashemi Rafsanjani, M. Mirhosseini, O. S. Magana-Loaiza, J. P. Dowling, C. C. Gerry, R. W. Boyd, and B. C. Capron, “Enhanced thermal object imaging by photon addition and subtraction,” arXiv:1609:02780 (2016).

Braun, D.

D. Braun, P. Jian, O. Pinel, and N. Treps, “Precision measurements with photon-subtracted or photon-added Gaussian states,” Phys. Rev. A 90, 013821 (2014).
[Crossref]

Bruno, N.

C. I. Osorio, N. Bruno, N. Sangouard, H. Zbinden, N. Gisin, and R. T. Thew, “Heralded photon amplification for quantum communication,” Phys. Rev. A 86, 023815 (2012).
[Crossref]

Büchler, H. P.

J. Honer, R. Löw, H. Weimer, T. Pfau, and H. P. Büchler, “Artificial atoms can do more than atoms: deterministic single photon subtraction from arbitrary light fields,” Phys. Rev. Lett. 107, 093601 (2011).
[Crossref]

Calkins, B.

Campos, R. A.

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]

Capron, B. C.

C. G. Parazzoli, B. E. Koltenbah, D. R. Gerwe, P. S. Idell, B. T. Gard, R. Birrittella, S. M. Hashemi Rafsanjani, M. Mirhosseini, O. S. Magana-Loaiza, J. P. Dowling, C. C. Gerry, R. W. Boyd, and B. C. Capron, “Enhanced thermal object imaging by photon addition and subtraction,” arXiv:1609:02780 (2016).

Carranza, R.

Caves, C. M.

C. M. Caves, “Quantum limits on noise in linear-amplifiers,” Phys. Rev. D 26, 1817–1839 (1982).
[Crossref]

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

Chekhova, M. V.

M. D’Angelo, M. V. Chekhova, and Y. Shih, “Two-photon diffraction and quantum lithography,” Phys. Rev. Lett. 87, 013602 (2001).
[Crossref]

D’Angelo, M.

M. D’Angelo, M. V. Chekhova, and Y. Shih, “Two-photon diffraction and quantum lithography,” Phys. Rev. Lett. 87, 013602 (2001).
[Crossref]

Dahlsten, O.

M. D. Vidrighin, O. Dahlsten, M. Barbieri, M. S. Kim, V. Vedral, and I. A. Walmsley, “Photonic Maxwell’s demon,” Phys. Rev. Lett. 116, 050401 (2016).
[Crossref]

Dayan, B.

S. Rosenblum, O. Bechler, I. Shomroni, Y. Lovsky, G. Guendelman, and B. Dayan, “Extraction of a single photon from an optical pulse,” Nat. Photonics 10, 19–22 (2016).
[Crossref]

Demkowicz-Dobrzanski, R.

M. Kacprowicz, R. Demkowicz-Dobrzański, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Experimental quantum-enhanced estimation of a lossy phase shift,” Nat. Photonics 4, 357–360 (2010).
[Crossref]

Dowling, J. P.

C. G. Parazzoli, B. E. Koltenbah, D. R. Gerwe, P. S. Idell, B. T. Gard, R. Birrittella, S. M. Hashemi Rafsanjani, M. Mirhosseini, O. S. Magana-Loaiza, J. P. Dowling, C. C. Gerry, R. W. Boyd, and B. C. Capron, “Enhanced thermal object imaging by photon addition and subtraction,” arXiv:1609:02780 (2016).

Fan, J.

Ferreyrol, F.

F. Ferreyrol, M. Barbieri, R. Blandino, S. Fossier, R. Tualle-Brouri, and P. Grangier, “Implementation of a nondeterministic optical noiseless amplifier,” Phys. Rev. Lett. 104, 123603 (2010).
[Crossref]

A. Ourjoumtsev, F. Ferreyrol, R. Tualle-Brouri, and P. Grangier, “Preparation of non-local superpositions of quasi-classical light states,” Nat. Phys. 5, 189–192 (2009).
[Crossref]

Filip, R.

