Abstract

Balanced homodyne detection relies on a beam splitter to superpose the weak signal input and strong local oscillator. However, recent investigation shows that a high gain phase sensitive amplifier (PSA) can be viewed as homodyne detector, in which the strong pump of PSA serves as the local oscillator [1]. Here, we analyze a new method of measuring the continuous variable entanglement by assisting a balanced homodyne detector with the PSA and implement it experimentally. Before measuring quadrature amplitude with the balanced homodyne detectors, two entangled fields generated from a pulse pumped fiber optical parametric amplifier are simultaneously coupled into the PSA. We find that the normalized noise for both the difference and sum of the quadrature amplitudes of the two entangled fields fall below the shot noise limit by about 4.6 dB, which is the record degree of entanglement measured in optical fiber systems. The experimental results illustrate that the advantages of the new measurement method include but not limit to tolerance to detection loss and characterizing entanglement with only one homodyne detector. The influence of mode-mismatching due to multi-mode property of entanglement on the measured noise reduction can also be greatly mitigated, indicating the new method is advantageous over the traditional measurement in multi-mode case.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]

2018 (3)

Y. Shaked, Y. Michael, R. Z. Vered, L. Bello, M. Rosenbluh, and A. Pe’er, “Lifting the bandwidth limit of optical homodyne measurement with broadband parametric amplification,” Nat. Commun. 9(1), 609 (2018).
[Crossref]

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(5), 052127 (2018).
[Crossref]

Y. Liu, J. Li, L. Cui, N. Huo, S. M. Assad, X. Li, and Z. Y. Ou, “Loss-tolerant quantum dense metrology with SU(1,1) interferometer,” Opt. Express 26(21), 27705 (2018).
[Crossref]

2017 (3)

2016 (4)

N. Liu, Y. Liu, X. Guo, L. Yang, X. Li, and Z. Y. Ou, “Approaching single temporal mode operation in twin beams generated by pulse pumped high gain spontaneous four wave mixing,” Opt. Express 24(2), 1096–1108 (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(3), 653–656 (2016).
[Crossref]

C. Sparaciari, S. Olivares, and M. G. A. Paris, “Gaussian-state interferometry with passive and active elements,” Phys. Rev. A 93(2), 023810 (2016).
[Crossref]

S. Cialdi, C. Porto, D. Cipriani, S. Olivares, and M. G. A. Paris, “Full quantum state reconstruction of symmetric two-mode squeezed thermal states via spectral homodyne detection and a state-balancing detector,” Phys. Rev. A 93(4), 043805 (2016).
[Crossref]

2015 (2)

2014 (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(1), 3049 (2014).
[Crossref]

2013 (2)

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(3), 033608 (2013).
[Crossref]

S. Steinlechner, J. Bauchrowitz, M. Meinders, H. Muller-Ebhardt, K. Danzmann, and R. Schnabel, “Quantum-dense metrology,” Nat. Photonics 7(8), 626–630 (2013).
[Crossref]

2012 (2)

X. Guo, X. Li, N. Liu, L. Yang, and Z. Y. Ou, “An all-fiber source of pulsed twin beams for quantum communication,” Appl. Phys. Lett. 101(26), 261111 (2012).
[Crossref]

E. Flurin, N. Roch, F. Mallet, M. H. Devoret, and B. Huard, “Generating entangled microwave radiation over two transmission lines,” Phys. Rev. Lett. 109(18), 183901 (2012).
[Crossref]

2011 (1)

Y.-M. Chi, B. Qi, W. Zhu, L. Qian, H.-K. Lo, S.-H. Youn, A. I. Lvovsky, and L. Tian, “A balanced homodyne detector for high-rate Gaussian-modulated coherent-state quantum key distribution,” New J. Phys. 13(1), 013003 (2011).
[Crossref]

2009 (1)

M. Reid, P. Drummond, W. Bowen, E. G. Cavalcanti, P. K. Lam, H. Bachor, U. L. Andersen, and G. Leuchs, “Colloquium: the Einstein-Podolsky-Rosen paradox: from concepts to applications,” Rev. Mod. Phys. 81(4), 1727–1751 (2009).
[Crossref]

2008 (2)

2006 (1)

W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulsed squeezed light: Simultaneous squeezing of multiple modes,” Phys. Rev. A 73(6), 063819 (2006).
[Crossref]

2005 (1)

S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77(2), 513–577 (2005).
[Crossref]

2004 (1)

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(4), 047904 (2002).
[Crossref]

2001 (1)

C. Silberhorn, P. K. Lam, O. Weiß, F. König, N. Korolkova, and G. Leuchs, “Generation of continuous variable Einstein-Podolsky-Rosen entanglement via the kerr nonlinearity in an optical fiber,” Phys. Rev. Lett. 86(19), 4267–4270 (2001).
[Crossref]

2000 (1)

L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84(12), 2722–2725 (2000).
[Crossref]

1998 (1)

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282(5389), 706–709 (1998).
[Crossref]

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(25), 3663–3666 (1992).
[Crossref]

1990 (1)

O. Aytür and P. Kumar, “Pulsed twin beams of light,” Phys. Rev. Lett. 65(13), 1551–1554 (1990).
[Crossref]

1989 (1)

A. La Porta, R. E. Slusher, and B. Yurke, “Back-action evading measurements of an optical field using parametric down conversion,” Phys. Rev. Lett. 62(1), 28–31 (1989).
[Crossref]

1987 (1)

R. E. Slusher, P. Grangier, A. LaPorta, B. Yurke, and M. J. Potasek, “Pulsed squeezed light,” Phys. Rev. Lett. 59(22), 2566–2569 (1987).
[Crossref]

1985 (1)

1935 (1)

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47(10), 777–780 (1935).
[Crossref]

Andersen, U. L.

