Abstract

Heisenberg uncertainty relation in quantum mechanics sets the limit on the measurement precision of non-commuting observables in one system, which prevents us from measuring them accurately at the same time. However, quantum entanglement between two systems allows us to infer through Einstein-Podolsky-Rosen correlations two conjugate observables with precision better than what is allowed by Heisenberg uncertainty relation. With the help of the newly developed SU(1,) interferometer, we implement a scheme to jointly measure information encoded in multiple non-commuting observables of an optical field with a signal-to-noise ratio improvement of about 20% over the classical limit on all measured quantities simultaneously. This scheme can be generalized to the joint measurement of information in arbitrary number of non-commuting observables.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  29. X. Guo, X. Li, N. Liu, and Z. Y. Ou, “Quantum information tapping using a fiber optical parametric amplifier with noise figure improved by correlated inputs,” Sci. Reports 6, 30214 (2016).
    [Crossref]
  30. C. Sparaciari, S. Olivares, and M. G. A. Paris, “Gaussian-state interferometry with passive and active elements,” Phys. Rev. A 93, 023810 (2016).
    [Crossref]
  31. X. Guo, N. Liu, Y. Liu, X. Li, and Z. Y. Ou, “Generation of continuous variable quantum entanglement using a fiber optical parametric amplifier,” Opt. Lett. 41, 653 (2016).
    [Crossref] [PubMed]
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  33. Y. Liu, J. Li, L. Cui, N. Huo, S. M. Assad, X. Li, and Z. Ou, In preparation for publication, see fig. 5 at preprint arXiv:1712.01553v2 (2017).
  34. D. Linnemann, H. Strobel, W. Muessel, J. Schulz, R. Lewis-Swan, K. V. Kheruntsyan, and M. K. Oberthaler, “Quantum-enhanced sensing based on time reversal of nonlinear dynamics,” Phys. Rev. Lett. 117, 013001 (2016).
    [Crossref] [PubMed]
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  36. B. Chen, C. Qiu, S. Chen, J. Guo, L. Chen, Z. Y. Ou, and W. Zhang, “Atom-light hybrid interferometer,” Phys. Rev. Lett. 115, 043602 (2015).
    [Crossref] [PubMed]

2018 (1)

J. Li, Y. Liu, L. Cui, N. Huo, S. M. Assad, X. Li, and Z. Y. Ou, “Joint measurement of multiple noncommuting parameters,” Phys. Rev. A 97, 052127 (2018).
[Crossref]

2017 (3)

B. E. Anderson, P. Gupta, B. L. Schmittberger, T. Horrom, C. Hermann-Avigliano, K. M. Jones, and P. D. Lett, “Phase sensing beyond the standard quantum limit with a variation on the su(1,1) interferometer,” Optica 4, 752–756 (2017).
[Crossref]

M. Manceau, G. Leuchs, F. Khalili, and M. Chekhova, “Detection loss tolerant supersensitive phase measurement with an su(1,1) interferometer,” Phys. Rev. Lett. 119, 223604 (2017).
[Crossref] [PubMed]

M. Manceau, F. Khalili, and M. Chekhova, “Improving the phase super-sensitivity of squeezing-assisted interferometers by squeeze factor unbalancing,” New J. Phys.  19, 013014 (2017).
[Crossref]

2016 (6)

X. Guo, X. Li, N. Liu, and Z. Y. Ou, “Quantum information tapping using a fiber optical parametric amplifier with noise figure improved by correlated inputs,” Sci. Reports 6, 30214 (2016).
[Crossref]

C. Sparaciari, S. Olivares, and M. G. A. Paris, “Gaussian-state interferometry with passive and active elements,” Phys. Rev. A 93, 023810 (2016).
[Crossref]

X. Guo, N. Liu, Y. Liu, X. Li, and Z. Y. Ou, “Generation of continuous variable quantum entanglement using a fiber optical parametric amplifier,” Opt. Lett. 41, 653 (2016).
[Crossref] [PubMed]

D. Linnemann, H. Strobel, W. Muessel, J. Schulz, R. Lewis-Swan, K. V. Kheruntsyan, and M. K. Oberthaler, “Quantum-enhanced sensing based on time reversal of nonlinear dynamics,” Phys. Rev. Lett. 117, 013001 (2016).
[Crossref] [PubMed]

T. Baumgratz and A. Datta, “Quantum enhanced estimation of a multidimensional field,” Phys. Rev. Lett. 116, 030801 (2016).
[Crossref] [PubMed]

M. Szczykulska, T. Baumgratz, and A. Datta, “Multi-parameter quantum metrology,” Adv. Physics:  X1, 621–639 (2016).

2015 (2)

2014 (2)

F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, and W. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun.  5, 3049 (2014).
[Crossref] [PubMed]

P. J. D. Crowley, A. Datta, M. Barbieri, and I. A. Walmsley, “Tradeoff in simultaneous quantum-limited phase and loss estimation in interferometry,” Phys. Rev. A 89, 023845 (2014).
[Crossref]

2013 (4)

S. Steinlechner, J. Bauchrowitz, M. Meinders, H. Müller-Ebhardt, K. Danzmann, and R. Schnabel, “Quantum-dense metrology,” Nat. Photonics 7, 626–630 (2013).
[Crossref]

P. C. Humphreys, M. Barbieri, A. Datta, and I. A. Walmsley, “Quantum enhanced multiple phase estimation,” Phys. Rev. Lett. 111, 070403 (2013).
[Crossref] [PubMed]

M. G. Genoni, M. G. A. Paris, G. Adesso, H. Nha, P. L. Knight, and M. S. Kim, “Optimal estimation of joint parameters in phase space,” Phys. Rev. A 87, 012107 (2013).
[Crossref]

J. Kong, F. Hudelist, Z. Y. Ou, and W. Zhang, “Cancellation of internal quantum noise of an amplifier by quantum correlation,” Phys. Rev. Lett. 111, 033608 (2013).
[Crossref] [PubMed]

2012 (3)

E. Flurin, N. Roch, F. Mallet, M. H. Devoret, and B. Huard, “Generating entangled microwave radiation over two transmission lines,” Phys. Rev. Lett. 109, 183901 (2012).
[Crossref] [PubMed]