M. A. Usuga, C. R. Müller, C. Wittmann, P. Marek, R. Filip, C. Marquardt, G. Leuchs, and U. L. Andersen, “Noise-powered probabilistic concentration of phase information,” Nat. Phys. 6, 767–771 (2010).
[Crossref]

Fiurášek, J.

A. Zavatta, J. Fiurášek, and M. Bellini, “A high-fidelity noiseless amplifier for quantum light states,” Nat. Photonics 5, 52–60 (2011).
[Crossref]

Fossier, S.

F. Ferreyrol, M. Barbieri, R. Blandino, S. Fossier, R. Tualle-Brouri, and P. Grangier, “Implementation of a nondeterministic optical noiseless amplifier,” Phys. Rev. Lett. 104, 123603 (2010).
[Crossref]

Furusawa, A.

H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Entanglement distillation from Gaussian input states,” Nat. Photonics 4, 178–181 (2010).

Gard, B. T.

C. G. Parazzoli, B. E. Koltenbah, D. R. Gerwe, P. S. Idell, B. T. Gard, R. Birrittella, S. M. Hashemi Rafsanjani, M. Mirhosseini, O. S. Magana-Loaiza, J. P. Dowling, C. C. Gerry, R. W. Boyd, and B. C. Capron, “Enhanced thermal object imaging by photon addition and subtraction,” arXiv:1609:02780 (2016).

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]

Gerrits, T.

Gerry, C. C.

R. Birrittella and C. C. Gerry, “Quantum optical interferometry via the mixing of coherent and photon-subtracted squeezed vacuum states of light,” J. Opt. Soc. Am. B 31, 586–588 (2014).
[Crossref]

R. Carranza and C. C. Gerry, “Photon-subtracted two-mode squeezed vacuum states and applications to quantum optical interferometry,” J. Opt. Soc. Am. B 29, 2581–2587 (2012).
[Crossref]

C. C. Gerry and J. Mimih, “The parity operator in quantum optical metrology,” Contemp. Phys. 51, 497–511 (2010).
[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. G. Parazzoli, B. E. Koltenbah, D. R. Gerwe, P. S. Idell, B. T. Gard, R. Birrittella, S. M. Hashemi Rafsanjani, M. Mirhosseini, O. S. Magana-Loaiza, J. P. Dowling, C. C. Gerry, R. W. Boyd, and B. C. Capron, “Enhanced thermal object imaging by photon addition and subtraction,” arXiv:1609:02780 (2016).

e.g., Eq. (2) can be derived using a procedure described in Chap. 6 of C. C. Gerry and P. Knight, Introductory Quantum Optics (Cambridge University, 2004).

Gerwe, D. R.

C. G. Parazzoli, B. E. Koltenbah, D. R. Gerwe, P. S. Idell, B. T. Gard, R. Birrittella, S. M. Hashemi Rafsanjani, M. Mirhosseini, O. S. Magana-Loaiza, J. P. Dowling, C. C. Gerry, R. W. Boyd, and B. C. Capron, “Enhanced thermal object imaging by photon addition and subtraction,” arXiv:1609:02780 (2016).

Giovannetti, V.

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
[Crossref]

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

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

Gisin, N.

C. I. Osorio, N. Bruno, N. Sangouard, H. Zbinden, N. Gisin, and R. T. Thew, “Heralded photon amplification for quantum communication,” Phys. Rev. A 86, 023815 (2012).
[Crossref]

Glebov, B. L.

Grangier, P.

F. Ferreyrol, M. Barbieri, R. Blandino, S. Fossier, R. Tualle-Brouri, and P. Grangier, “Implementation of a nondeterministic optical noiseless amplifier,” Phys. Rev. Lett. 104, 123603 (2010).
[Crossref]

A. Ourjoumtsev, F. Ferreyrol, R. Tualle-Brouri, and P. Grangier, “Preparation of non-local superpositions of quasi-classical light states,” Nat. Phys. 5, 189–192 (2009).
[Crossref]

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312, 83–86 (2006).
[Crossref]

J. Wenger, R. Tualle-Brouri, and P. Grangier, “Non-Gaussian statistics from individual pulses of squeezed light,” Phys. Rev. Lett. 92, 153601 (2004).
[Crossref]

P. Grangier, R. E. Slusher, B. Yurke, and A. LaPorta, “Squeezed-light-enhanced polarization interferometer,” Phys. Rev. Lett. 59, 2153–2156 (1987).
[Crossref]

Guendelman, G.