M. Reid, P. Drummond, W. Bowen, E. G. Cavalcanti, P. K. Lam, H. Bachor, U. L. Andersen, and G. Leuchs, “Colloquium: the Einstein-Podolsky-Rosen paradox: from concepts to applications,” Rev. Mod. Phys. 81(4), 1727–1751 (2009).
[Crossref]

Anderson, B. E.

Assad, S. M.

Y. Liu, J. Li, L. Cui, N. Huo, S. M. Assad, X. Li, and Z. Y. Ou, “Loss-tolerant quantum dense metrology with SU(1,1) interferometer,” Opt. Express 26(21), 27705 (2018).
[Crossref]

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(5), 052127 (2018).
[Crossref]

Aytür, O.

O. Aytür and P. Kumar, “Pulsed twin beams of light,” Phys. Rev. Lett. 65(13), 1551–1554 (1990).
[Crossref]

Bachor, H.

M. Reid, P. Drummond, W. Bowen, E. G. Cavalcanti, P. K. Lam, H. Bachor, U. L. Andersen, and G. Leuchs, “Colloquium: the Einstein-Podolsky-Rosen paradox: from concepts to applications,” Rev. Mod. Phys. 81(4), 1727–1751 (2009).
[Crossref]

Banaszek, K.

W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulsed squeezed light: Simultaneous squeezing of multiple modes,” Phys. Rev. A 73(6), 063819 (2006).
[Crossref]

Bauchrowitz, J.

S. Steinlechner, J. Bauchrowitz, M. Meinders, H. Muller-Ebhardt, K. Danzmann, and R. Schnabel, “Quantum-dense metrology,” Nat. Photonics 7(8), 626–630 (2013).
[Crossref]

Bello, L.

Y. Shaked, Y. Michael, R. Z. Vered, L. Bello, M. Rosenbluh, and A. Pe’er, “Lifting the bandwidth limit of optical homodyne measurement with broadband parametric amplification,” Nat. Commun. 9(1), 609 (2018).
[Crossref]

Bowen, W.

M. Reid, P. Drummond, W. Bowen, E. G. Cavalcanti, P. K. Lam, H. Bachor, U. L. Andersen, and G. Leuchs, “Colloquium: the Einstein-Podolsky-Rosen paradox: from concepts to applications,” Rev. Mod. Phys. 81(4), 1727–1751 (2009).
[Crossref]

Braunstein, S. L.

S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77(2), 513–577 (2005).
[Crossref]

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282(5389), 706–709 (1998).
[Crossref]

Cavalcanti, E. G.

M. Reid, P. Drummond, W. Bowen, E. G. Cavalcanti, P. K. Lam, H. Bachor, U. L. Andersen, and G. Leuchs, “Colloquium: the Einstein-Podolsky-Rosen paradox: from concepts to applications,” Rev. Mod. Phys. 81(4), 1727–1751 (2009).
[Crossref]

Chekhova, M.

M. Manceau, F. Khalili, and M. Chekhova, “Improving the phase super-sensitivity of squeezing-assisted interferometers by squeeze factor unbalancing,” New J. Phys. 19(1), 013014 (2017).
[Crossref]

Chi, Y.-M.

Y.-M. Chi, B. Qi, W. Zhu, L. Qian, H.-K. Lo, S.-H. Youn, A. I. Lvovsky, and L. Tian, “A balanced homodyne detector for high-rate Gaussian-modulated coherent-state quantum key distribution,” New J. Phys. 13(1), 013003 (2011).
[Crossref]

Cialdi, S.

S. Cialdi, C. Porto, D. Cipriani, S. Olivares, and M. G. A. Paris, “Full quantum state reconstruction of symmetric two-mode squeezed thermal states via spectral homodyne detection and a state-balancing detector,” Phys. Rev. A 93(4), 043805 (2016).
[Crossref]

Cipriani, D.

S. Cialdi, C. Porto, D. Cipriani, S. Olivares, and M. G. A. Paris, “Full quantum state reconstruction of symmetric two-mode squeezed thermal states via spectral homodyne detection and a state-balancing detector,” Phys. Rev. A 93(4), 043805 (2016).
[Crossref]

Cirac, J. I.