Z. Y. Ou, “Enhancement of the phase-measurement sensitivity beyond the standard quantum limit by a nonlinear interferometer,” Phys. Rev. A 85, 023815 (2012).
[Crossref]

A. M. Marino, N. V. Corzo Trejo, and P. D. Lett, “Effect of losses on the performance of an su(1,1) interferometer,” Phys. Rev. A 86, 023844 (2012).
[Crossref]

2011 (1)

J. Jing, C. Liu, Z. Zhou, Z. Y. Ou, and W. Zhang, “Realization of a nonlinear interferometer with parametric amplifiers,” Appl. Phys. Lett. 99, 011110 (2011).
[Crossref]

2010 (1)

W. N. Plick, J. P. Dowling, and G. S. Agarwal, “Coherent-light-boosted, sub-shot noise, quantum interferometry,” New J. Phys.  12, 083014 (2010).
[Crossref]

2002 (1)

X. Li, Q. Pan, J. Jing, J. Zhang, C. Xie, and K. Peng, “Quantum dense coding exploiting a bright einstein-podolsky-rosen beam,” Phys. Rev. Lett. 88, 047904 (2002).
[Crossref] [PubMed]

2000 (2)

S. L. Braunstein and H. J. Kimble, “Dense coding for continuous variables,” Phys. Rev. A 61, 042302 (2000).
[Crossref]

J. Zhang and K. Peng, “Quantum teleportation and dense coding by means of bright amplitude-squeezed light and direct measurement of a bell state,” Phys. Rev. A 62, 064302 (2000).
[Crossref]

1993 (1)

Z. Y. Ou, “Quantum amplification with correlated quantum fields,” Phys. Rev. A 48, R1761–R1764 (1993).
[Crossref] [PubMed]

1992 (1)

Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the einstein-podolsky-rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992).
[Crossref] [PubMed]

1987 (2)

M. Xiao, L.-A. Wu, and H. J. Kimble, “Precision measurement beyond the shot-noise limit,” Phys. Rev. Lett. 59, 278–281 (1987).
[Crossref] [PubMed]

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

1986 (1)

B. Yurke, S. L. McCall, and J. R. Klauder, “Su(2) and su(1,1) interferometers,” Phys. Rev. A 33, 4033–4054 (1986).
[Crossref]

1981 (1)

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

1935 (1)

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[Crossref]

Adesso, G.

M. G. Genoni, M. G. A. Paris, G. Adesso, H. Nha, P. L. Knight, and M. S. Kim, “Optimal estimation of joint parameters in phase space,” Phys. Rev. A 87, 012107 (2013).
[Crossref]

Agarwal, G. S.

W. N. Plick, J. P. Dowling, and G. S. Agarwal, “Coherent-light-boosted, sub-shot noise, quantum interferometry,” New J. Phys.  12, 083014 (2010).
[Crossref]

Anderson, B. E.

Assad, S. M.

J. Li, Y. Liu, L. Cui, N. Huo, S. M. Assad, X. Li, and Z. Y. Ou, “Joint measurement of multiple noncommuting parameters,” Phys. Rev. A 97, 052127 (2018).
[Crossref]

Y. Liu, J. Li, L. Cui, N. Huo, S. M. Assad, X. Li, and Z. Ou, In preparation for publication, see fig. 5 at preprint arXiv:1712.01553v2 (2017).

Barbieri, M.

P. J. D. Crowley, A. Datta, M. Barbieri, and I. A. Walmsley, “Tradeoff in simultaneous quantum-limited phase and loss estimation in interferometry,” Phys. Rev. A 89, 023845 (2014).
[Crossref]

P. C. Humphreys, M. Barbieri, A. Datta, and I. A. Walmsley, “Quantum enhanced multiple phase estimation,” Phys. Rev. Lett. 111, 070403 (2013).
[Crossref] [PubMed]

Bauchrowitz, J.

S. Steinlechner, J. Bauchrowitz, M. Meinders, H. Müller-Ebhardt, K. Danzmann, and R. Schnabel, “Quantum-dense metrology,” Nat. Photonics 7, 626–630 (2013).
[Crossref]

Baumgratz, T.

T. Baumgratz and A. Datta, “Quantum enhanced estimation of a multidimensional field,” Phys. Rev. Lett. 116, 030801 (2016).
[Crossref] [PubMed]

M. Szczykulska, T. Baumgratz, and A. Datta, “Multi-parameter quantum metrology,” Adv. Physics:  X1, 621–639 (2016).

Braunstein, S. L.

S. L. Braunstein and H. J. Kimble, “Dense coding for continuous variables,” Phys. Rev. A 61, 042302 (2000).
[Crossref]

Caves, C. M.

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

Chakram, S.

H. F. Cheung, Y. S. Patil, L. Chang, S. Chakram, and M. Vengalattore, “Nonlinear phonon interferometry at the heisenberg limit,” arXiv preprint arXiv:1601.02324 (2016).

Chang, L.

H. F. Cheung, Y. S. Patil, L. Chang, S. Chakram, and M. Vengalattore, “Nonlinear phonon interferometry at the heisenberg limit,” arXiv preprint arXiv:1601.02324 (2016).

Chekhova, M.

M. Manceau, G. Leuchs, F. Khalili, and M. Chekhova, “Detection loss tolerant supersensitive phase measurement with an su(1,1) interferometer,” Phys. Rev. Lett. 119, 223604 (2017).
[Crossref] [PubMed]

M. Manceau, F. Khalili, and M. Chekhova, “Improving the phase super-sensitivity of squeezing-assisted interferometers by squeeze factor unbalancing,” New J. Phys.  19, 013014 (2017).
[Crossref]

Chen, B.

B. Chen, C. Qiu, S. Chen, J. Guo, L. Chen, Z. Y. Ou, and W. Zhang, “Atom-light hybrid interferometer,” Phys. Rev. Lett. 115, 043602 (2015).
[Crossref] [PubMed]

Chen, L.

B. Chen, C. Qiu, S. Chen, J. Guo, L. Chen, Z. Y. Ou, and W. Zhang, “Atom-light hybrid interferometer,” Phys. Rev. Lett. 115, 043602 (2015).
[Crossref] [PubMed]

Chen, S.