S. Rosenblum, O. Bechler, I. Shomroni, Y. Lovsky, G. Guendelman, and B. Dayan, “Extraction of a single photon from an optical pulse,” Nat. Photonics 10, 19–22 (2016).
[Crossref]

Hashemi Rafsanjani, S. M.

C. G. Parazzoli, B. E. Koltenbah, D. R. Gerwe, P. S. Idell, B. T. Gard, R. Birrittella, S. M. Hashemi Rafsanjani, M. Mirhosseini, O. S. Magana-Loaiza, J. P. Dowling, C. C. Gerry, R. W. Boyd, and B. C. Capron, “Enhanced thermal object imaging by photon addition and subtraction,” arXiv:1609:02780 (2016).

Hayasaka, K.

H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Entanglement distillation from Gaussian input states,” Nat. Photonics 4, 178–181 (2010).

Hettich, C.

J. S. Neergaard-Nielsen, B. M. Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik, “Generation of a superposition of odd photon number states for quantum information networks,” Phys. Rev. Lett. 97, 083604 (2006).
[Crossref]

Honer, J.

J. Honer, R. Löw, H. Weimer, T. Pfau, and H. P. Büchler, “Artificial atoms can do more than atoms: deterministic single photon subtraction from arbitrary light fields,” Phys. Rev. Lett. 107, 093601 (2011).
[Crossref]

Idell, P. S.

C. G. Parazzoli, B. E. Koltenbah, D. R. Gerwe, P. S. Idell, B. T. Gard, R. Birrittella, S. M. Hashemi Rafsanjani, M. Mirhosseini, O. S. Magana-Loaiza, J. P. Dowling, C. C. Gerry, R. W. Boyd, and B. C. Capron, “Enhanced thermal object imaging by photon addition and subtraction,” arXiv:1609:02780 (2016).

Jeong, H.

J. Park, J. Joo, A. Zavatta, M. Bellini, and H. Jeong, “Efficient noiseless linear amplification for light fields with larger amplitudes,” Opt. Express 24, 1331–1346 (2016).
[Crossref]

A. Zavatta, V. Parigi, M. S. Kim, H. Jeong, and M. Bellini, “Experimental demonstration of the bosonic commutation relation via superpositions of quantum operations on thermal light fields,” Phys. Rev. Lett. 103, 140406 (2009).
[Crossref]

Jian, P.

D. Braun, P. Jian, O. Pinel, and N. Treps, “Precision measurements with photon-subtracted or photon-added Gaussian states,” Phys. Rev. A 90, 013821 (2014).
[Crossref]

Joo, J.

Kacprowicz, M.

M. Kacprowicz, R. Demkowicz-Dobrzański, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Experimental quantum-enhanced estimation of a lossy phase shift,” Nat. Photonics 4, 357–360 (2010).
[Crossref]

Kim, M.

V. Parigi, A. Zavatta, M. Kim, and M. Bellini, “Probing quantum commutation rules by addition and subtraction of single photons to/from a light field,” Science 317, 1890–1893 (2007).
[Crossref]

Kim, M. S.

M. D. Vidrighin, O. Dahlsten, M. Barbieri, M. S. Kim, V. Vedral, and I. A. Walmsley, “Photonic Maxwell’s demon,” Phys. Rev. Lett. 116, 050401 (2016).
[Crossref]

A. Zavatta, V. Parigi, M. S. Kim, H. Jeong, and M. Bellini, “Experimental demonstration of the bosonic commutation relation via superpositions of quantum operations on thermal light fields,” Phys. Rev. Lett. 103, 140406 (2009).
[Crossref]

Knight, P.

e.g., Eq. (2) can be derived using a procedure described in Chap. 6 of C. C. Gerry and P. Knight, Introductory Quantum Optics (Cambridge University, 2004).

