Abstract

We identify and discuss nonlinear phase noise arising in Kerr self-phase modulation of a coherent light pulse propagating through an attenuating medium with third-order nonlinearity in a dispersion-free setting. This phenomenon, accompanying the standard unitary Kerr transformation of the optical field, is described with high accuracy as Gaussian phase diffusion with parameters given by closed expressions in terms of system properties. We show that the irreversibility of the nonlinear phase noise ultimately limits the ability to transmit classical information in the phase variable over a lossy single-mode bosonic channel with Kerr-type nonlinearity. Our model can be also used to estimate the amount of squeezing attainable through self-phase modulation in a Kerr medium with distributed attenuation.

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

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References

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  25. A. Mandarino, M. Bina, C. Porto, S. Cialdi, S. Olivares, and M. G. A. Paris, “Assessing the significance of fidelity as a figure of merit in quantum state reconstruction of discrete and continuous-variable systems,” Phys. Rev. A 93, 062118 (2016).
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    [Crossref]
  31. J. F. Corney, J. Heersink, R. Dong, V. Josse, P. D. Drummond, G. Leuchs, and U. L. Andersen, “Simulations and experiments on polarization squeezing in optical fiber,” Phys. Rev. A 78, 023831 (2008).
    [Crossref]
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    [Crossref]
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    [Crossref]
  34. J. R. Anglin, J. P. Paz, and W. H. Zurek, “Deconstructing decoherence,” Phys. Rev. A 55, 4041–4053 (1997).
    [Crossref]
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    [Crossref]
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    [Crossref]

2016 (3)

A. Mandarino, M. Bina, C. Porto, S. Cialdi, S. Olivares, and M. G. A. Paris, “Assessing the significance of fidelity as a figure of merit in quantum state reconstruction of discrete and continuous-variable systems,” Phys. Rev. A 93, 062118 (2016).
[Crossref]

U. L. Andersen, T. Gehring, C. Marquardt, and G. Leuchs, “30 years of squeezed light generation,” Phys. Scripta 91, 053001 (2016).
[Crossref]

K. Kikuchi, “Fundamentals of coherent optical fiber communications,” J. Lightwave Technol. 34, 157–179 (2016).
[Crossref]

2015 (1)

E. Temprana, E. Myslivets, B.-P. Kuo, L. Liu, V. Ataie, N. Alic, and S. Radic, “Overcoming Kerr-induced capacity limit in optical fiber transmission,” Science 348, 1445–1448 (2015).
[Crossref]

2014 (2)

V. Giovannetti, R. García-Patrón, N. J. Cerf, and A. S. Holevo, “Ultimate classical communication rates of quantum optical channels,” Nat. Photonics 8, 796–800 (2014).
[Crossref]

M. Bina, A. Mandarino, S. Olivares, and M. G. A. Paris, “Drawbacks of the use of fidelity to assess quantum resources,” Phys. Rev. A 89, 012305 (2014).
[Crossref]

2013 (1)

C. Benedetti, A. P. Shurupov, M. G. A. Paris, G. Brida, and M. Genovese, “Experimental estimation of quantum discord for a polarization qubit and the use of fidelity to assess quantum correlations,” Phys. Rev. A 87, 052136 (2013).
[Crossref]

2010 (2)

R. Chrapkiewicz and W. Wasilewski, “Multimode spontaneous parametric down-conversion in a lossy medium,” J. Mod. Opt. 57, 345–355 (2010).
[Crossref]

A. Ellis, J. Zhao, and D. Cotter, “Approaching the non-linear Shannon limit,” J. Lightwave Technol. 28, 423–433 (2010).
[Crossref]

2008 (1)

J. F. Corney, J. Heersink, R. Dong, V. Josse, P. D. Drummond, G. Leuchs, and U. L. Andersen, “Simulations and experiments on polarization squeezing in optical fiber,” Phys. Rev. A 78, 023831 (2008).
[Crossref]

2007 (2)

2006 (2)

L. G. Kazovsky, G. Kalogerakis, and W. T. Shaw, “Homodyne phase-shift-keying systems: past challenges and future opportunities,” J. Lightwave Technol. 24, 4876–4884 (2006).
[Crossref]

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

2004 (2)

J. M. Kahn and K.-P. Ho, “Spectral efficiency limits and modulation/detection techniques for DWDM systems,” IEEE J. Sel. Top. Quantum Electron. 10, 259–272 (2004).
[Crossref]

L. Wegener, M. Povinelli, A. Green, P. Mitra, J. Stark, and P. Littlewood, “The effect of propagation nonlinearities on the information capacity of WDM optical fiber systems: cross-phase modulation and four-wave mixing,” Physica D 189, 81–99 (2004).
[Crossref]

2002 (1)

2001 (1)

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature 411, 1027–1030 (2001).
[Crossref]