B. Chen, C. Qiu, S. Chen, J. Guo, L. Chen, Z. Y. Ou, and W. Zhang, “Atom-light hybrid interferometer,” Phys. Rev. Lett. 115, 043602 (2015).
[Crossref] [PubMed]

Cheung, H. F.

H. F. Cheung, Y. S. Patil, L. Chang, S. Chakram, and M. Vengalattore, “Nonlinear phonon interferometry at the heisenberg limit,” arXiv preprint arXiv:1601.02324 (2016).

Crowley, P. J. D.

P. J. D. Crowley, A. Datta, M. Barbieri, and I. A. Walmsley, “Tradeoff in simultaneous quantum-limited phase and loss estimation in interferometry,” Phys. Rev. A 89, 023845 (2014).
[Crossref]

Cui, L.

J. Li, Y. Liu, L. Cui, N. Huo, S. M. Assad, X. Li, and Z. Y. Ou, “Joint measurement of multiple noncommuting parameters,” Phys. Rev. A 97, 052127 (2018).
[Crossref]

J. Li, Y. Liu, N. Huo, L. Cui, X. Li, and Z. Y. Ou, “Loss-tolerant measurement of continuous-variable quantum entanglement with the aid of a high gain parametric amplifier,” arXiv preprint arXiv:1808.10258 (2018).

Y. Liu, J. Li, L. Cui, N. Huo, S. M. Assad, X. Li, and Z. Ou, In preparation for publication, see fig. 5 at preprint arXiv:1712.01553v2 (2017).

Danzmann, K.

S. Steinlechner, J. Bauchrowitz, M. Meinders, H. Müller-Ebhardt, K. Danzmann, and R. Schnabel, “Quantum-dense metrology,” Nat. Photonics 7, 626–630 (2013).
[Crossref]

Datta, A.

M. Szczykulska, T. Baumgratz, and A. Datta, “Multi-parameter quantum metrology,” Adv. Physics:  X1, 621–639 (2016).

T. Baumgratz and A. Datta, “Quantum enhanced estimation of a multidimensional field,” Phys. Rev. Lett. 116, 030801 (2016).
[Crossref] [PubMed]

P. J. D. Crowley, A. Datta, M. Barbieri, and I. A. Walmsley, “Tradeoff in simultaneous quantum-limited phase and loss estimation in interferometry,” Phys. Rev. A 89, 023845 (2014).
[Crossref]

P. C. Humphreys, M. Barbieri, A. Datta, and I. A. Walmsley, “Quantum enhanced multiple phase estimation,” Phys. Rev. Lett. 111, 070403 (2013).
[Crossref] [PubMed]

Devoret, M. H.

E. Flurin, N. Roch, F. Mallet, M. H. Devoret, and B. Huard, “Generating entangled microwave radiation over two transmission lines,” Phys. Rev. Lett. 109, 183901 (2012).
[Crossref] [PubMed]

Dowling, J. P.

W. N. Plick, J. P. Dowling, and G. S. Agarwal, “Coherent-light-boosted, sub-shot noise, quantum interferometry,” New J. Phys.  12, 083014 (2010).
[Crossref]

Einstein, A.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[Crossref]

Flurin, E.

E. Flurin, N. Roch, F. Mallet, M. H. Devoret, and B. Huard, “Generating entangled microwave radiation over two transmission lines,” Phys. Rev. Lett. 109, 183901 (2012).
[Crossref] [PubMed]

Genoni, M. G.

M. G. Genoni, M. G. A. Paris, G. Adesso, H. Nha, P. L. Knight, and M. S. Kim, “Optimal estimation of joint parameters in phase space,” Phys. Rev. A 87, 012107 (2013).
[Crossref]

Grangier, P.

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

Guo, J.

B. Chen, C. Qiu, S. Chen, J. Guo, L. Chen, Z. Y. Ou, and W. Zhang, “Atom-light hybrid interferometer,” Phys. Rev. Lett. 115, 043602 (2015).
[Crossref] [PubMed]

Guo, X.

Gupta, P.

Hermann-Avigliano, C.

Horrom, T.

Huard, B.

E. Flurin, N. Roch, F. Mallet, M. H. Devoret, and B. Huard, “Generating entangled microwave radiation over two transmission lines,” Phys. Rev. Lett. 109, 183901 (2012).
[Crossref] [PubMed]

Hudelist, F.

F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, and W. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun.  5, 3049 (2014).
[Crossref] [PubMed]

J. Kong, F. Hudelist, Z. Y. Ou, and W. Zhang, “Cancellation of internal quantum noise of an amplifier by quantum correlation,” Phys. Rev. Lett. 111, 033608 (2013).
[Crossref] [PubMed]

Humphreys, P. C.

P. C. Humphreys, M. Barbieri, A. Datta, and I. A. Walmsley, “Quantum enhanced multiple phase estimation,” Phys. Rev. Lett. 111, 070403 (2013).
[Crossref] [PubMed]

Huo, N.

J. Li, Y. Liu, L. Cui, N. Huo, S. M. Assad, X. Li, and Z. Y. Ou, “Joint measurement of multiple noncommuting parameters,” Phys. Rev. A 97, 052127 (2018).
[Crossref]

J. Li, Y. Liu, N. Huo, L. Cui, X. Li, and Z. Y. Ou, “Loss-tolerant measurement of continuous-variable quantum entanglement with the aid of a high gain parametric amplifier,” arXiv preprint arXiv:1808.10258 (2018).

Y. Liu, J. Li, L. Cui, N. Huo, S. M. Assad, X. Li, and Z. Ou, In preparation for publication, see fig. 5 at preprint arXiv:1712.01553v2 (2017).

Jing, J.

F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, and W. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun.  5, 3049 (2014).
[Crossref] [PubMed]

J. Jing, C. Liu, Z. Zhou, Z. Y. Ou, and W. Zhang, “Realization of a nonlinear interferometer with parametric amplifiers,” Appl. Phys. Lett. 99, 011110 (2011).
[Crossref]

X. Li, Q. Pan, J. Jing, J. Zhang, C. Xie, and K. Peng, “Quantum dense coding exploiting a bright einstein-podolsky-rosen beam,” Phys. Rev. Lett. 88, 047904 (2002).
[Crossref] [PubMed]

Jones, K. M.