Koltenbah, B. E.

C. G. Parazzoli, B. E. Koltenbah, D. R. Gerwe, P. S. Idell, B. T. Gard, R. Birrittella, S. M. Hashemi Rafsanjani, M. Mirhosseini, O. S. Magana-Loaiza, J. P. Dowling, C. C. Gerry, R. W. Boyd, and B. C. Capron, “Enhanced thermal object imaging by photon addition and subtraction,” arXiv:1609:02780 (2016).

LaPorta, A.

P. Grangier, R. E. Slusher, B. Yurke, and A. LaPorta, “Squeezed-light-enhanced polarization interferometer,” Phys. Rev. Lett. 59, 2153–2156 (1987).
[Crossref]

Laurat, J.

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312, 83–86 (2006).
[Crossref]

Leuchs, G.

M. A. Usuga, C. R. Müller, C. Wittmann, P. Marek, R. Filip, C. Marquardt, G. Leuchs, and U. L. Andersen, “Noise-powered probabilistic concentration of phase information,” Nat. Phys. 6, 767–771 (2010).
[Crossref]

Lita, A. E.

Lloyd, S.

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
[Crossref]

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

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

Lovsky, Y.

S. Rosenblum, O. Bechler, I. Shomroni, Y. Lovsky, G. Guendelman, and B. Dayan, “Extraction of a single photon from an optical pulse,” Nat. Photonics 10, 19–22 (2016).
[Crossref]

Löw, R.

J. Honer, R. Löw, H. Weimer, T. Pfau, and H. P. Büchler, “Artificial atoms can do more than atoms: deterministic single photon subtraction from arbitrary light fields,” Phys. Rev. Lett. 107, 093601 (2011).
[Crossref]

Lund, A. P.

G. Y. Xiang, T. C. Ralph, A. P. Lund, N. Walk, and G. J. Pryde, “Heralded noiseless linear amplification and distillation of entanglement,” Nat. Photonics 4, 316–319 (2010).
[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]

Maccone, L.

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
[Crossref]

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

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

Magana-Loaiza, O. S.

C. G. Parazzoli, B. E. Koltenbah, D. R. Gerwe, P. S. Idell, B. T. Gard, R. Birrittella, S. M. Hashemi Rafsanjani, M. Mirhosseini, O. S. Magana-Loaiza, J. P. Dowling, C. C. Gerry, R. W. Boyd, and B. C. Capron, “Enhanced thermal object imaging by photon addition and subtraction,” arXiv:1609:02780 (2016).

Marek, P.

M. A. Usuga, C. R. Müller, C. Wittmann, P. Marek, R. Filip, C. Marquardt, G. Leuchs, and U. L. Andersen, “Noise-powered probabilistic concentration of phase information,” Nat. Phys. 6, 767–771 (2010).
[Crossref]

Marquardt, C.

M. A. Usuga, C. R. Müller, C. Wittmann, P. Marek, R. Filip, C. Marquardt, G. Leuchs, and U. L. Andersen, “Noise-powered probabilistic concentration of phase information,” Nat. Phys. 6, 767–771 (2010).
[Crossref]

Migdall, A.

Mimih, J.

C. C. Gerry and J. Mimih, “The parity operator in quantum optical metrology,” Contemp. Phys. 51, 497–511 (2010).
[Crossref]

Mirhosseini, M.

C. G. Parazzoli, B. E. Koltenbah, D. R. Gerwe, P. S. Idell, B. T. Gard, R. Birrittella, S. M. Hashemi Rafsanjani, M. Mirhosseini, O. S. Magana-Loaiza, J. P. Dowling, C. C. Gerry, R. W. Boyd, and B. C. Capron, “Enhanced thermal object imaging by photon addition and subtraction,” arXiv:1609:02780 (2016).

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]

Mølmer, K.

J. S. Neergaard-Nielsen, B. M. Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik, “Generation of a superposition of odd photon number states for quantum information networks,” Phys. Rev. Lett. 97, 083604 (2006).
[Crossref]

Müller, C. R.