L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84(12), 2722–2725 (2000).
[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(5), 052127 (2018).
[Crossref]

Y. Liu, J. Li, L. Cui, N. Huo, S. M. Assad, X. Li, and Z. Y. Ou, “Loss-tolerant quantum dense metrology with SU(1,1) interferometer,” Opt. Express 26(21), 27705 (2018).
[Crossref]

J. Li, Y. Liu, L. Cui, N. Huo, X. Li, and Z. Y. Ou, “Loss-tolerant measurement of continuous-variable quantum entanglement with the aid of a high gain parametric amplifier,” arXiv1808, 10258 (2018).

Danzmann, K.

S. Steinlechner, J. Bauchrowitz, M. Meinders, H. Muller-Ebhardt, K. Danzmann, and R. Schnabel, “Quantum-dense metrology,” Nat. Photonics 7(8), 626–630 (2013).
[Crossref]

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(18), 183901 (2012).
[Crossref]

Drummond, P.

M. Reid, P. Drummond, W. Bowen, E. G. Cavalcanti, P. K. Lam, H. Bachor, U. L. Andersen, and G. Leuchs, “Colloquium: the Einstein-Podolsky-Rosen paradox: from concepts to applications,” Rev. Mod. Phys. 81(4), 1727–1751 (2009).
[Crossref]

Duan, L.-M.

L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84(12), 2722–2725 (2000).
[Crossref]

Einstein, A.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47(10), 777–780 (1935).
[Crossref]

Eto, Y.

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(18), 183901 (2012).
[Crossref]

Fuchs, C. A.

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282(5389), 706–709 (1998).
[Crossref]

Furusawa, A.

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282(5389), 706–709 (1998).
[Crossref]

Giedke, G.

L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84(12), 2722–2725 (2000).
[Crossref]

Grangier, P.

Guo, X.

Gupta, P.

Hermann-Avigliano, C.

Hirano, M.

Hirano, T.

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(18), 183901 (2012).
[Crossref]

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(1), 3049 (2014).
[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(3), 033608 (2013).
[Crossref]

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(5), 052127 (2018).
[Crossref]

Y. Liu, J. Li, L. Cui, N. Huo, S. M. Assad, X. Li, and Z. Y. Ou, “Loss-tolerant quantum dense metrology with SU(1,1) interferometer,” Opt. Express 26(21), 27705 (2018).
[Crossref]

J. Li, Y. Liu, L. Cui, N. Huo, X. Li, and Z. Y. Ou, “Loss-tolerant measurement of continuous-variable quantum entanglement with the aid of a high gain parametric amplifier,” arXiv1808, 10258 (2018).

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(1), 3049 (2014).
[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(4), 047904 (2002).
[Crossref]

Jones, K. M.

Khalili, F.

M. Manceau, F. Khalili, and M. Chekhova, “Improving the phase super-sensitivity of squeezing-assisted interferometers by squeeze factor unbalancing,” New J. Phys. 19(1), 013014 (2017).
[Crossref]

Kimble, H. J.

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282(5389), 706–709 (1998).
[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(25), 3663–3666 (1992).
[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(1), 3049 (2014).
[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(3), 033608 (2013).
[Crossref]

König, F.

C. Silberhorn, P. K. Lam, O. Weiß, F. König, N. Korolkova, and G. Leuchs, “Generation of continuous variable Einstein-Podolsky-Rosen entanglement via the kerr nonlinearity in an optical fiber,” Phys. Rev. Lett. 86(19), 4267–4270 (2001).
[Crossref]

Korolkova, N.

C. Silberhorn, P. K. Lam, O. Weiß, F. König, N. Korolkova, and G. Leuchs, “Generation of continuous variable Einstein-Podolsky-Rosen entanglement via the kerr nonlinearity in an optical fiber,” Phys. Rev. Lett. 86(19), 4267–4270 (2001).
[Crossref]

Kumar, P.

O. Aytür and P. Kumar, “Pulsed twin beams of light,” Phys. Rev. Lett. 65(13), 1551–1554 (1990).
[Crossref]

La Porta, A.

A. La Porta, R. E. Slusher, and B. Yurke, “Back-action evading measurements of an optical field using parametric down conversion,” Phys. Rev. Lett. 62(1), 28–31 (1989).
[Crossref]

Lam, P. K.

M. Reid, P. Drummond, W. Bowen, E. G. Cavalcanti, P. K. Lam, H. Bachor, U. L. Andersen, and G. Leuchs, “Colloquium: the Einstein-Podolsky-Rosen paradox: from concepts to applications,” Rev. Mod. Phys. 81(4), 1727–1751 (2009).
[Crossref]

C. Silberhorn, P. K. Lam, O. Weiß, F. König, N. Korolkova, and G. Leuchs, “Generation of continuous variable Einstein-Podolsky-Rosen entanglement via the kerr nonlinearity in an optical fiber,” Phys. Rev. Lett. 86(19), 4267–4270 (2001).
[Crossref]

LaPorta, A.

R. E. Slusher, P. Grangier, A. LaPorta, B. Yurke, and M. J. Potasek, “Pulsed squeezed light,” Phys. Rev. Lett. 59(22), 2566–2569 (1987).
[Crossref]

Lett, P. D.

Leuchs, G.

