Abstract

Multipartite entanglement is a resource for quantum communication and computation. Vector four-wave mixing (FWM) in a fiber, driven by two strong optical pumps, couples the evolution of four weak optical sidebands (modes). Depending on the fiber dispersion and pump frequencies, the mode frequencies can be similar (separated by less than 1 THz) or dissimilar (separated by more than 10 THz). In this report, the discrete- and continuous-variable entanglement produced by vector FWM is studied in detail. Formulas are derived for the variances of, and correlations between, the mode quadratures and photon numbers. These formulas and related results show that the modes are four-partite entangled.

© 2008 Optical Society of America

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2007 (3)

2006 (5)

C. J. McKinstrie and M. G. Raymer, "Four-wave mixing cascades near the zero-dispersion frequency," Opt. Express 14, 9600-9610 (2006).
[CrossRef] [PubMed]

Z. G. Lu, P. J. Bock, J. R. Liu, F. G. Sun and T. J. Hall, "All-optical 1550 to 1310 nm wavelength converter," Electron. Lett. 42, 937-938 (2006).
[CrossRef]

M. Bondani, A. Allevi, E. Gevinti, A. Agliati and A. Andreoni, "3D phase-matching conditions for the generation of entangled triplets by ?(2) interlinked interactions," Opt. Express 14, 9838-9843 (2006).
[CrossRef] [PubMed]

S. Gröblacher, T. Jennewein, A. Varizi, G. Weihs and A. Zeilinger, "Experimental quantum cryptography with qutrits," New J. Phys. 8, 75 (2006).
[CrossRef] [PubMed]

N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph and M. A. Nielsen, "Universal quantum computation with continuous-variable cluster states," Phys. Rev. Lett. 97, 110501 (2006).
[CrossRef] [PubMed]

2005 (7)

2004 (10)

C. J. McKinstrie, S. Radic and M. G. Raymer, "Quantum noise properties of parametric amplifiers driven by two pump waves," Opt. Express 12, 5037-5066 (2004).
[CrossRef] [PubMed]

C. J. McKinstrie and S. Radic, "Phase-sensitive amplification in a fiber," Opt. Express 12, 4973-4979 (2004).
[CrossRef] [PubMed]

J. E. Sharping, J. Chen, X. Li, P. Kumar, "Quantum-correlated twin photons from microstructure fiber," Opt. Express 12, 3086-3094 (2004).
[CrossRef]

H. Takesue and K. Inoue, "Generation of polarization-entangled photon pairs and violation of Bell’s inequality using spontaneous four-wave mixing in a fiber loop," Phys. Rev. A 70, 031802R (2004).
[CrossRef] [PubMed]

A. Ferraro, M. G. A. Paris, M. Bondani, A. Allevi, E. Puddu and A. Andreoni, "Three-mode entanglement by interlinked nonlinear interactions in optical ?(2) media," J. Opt. Soc. Am. B 21, 1241-1249 (2004).
[CrossRef] [PubMed]

A. V. Rodionov and A. S. Chirkin, "Entangled photon states in consecutive nonlinear optical interactions," JETP Lett. 79, 253-256 (2004).
[CrossRef]

O. Pfister, S. Feng, G. Jennings, R. Pooser and D. Xie, "Multipartite continuous-variable entanglement from concurrent nonlinearities," Phys. Rev. A 70, 020302R (2004).
[CrossRef]

A. M. Lance, T. Symul, W. P. Bowen, B. C. Sanders and P. K. Lam, "Tripartite quantum state sharing," Phys. Rev. Lett. 92, 177903 (2004).
[CrossRef] [PubMed]

H. Yonezawa, T. Aoki and A. Furusawa, "Demonstration of a quantum teleportation network for continuous variabes," Nature 431, 430-434 (2004).
[CrossRef] [PubMed]

M. A. Nielsen, "Optical quantum computing using cluster states," Phys. Rev. Lett. 93, 040503 (2004).
[CrossRef] [PubMed]

2003 (10)

J. Jing, J. Zhang, Y. Fan, F. Zhao, C. Xie and K. Peng, "Experimental demonstration of tripartite entanglement and controlled dense coding for continuous variables," Phys. Rev. Lett. 90, 167903 (2003).
[CrossRef]

T. Aoki, N. Takei, H. Yonezawa, K. Wakui, T. Hiraoka and A. Furusawa, "Experimental creation of a fully inseparable tripartite continuous-variable state," Phys. Rev. Lett. 91, 080404 (2003).
[CrossRef] [PubMed]