2000 (1)

C.-C. Cheng and M. G. Raymer, “Propagation of transverse optical coherence in random multiple-scattering media,” Phys. Rev. A 62, 023811 (2000).
[Crossref]

1999 (1)

M. G. A. Paris, “Generation of mesoscopic quantum superpositions through Kerr-stimulated degenerate downconversion,” J. Opt. B 1, 662–667 (1999).
[Crossref]

1998 (3)

S. Schmitt, J. Ficker, M. Wolff, F. König, A. Sizmann, and G. Leuchs, “Photon-number squeezed solitons from an asymmetric fiber-optic Sagnac interferometer,” Phys. Rev. Lett. 81, 2446–2449 (1998).
[Crossref]

M. J. Werner, “Quantum soliton generation using an interferometer,” Phys. Rev. Lett. 81, 4132–4135 (1998).
[Crossref]

A. S. Holevo, “The capacity of the quantum channel with general signal states,” IEEE Trans. Inf. Theory 44, 269–273 (1998).
[Crossref]

1997 (3)

B. Schumacher and M. D. Westmoreland, “Sending classical information via noisy quantum channels,” Phys. Rev. A 56, 131–138 (1997).
[Crossref]

K. Banaszek and K. Wódkiewicz, “Operational theory of homodyne detection,” Phys. Rev. A 55, 3117–3123 (1997).
[Crossref]

J. R. Anglin, J. P. Paz, and W. H. Zurek, “Deconstructing decoherence,” Phys. Rev. A 55, 4041–4053 (1997).
[Crossref]

1995 (1)

V. Bužek and P. L. Knight, “Quantum interference, superposition states of light, and nonclassical effects,” Prog. Opt. 34, 1–158 (1995).
[Crossref]

1993 (2)

W. Vogel and J. Grabow, “Statistics of difference events in homodyne detection,” Phys. Rev. A 47, 4227–4235 (1993).
[Crossref]

U. Leonhardt and H. Paul, “Realistic optical homodyne measurements and quasiprobability distributions,” Phys. Rev. A 48, 4598–4604 (1993).
[Crossref]

1991 (1)

M. Rosenbluh and R. M. Shelby, “Squeezed optical solitons,” Phys. Rev. Lett. 66, 153–156 (1991).
[Crossref]

1990 (1)

1989 (1)

D. J. Daniel and G. J. Milburn, “Destruction of quantum coherence in a nonlinear oscillator via attenuation and amplification,” Phys. Rev. A 39, 4628–4640 (1989).
[Crossref]

1986 (2)

G. J. Milburn and C. A. Holmes, “Dissipative quantum and classical Liouville mechanics of the anharmonic oscillator,” Phys. Rev. Lett. 56, 2237–2240 (1986).
[Crossref]

B. Yurke and D. Stoler, “Generating quantum mechanical superpositions of macroscopically distinguishable states via amplitude dispersion,” Phys. Rev. Lett. 57, 13–16 (1986).
[Crossref]

1981 (1)

M. G. Raymer and J. Mostowski, “Stimulated Raman scattering: unified treatment of spontaneous initiation and spatial propagation,” Phys. Rev. A 24, 1980–1993 (1981).
[Crossref]

1973 (1)

A. Holevo, “Some estimates of information transmitted through quantum communication channel,” Problemy Peredachi Informatsii 9, 3–11 (1973).

1963 (2)

E. C. G. Sudarshan, “Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams,” Phys. Rev. Lett. 10, 277–279 (1963).
[Crossref]

R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).
[Crossref]

Alic, N.

E. Temprana, E. Myslivets, B.-P. Kuo, L. Liu, V. Ataie, N. Alic, and S. Radic, “Overcoming Kerr-induced capacity limit in optical fiber transmission,” Science 348, 1445–1448 (2015).
[Crossref]

Andersen, U. L.

U. L. Andersen, T. Gehring, C. Marquardt, and G. Leuchs, “30 years of squeezed light generation,” Phys. Scripta 91, 053001 (2016).
[Crossref]

J. F. Corney, J. Heersink, R. Dong, V. Josse, P. D. Drummond, G. Leuchs, and U. L. Andersen, “Simulations and experiments on polarization squeezing in optical fiber,” Phys. Rev. A 78, 023831 (2008).
[Crossref]

Anglin, J. R.

J. R. Anglin, J. P. Paz, and W. H. Zurek, “Deconstructing decoherence,” Phys. Rev. A 55, 4041–4053 (1997).
[Crossref]

Ataie, V.

E. Temprana, E. Myslivets, B.-P. Kuo, L. Liu, V. Ataie, N. Alic, and S. Radic, “Overcoming Kerr-induced capacity limit in optical fiber transmission,” Science 348, 1445–1448 (2015).
[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, 063819 (2006).
[Crossref]

K. Banaszek and K. Wódkiewicz, “Operational theory of homodyne detection,” Phys. Rev. A 55, 3117–3123 (1997).
[Crossref]

Benedetti, C.