Khalili, F.

M. Manceau, G. Leuchs, F. Khalili, and M. Chekhova, “Detection loss tolerant supersensitive phase measurement with an su(1,1) interferometer,” Phys. Rev. Lett. 119, 223604 (2017).
[Crossref] [PubMed]

M. Manceau, F. Khalili, and M. Chekhova, “Improving the phase super-sensitivity of squeezing-assisted interferometers by squeeze factor unbalancing,” New J. Phys.  19, 013014 (2017).
[Crossref]

Kheruntsyan, K. V.

D. Linnemann, H. Strobel, W. Muessel, J. Schulz, R. Lewis-Swan, K. V. Kheruntsyan, and M. K. Oberthaler, “Quantum-enhanced sensing based on time reversal of nonlinear dynamics,” Phys. Rev. Lett. 117, 013001 (2016).
[Crossref] [PubMed]

Kim, M. S.

M. G. Genoni, M. G. A. Paris, G. Adesso, H. Nha, P. L. Knight, and M. S. Kim, “Optimal estimation of joint parameters in phase space,” Phys. Rev. A 87, 012107 (2013).
[Crossref]

Kimble, H. J.

S. L. Braunstein and H. J. Kimble, “Dense coding for continuous variables,” Phys. Rev. A 61, 042302 (2000).
[Crossref]

Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the einstein-podolsky-rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992).
[Crossref] [PubMed]

M. Xiao, L.-A. Wu, and H. J. Kimble, “Precision measurement beyond the shot-noise limit,” Phys. Rev. Lett. 59, 278–281 (1987).
[Crossref] [PubMed]

Klauder, J. R.

B. Yurke, S. L. McCall, and J. R. Klauder, “Su(2) and su(1,1) interferometers,” Phys. Rev. A 33, 4033–4054 (1986).
[Crossref]

Knight, P. L.

M. G. Genoni, M. G. A. Paris, G. Adesso, H. Nha, P. L. Knight, and M. S. Kim, “Optimal estimation of joint parameters in phase space,” Phys. Rev. A 87, 012107 (2013).
[Crossref]

Kong, J.

F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, and W. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun.  5, 3049 (2014).
[Crossref] [PubMed]

J. Kong, F. Hudelist, Z. Y. Ou, and W. Zhang, “Cancellation of internal quantum noise of an amplifier by quantum correlation,” Phys. Rev. Lett. 111, 033608 (2013).
[Crossref] [PubMed]

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] [PubMed]

Lett, P. D.

Leuchs, G.

M. Manceau, G. Leuchs, F. Khalili, and M. Chekhova, “Detection loss tolerant supersensitive phase measurement with an su(1,1) interferometer,” Phys. Rev. Lett. 119, 223604 (2017).
[Crossref] [PubMed]

Lewis-Swan, R.

D. Linnemann, H. Strobel, W. Muessel, J. Schulz, R. Lewis-Swan, K. V. Kheruntsyan, and M. K. Oberthaler, “Quantum-enhanced sensing based on time reversal of nonlinear dynamics,” Phys. Rev. Lett. 117, 013001 (2016).
[Crossref] [PubMed]

Li, J.

J. Li, Y. Liu, L. Cui, N. Huo, S. M. Assad, X. Li, and Z. Y. Ou, “Joint measurement of multiple noncommuting parameters,” Phys. Rev. A 97, 052127 (2018).
[Crossref]

J. Li, Y. Liu, N. Huo, L. Cui, X. Li, and Z. Y. Ou, “Loss-tolerant measurement of continuous-variable quantum entanglement with the aid of a high gain parametric amplifier,” arXiv preprint arXiv:1808.10258 (2018).

Y. Liu, J. Li, L. Cui, N. Huo, S. M. Assad, X. Li, and Z. Ou, In preparation for publication, see fig. 5 at preprint arXiv:1712.01553v2 (2017).

Li, X.

J. Li, Y. Liu, L. Cui, N. Huo, S. M. Assad, X. Li, and Z. Y. Ou, “Joint measurement of multiple noncommuting parameters,” Phys. Rev. A 97, 052127 (2018).
[Crossref]

X. Guo, X. Li, N. Liu, and Z. Y. Ou, “Quantum information tapping using a fiber optical parametric amplifier with noise figure improved by correlated inputs,” Sci. Reports 6, 30214 (2016).
[Crossref]

X. Guo, N. Liu, Y. Liu, X. Li, and Z. Y. Ou, “Generation of continuous variable quantum entanglement using a fiber optical parametric amplifier,” Opt. Lett. 41, 653 (2016).
[Crossref] [PubMed]

X. Guo, N. Liu, X. Li, and Z. Y. Ou, “Complete temporal mode analysis in pulse-pumped fiber-optical parametric amplifier for continuous variable entanglement generation,” Opt. Express 23, 29369 (2015).
[Crossref] [PubMed]

X. Li, Q. Pan, J. Jing, J. Zhang, C. Xie, and K. Peng, “Quantum dense coding exploiting a bright einstein-podolsky-rosen beam,” Phys. Rev. Lett. 88, 047904 (2002).
[Crossref] [PubMed]

Y. Liu, J. Li, L. Cui, N. Huo, S. M. Assad, X. Li, and Z. Ou, In preparation for publication, see fig. 5 at preprint arXiv:1712.01553v2 (2017).

J. Li, Y. Liu, N. Huo, L. Cui, X. Li, and Z. Y. Ou, “Loss-tolerant measurement of continuous-variable quantum entanglement with the aid of a high gain parametric amplifier,” arXiv preprint arXiv:1808.10258 (2018).

Linnemann, D.

D. Linnemann, H. Strobel, W. Muessel, J. Schulz, R. Lewis-Swan, K. V. Kheruntsyan, and M. K. Oberthaler, “Quantum-enhanced sensing based on time reversal of nonlinear dynamics,” Phys. Rev. Lett. 117, 013001 (2016).
[Crossref] [PubMed]

Liu, C.