M. Reid, P. Drummond, W. Bowen, E. G. Cavalcanti, P. K. Lam, H. Bachor, U. L. Andersen, and G. Leuchs, “Colloquium: the Einstein-Podolsky-Rosen paradox: from concepts to applications,” Rev. Mod. Phys. 81(4), 1727–1751 (2009).
[Crossref]

C. Silberhorn, P. K. Lam, O. Weiß, F. König, N. Korolkova, and G. Leuchs, “Generation of continuous variable Einstein-Podolsky-Rosen entanglement via the kerr nonlinearity in an optical fiber,” Phys. Rev. Lett. 86(19), 4267–4270 (2001).
[Crossref]

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(5), 052127 (2018).
[Crossref]

Y. Liu, J. Li, L. Cui, N. Huo, S. M. Assad, X. Li, and Z. Y. Ou, “Loss-tolerant quantum dense metrology with SU(1,1) interferometer,” Opt. Express 26(21), 27705 (2018).
[Crossref]

J. Li, Y. Liu, L. Cui, N. Huo, X. Li, and Z. Y. Ou, “Loss-tolerant measurement of continuous-variable quantum entanglement with the aid of a high gain parametric amplifier,” arXiv1808, 10258 (2018).

Li, T.

Li, X.

Y. Liu, J. Li, L. Cui, N. Huo, S. M. Assad, X. Li, and Z. Y. Ou, “Loss-tolerant quantum dense metrology with SU(1,1) interferometer,” Opt. Express 26(21), 27705 (2018).
[Crossref]

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(5), 052127 (2018).
[Crossref]

N. Liu, Y. Liu, X. Guo, L. Yang, X. Li, and Z. Y. Ou, “Approaching single temporal mode operation in twin beams generated by pulse pumped high gain spontaneous four wave mixing,” Opt. Express 24(2), 1096–1108 (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(3), 653–656 (2016).
[Crossref]

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(23), 29369–83 (2015).
[Crossref]

X. Guo, X. Li, N. Liu, L. Yang, and Z. Y. Ou, “An all-fiber source of pulsed twin beams for quantum communication,” Appl. Phys. Lett. 101(26), 261111 (2012).
[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(4), 047904 (2002).
[Crossref]

J. Li, Y. Liu, L. Cui, N. Huo, X. Li, and Z. Y. Ou, “Loss-tolerant measurement of continuous-variable quantum entanglement with the aid of a high gain parametric amplifier,” arXiv1808, 10258 (2018).

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(1), 3049 (2014).
[Crossref]

Liu, N.

Liu, Y.

Lo, H.-K.

Y.-M. Chi, B. Qi, W. Zhu, L. Qian, H.-K. Lo, S.-H. Youn, A. I. Lvovsky, and L. Tian, “A balanced homodyne detector for high-rate Gaussian-modulated coherent-state quantum key distribution,” New J. Phys. 13(1), 013003 (2011).
[Crossref]

Lvovsky, A. I.

Y.-M. Chi, B. Qi, W. Zhu, L. Qian, H.-K. Lo, S.-H. Youn, A. I. Lvovsky, and L. Tian, “A balanced homodyne detector for high-rate Gaussian-modulated coherent-state quantum key distribution,” New J. Phys. 13(1), 013003 (2011).
[Crossref]

W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulsed squeezed light: Simultaneous squeezing of multiple modes,” Phys. Rev. A 73(6), 063819 (2006).
[Crossref]

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(18), 183901 (2012).
[Crossref]

Manceau, M.

M. Manceau, F. Khalili, and M. Chekhova, “Improving the phase super-sensitivity of squeezing-assisted interferometers by squeeze factor unbalancing,” New J. Phys. 19(1), 013014 (2017).
[Crossref]

Meinders, M.

S. Steinlechner, J. Bauchrowitz, M. Meinders, H. Muller-Ebhardt, K. Danzmann, and R. Schnabel, “Quantum-dense metrology,” Nat. Photonics 7(8), 626–630 (2013).
[Crossref]

Michael, Y.

Y. Shaked, Y. Michael, R. Z. Vered, L. Bello, M. Rosenbluh, and A. Pe’er, “Lifting the bandwidth limit of optical homodyne measurement with broadband parametric amplification,” Nat. Commun. 9(1), 609 (2018).
[Crossref]

Muller-Ebhardt, H.

S. Steinlechner, J. Bauchrowitz, M. Meinders, H. Muller-Ebhardt, K. Danzmann, and R. Schnabel, “Quantum-dense metrology,” Nat. Photonics 7(8), 626–630 (2013).
[Crossref]

Okubo, R.

Olivares, S.