W. P. Bowen, N. Treps, B. C. Buchler, R. Schnabel, T. C. Ralph, H. A. Bachor, T. Symul and P. K. Lam, "Experimental investigation of continuous-variable quantum teleportation," Phys. Rev. A 67, 032302 (2003).
[CrossRef] [PubMed]

T. C. Zhang, K. W. Goh, C. W. Chou, P. Lodahl and H. J. Kimble, "Quantum teleportation of light beams," Phys. Rev. A 67, 033802 (2003).
[CrossRef]

O. Gl¨ockl, S. Lorenz, C. Marquardt, J. Heersink, M. Brownnutt, C. Silberhorn, Q. Pan, P. van Loock, N. Korolkova and G. Leuchs, "Experiment towards continuous-variable entanglement swapping: Highly correlated four-partite quantum state," Phys. Rev. A 68, 012319 (2003).
[CrossRef] [PubMed]

C. J. McKinstrie, S. Radic and C. Xie, "Parametric instabilities driven by orthogonal pump waves in birefringent fibers," Opt. Express 11, 2619-2633 (2003).
[CrossRef]

S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, Q. Lin and G. P. Agrawal, "Record performance of a parametric amplifier constructed with highly-nonlinear fiber," Electron. Lett. 39, 838-839 (2003).
[CrossRef] [PubMed]

S. J. van Enk, "Entanglement of electromagnetic fields," Phys. Rev. A 67, 022303 (2003).
[CrossRef] [PubMed]

P. van Loock and A. Furusawa, "Detecting genuine multipartite continuous-variable entanglement," Phys. Rev. A 67, 052315 (2003).
[CrossRef] [PubMed]

K. P. Hansen, "Dispersion flattened hybrid-core nonlinear photonic crystal fiber," Opt. Express 11, 1503-1509 (2003).
[PubMed]

2002 (5)

W. H. Reeves, J. C. Knight, P. S. J. Russell and P. J. Roberts, "Demonstration of ultra-flattened dispersion in photonic crystal fibers," Opt. Express 10, 609-613 (2002).
[CrossRef]

C. J. McKinstrie, S. Radic and A. R. Chraplyvy, "Parametric amplifiers driven by two pump waves," IEEE J. Sel. Top. Quantum Electron. 8, 538-547 (2002).
[CrossRef] [PubMed]

M. Fiorentino, P. L. Voss, J. E. Sharping and P. Kumar, "All-fiber photon-pair source for quantum communications," IEEE Photon. Technol. Lett. 14, 983-985 (2002).
[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]

J. Zhang, C. Xie and K. Peng, "Controlled dense coding for continuous variables using three-partite entangled states," Phys. Rev. A 66, 032318 (2002).
[CrossRef] [PubMed]

2001 (2)

R. Raussendorf and H. J. Briegel, "A one-way quantum computer," Phys. Rev. Lett. 86, 5188-5191 (2001).
[CrossRef]

J. W. Pan, M. Daniell, S. Gasparoni, G. Weihs and A. Zeilinger, "Experimental demonstration of four-photon entanglement and high-fidelity teleportation," Phys. Rev. Lett. 86, 4435-4438 (2001).
[CrossRef]

2000 (5)

P. van Loock and S. L. Braunstein, "Multipartite entanglement for continuous variables: A quantum teleportation network," Phys. Rev. Lett. 84, 3482-3485 (2000).
[CrossRef]

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

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter and A. Zeilinger, "Quantum cryptography with entangled photons," Phys. Rev. Lett. 84, 4729-4732 (2000).
[CrossRef] [PubMed]

D. S. Naik, C. G. Peterson, A. G. White, A. J. Berglund and P. G. Kwiat, "Entangled state quantum cryptography: Eavesdropping on the Ekert protocol," Phys. Rev. Lett. 84, 4733-4736 (2000).
[CrossRef] [PubMed]

W. Tittel, J. Brendel, H. Zbinden and N. Gisin, "Quantum cryptography using entangled photons in energy-time Bell states," Phys. Rev. Lett. 84, 4737-4740 (2000).
[CrossRef] [PubMed]

1999 (1)

D. Bouwmeester, J. W. Pan, M. Daniell, H. Weinfurter and A. Zeilinger, "Observation of three-photon Greenburger-Horne-Zeilinger entanglement," Phys. Rev. Lett. 82, 1345-1349 (1999).
[CrossRef] [PubMed]

1998 (2)

S. L. Braunstein and H. J. Kimble, "Teleportation of continuous quantum variables," Phys. Rev. Lett. 80, 869-872 (1998).
[CrossRef]

A. Furusawa, J. L. Sorensen, S. J. Braunstein, C. A. Fuchs, H. J. Kimble and E. S. Polzik, "Unconditional quantum teleportation," Science 282, 706-709 (1998).
[CrossRef]