M. A. Usuga, C. R. Müller, C. Wittmann, P. Marek, R. Filip, C. Marquardt, G. Leuchs, and U. L. Andersen, “Noise-powered probabilistic concentration of phase information,” Nat. Phys. 6, 767–771 (2010).
[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]

Nam, S. W.

Neergaard-Nielsen, J. S.

H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Entanglement distillation from Gaussian input states,” Nat. Photonics 4, 178–181 (2010).

J. S. Neergaard-Nielsen, B. M. Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik, “Generation of a superposition of odd photon number states for quantum information networks,” Phys. Rev. Lett. 97, 083604 (2006).
[Crossref]

Nielsen, B. M.

J. S. Neergaard-Nielsen, B. M. Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik, “Generation of a superposition of odd photon number states for quantum information networks,” Phys. Rev. Lett. 97, 083604 (2006).
[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]

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]

Osorio, C. I.

C. I. Osorio, N. Bruno, N. Sangouard, H. Zbinden, N. Gisin, and R. T. Thew, “Heralded photon amplification for quantum communication,” Phys. Rev. A 86, 023815 (2012).
[Crossref]

Ourjoumtsev, A.

A. Ourjoumtsev, F. Ferreyrol, R. Tualle-Brouri, and P. Grangier, “Preparation of non-local superpositions of quasi-classical light states,” Nat. Phys. 5, 189–192 (2009).
[Crossref]

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312, 83–86 (2006).
[Crossref]

Pan, J. W.

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

Parazzoli, C. G.

C. G. Parazzoli, B. E. Koltenbah, D. R. Gerwe, P. S. Idell, B. T. Gard, R. Birrittella, S. M. Hashemi Rafsanjani, M. Mirhosseini, O. S. Magana-Loaiza, J. P. Dowling, C. C. Gerry, R. W. Boyd, and B. C. Capron, “Enhanced thermal object imaging by photon addition and subtraction,” arXiv:1609:02780 (2016).

Parigi, V.

A. Zavatta, V. Parigi, M. S. Kim, H. Jeong, and M. Bellini, “Experimental demonstration of the bosonic commutation relation via superpositions of quantum operations on thermal light fields,” Phys. Rev. Lett. 103, 140406 (2009).
[Crossref]

V. Parigi, A. Zavatta, M. Kim, and M. Bellini, “Probing quantum commutation rules by addition and subtraction of single photons to/from a light field,” Science 317, 1890–1893 (2007).
[Crossref]

Park, J.

Pfau, T.

J. Honer, R. Löw, H. Weimer, T. Pfau, and H. P. Büchler, “Artificial atoms can do more than atoms: deterministic single photon subtraction from arbitrary light fields,” Phys. Rev. Lett. 107, 093601 (2011).
[Crossref]

Pinel, O.

D. Braun, P. Jian, O. Pinel, and N. Treps, “Precision measurements with photon-subtracted or photon-added Gaussian states,” Phys. Rev. A 90, 013821 (2014).
[Crossref]

Polzik, E. S.

J. S. Neergaard-Nielsen, B. M. Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik, “Generation of a superposition of odd photon number states for quantum information networks,” Phys. Rev. Lett. 97, 083604 (2006).
[Crossref]

Pryde, G. J.

G. Y. Xiang, T. C. Ralph, A. P. Lund, N. Walk, and G. J. Pryde, “Heralded noiseless linear amplification and distillation of entanglement,” Nat. Photonics 4, 316–319 (2010).
[Crossref]

Ralph, T. C.

G. Y. Xiang, T. C. Ralph, A. P. Lund, N. Walk, and G. J. Pryde, “Heralded noiseless linear amplification and distillation of entanglement,” Nat. Photonics 4, 316–319 (2010).
[Crossref]

Rosenblum, S.

S. Rosenblum, O. Bechler, I. Shomroni, Y. Lovsky, G. Guendelman, and B. Dayan, “Extraction of a single photon from an optical pulse,” Nat. Photonics 10, 19–22 (2016).
[Crossref]

Sangouard, N.