F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, and W. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun.  5, 3049 (2014).
[Crossref] [PubMed]

J. Jing, C. Liu, Z. Zhou, Z. Y. Ou, and W. Zhang, “Realization of a nonlinear interferometer with parametric amplifiers,” Appl. Phys. Lett. 99, 011110 (2011).
[Crossref]

Liu, N.

Liu, Y.

J. Li, Y. Liu, L. Cui, N. Huo, S. M. Assad, X. Li, and Z. Y. Ou, “Joint measurement of multiple noncommuting parameters,” Phys. Rev. A 97, 052127 (2018).
[Crossref]

X. Guo, N. Liu, Y. Liu, X. Li, and Z. Y. Ou, “Generation of continuous variable quantum entanglement using a fiber optical parametric amplifier,” Opt. Lett. 41, 653 (2016).
[Crossref] [PubMed]

Y. Liu, J. Li, L. Cui, N. Huo, S. M. Assad, X. Li, and Z. Ou, In preparation for publication, see fig. 5 at preprint arXiv:1712.01553v2 (2017).

J. Li, Y. Liu, N. Huo, L. Cui, X. Li, and Z. Y. Ou, “Loss-tolerant measurement of continuous-variable quantum entanglement with the aid of a high gain parametric amplifier,” arXiv preprint arXiv:1808.10258 (2018).

Mallet, F.

E. Flurin, N. Roch, F. Mallet, M. H. Devoret, and B. Huard, “Generating entangled microwave radiation over two transmission lines,” Phys. Rev. Lett. 109, 183901 (2012).
[Crossref] [PubMed]

Manceau, M.

M. Manceau, G. Leuchs, F. Khalili, and M. Chekhova, “Detection loss tolerant supersensitive phase measurement with an su(1,1) interferometer,” Phys. Rev. Lett. 119, 223604 (2017).
[Crossref] [PubMed]

M. Manceau, F. Khalili, and M. Chekhova, “Improving the phase super-sensitivity of squeezing-assisted interferometers by squeeze factor unbalancing,” New J. Phys.  19, 013014 (2017).
[Crossref]

Marino, A. M.

A. M. Marino, N. V. Corzo Trejo, and P. D. Lett, “Effect of losses on the performance of an su(1,1) interferometer,” Phys. Rev. A 86, 023844 (2012).
[Crossref]

McCall, S. L.

B. Yurke, S. L. McCall, and J. R. Klauder, “Su(2) and su(1,1) interferometers,” Phys. Rev. A 33, 4033–4054 (1986).
[Crossref]

Meinders, M.

S. Steinlechner, J. Bauchrowitz, M. Meinders, H. Müller-Ebhardt, K. Danzmann, and R. Schnabel, “Quantum-dense metrology,” Nat. Photonics 7, 626–630 (2013).
[Crossref]

Muessel, W.

D. Linnemann, H. Strobel, W. Muessel, J. Schulz, R. Lewis-Swan, K. V. Kheruntsyan, and M. K. Oberthaler, “Quantum-enhanced sensing based on time reversal of nonlinear dynamics,” Phys. Rev. Lett. 117, 013001 (2016).
[Crossref] [PubMed]

Müller-Ebhardt, H.

S. Steinlechner, J. Bauchrowitz, M. Meinders, H. Müller-Ebhardt, K. Danzmann, and R. Schnabel, “Quantum-dense metrology,” Nat. Photonics 7, 626–630 (2013).
[Crossref]

Nha, H.

M. G. Genoni, M. G. A. Paris, G. Adesso, H. Nha, P. L. Knight, and M. S. Kim, “Optimal estimation of joint parameters in phase space,” Phys. Rev. A 87, 012107 (2013).
[Crossref]

Oberthaler, M. K.

D. Linnemann, H. Strobel, W. Muessel, J. Schulz, R. Lewis-Swan, K. V. Kheruntsyan, and M. K. Oberthaler, “Quantum-enhanced sensing based on time reversal of nonlinear dynamics,” Phys. Rev. Lett. 117, 013001 (2016).
[Crossref] [PubMed]

Olivares, S.

C. Sparaciari, S. Olivares, and M. G. A. Paris, “Gaussian-state interferometry with passive and active elements,” Phys. Rev. A 93, 023810 (2016).
[Crossref]

Ou, Z.

Y. Liu, J. Li, L. Cui, N. Huo, S. M. Assad, X. Li, and Z. Ou, In preparation for publication, see fig. 5 at preprint arXiv:1712.01553v2 (2017).

Ou, Z. Y.

J. Li, Y. Liu, L. Cui, N. Huo, S. M. Assad, X. Li, and Z. Y. Ou, “Joint measurement of multiple noncommuting parameters,” Phys. Rev. A 97, 052127 (2018).
[Crossref]

X. Guo, X. Li, N. Liu, and Z. Y. Ou, “Quantum information tapping using a fiber optical parametric amplifier with noise figure improved by correlated inputs,” Sci. Reports 6, 30214 (2016).
[Crossref]

X. Guo, N. Liu, Y. Liu, X. Li, and Z. Y. Ou, “Generation of continuous variable quantum entanglement using a fiber optical parametric amplifier,” Opt. Lett. 41, 653 (2016).
[Crossref] [PubMed]

X. Guo, N. Liu, X. Li, and Z. Y. Ou, “Complete temporal mode analysis in pulse-pumped fiber-optical parametric amplifier for continuous variable entanglement generation,” Opt. Express 23, 29369 (2015).
[Crossref] [PubMed]

B. Chen, C. Qiu, S. Chen, J. Guo, L. Chen, Z. Y. Ou, and W. Zhang, “Atom-light hybrid interferometer,” Phys. Rev. Lett. 115, 043602 (2015).
[Crossref] [PubMed]

F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, and W. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun.  5, 3049 (2014).
[Crossref] [PubMed]

J. Kong, F. Hudelist, Z. Y. Ou, and W. Zhang, “Cancellation of internal quantum noise of an amplifier by quantum correlation,” Phys. Rev. Lett. 111, 033608 (2013).
[Crossref] [PubMed]

Z. Y. Ou, “Enhancement of the phase-measurement sensitivity beyond the standard quantum limit by a nonlinear interferometer,” Phys. Rev. A 85, 023815 (2012).
[Crossref]

J. Jing, C. Liu, Z. Zhou, Z. Y. Ou, and W. Zhang, “Realization of a nonlinear interferometer with parametric amplifiers,” Appl. Phys. Lett. 99, 011110 (2011).
[Crossref]

Z. Y. Ou, “Quantum amplification with correlated quantum fields,” Phys. Rev. A 48, R1761–R1764 (1993).
[Crossref] [PubMed]

Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the einstein-podolsky-rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992).
[Crossref] [PubMed]

J. Li, Y. Liu, N. Huo, L. Cui, X. Li, and Z. Y. Ou, “Loss-tolerant measurement of continuous-variable quantum entanglement with the aid of a high gain parametric amplifier,” arXiv preprint arXiv:1808.10258 (2018).