1997 (1)

D. Bouwmeester, J.W. Pan, K. Mattle, M. Eibl, H. Weinfurter and A. Zeilinger, "Experimental quantum teleportation," Nature 390, 575-579 (1997).
[CrossRef] [PubMed]

1996 (1)

K. Mattle, H. Weinfurter, P. G. Kwiat and A. Zeilinger, "Dense coding in experimental quantum communications," Phys. Rev. Lett. 76, 4656-4659 (1996).
[CrossRef] [PubMed]

1995 (1)

P. G. Kwiat, K. Mattle, H. Weinfurter and A. Zeilinger, A. V. Sergienko and Y. Shih, "New high-intensity source of polarization-entangled photon pairs," Phys. Rev. Lett. 75, 4337-4341 (1995).
[CrossRef]

1993 (1)

C. H. Bennett, G. Brassard, C. Crepeau, R. Josza, A. Peres and W. K. Wootters, "Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels," Phys. Rev. Lett. 70, 1895-1899 (1993).
[CrossRef] [PubMed]

1992 (1)

C. H. Bennett and S. J. Wiesner, "Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states," Phys. Rev. Lett. 69, 2881-2884 (1992).

1991 (1)

A. K. Ekert, "Quantum cryptography based on Bell’s theorem," Phys. Rev. Lett. 67, 661-663 (1991).
[CrossRef] [PubMed]

1986 (1)

L. A. Wu, H. J. Kimble, J. L. Hall and H. Wu, "Generation of squeezed states by parametric down conversion," Phys. Rev. Lett. 57, 2520-2523 (1986).
[CrossRef] [PubMed]

1970 (1)

D. C. Burnham and D. L. Weinberg, "Observation of simultaneity in parametric production of optical photon pairs," Phys. Rev. Lett. 25, 84-87 (1970).
[CrossRef]

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Agrawal, G. P.

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T. C. Zhang, K. W. Goh, C. W. Chou, P. Lodahl and H. J. Kimble, "Quantum teleportation of light beams," Phys. Rev. A 67, 033802 (2003).
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S. L. Braunstein and H. J. Kimble, "Dense coding for continuous variables," Phys. Rev. A 61, 042302 (2000).
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D. S. Naik, C. G. Peterson, A. G. White, A. J. Berglund and P. G. Kwiat, "Entangled state quantum cryptography: Eavesdropping on the Ekert protocol," Phys. Rev. Lett. 84, 4733-4736 (2000).
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A. M. Lance, T. Symul, W. P. Bowen, B. C. Sanders and P. K. Lam, "Tripartite quantum state sharing," Phys. Rev. Lett. 92, 177903 (2004).
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O. Gl¨ockl, S. Lorenz, C. Marquardt, J. Heersink, M. Brownnutt, C. Silberhorn, Q. Pan, P. van Loock, N. Korolkova and G. Leuchs, "Experiment towards continuous-variable entanglement swapping: Highly correlated four-partite quantum state," Phys. Rev. A 68, 012319 (2003).
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J. E. Sharping, J. Chen, X. Li, P. Kumar, "Quantum-correlated twin photons from microstructure fiber," Opt. Express 12, 3086-3094 (2004).
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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).
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S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, Q. Lin and G. P. Agrawal, "Record performance of a parametric amplifier constructed with highly-nonlinear fiber," Electron. Lett. 39, 838-839 (2003).
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Z. G. Lu, P. J. Bock, J. R. Liu, F. G. Sun and T. J. Hall, "All-optical 1550 to 1310 nm wavelength converter," Electron. Lett. 42, 937-938 (2006).
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T. C. Zhang, K. W. Goh, C. W. Chou, P. Lodahl and H. J. Kimble, "Quantum teleportation of light beams," Phys. Rev. A 67, 033802 (2003).
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O. Gl¨ockl, S. Lorenz, C. Marquardt, J. Heersink, M. Brownnutt, C. Silberhorn, Q. Pan, P. van Loock, N. Korolkova and G. Leuchs, "Experiment towards continuous-variable entanglement swapping: Highly correlated four-partite quantum state," Phys. Rev. A 68, 012319 (2003).
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Z. G. Lu, P. J. Bock, J. R. Liu, F. G. Sun and T. J. Hall, "All-optical 1550 to 1310 nm wavelength converter," Electron. Lett. 42, 937-938 (2006).
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O. Gl¨ockl, S. Lorenz, C. Marquardt, J. Heersink, M. Brownnutt, C. Silberhorn, Q. Pan, P. van Loock, N. Korolkova and G. Leuchs, "Experiment towards continuous-variable entanglement swapping: Highly correlated four-partite quantum state," Phys. Rev. A 68, 012319 (2003).
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Mattle, K.