C. I. Osorio, N. Bruno, N. Sangouard, H. Zbinden, N. Gisin, and R. T. Thew, “Heralded photon amplification for quantum communication,” Phys. Rev. A 86, 023815 (2012).
[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]

Sasaki, M.

H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Entanglement distillation from Gaussian input states,” Nat. Photonics 4, 178–181 (2010).

Shih, Y.

M. D’Angelo, M. V. Chekhova, and Y. Shih, “Two-photon diffraction and quantum lithography,” Phys. Rev. Lett. 87, 013602 (2001).
[Crossref]

Shomroni, I.

S. Rosenblum, O. Bechler, I. Shomroni, Y. Lovsky, G. Guendelman, and B. Dayan, “Extraction of a single photon from an optical pulse,” Nat. Photonics 10, 19–22 (2016).
[Crossref]

Slusher, R. E.

P. Grangier, R. E. Slusher, B. Yurke, and A. LaPorta, “Squeezed-light-enhanced polarization interferometer,” Phys. Rev. Lett. 59, 2153–2156 (1987).
[Crossref]

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]

Takahashi, H.

H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Entanglement distillation from Gaussian input states,” Nat. Photonics 4, 178–181 (2010).

Takeoka, M.

H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Entanglement distillation from Gaussian input states,” Nat. Photonics 4, 178–181 (2010).

Takeuchi, M.

H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Entanglement distillation from Gaussian input states,” Nat. Photonics 4, 178–181 (2010).

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]

Thew, R. T.

C. I. Osorio, N. Bruno, N. Sangouard, H. Zbinden, N. Gisin, and R. T. Thew, “Heralded photon amplification for quantum communication,” Phys. Rev. A 86, 023815 (2012).
[Crossref]

Treps, N.

D. Braun, P. Jian, O. Pinel, and N. Treps, “Precision measurements with photon-subtracted or photon-added Gaussian states,” Phys. Rev. A 90, 013821 (2014).
[Crossref]

Tualle-Brouri, R.

F. Ferreyrol, M. Barbieri, R. Blandino, S. Fossier, R. Tualle-Brouri, and P. Grangier, “Implementation of a nondeterministic optical noiseless amplifier,” Phys. Rev. Lett. 104, 123603 (2010).
[Crossref]

A. Ourjoumtsev, F. Ferreyrol, R. Tualle-Brouri, and P. Grangier, “Preparation of non-local superpositions of quasi-classical light states,” Nat. Phys. 5, 189–192 (2009).
[Crossref]

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312, 83–86 (2006).
[Crossref]

J. Wenger, R. Tualle-Brouri, and P. Grangier, “Non-Gaussian statistics from individual pulses of squeezed light,” Phys. Rev. Lett. 92, 153601 (2004).
[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]

Usuga, M. A.

M. A. Usuga, C. R. Müller, C. Wittmann, P. Marek, R. Filip, C. Marquardt, G. Leuchs, and U. L. Andersen, “Noise-powered probabilistic concentration of phase information,” Nat. Phys. 6, 767–771 (2010).
[Crossref]

Vedral, V.

M. D. Vidrighin, O. Dahlsten, M. Barbieri, M. S. Kim, V. Vedral, and I. A. Walmsley, “Photonic Maxwell’s demon,” Phys. Rev. Lett. 116, 050401 (2016).
[Crossref]

Vidrighin, M. D.

M. D. Vidrighin, O. Dahlsten, M. Barbieri, M. S. Kim, V. Vedral, and I. A. Walmsley, “Photonic Maxwell’s demon,” Phys. Rev. Lett. 116, 050401 (2016).
[Crossref]

Walk, N.

G. Y. Xiang, T. C. Ralph, A. P. Lund, N. Walk, and G. J. Pryde, “Heralded noiseless linear amplification and distillation of entanglement,” Nat. Photonics 4, 316–319 (2010).
[Crossref]

Walmsley, I. A.