C. Sparaciari, S. Olivares, and M. G. A. Paris, “Gaussian-state interferometry with passive and active elements,” Phys. Rev. A 93(2), 023810 (2016).
[Crossref]

S. Cialdi, C. Porto, D. Cipriani, S. Olivares, and M. G. A. Paris, “Full quantum state reconstruction of symmetric two-mode squeezed thermal states via spectral homodyne detection and a state-balancing detector,” Phys. Rev. A 93(4), 043805 (2016).
[Crossref]

C. Sparaciari, S. Olivares, and M. G. A. Paris, “Bounds to precision for quantum interferometry with gaussian states and operations,” J. Opt. Soc. Am. B 32(7), 1354–1359 (2015).
[Crossref]

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(5), 052127 (2018).
[Crossref]

Y. Liu, J. Li, L. Cui, N. Huo, S. M. Assad, X. Li, and Z. Y. Ou, “Loss-tolerant quantum dense metrology with SU(1,1) interferometer,” Opt. Express 26(21), 27705 (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(3), 653–656 (2016).
[Crossref]

N. Liu, Y. Liu, X. Guo, L. Yang, X. Li, and Z. Y. Ou, “Approaching single temporal mode operation in twin beams generated by pulse pumped high gain spontaneous four wave mixing,” Opt. Express 24(2), 1096–1108 (2016).
[Crossref]

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(23), 29369–83 (2015).
[Crossref]

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(1), 3049 (2014).
[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(3), 033608 (2013).
[Crossref]

X. Guo, X. Li, N. Liu, L. Yang, and Z. Y. Ou, “An all-fiber source of pulsed twin beams for quantum communication,” Appl. Phys. Lett. 101(26), 261111 (2012).
[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(25), 3663–3666 (1992).
[Crossref]

J. Li, Y. Liu, L. Cui, N. Huo, X. Li, and Z. Y. Ou, “Loss-tolerant measurement of continuous-variable quantum entanglement with the aid of a high gain parametric amplifier,” arXiv1808, 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(4), 047904 (2002).
[Crossref]

Paris, M. G. A.

S. Cialdi, C. Porto, D. Cipriani, S. Olivares, and M. G. A. Paris, “Full quantum state reconstruction of symmetric two-mode squeezed thermal states via spectral homodyne detection and a state-balancing detector,” Phys. Rev. A 93(4), 043805 (2016).
[Crossref]

C. Sparaciari, S. Olivares, and M. G. A. Paris, “Gaussian-state interferometry with passive and active elements,” Phys. Rev. A 93(2), 023810 (2016).
[Crossref]

C. Sparaciari, S. Olivares, and M. G. A. Paris, “Bounds to precision for quantum interferometry with gaussian states and operations,” J. Opt. Soc. Am. B 32(7), 1354–1359 (2015).
[Crossref]

Pe’er, A.

Y. Shaked, Y. Michael, R. Z. Vered, L. Bello, M. Rosenbluh, and A. Pe’er, “Lifting the bandwidth limit of optical homodyne measurement with broadband parametric amplification,” Nat. Commun. 9(1), 609 (2018).
[Crossref]

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(4), 047904 (2002).
[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(25), 3663–3666 (1992).
[Crossref]

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(25), 3663–3666 (1992).
[Crossref]

Podolsky, B.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47(10), 777–780 (1935).
[Crossref]

Polzik, E. S.

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282(5389), 706–709 (1998).
[Crossref]

Porto, C.

S. Cialdi, C. Porto, D. Cipriani, S. Olivares, and M. G. A. Paris, “Full quantum state reconstruction of symmetric two-mode squeezed thermal states via spectral homodyne detection and a state-balancing detector,” Phys. Rev. A 93(4), 043805 (2016).
[Crossref]

Potasek, M. J.

R. E. Slusher, P. Grangier, A. LaPorta, B. Yurke, and M. J. Potasek, “Pulsed squeezed light,” Phys. Rev. Lett. 59(22), 2566–2569 (1987).
[Crossref]

Qi, B.

Y.-M. Chi, B. Qi, W. Zhu, L. Qian, H.-K. Lo, S.-H. Youn, A. I. Lvovsky, and L. Tian, “A balanced homodyne detector for high-rate Gaussian-modulated coherent-state quantum key distribution,” New J. Phys. 13(1), 013003 (2011).
[Crossref]

Qian, L.

Y.-M. Chi, B. Qi, W. Zhu, L. Qian, H.-K. Lo, S.-H. Youn, A. I. Lvovsky, and L. Tian, “A balanced homodyne detector for high-rate Gaussian-modulated coherent-state quantum key distribution,” New J. Phys. 13(1), 013003 (2011).
[Crossref]

Radzewicz, C.

W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulsed squeezed light: Simultaneous squeezing of multiple modes,” Phys. Rev. A 73(6), 063819 (2006).
[Crossref]

Reid, M.

M. Reid, P. Drummond, W. Bowen, E. G. Cavalcanti, P. K. Lam, H. Bachor, U. L. Andersen, and G. Leuchs, “Colloquium: the Einstein-Podolsky-Rosen paradox: from concepts to applications,” Rev. Mod. Phys. 81(4), 1727–1751 (2009).
[Crossref]

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(18), 183901 (2012).
[Crossref]

Rosen, N.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47(10), 777–780 (1935).
[Crossref]

Rosenbluh, M.

Y. Shaked, Y. Michael, R. Z. Vered, L. Bello, M. Rosenbluh, and A. Pe’er, “Lifting the bandwidth limit of optical homodyne measurement with broadband parametric amplification,” Nat. Commun. 9(1), 609 (2018).
[Crossref]

Schmittberger, B. L.