D. Bouwmeester, J.W. Pan, K. Mattle, M. Eibl, H. Weinfurter and A. Zeilinger, "Experimental quantum teleportation," Nature 390, 575-579 (1997).
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C. J. McKinstrie, S. Radic, M. G. Raymer and L. Schenato, "Unimpaired phase-sensitive amplification by vector four-wave mixing near the zero-dispersion frequency," Opt. Express 15, 2178-2189 (2007).
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C. J. McKinstrie, S. Radic and M. G. Raymer, "Quantum noise properties of parametric amplifiers driven by two pump waves," Opt. Express 12, 5037-5066 (2004).
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C. J. McKinstrie and S. Radic, "Phase-sensitive amplification in a fiber," Opt. Express 12, 4973-4979 (2004).
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S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, Q. Lin and G. P. Agrawal, "Record performance of a parametric amplifier constructed with highly-nonlinear fiber," Electron. Lett. 39, 838-839 (2003).
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C. J. McKinstrie, S. Radic and A. R. Chraplyvy, "Parametric amplifiers driven by two pump waves," IEEE J. Sel. Top. Quantum Electron. 8, 538-547 (2002).
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Menicucci, N. C.

N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph and M. A. Nielsen, "Universal quantum computation with continuous-variable cluster states," Phys. Rev. Lett. 97, 110501 (2006).
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Migdall, A.

Naik, D. S.

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IEEE Photon. Technol. Lett. (1)

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S. Gröblacher, T. Jennewein, A. Varizi, G. Weihs and A. Zeilinger, "Experimental quantum cryptography with qutrits," New J. Phys. 8, 75 (2006).
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J. E. Sharping, J. Chen, X. Li, P. Kumar, "Quantum-correlated twin photons from microstructure fiber," Opt. Express 12, 3086-3094 (2004).
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C. J. McKinstrie and S. Radic, "Phase-sensitive amplification in a fiber," Opt. Express 12, 4973-4979 (2004).
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C. J. McKinstrie, S. Radic and M. G. Raymer, "Quantum noise properties of parametric amplifiers driven by two pump waves," Opt. Express 12, 5037-5066 (2004).
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J. G. Rarity, J. Fulconis, J. Duligall, W. J. Wadsworth and P. S. J. Russell, "Photonic crystal fiber source of correlated photon pairs," Opt. Express 13, 534-544 (2005).
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C. J. McKinstrie, M. Yu, M. G. Raymer and S. Radic, "Quantum noise properties of parametric processes in fibers," Opt. Express 13, 4986-5012 (2005).

J. Fan and A. Migdall, "Generation of cross-polarized photon pairs in a microstructure fiber with frequencyconjugate laser pump pulses," Opt. Express 13, 5777-5782 (2005).
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W. H. Reeves, J. C. Knight, P. S. J. Russell and P. J. Roberts, "Demonstration of ultra-flattened dispersion in photonic crystal fibers," Opt. Express 10, 609-613 (2002).
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K. P. Hansen, "Dispersion flattened hybrid-core nonlinear photonic crystal fiber," Opt. Express 11, 1503-1509 (2003).
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C. J. McKinstrie, S. Radic and C. Xie, "Parametric instabilities driven by orthogonal pump waves in birefringent fibers," Opt. Express 11, 2619-2633 (2003).
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C. J. McKinstrie, S. Radic, M. G. Raymer and L. Schenato, "Unimpaired phase-sensitive amplification by vector four-wave mixing near the zero-dispersion frequency," Opt. Express 15, 2178-2189 (2007).
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C. J. McKinstrie, J. D. Harvey, S. Radic and M. G. Raymer, "Translation of quantum states by four-wave mixing in fibers," Opt. Express 13, 9131-9142 (2005).
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C. J. McKinstrie and M. G. Raymer, "Four-wave mixing cascades near the zero-dispersion frequency," Opt. Express 14, 9600-9610 (2006).
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M. Bondani, A. Allevi, E. Gevinti, A. Agliati and A. Andreoni, "3D phase-matching conditions for the generation of entangled triplets by ?(2) interlinked interactions," Opt. Express 14, 9838-9843 (2006).
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W. P. Bowen, N. Treps, B. C. Buchler, R. Schnabel, T. C. Ralph, H. A. Bachor, T. Symul and P. K. Lam, "Experimental investigation of continuous-variable quantum teleportation," Phys. Rev. A 67, 032302 (2003).
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T. C. Zhang, K. W. Goh, C. W. Chou, P. Lodahl and H. J. Kimble, "Quantum teleportation of light beams," Phys. Rev. A 67, 033802 (2003).
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O. Pfister, S. Feng, G. Jennings, R. Pooser and D. Xie, "Multipartite continuous-variable entanglement from concurrent nonlinearities," Phys. Rev. A 70, 020302R (2004).
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H. Takesue and K. Inoue, "Generation of polarization-entangled photon pairs and violation of Bell’s inequality using spontaneous four-wave mixing in a fiber loop," Phys. Rev. A 70, 031802R (2004).
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X. Su, A. Tan, X. Jia, J. Zhang, C. Xie and K. Peng, "Experimental preparation of quadripartite cluster and Greenberger-Horne-Zeilinger entangled states for continuous variables," Phys. Rev. Lett. 98, 707502 (2007).
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T. Aoki, N. Takei, H. Yonezawa, K. Wakui, T. Hiraoka and A. Furusawa, "Experimental creation of a fully inseparable tripartite continuous-variable state," Phys. Rev. Lett. 91, 080404 (2003).
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R. Raussendorf and H. J. Briegel, "A one-way quantum computer," Phys. Rev. Lett. 86, 5188-5191 (2001).
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Figures (8)