M. D. Vidrighin, O. Dahlsten, M. Barbieri, M. S. Kim, V. Vedral, and I. A. Walmsley, “Photonic Maxwell’s demon,” Phys. Rev. Lett. 116, 050401 (2016).
[Crossref]

M. Kacprowicz, R. Demkowicz-Dobrzański, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Experimental quantum-enhanced estimation of a lossy phase shift,” Nat. Photonics 4, 357–360 (2010).
[Crossref]

Walther, P.

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

Wasilewski, W.

M. Kacprowicz, R. Demkowicz-Dobrzański, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Experimental quantum-enhanced estimation of a lossy phase shift,” Nat. Photonics 4, 357–360 (2010).
[Crossref]

Weimer, H.

J. Honer, R. Löw, H. Weimer, T. Pfau, and H. P. Büchler, “Artificial atoms can do more than atoms: deterministic single photon subtraction from arbitrary light fields,” Phys. Rev. Lett. 107, 093601 (2011).
[Crossref]

Wen, J.

Wenger, J.

J. Wenger, R. Tualle-Brouri, and P. Grangier, “Non-Gaussian statistics from individual pulses of squeezed light,” Phys. Rev. Lett. 92, 153601 (2004).
[Crossref]

Wittmann, C.

M. A. Usuga, C. R. Müller, C. Wittmann, P. Marek, R. Filip, C. Marquardt, G. Leuchs, and U. L. Andersen, “Noise-powered probabilistic concentration of phase information,” Nat. Phys. 6, 767–771 (2010).
[Crossref]

Xiang, G. Y.

G. Y. Xiang, T. C. Ralph, A. P. Lund, N. Walk, and G. J. Pryde, “Heralded noiseless linear amplification and distillation of entanglement,” Nat. Photonics 4, 316–319 (2010).
[Crossref]

Yurke, B.

P. Grangier, R. E. Slusher, B. Yurke, and A. LaPorta, “Squeezed-light-enhanced polarization interferometer,” Phys. Rev. Lett. 59, 2153–2156 (1987).
[Crossref]

Zavatta, A.

J. Park, J. Joo, A. Zavatta, M. Bellini, and H. Jeong, “Efficient noiseless linear amplification for light fields with larger amplitudes,” Opt. Express 24, 1331–1346 (2016).
[Crossref]

A. Zavatta, J. Fiurášek, and M. Bellini, “A high-fidelity noiseless amplifier for quantum light states,” Nat. Photonics 5, 52–60 (2011).
[Crossref]

A. Zavatta, V. Parigi, M. S. Kim, H. Jeong, and M. Bellini, “Experimental demonstration of the bosonic commutation relation via superpositions of quantum operations on thermal light fields,” Phys. Rev. Lett. 103, 140406 (2009).
[Crossref]

V. Parigi, A. Zavatta, M. Kim, and M. Bellini, “Probing quantum commutation rules by addition and subtraction of single photons to/from a light field,” Science 317, 1890–1893 (2007).
[Crossref]

Zbinden, H.

C. I. Osorio, N. Bruno, N. Sangouard, H. Zbinden, N. Gisin, and R. T. Thew, “Heralded photon amplification for quantum communication,” Phys. Rev. A 86, 023815 (2012).
[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]

Zhai, Y.

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Supplementary Material (1)

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Figures (5)

Fig. 1.
Fig. 1. Schematic of the experimental setup used to observe increased measurement sensitivity through photon subtraction. The output beam from a narrow-bandwidth cw laser is focused onto a rotating ground-glass plate and is then coupled into a single-mode optical fiber (SMF). The single-transverse-mode, thermal light exiting the SMF is then sent to one input port ( a ^ ) of a Mach–Zehnder interferometer (MZI). Light from one output port ( c ^ ) is then sent to a combination of a half-wave plate (HWP) and polarizing beam splitter (PBS2) to perform the process of photon subtraction. Detector APD2 counts the number of photons in a time window of fixed length, conditioned on a detection event in APD1. Similarly, light from the other output port ( d ^ ) is sent to detector APD3. In our actual implementation (see inset), we use a common-path MZI to increase the stability. In this case a rotatable HWP is used to control the phase difference between two orthogonal polarization states of the light beam.