Schnabel, R.

S. Steinlechner, J. Bauchrowitz, M. Meinders, H. Muller-Ebhardt, K. Danzmann, and R. Schnabel, “Quantum-dense metrology,” Nat. Photonics 7(8), 626–630 (2013).
[Crossref]

Shaked, Y.

Y. Shaked, Y. Michael, R. Z. Vered, L. Bello, M. Rosenbluh, and A. Pe’er, “Lifting the bandwidth limit of optical homodyne measurement with broadband parametric amplification,” Nat. Commun. 9(1), 609 (2018).
[Crossref]

Silberhorn, C.

C. Silberhorn, P. K. Lam, O. Weiß, F. König, N. Korolkova, and G. Leuchs, “Generation of continuous variable Einstein-Podolsky-Rosen entanglement via the kerr nonlinearity in an optical fiber,” Phys. Rev. Lett. 86(19), 4267–4270 (2001).
[Crossref]

Slusher, R. E.

A. La Porta, R. E. Slusher, and B. Yurke, “Back-action evading measurements of an optical field using parametric down conversion,” Phys. Rev. Lett. 62(1), 28–31 (1989).
[Crossref]

R. E. Slusher, P. Grangier, A. LaPorta, B. Yurke, and M. J. Potasek, “Pulsed squeezed light,” Phys. Rev. Lett. 59(22), 2566–2569 (1987).
[Crossref]

Sørensen, J. L.

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282(5389), 706–709 (1998).
[Crossref]

Sparaciari, C.

C. Sparaciari, S. Olivares, and M. G. A. Paris, “Gaussian-state interferometry with passive and active elements,” Phys. Rev. A 93(2), 023810 (2016).
[Crossref]

C. Sparaciari, S. Olivares, and M. G. A. Paris, “Bounds to precision for quantum interferometry with gaussian states and operations,” J. Opt. Soc. Am. B 32(7), 1354–1359 (2015).
[Crossref]

Steinlechner, S.

S. Steinlechner, J. Bauchrowitz, M. Meinders, H. Muller-Ebhardt, K. Danzmann, and R. Schnabel, “Quantum-dense metrology,” Nat. Photonics 7(8), 626–630 (2013).
[Crossref]

Tajima, T.

Tian, L.

Y.-M. Chi, B. Qi, W. Zhu, L. Qian, H.-K. Lo, S.-H. Youn, A. I. Lvovsky, and L. Tian, “A balanced homodyne detector for high-rate Gaussian-modulated coherent-state quantum key distribution,” New J. Phys. 13(1), 013003 (2011).
[Crossref]

Tualle-Brouri, R.

van Loock, P.

S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77(2), 513–577 (2005).
[Crossref]

Vered, R. Z.

Y. Shaked, Y. Michael, R. Z. Vered, L. Bello, M. Rosenbluh, and A. Pe’er, “Lifting the bandwidth limit of optical homodyne measurement with broadband parametric amplification,” Nat. Commun. 9(1), 609 (2018).
[Crossref]

Wasilewski, W.

W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulsed squeezed light: Simultaneous squeezing of multiple modes,” Phys. Rev. A 73(6), 063819 (2006).
[Crossref]

Weiß, O.

C. Silberhorn, P. K. Lam, O. Weiß, F. König, N. Korolkova, and G. Leuchs, “Generation of continuous variable Einstein-Podolsky-Rosen entanglement via the kerr nonlinearity in an optical fiber,” Phys. Rev. Lett. 86(19), 4267–4270 (2001).
[Crossref]

Wenger, J.

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(4), 047904 (2002).
[Crossref]

Yang, L.

N. Liu, Y. Liu, X. Guo, L. Yang, X. Li, and Z. Y. Ou, “Approaching single temporal mode operation in twin beams generated by pulse pumped high gain spontaneous four wave mixing,” Opt. Express 24(2), 1096–1108 (2016).
[Crossref]

X. Guo, X. Li, N. Liu, L. Yang, and Z. Y. Ou, “An all-fiber source of pulsed twin beams for quantum communication,” Appl. Phys. Lett. 101(26), 261111 (2012).
[Crossref]

Youn, S.-H.

Y.-M. Chi, B. Qi, W. Zhu, L. Qian, H.-K. Lo, S.-H. Youn, A. I. Lvovsky, and L. Tian, “A balanced homodyne detector for high-rate Gaussian-modulated coherent-state quantum key distribution,” New J. Phys. 13(1), 013003 (2011).
[Crossref]

Yurke, B.

A. La Porta, R. E. Slusher, and B. Yurke, “Back-action evading measurements of an optical field using parametric down conversion,” Phys. Rev. Lett. 62(1), 28–31 (1989).
[Crossref]

R. E. Slusher, P. Grangier, A. LaPorta, B. Yurke, and M. J. Potasek, “Pulsed squeezed light,” Phys. Rev. Lett. 59(22), 2566–2569 (1987).
[Crossref]

B. Yurke, “Optical back-action-evading amplifiers,” J. Opt. Soc. Am. B 2(5), 732–738 (1985).
[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(4), 047904 (2002).
[Crossref]

Zhang, W.