Pan, Q.

X. Li, Q. Pan, J. Jing, J. Zhang, C. Xie, and K. Peng, “Quantum dense coding exploiting a bright einstein-podolsky-rosen beam,” Phys. Rev. Lett. 88, 047904 (2002).
[Crossref] [PubMed]

Paris, M. G. A.

C. Sparaciari, S. Olivares, and M. G. A. Paris, “Gaussian-state interferometry with passive and active elements,” Phys. Rev. A 93, 023810 (2016).
[Crossref]

M. G. Genoni, M. G. A. Paris, G. Adesso, H. Nha, P. L. Knight, and M. S. Kim, “Optimal estimation of joint parameters in phase space,” Phys. Rev. A 87, 012107 (2013).
[Crossref]

Patil, Y. S.

H. F. Cheung, Y. S. Patil, L. Chang, S. Chakram, and M. Vengalattore, “Nonlinear phonon interferometry at the heisenberg limit,” arXiv preprint arXiv:1601.02324 (2016).

Peng, K.

X. Li, Q. Pan, J. Jing, J. Zhang, C. Xie, and K. Peng, “Quantum dense coding exploiting a bright einstein-podolsky-rosen beam,” Phys. Rev. Lett. 88, 047904 (2002).
[Crossref] [PubMed]

J. Zhang and K. Peng, “Quantum teleportation and dense coding by means of bright amplitude-squeezed light and direct measurement of a bell state,” Phys. Rev. A 62, 064302 (2000).
[Crossref]

Peng, K. C.

Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the einstein-podolsky-rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992).
[Crossref] [PubMed]

Pereira, S. F.

Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the einstein-podolsky-rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992).
[Crossref] [PubMed]

Plick, W. N.

W. N. Plick, J. P. Dowling, and G. S. Agarwal, “Coherent-light-boosted, sub-shot noise, quantum interferometry,” New J. Phys.  12, 083014 (2010).
[Crossref]

Podolsky, B.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[Crossref]

Qiu, C.

B. Chen, C. Qiu, S. Chen, J. Guo, L. Chen, Z. Y. Ou, and W. Zhang, “Atom-light hybrid interferometer,” Phys. Rev. Lett. 115, 043602 (2015).
[Crossref] [PubMed]

Roch, N.

E. Flurin, N. Roch, F. Mallet, M. H. Devoret, and B. Huard, “Generating entangled microwave radiation over two transmission lines,” Phys. Rev. Lett. 109, 183901 (2012).
[Crossref] [PubMed]

Rosen, N.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[Crossref]

Schmittberger, B. L.

Schnabel, R.

S. Steinlechner, J. Bauchrowitz, M. Meinders, H. Müller-Ebhardt, K. Danzmann, and R. Schnabel, “Quantum-dense metrology,” Nat. Photonics 7, 626–630 (2013).
[Crossref]

Schulz, J.

D. Linnemann, H. Strobel, W. Muessel, J. Schulz, R. Lewis-Swan, K. V. Kheruntsyan, and M. K. Oberthaler, “Quantum-enhanced sensing based on time reversal of nonlinear dynamics,” Phys. Rev. Lett. 117, 013001 (2016).
[Crossref] [PubMed]

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] [PubMed]

Sparaciari, C.

C. Sparaciari, S. Olivares, and M. G. A. Paris, “Gaussian-state interferometry with passive and active elements,” Phys. Rev. A 93, 023810 (2016).
[Crossref]

Steinlechner, S.

S. Steinlechner, J. Bauchrowitz, M. Meinders, H. Müller-Ebhardt, K. Danzmann, and R. Schnabel, “Quantum-dense metrology,” Nat. Photonics 7, 626–630 (2013).
[Crossref]

Strobel, H.

D. Linnemann, H. Strobel, W. Muessel, J. Schulz, R. Lewis-Swan, K. V. Kheruntsyan, and M. K. Oberthaler, “Quantum-enhanced sensing based on time reversal of nonlinear dynamics,” Phys. Rev. Lett. 117, 013001 (2016).
[Crossref] [PubMed]

Szczykulska, M.

M. Szczykulska, T. Baumgratz, and A. Datta, “Multi-parameter quantum metrology,” Adv. Physics:  X1, 621–639 (2016).

Trejo, N. V. Corzo

A. M. Marino, N. V. Corzo Trejo, and P. D. Lett, “Effect of losses on the performance of an su(1,1) interferometer,” Phys. Rev. A 86, 023844 (2012).
[Crossref]

Vengalattore, M.

H. F. Cheung, Y. S. Patil, L. Chang, S. Chakram, and M. Vengalattore, “Nonlinear phonon interferometry at the heisenberg limit,” arXiv preprint arXiv:1601.02324 (2016).

Walmsley, I. A.

P. J. D. Crowley, A. Datta, M. Barbieri, and I. A. Walmsley, “Tradeoff in simultaneous quantum-limited phase and loss estimation in interferometry,” Phys. Rev. A 89, 023845 (2014).
[Crossref]

P. C. Humphreys, M. Barbieri, A. Datta, and I. A. Walmsley, “Quantum enhanced multiple phase estimation,” Phys. Rev. Lett. 111, 070403 (2013).
[Crossref] [PubMed]

Wu, L.-A.

M. Xiao, L.-A. Wu, and H. J. Kimble, “Precision measurement beyond the shot-noise limit,” Phys. Rev. Lett. 59, 278–281 (1987).
[Crossref] [PubMed]

Xiao, M.