Fig. 1.
Fig. 1.

Frequency diagram for the interaction of two pumps (1 and 2) and four sidebands (1± and 2±). Depending on the fiber dispersion and pump frequencies, six different four-wave mixing (FWM) processes can occur, separately or simultaneously. The red, blue and green dashed lines denote modulation interaction (MI), phase conjugation (PC) and Bragg scattering (BS), respectively.

Fig. 2.
Fig. 2.

Polarization diagram for the four-sideband interaction driven by perpendicular pumps. (a) Special case in which the pump-pump frequency difference is twice the pump-sideband difference. (b) General case in which the pump-pump difference is (much) larger than the pump-sideband difference.

Fig. 3.
Fig. 3.

Quadrature variances and correlations, normalized to the input variance 1/2 and measured in dB, plotted as functions of distance. (a) MI of pump 1, which involves modes 1- and 1+. The solid curve denotes the variance of either mode, whereas the dashed curve denotes the correlation between the modes [Eqs. (23) and (24)]. The local-oscillator phase θ=π/2 and the distance parameter is γKP1z. Similar results apply to the interaction between the superposition modes b + and c +, for which the distance parameter is γK (P 1+P 2)z [Eqs. (37)–(39)]. (b) Four-mode interaction driven by pumps with equal powers. The solid curve denotes the variance of any mode, whereas the dashed curve denotes the correlation between any pair of modes [Eqs. (25)–(32)]. The phase is π/2 and the distance parameter is γKPz.

Fig. 4.
Fig. 4.

(a) Probability (in dB) that there are n photons in each of modes b+ and c+ [Eq. (62)]. The dashed, dot-dashed and solid lines represent the distance parameters γK (P 1+P 2)z=0.3, 1.0 and 3.0, respectively. (b) Joint probability distribution (PD) of modes b+ and c+ [Eq. (63)] for the intermediate distance. These modes are correlated.

Fig. 5.
Fig. 5.

Total probability (in dB) that there are n photons in mode 1-[Eq. (67)]. The dashed, dot-dashed and solid lines represent the distance parameters 2γKPz=0.3, 1.0 and 3.0, respectively. The PDs of modes 1+, 2- and 2+ are identical.

Fig. 6.
Fig. 6.

(a) Joint PD (in dB) of modes 1- and 1+ [Eq. (69)] for the n=4 state and the intermediate distance-parameter 2γKPz=1.0. The joint PD of modes 1- and 2+ is identical. (b) Joint PD of modes 1- and 2- [Eq. (70)] for the same state and distance. These modes are anti-correlated.

Fig. 7.
Fig. 7.

(a) Joint PD (in dB) of modes 1- and 1+ [Eq. (71)] for the intermediate distance-parameter 2γKPz=1.0, which should be compared to the PD shown in Fig. 4(b). The joint PD of modes 1- and 2+ is identical. (b) Joint PD of modes 1- and 2- [Eq. (72)] for the same distance.

Fig. 8.
Fig. 8.

Entanglement (entropy) of mode 1- plotted as a function of the distance parameter 2γKPz. The dashed and solid curves denote the two-mode interaction with mode 1+ [Eq. (73)] and the four-mode interaction with 1+, 2- and 2+ [Eq. (74)], respectively. For comparison, the dot-dashed curve denotes the entropy of mode b +, which interacts with mode c + [also Eq. (73)].