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Fig. 2.
Fig. 2. (a) Probability distribution of the photon number distribution. Each column corresponds to a given value of the phase, and the probability is encoded in the color coding. (b) Mean photon number and (c) signal-to-noise ratio measured at the output of the interferometer by APD2 as a function of the phase difference between the two arms of the interferometer. Each column in panel (a) corresponds to a dot in panels (b) and (c). Dots represent experimental results, and the solid lines describe what is expected from theory. Here the average photon number before the interferometer is n ¯ = 4.1 , the transmission of the subtracting PBS is T = 0.9 , the detection efficiency of APD2 is η 2 ≈ 0.3 , and N ¯ c = 1.1 .

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Fig. 3.
Fig. 3. (a) Probability distribution of the photon number distribution. Each column corresponds to a given value of the phase, and the probability is encoded in the color coding. (b) Mean photon number (blue) and (c) signal-to-noise ratio (blue) measured at the output of the interferometer by APD2 as a function of the phase difference between the two arms of the interferometer. Each column in panel (a) corresponds to a dot in panels (b) and (c). Dots represent experimental results, and the solid lines describe what is expected from theory. The parameters are the same as in Fig. 2, and the detection efficiency of APD1 is η 1 ≈ 0.33 . Also in red are the results for thermal light without subtraction for comparison.

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Fig. 4.
Fig. 4. Same as Fig. 3, except for two-photon-subtracted thermal light. Also shown are the results for thermal light without subtraction (in red) and one-photon-subtracted thermal light (in blue) for comparison.

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Fig. 5.
Fig. 5. Average occupation number (top) and signal-to-noise ratio (bottom) at the other output port d ^ of the MZI measured as a function of the phase shift introduced within the interferometer. The dots represent the experimental results, and the lines are the theoretical predictions. Here n ¯ = 4.1 , T = 0.9 , η 3 ∼ 0.42 , and η 1 ∼ 0.33 . Also included are the non-conditioned and one-photon-subtracted thermal light for comparison.

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Equations (8)

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c ^ = 1 2 [ ( e i φ − 1 ) b ^ + i ( e i φ + 1 ) a ^ ] , d ^ = 1 2 [ i ( e i φ + 1 ) b ^ + ( 1 − e i φ ) a ^ ] ,
n ¯ c = Tr [ c ^ † c ^ ρ ^ 0 ] = n ¯   cos 2   φ 2 , Δ n c = n ¯ c 2 + n ¯ c , n ¯ d = Tr [ d ^ † d ^ ρ ^ 0 ] = n ¯   sin 2   φ 2 , Δ n d = n ¯ d 2 + n ¯ d .
ρ ^ 1 c = ∑ n ( n + 1 ) n ¯ c n ( 1 + n ¯ c ) n + 2 | n ⟩ c ⟨ n | c .
Tr [ c ^ † c ^ ρ ^ 1 c ] = 2 n ¯   cos 2   φ 2 = 2 n ¯ c , ( Tr [ ( c ^ † c ^ ) 2 ρ ^ 1 c ] − Tr [ c ^ † c ^ ρ ^ 1 c ] 2 ) 1 / 2 = 2 Δ n c .
N ¯ c = n ¯ T η 2   cos 2   φ 2 ( δ 0 m + ( 1 − δ 0 m ) ( m + 1 ) 1 + n ¯ ( 1 − T ) η 1   cos 2   φ 2 ) , Δ N c = N ¯ c / ( 1 + m ) n ¯ T η 2   cos 2   φ 2 1 + n ¯ ( T η 2 + ( 1 − T ) η 1 ) cos 2   φ 2 .
δ ϕ = Δ s ( ∂ s ∂ ϕ ) − 1 ,
δ ϕ sub = Δ N sub ( ∂ N ¯ sub ∂ ϕ ) − 1 = 2 Δ N ( 2 ∂ N ¯ ∂ ϕ ) − 1 = 1 2 Δ N ( ∂ N ¯ ∂ ϕ ) − 1 = δ ϕ 2 .
δ ϕ sub = δ ϕ 2 ξ .

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