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(1), 3049 (2014).
[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(3), 033608 (2013).
[Crossref]

Zhang, Y.

Zhu, W.

Y.-M. Chi, B. Qi, W. Zhu, L. Qian, H.-K. Lo, S.-H. Youn, A. I. Lvovsky, and L. Tian, “A balanced homodyne detector for high-rate Gaussian-modulated coherent-state quantum key distribution,” New J. Phys. 13(1), 013003 (2011).
[Crossref]

Zoller, P.

L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84(12), 2722–2725 (2000).
[Crossref]

Appl. Phys. Lett. (1)

X. Guo, X. Li, N. Liu, L. Yang, and Z. Y. Ou, “An all-fiber source of pulsed twin beams for quantum communication,” Appl. Phys. Lett. 101(26), 261111 (2012).
[Crossref]

J. Opt. Soc. Am. B (2)

Nat. Commun. (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(1), 3049 (2014).
[Crossref]

Y. Shaked, Y. Michael, R. Z. Vered, L. Bello, M. Rosenbluh, and A. Pe’er, “Lifting the bandwidth limit of optical homodyne measurement with broadband parametric amplification,” Nat. Commun. 9(1), 609 (2018).
[Crossref]

Nat. Photonics (1)

S. Steinlechner, J. Bauchrowitz, M. Meinders, H. Muller-Ebhardt, K. Danzmann, and R. Schnabel, “Quantum-dense metrology,” Nat. Photonics 7(8), 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(1), 013014 (2017).
[Crossref]

Y.-M. Chi, B. Qi, W. Zhu, L. Qian, H.-K. Lo, S.-H. Youn, A. I. Lvovsky, and L. Tian, “A balanced homodyne detector for high-rate Gaussian-modulated coherent-state quantum key distribution,” New J. Phys. 13(1), 013003 (2011).
[Crossref]

Opt. Express (5)

Opt. Lett. (3)

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(10), 777–780 (1935).
[Crossref]

Phys. Rev. A (4)

S. Cialdi, C. Porto, D. Cipriani, S. Olivares, and M. G. A. Paris, “Full quantum state reconstruction of symmetric two-mode squeezed thermal states via spectral homodyne detection and a state-balancing detector,” Phys. Rev. A 93(4), 043805 (2016).
[Crossref]

W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulsed squeezed light: Simultaneous squeezing of multiple modes,” Phys. Rev. A 73(6), 063819 (2006).
[Crossref]

C. Sparaciari, S. Olivares, and M. G. A. Paris, “Gaussian-state interferometry with passive and active elements,” Phys. Rev. A 93(2), 023810 (2016).
[Crossref]

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(5), 052127 (2018).
[Crossref]

Phys. Rev. Lett. (9)

C. Silberhorn, P. K. Lam, O. Weiß, F. König, N. Korolkova, and G. Leuchs, “Generation of continuous variable Einstein-Podolsky-Rosen entanglement via the kerr nonlinearity in an optical fiber,” Phys. Rev. Lett. 86(19), 4267–4270 (2001).
[Crossref]

A. La Porta, R. E. Slusher, and B. Yurke, “Back-action evading measurements of an optical field using parametric down conversion,” Phys. Rev. Lett. 62(1), 28–31 (1989).
[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(4), 047904 (2002).
[Crossref]

R. E. Slusher, P. Grangier, A. LaPorta, B. Yurke, and M. J. Potasek, “Pulsed squeezed light,” Phys. Rev. Lett. 59(22), 2566–2569 (1987).
[Crossref]

O. Aytür and P. Kumar, “Pulsed twin beams of light,” Phys. Rev. Lett. 65(13), 1551–1554 (1990).
[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(25), 3663–3666 (1992).
[Crossref]

L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84(12), 2722–2725 (2000).
[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(3), 033608 (2013).
[Crossref]

E. Flurin, N. Roch, F. Mallet, M. H. Devoret, and B. Huard, “Generating entangled microwave radiation over two transmission lines,” Phys. Rev. Lett. 109(18), 183901 (2012).
[Crossref]

Rev. Mod. Phys. (2)

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J. Li, Y. Liu, L. Cui, N. Huo, X. Li, and Z. Y. Ou, “Loss-tolerant measurement of continuous-variable quantum entanglement with the aid of a high gain parametric amplifier,” arXiv1808, 10258 (2018).