M. Xiao, L.-A. Wu, and H. J. Kimble, “Precision measurement beyond the shot-noise limit,” Phys. Rev. Lett. 59, 278–281 (1987).
[Crossref] [PubMed]

Xie, C.

X. Li, Q. Pan, J. Jing, J. Zhang, C. Xie, and K. Peng, “Quantum dense coding exploiting a bright einstein-podolsky-rosen beam,” Phys. Rev. Lett. 88, 047904 (2002).
[Crossref] [PubMed]

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] [PubMed]

B. Yurke, S. L. McCall, and J. R. Klauder, “Su(2) and su(1,1) interferometers,” Phys. Rev. A 33, 4033–4054 (1986).
[Crossref]

Zhang, J.

X. Li, Q. Pan, J. Jing, J. Zhang, C. Xie, and K. Peng, “Quantum dense coding exploiting a bright einstein-podolsky-rosen beam,” Phys. Rev. Lett. 88, 047904 (2002).
[Crossref] [PubMed]

J. Zhang and K. Peng, “Quantum teleportation and dense coding by means of bright amplitude-squeezed light and direct measurement of a bell state,” Phys. Rev. A 62, 064302 (2000).
[Crossref]

Zhang, W.

B. Chen, C. Qiu, S. Chen, J. Guo, L. Chen, Z. Y. Ou, and W. Zhang, “Atom-light hybrid interferometer,” Phys. Rev. Lett. 115, 043602 (2015).
[Crossref] [PubMed]

F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, and W. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun.  5, 3049 (2014).
[Crossref] [PubMed]

J. Kong, F. Hudelist, Z. Y. Ou, and W. Zhang, “Cancellation of internal quantum noise of an amplifier by quantum correlation,” Phys. Rev. Lett. 111, 033608 (2013).
[Crossref] [PubMed]

J. Jing, C. Liu, Z. Zhou, Z. Y. Ou, and W. Zhang, “Realization of a nonlinear interferometer with parametric amplifiers,” Appl. Phys. Lett. 99, 011110 (2011).
[Crossref]

Zhou, Z.

J. Jing, C. Liu, Z. Zhou, Z. Y. Ou, and W. Zhang, “Realization of a nonlinear interferometer with parametric amplifiers,” Appl. Phys. Lett. 99, 011110 (2011).
[Crossref]

Adv. Physics (1)

M. Szczykulska, T. Baumgratz, and A. Datta, “Multi-parameter quantum metrology,” Adv. Physics:  X1, 621–639 (2016).

Appl. Phys. Lett. (1)

J. Jing, C. Liu, Z. Zhou, Z. Y. Ou, and W. Zhang, “Realization of a nonlinear interferometer with parametric amplifiers,” Appl. Phys. Lett. 99, 011110 (2011).
[Crossref]

Nat. Commun (1)

F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, and W. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun.  5, 3049 (2014).
[Crossref] [PubMed]

Nat. Photonics (1)

S. Steinlechner, J. Bauchrowitz, M. Meinders, H. Müller-Ebhardt, K. Danzmann, and R. Schnabel, “Quantum-dense metrology,” Nat. Photonics 7, 626–630 (2013).
[Crossref]

New J. Phys (2)

M. Manceau, F. Khalili, and M. Chekhova, “Improving the phase super-sensitivity of squeezing-assisted interferometers by squeeze factor unbalancing,” New J. Phys.  19, 013014 (2017).
[Crossref]

W. N. Plick, J. P. Dowling, and G. S. Agarwal, “Coherent-light-boosted, sub-shot noise, quantum interferometry,” New J. Phys.  12, 083014 (2010).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Optica (1)

Phys. Rev. (1)

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[Crossref]

Phys. Rev. A (10)

J. Li, Y. Liu, L. Cui, N. Huo, S. M. Assad, X. Li, and Z. Y. Ou, “Joint measurement of multiple noncommuting parameters,” Phys. Rev. A 97, 052127 (2018).
[Crossref]

C. Sparaciari, S. Olivares, and M. G. A. Paris, “Gaussian-state interferometry with passive and active elements,” Phys. Rev. A 93, 023810 (2016).
[Crossref]

Z. Y. Ou, “Quantum amplification with correlated quantum fields,” Phys. Rev. A 48, R1761–R1764 (1993).
[Crossref] [PubMed]

Z. Y. Ou, “Enhancement of the phase-measurement sensitivity beyond the standard quantum limit by a nonlinear interferometer,” Phys. Rev. A 85, 023815 (2012).
[Crossref]

A. M. Marino, N. V. Corzo Trejo, and P. D. Lett, “Effect of losses on the performance of an su(1,1) interferometer,” Phys. Rev. A 86, 023844 (2012).
[Crossref]

B. Yurke, S. L. McCall, and J. R. Klauder, “Su(2) and su(1,1) interferometers,” Phys. Rev. A 33, 4033–4054 (1986).
[Crossref]

M. G. Genoni, M. G. A. Paris, G. Adesso, H. Nha, P. L. Knight, and M. S. Kim, “Optimal estimation of joint parameters in phase space,” Phys. Rev. A 87, 012107 (2013).
[Crossref]

P. J. D. Crowley, A. Datta, M. Barbieri, and I. A. Walmsley, “Tradeoff in simultaneous quantum-limited phase and loss estimation in interferometry,” Phys. Rev. A 89, 023845 (2014).
[Crossref]

S. L. Braunstein and H. J. Kimble, “Dense coding for continuous variables,” Phys. Rev. A 61, 042302 (2000).
[Crossref]

J. Zhang and K. Peng, “Quantum teleportation and dense coding by means of bright amplitude-squeezed light and direct measurement of a bell state,” Phys. Rev. A 62, 064302 (2000).
[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. (11)

M. Xiao, L.-A. Wu, and H. J. Kimble, “Precision measurement beyond the shot-noise limit,” Phys. Rev. Lett. 59, 278–281 (1987).
[Crossref] [PubMed]

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

P. C. Humphreys, M. Barbieri, A. Datta, and I. A. Walmsley, “Quantum enhanced multiple phase estimation,” Phys. Rev. Lett. 111, 070403 (2013).
[Crossref] [PubMed]