Equations (121)

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H a = α ( a 1 a 1 + a 1 + a 1 + ) + α ( a 1 a 1 + + a 1 a 1 + )
  + β ( a 1 a 2 + a 1 a 2 ) + β ( a 1 + a 2 + + a 1 + a 2 + )
+ β ( a 1 a 2 + + a 1 a 2 + ) + β ( a 1 + a 2 + a 1 + a 2 )
+ γ ( a 2 a 2 + a 2 + a 2 + ) + γ ( a 2 a 2 + + a 2 a 2 + ) ,
da j dz = i [ a j , H a ]
da 1 dz = i α a 1 i α a 1 + i β a 2 i β a 2 + ,
da 1 + dz = i α a 1 + i α a 1 + + i β a 2 + i β a 2 + ,
da 2 dz = i β a 1 i β a 1 + i γ a 2 i γ a 2 + ,
da 2 + dz = i β a 1 + i β a 1 + + i γ a 2 + i γ a 2 + .
a 1 ( z ) = ( 1 i α z ) a 1 ( 0 ) i α za 1 + ( 0 ) i β za 2 ( 0 ) i β za 2 + ( 0 ) ,
a 1 + ( z ) = i α za 1 ( 0 ) + ( 1 + i α z ) a 1 + ( 0 ) + i β za 2 ( 0 ) + i β za 2 + ( 0 ) ,
a 2 ( z ) = i β za 1 ( 0 ) i β za 1 + ( 0 ) + ( 1 i γ z ) a 2 ( 0 ) i γ za 2 + ( 0 ) ,
a 2 + ( z ) = i β za 1 ( 0 ) + i β za 1 + ( 0 ) + i γ za 2 ( 0 ) + ( 1 + i γ z ) a 2 + ( 0 ) ,
q j ( θ j ) = ( a j e i θ j + a j e i θ j ) 2 1 2 ,
δ q j ( θ j ) = q j ( θ j ) q j ( θ j ) ,
a j ( z ) = k [ μ j k ( z ) a k ( 0 ) + ν j k ( z ) a k ( 0 ) ] .
δ q j ( θ j ) δ q k ( θ k ) = l ( μ j l e i θ j + ν j l * e i θ j ) ( μ kl * e i θ k + ν k l e i θ k ) 2 .
δ q 1 ± 2 ( θ 1 ± ) = [ 1 + 2 ( α 2 + β 2 ) z 2 ] 2 ,
δ q 2 ± 2 ( θ 2 ± ) = [ 1 + 2 ( β 2 + γ 2 ) z 2 ] 2
δ q 1 ( θ 1 ) δ q 1 + ( θ 1 + ) = α z sin ( θ 1 + θ 1 + ) ( α 2 + β 2 ) z 2 cos ( θ 1 + θ 1 + ) ,
δ q 1 ( θ 1 ) δ q 2 ( θ 2 ) = β ( α + γ ) z 2 cos ( θ 1 θ 2 ) ,
δ q 1 ( θ 1 ) δ q 2 + ( θ 2 + ) = β z sin ( θ 1 + θ 2 + ) β ( α + γ ) z 2 cos ( θ 1 + θ 2 + ) ,
δ q 1 + ( θ 1 + ) δ q 2 ( θ 2 ) = β z sin ( θ 1 + + θ 2 ) β ( α + γ ) z 2 cos ( θ 1 + + θ 2 ) ,
δ q 1 + ( θ 1 + ) δ q 2 + ( θ 2 + ) = β ( α + γ ) z 2 cos ( θ 1 + θ 2 + ) ,
δ q 2 ( θ 2 ) δ q 2 + ( θ 2 + ) = γz sin ( θ 2 + θ 2 + ) ( β 2 + γ 2 ) z 2 cos ( θ 2 + θ 2 + ) .
δ q 1 ± 2 ( θ ) [ 1 + 2 ( α z ) 2 ] 2 ,
δ q 1 ( θ ) δ q 1 + ( θ ) ( α z ) sin ( 2 θ ) ( α z ) 2 cos ( 2 θ ) .
δ q 1 ± 2 ( θ ) = [ 1 + 4 ( z ) 2 ] 2 ,
δ q 2 ± 2 ( θ ) = [ 1 + 4 ( z ) 2 ] 2 ,
δ q 1 ( θ ) δ q 1 + ( θ ) = z sin ( 2 θ ) 2 ( z ) 2 cos ( 2 θ ) ,
δ q 1 ( θ ) δ q 2 ( θ ) = 2 ( z ) 2 ,
δ q 1 ( θ ) δ q 2 + ( θ ) = z sin ( 2 θ ) 2 ( z ) 2 cos ( 2 θ ) ,
δ q 1 + ( θ ) δ q 2 ( θ ) = z sin ( 2 θ ) 2 ( z ) 2 cos ( 2 θ ) ,
δ q 1 + ( θ ) δ q 2 + ( θ ) = 2 ( z ) 2 ,