C. Benedetti, A. P. Shurupov, M. G. A. Paris, G. Brida, and M. Genovese, “Experimental estimation of quantum discord for a polarization qubit and the use of fidelity to assess quantum correlations,” Phys. Rev. A 87, 052136 (2013).
[Crossref]

Bina, M.

A. Mandarino, M. Bina, C. Porto, S. Cialdi, S. Olivares, and M. G. A. Paris, “Assessing the significance of fidelity as a figure of merit in quantum state reconstruction of discrete and continuous-variable systems,” Phys. Rev. A 93, 062118 (2016).
[Crossref]

M. Bina, A. Mandarino, S. Olivares, and M. G. A. Paris, “Drawbacks of the use of fidelity to assess quantum resources,” Phys. Rev. A 89, 012305 (2014).
[Crossref]

Brida, G.

C. Benedetti, A. P. Shurupov, M. G. A. Paris, G. Brida, and M. Genovese, “Experimental estimation of quantum discord for a polarization qubit and the use of fidelity to assess quantum correlations,” Phys. Rev. A 87, 052136 (2013).
[Crossref]

Bužek, V.

V. Bužek and P. L. Knight, “Quantum interference, superposition states of light, and nonclassical effects,” Prog. Opt. 34, 1–158 (1995).
[Crossref]

Cerf, N. J.

V. Giovannetti, R. García-Patrón, N. J. Cerf, and A. S. Holevo, “Ultimate classical communication rates of quantum optical channels,” Nat. Photonics 8, 796–800 (2014).
[Crossref]

Cheng, C.-C.

C.-C. Cheng and M. G. Raymer, “Propagation of transverse optical coherence in random multiple-scattering media,” Phys. Rev. A 62, 023811 (2000).
[Crossref]

Chrapkiewicz, R.

R. Chrapkiewicz and W. Wasilewski, “Multimode spontaneous parametric down-conversion in a lossy medium,” J. Mod. Opt. 57, 345–355 (2010).
[Crossref]

Cialdi, S.

A. Mandarino, M. Bina, C. Porto, S. Cialdi, S. Olivares, and M. G. A. Paris, “Assessing the significance of fidelity as a figure of merit in quantum state reconstruction of discrete and continuous-variable systems,” Phys. Rev. A 93, 062118 (2016).
[Crossref]

Corney, J. F.

J. F. Corney, J. Heersink, R. Dong, V. Josse, P. D. Drummond, G. Leuchs, and U. L. Andersen, “Simulations and experiments on polarization squeezing in optical fiber,” Phys. Rev. A 78, 023831 (2008).
[Crossref]

Cotter, D.

Daniel, D. J.

D. J. Daniel and G. J. Milburn, “Destruction of quantum coherence in a nonlinear oscillator via attenuation and amplification,” Phys. Rev. A 39, 4628–4640 (1989).
[Crossref]

Demir, A.

Dong, R.

J. F. Corney, J. Heersink, R. Dong, V. Josse, P. D. Drummond, G. Leuchs, and U. L. Andersen, “Simulations and experiments on polarization squeezing in optical fiber,” Phys. Rev. A 78, 023831 (2008).
[Crossref]

Drummond, P. D.

J. F. Corney, J. Heersink, R. Dong, V. Josse, P. D. Drummond, G. Leuchs, and U. L. Andersen, “Simulations and experiments on polarization squeezing in optical fiber,” Phys. Rev. A 78, 023831 (2008).
[Crossref]

Ellis, A.

Ficker, J.

S. Schmitt, J. Ficker, M. Wolff, F. König, A. Sizmann, and G. Leuchs, “Photon-number squeezed solitons from an asymmetric fiber-optic Sagnac interferometer,” Phys. Rev. Lett. 81, 2446–2449 (1998).
[Crossref]

Fiorentino, M.

García-Patrón, R.

V. Giovannetti, R. García-Patrón, N. J. Cerf, and A. S. Holevo, “Ultimate classical communication rates of quantum optical channels,” Nat. Photonics 8, 796–800 (2014).
[Crossref]

Gehring, T.

U. L. Andersen, T. Gehring, C. Marquardt, and G. Leuchs, “30 years of squeezed light generation,” Phys. Scripta 91, 053001 (2016).
[Crossref]

Genovese, M.

C. Benedetti, A. P. Shurupov, M. G. A. Paris, G. Brida, and M. Genovese, “Experimental estimation of quantum discord for a polarization qubit and the use of fidelity to assess quantum correlations,” Phys. Rev. A 87, 052136 (2013).
[Crossref]

Giovannetti, V.