X. Li, Q. Pan, J. Jing, J. Zhang, C. Xie, and K. Peng, “Quantum dense coding exploiting a bright einstein-podolsky-rosen beam,” Phys. Rev. Lett. 88, 047904 (2002).
[Crossref] [PubMed]

T. Baumgratz and A. Datta, “Quantum enhanced estimation of a multidimensional field,” Phys. Rev. Lett. 116, 030801 (2016).
[Crossref] [PubMed]

M. Manceau, G. Leuchs, F. Khalili, and M. Chekhova, “Detection loss tolerant supersensitive phase measurement with an su(1,1) interferometer,” Phys. Rev. Lett. 119, 223604 (2017).
[Crossref] [PubMed]

J. Kong, F. Hudelist, Z. Y. Ou, and W. Zhang, “Cancellation of internal quantum noise of an amplifier by quantum correlation,” Phys. Rev. Lett. 111, 033608 (2013).
[Crossref] [PubMed]

E. Flurin, N. Roch, F. Mallet, M. H. Devoret, and B. Huard, “Generating entangled microwave radiation over two transmission lines,” Phys. Rev. Lett. 109, 183901 (2012).
[Crossref] [PubMed]

Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the einstein-podolsky-rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992).
[Crossref] [PubMed]

B. Chen, C. Qiu, S. Chen, J. Guo, L. Chen, Z. Y. Ou, and W. Zhang, “Atom-light hybrid interferometer,” Phys. Rev. Lett. 115, 043602 (2015).
[Crossref] [PubMed]

D. Linnemann, H. Strobel, W. Muessel, J. Schulz, R. Lewis-Swan, K. V. Kheruntsyan, and M. K. Oberthaler, “Quantum-enhanced sensing based on time reversal of nonlinear dynamics,” Phys. Rev. Lett. 117, 013001 (2016).
[Crossref] [PubMed]

Sci. Reports (1)

X. Guo, X. Li, N. Liu, and Z. Y. Ou, “Quantum information tapping using a fiber optical parametric amplifier with noise figure improved by correlated inputs,” Sci. Reports 6, 30214 (2016).
[Crossref]

Other (3)

J. Li, Y. Liu, N. Huo, L. Cui, X. Li, and Z. Y. Ou, “Loss-tolerant measurement of continuous-variable quantum entanglement with the aid of a high gain parametric amplifier,” arXiv preprint arXiv:1808.10258 (2018).

H. F. Cheung, Y. S. Patil, L. Chang, S. Chakram, and M. Vengalattore, “Nonlinear phonon interferometry at the heisenberg limit,” arXiv preprint arXiv:1601.02324 (2016).

Y. Liu, J. Li, L. Cui, N. Huo, S. M. Assad, X. Li, and Z. Ou, In preparation for publication, see fig. 5 at preprint arXiv:1712.01553v2 (2017).

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Figures (5)

Fig. 1
Fig. 1 Schematics for joint measurement of information encoded in multiple non-commuting observables through weak modulations on the probe beam by an amplitude modulator (AM) and a phase modulator (PM). (a) An SU(1,1) interferometer with optical parametric amplifiers (OPA1, OPA2). (b) Classical scheme with a beam splitter (BS). (c) Classical scheme with an optical parametric amplifier (OPA). HD: homodyne detection.
Fig. 2
Fig. 2 Experimental setup. The SU(1,1) interferometer (SUI) is formed with OPA1 and OPA2 with seed injection at OPA1. AM, amplitude modulator; PM, phase modulator; P1, P2, pulsed pumps; DSF, dispersion shifted fiber; CWDM, coarse wavelength division multiplexer; BS, 50/50 beam splitter; DL, delay line; OPA2 lock, locking signal for OPA2; PLL, phase locking loops; LOs, LOi, local oscillators; PZT, Piezo-Electric ceramic Transducer; HD, homodyne detection; DAQ, data acquisition system.
Fig. 3
Fig. 3 Joint measurement of the amplitude and phase modulations ϵ and δ under different situations. (a) and (b) are the results of simultaneous measurement by HD1 and HD2 on X ^ s ( o u t ) , Y ^ i ( o u t ) , respectively, for the SUI scheme in Fig. 1(a) (red traces) and classical amplifier scheme 1(c) (blue traces). (c) and (d) are results from the beam splitter scheme in Fig. 1(b). The peaks at 0.8 MHz and 1.2 MHz correspond to the power of AM and PM modulation signals ϵ and δ, respectively. The measurement results are normalized to the shot noise level at same local oscillator power.
Fig. 4
Fig. 4 Information of ϵ (at 0.8MHz) and ξ(π/4) (at 1.0MHz) encoded in non-orthogonal quadrature-phase amplitudes X ^ s ( 0 ) and X ^ s ( π / 4 ) simultaneously measured by (a)HD1 and (b)HD2, respectively. Blue and red traces represent the results of SUI (Fig. 1(a)) and conventional OPA (Fig. 1(c)), respectively.
Fig. 5
Fig. 5 Information of modulations ϵ (at 0.8MHz), δ (at 1.2MHz) and ξ(π/4) (at 1.0MHz) encoded in three non-commuting quadrature-phase amplitudes of X ^ s , Y ^ s , X ^ s ( π / 4 ) jointly measured by (a)HD1, (b)HD2, and (c)HD3, respectively. Blue and red traces represent the results of SUI (Fig. 1(a)) and conventional OPA (Fig. 1(c)), respectively.

Equations (5)

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a ^ s ( o u t ) = G 2 a ^ s g 2 a ^ i , a ^ i ( o u t ) = G 2 a ^ i g 2 a ^ s ,
S N R B S ( X ^ b 1 ) = 2 I p s ϵ 2 , S N R B S ( Y ^ b 2 ) = 2 I p s δ 2 ;
S N R A m p ( X ^ s ( o u t ) ) = 4 G 2 I p s ϵ 2 G 2 + g 2 , S N R A m p ( Y ^ i ( o u t ) ) = 4 g 2 I p s δ 2 G 2 + g 2 ,
S N R S U I ( X ^ s ( o u t ) ) = 2 ( G 1 + g 1 ) 2 I p s ϵ 2 ,
S N R S U I ( Y ^ i ( o u t ) ) = 2 ( G 1 + g 1 ) 2 I p s δ 2 ,

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