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Figures (4)

Fig. 1.
Fig. 1. Experimental setup of measuring entanglement generated from a fiber optical parametric amplifier (FOPA) by using the new method, realized by combining the PSA with balanced HDs. AM, amplitude modulator, PM, phase modulator; DSF, dispersion shifted fiber; CWDM, coarse wavelength division multiplexer; PSA, phase sensitive amplifier; P1, pump of FOPA; P2, pump of PSA; TIA, transimpedance amplifier; PZT, piezo mechanical transducer; HD, homodyne detector; LOs(i), local oscillator of HD1 (HD2).
Fig. 2.
Fig. 2. (a) The normalized noise fluctuation the quadrature amplitudes, $R_{jX}$ and $R_{jY}$, versus the average power of pump P2, $P_{2a}$. The data for $j=1,2$ (triangles and circles) is obtained by the measurement of individual HD1 and HD2, while the data for $j=0$ (squares) is obtained by the joint measurement of two HDs. (b) The power gain of PSA as a function of $P_{2a}$. The solid lines in the plots are only used for guiding eyes.
Fig. 3.
Fig. 3. The measured value of $I^{amp}_{JM}$ as a function of total detection loss ($L_d$) when the gain of PSA is 1, 5 and 16, respectively. The gray shadowed line is the linear fitting of the measured inseparability $I^{amp}_{JM}$ with $G=1$ (circles) according to the function of $I^{amp}_{JM}|_{G=1} = I'_{FOPA}\times (1-L_d)+2L_d$, where $I'_{FOPA}$ is the minimum inseparability by only correcting the detection loss. In the measurement, the gain of FOPA is fixed at 2.7.
Fig. 4.
Fig. 4. Jointly measured inseparability $I^{amp}_{JM}$ as a function average power of FOPA, $P_{1a}$. The data represented by squares and triangles is respectively obtained for the case of $P_{2a}$=0 and $P_{2a}$=7 mW. In the measurements, the detection loss of each HD is about $L_d$=0.22. The solid circles are the corrected data for the case $P_{2a}=0$ when the detection loss is taken into account. The inset shows the power gain of PSA as a function of average power $P_{1a}$.

Equations (15)

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R = Δ 2 ( X ^ 1 X ^ 2 ) Δ 2 ( X ^ 1 X ^ 2 ) S N L = Δ 2 ( Y ^ 1 + Y ^ 2 ) Δ 2 ( Y ^ 1 + Y ^ 2 ) S N L = ( μ ν ) 2 < 1 ,
I = Δ 2 ( X ^ 1 X ^ 2 ) Δ 2 ( X ^ 1 X ^ 2 ) S N L + Δ 2 ( Y ^ 1 + Y ^ 2 ) Δ 2 ( Y ^ 1 + Y ^ 2 ) S N L = 2 ( μ ν ) 2 < 2
Δ 2 X ^ 1 o u t = G 2 Δ 2 ( X ^ 1 g G X ^ 2 ) , Δ 2 Y ^ 1 o u t = G 2 Δ 2 ( Y ^ 1 + g G Y ^ 2 ) ,
Δ 2 X ^ 1 o u t S N L = Δ 2 Y ^ 1 o u t S N L = G 2 + g 2 ,
R 1 X = Δ 2 X ^ 1 o u t Δ X ^ 1 o u t S N L = G 2 Δ 2 ( X ^ 1 g G X ^ 2 ) G 2 + g 2 R 1 Y = Δ 2 Y ^ 1 o u t Δ Y ^ 1 o u t S N L = G 2 Δ 2 ( Y ^ 1 + g G Y ^ 2 ) G 2 + g 2
R 1 X Δ 2 ( X ^ 1 X ^ 2 ) Δ 2 ( X ^ 1 X ^ 2 ) S N L = ( μ v ) 2 < 1 R 1 Y Δ 2 ( Y ^ 1 + Y ^ 2 ) Δ 2 ( Y ^ 1 + Y ^ 2 ) S N L = ( μ v ) 2 < 1
Δ 2 X ^ o u t = ( G + k g ) 2 Δ 2 ( X ^ 1 g + k G G + k g X ^ 2 ) | Δ 2 Y ^ + o u t = ( G + k g ) 2 Δ 2 ( Y ^ 1 + g + k G G + k g Y ^ 2 )
R 0 X = Δ 2 X ^ out  Δ 2 X ^ out  S N L = Δ 2 ( X ^ 1 X ^ 2 ) Δ 2 ( X ^ 1 X ^ 2 ) S N L = ( μ v ) 2 R 0 Y = Δ 2 Y ^ + out  Δ 2 Y ^ + out  S N L = Δ 2 ( Y ^ 1 + Y ^ 2 ) Δ 2 ( Y ^ 1 + Y ^ 2 ) S N L = ( μ v ) 2
R j X , j Y R = ( μ ν ) 2 < 1     ( j = 0 , 1 , 2 )
( 1 L d ) ( G + g ) 2 1     and     ( 1 L d ) G 2 1
I J M a m p | G = 1 = I F O P A × ( 1 L d ) + 2 L d
A ^ j o u t = μ j A ^ j i n + ν j B ^ j i n ,     B ^ j o u t = μ j B ^ j i n + ν j A ^ j i n ,
A L O ( ω ) = j ξ j ϕ j ( ω ) ,     B L O ( ω ) = j ζ j ψ j ( ω ) ,
I m u l t i = j [ ( μ j | ξ j | ν j | ζ j | ) 2 + ( ν j | ξ j | μ j | ζ j | ) 2 ]
I J M a m p 2 ( | ξ 1 | + | ζ 1 | ) 2 G 1 2 ( μ 1 ν 1 ) 2 ( | ξ 1 | + | ζ 1 | ) 2 G 1 2 = 2 ( μ 1 ν 1 ) 2 ,

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