δ q 2 ( θ ) δ q 2 + ( θ ) = z sin ( 2 θ ) 2 ( z ) 2 cos ( 2 θ ) .
b ± = ( a 1 ± a 2 ) 2 1 2 ,
c ± = ( a 1 + ± a 2 + ) 2 1 2 ,
2 δ q b + δ q c ± = δ q 1 δ q 1 + ± δ q 1 δ q 2 + + δ q 2 δ q 1 + ± δ q 2 δ q 2 + ,
2 δ q b δ q c ± = δ q 1 δ q 1 + ± δ q 1 δ q 2 + δ q 2 δ q 1 + δ q 2 δ q 2 + .
δ q b + 2 ( θ ) = [ 1 + 2 ( 2 z ) 2 ] 2 ,
δ q c + 2 ( θ ) = [ 1 + 2 ( 2 z ) 2 ] 2 ,
δ q b + δ q c + ( θ ) = ( 2 z ) sin ( 2 θ ) ( 2 z ) 2 cos ( 2 θ ) .
δ q b 2 ( θ ) = 1 2 ,
δ q c 2 ( θ ) = 1 2 ,
δ q b ( θ ) δ q c ( θ ) = 0 .
b + = ε ( σ a 1 + a 2 ) , b = ε ( a 1 σ a 2 ) ,
c + = ε ( σ a 1 + + a 2 + ) , c = ε ( a 1 + σ a 2 + ) ,
b + ( z ) = [ 1 + i ( σ + 1 σ ) z ] b + ( 0 ) + i ( σ + 1 σ ) z c + ( 0 ) ,
c + ( z ) = i ( σ + 1 σ ) z b + ( 0 ) + [ 1 i ( σ + 1 σ ) z ] c + ( 0 ) .
b ( z ) = b ( 0 ) ,
c ( z ) = c ( 0 ) .
db + dz = ib + + ic + ,
dc + dz = ib + ic + ,
r = ( b + + c + ) 2 1 2 ,
s = ( b + c + ) 2 1 2
dr d z = ir + i r ,
ds d z = is i s ,
r ( z ) = ( 1 + i z ) r ( 0 ) + i z r ( 0 ) ,
s ( z ) = ( 1 + i z ) s ( 0 ) i z s ( 0 ) .
H bc = ( σ + 1 σ ) ( b + b + + c + c + + b + c + + b + c + ) .
H rs = r r + [ ( r ) 2 + r 2 ] 2 + s s [ ( s ) 2 + s 2 ] 2 .
exp ( i H r z ) = exp ( γ + K + ) exp ( γ 3 K 3 ) exp ( γ K ) ,
r = 1 ( 1 iz ) 1 2 n = 0 ( iz 1 iz ) n [ ( 2 n ) ! ] 1 2 2 n n ! 2 n ,
b + , c + = 1 1 i z n = 0 ( i z 1 i z ) n n , n ,
A ( n , z ) = z 2 n ( 1 + z 2 ) n + 1 .
Q ( k , l , z ) = A ( k , z ) δ ( k , l ) ,
ψ ( z ) = 1 1 i z n = 0 k = 0 n l = 0 n ( i z 1 i z ) n σ k + l n ! k , l , n k , n l ( 1 + σ 2 ) n [ k ! l ! ( n k ) ! ( n l ) ! ] 1 2 ,
ψ ( z ) 1 1 i z n = 0 ( i z 1 i z ) n n , n , 0 , 0 ,
ψ ( z ) 1 1 i z n = 0 k = 0 n l = 0 n ( i z 1 i z ) n n ! k , l , n k , n l 2 n [ k ! l ! ( n k ) ! ( n l ) ! ] 1 2 .
P t ( m , z ) = n = m A ( n , z ) B ( n , m , n m ) ,
B ( n , k , l ) = n ! ( 2 n k ! l ! ) .
Q c ( k , l ) = B ( n , k , n k ) B ( n , l , n l ) .
R c ( k , l ) = B ( n , k , n k ) δ ( n k , l ) .
Q t ( k , l , z ) = n = max ( k , l ) A ( n , z ) B ( n , k , n k ) B ( n , l , n l ) .
R t ( k , l , z ) = A ( k + l , z ) B ( k + l , k , l ) .
ρ b + ( z ) = n = 0 A n z n n ,
ρ 1 ( z ) = m = 0 P t m z m m ,
d z n 1 = i α ( a 1 a 1 + a 1 a 1 + ) + i β ( a 1 a 2 a 1 a 2 )
+ i β ( a 1 a 2 + a 1 a 2 + ) ,
d z n 1 + = i α ( a 1 a 1 + a 1 a 1 + ) + i β ( a 1 + a 2 a 1 + a 2 )
+ i β ( a 1 + a 2 + a 1 + a 2 + ) ,
d z n 2 = ( a 2 a 2 + a 2 a 1 + ) + i β ( a 1 a 2 a 1 a 2 )
+ i β ( a 1 + a 2 a 1 + a 2 ) ,
d z n 2 + = ( a 2 a 2 + a 2 a 2 + ) + i β ( a 1 a 2 + a 1 a 2 + )
            + i β ( a 1 + a 2 + a 1 + a 2 + ) .
d z ( n 1 n 1 + ) = d z ( n 2 + n 2 ) ,
d z ( n 1 + n 2 ) = d z ( n 1 + + n 2 + ) ,
d z ( n 1 n 2 + ) = d z ( n 1 + n 2 ) .
exp ( a ) b exp ( a ) = n = 0 [ a , b ] n n ! , ,
exp ( a ) b exp ( a ) = n = 0 m = 0 n a m b ( a ) n m m ! ( n n ) ! .
[ a , b ] n = m = 0 n n ! a m b ( a ) n m m ! ( n m ) ! .
a [ a , b ] n [ a , b ] n a = m = 0 n n ! [ a m + 1 b ( a ) n m a m b ( a ) n m a ] m ! ( n m ) ! ,
= m = 1 n + 1 n ! a m b ( a ) n + 1 m ( m 1 ) ! ( n + 1 m ) ! + m = 0 n n ! a m b ( a ) n + 1 m m ! ( n m ) ! .
n ! a m b ( a ) n + 1 m ( m 1 ) ! ( n m ) ! [ 1 n + 1 m + 1 m ] = ( n + 1 ) ! a m b ( a ) n + 1 m m ! ( n + 1 m ) ! .
F ( z ) = exp [ i ( K + + 2 K 3 + K ) z ] .
F ( z ) = exp [ ip ( z ) K + ] exp [ iq ( z ) K 3 ] exp [ ir ( z ) K ] ,
F = i ( K + + 2 K 3 + K ) F ,
F = ( i p K + + I q e i p K + K 3 e i p K + + i r e i p K + e i q K 3 K e i q K 3 e i p K + ) F .
e i p K + K 3 e i p K + = K 3 i p K + ,
e i q K 3 K e i q K 3 = K e i q ,
e i p K + K e i p K + = K 2 i p K 3 p 2 K + .
p i p q p 2 ( r e i q ) = 1 ,
q 2 i p ( r e i q ) = 2 ,
r e i q = 1 ,
i p ( z ) = i z ( 1 i z ) ,
i q ( z ) = 2 log ( 1 i z ) ,
i r ( z ) = i z ( 1 i z ) .
ψ = n = 0 a n n b n c ,
ρ = n = 0 n = 0 a n a n * n b n c n b n c .
ρ b = n = 0 a n 2 n b n b .
ψ = n = 0 k = 0 n l = 0 n a n b nk b nl k 1 l 2 n k 3 n l 4 ,
ρ = n = 0 k = 0 n l = 0 n n = 0 k = 0 n l = 0 n a n b nk b nl ( a n b n k b n l ) *
× k 1 l 2 n k 3 n l 4 k 1 l 2 n k 3 n l 4 .
ρ 234 = k = 0 n = k l = 0 n n = k l = 0 n a n b nk b nl ( a n b n k b n l ) *
× l 2 n k 3 n l 4 l 2 n k 3 n l 4 .
ρ 24 = n = 0 k = 0 n l = 0 n l = 0 n a n 2 b nk 2 b nl b nl * l 2 n l 4 l 2 n l 4 .
ρ 24 = n = 0 l = 0 n l = 0 n a n 2 b nl b nl * l 2 n - l 4 l 2 n - l 4 .
ρ 4 = l = 0 n = l a n 2 b nl 2 n l 4 n l 4 ,
ρ 4 = l = 0 n = l a n 2 b nl 2 l 4 l 4 ,
ρ 34 = l = 0 n = l n = l k = 0 k + ( a n b nk b nl ) ( a n b n k b n l ) * n k 3 n l 4 n k 3 n l 4 .
ρ 4 = n = 0 l = 0 n k = 0 n a n 2 b nk 2 b nl 2 n l 4 n l 4 .

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