V. Giovannetti, R. García-Patrón, N. J. Cerf, and A. S. Holevo, “Ultimate classical communication rates of quantum optical channels,” Nat. Photonics 8, 796–800 (2014).
[Crossref]

Glauber, R. J.

R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).
[Crossref]

Gordon, J. P.

Grabow, J.

W. Vogel and J. Grabow, “Statistics of difference events in homodyne detection,” Phys. Rev. A 47, 4227–4235 (1993).
[Crossref]

Green, A.

L. Wegener, M. Povinelli, A. Green, P. Mitra, J. Stark, and P. Littlewood, “The effect of propagation nonlinearities on the information capacity of WDM optical fiber systems: cross-phase modulation and four-wave mixing,” Physica D 189, 81–99 (2004).
[Crossref]

Heersink, J.

J. F. Corney, J. Heersink, R. Dong, V. Josse, P. D. Drummond, G. Leuchs, and U. L. Andersen, “Simulations and experiments on polarization squeezing in optical fiber,” Phys. Rev. A 78, 023831 (2008).
[Crossref]

Ho, K.-P.

J. M. Kahn and K.-P. Ho, “Spectral efficiency limits and modulation/detection techniques for DWDM systems,” IEEE J. Sel. Top. Quantum Electron. 10, 259–272 (2004).
[Crossref]

Holevo, A.

A. Holevo, “Some estimates of information transmitted through quantum communication channel,” Problemy Peredachi Informatsii 9, 3–11 (1973).

Holevo, A. S.

V. Giovannetti, R. García-Patrón, N. J. Cerf, and A. S. Holevo, “Ultimate classical communication rates of quantum optical channels,” Nat. Photonics 8, 796–800 (2014).
[Crossref]

A. S. Holevo, “The capacity of the quantum channel with general signal states,” IEEE Trans. Inf. Theory 44, 269–273 (1998).
[Crossref]

Holmes, C. A.

G. J. Milburn and C. A. Holmes, “Dissipative quantum and classical Liouville mechanics of the anharmonic oscillator,” Phys. Rev. Lett. 56, 2237–2240 (1986).
[Crossref]

Josse, V.

J. F. Corney, J. Heersink, R. Dong, V. Josse, P. D. Drummond, G. Leuchs, and U. L. Andersen, “Simulations and experiments on polarization squeezing in optical fiber,” Phys. Rev. A 78, 023831 (2008).
[Crossref]

Kahn, J. M.

J. M. Kahn and K.-P. Ho, “Spectral efficiency limits and modulation/detection techniques for DWDM systems,” IEEE J. Sel. Top. Quantum Electron. 10, 259–272 (2004).
[Crossref]

Kalogerakis, G.

Kazovsky, L. G.

Kikuchi, K.

Knight, P. L.

V. Bužek and P. L. Knight, “Quantum interference, superposition states of light, and nonclassical effects,” Prog. Opt. 34, 1–158 (1995).
[Crossref]

König, F.

S. Schmitt, J. Ficker, M. Wolff, F. König, A. Sizmann, and G. Leuchs, “Photon-number squeezed solitons from an asymmetric fiber-optic Sagnac interferometer,” Phys. Rev. Lett. 81, 2446–2449 (1998).
[Crossref]

Kumar, P.

Kuo, B.-P.

E. Temprana, E. Myslivets, B.-P. Kuo, L. Liu, V. Ataie, N. Alic, and S. Radic, “Overcoming Kerr-induced capacity limit in optical fiber transmission,” Science 348, 1445–1448 (2015).
[Crossref]

Leonhardt, U.

U. Leonhardt and H. Paul, “Realistic optical homodyne measurements and quasiprobability distributions,” Phys. Rev. A 48, 4598–4604 (1993).
[Crossref]

Leuchs, G.

U. L. Andersen, T. Gehring, C. Marquardt, and G. Leuchs, “30 years of squeezed light generation,” Phys. Scripta 91, 053001 (2016).
[Crossref]

J. F. Corney, J. Heersink, R. Dong, V. Josse, P. D. Drummond, G. Leuchs, and U. L. Andersen, “Simulations and experiments on polarization squeezing in optical fiber,” Phys. Rev. A 78, 023831 (2008).
[Crossref]

S. Schmitt, J. Ficker, M. Wolff, F. König, A. Sizmann, and G. Leuchs, “Photon-number squeezed solitons from an asymmetric fiber-optic Sagnac interferometer,” Phys. Rev. Lett. 81, 2446–2449 (1998).
[Crossref]

Littlewood, P.

L. Wegener, M. Povinelli, A. Green, P. Mitra, J. Stark, and P. Littlewood, “The effect of propagation nonlinearities on the information capacity of WDM optical fiber systems: cross-phase modulation and four-wave mixing,” Physica D 189, 81–99 (2004).
[Crossref]

Liu, L.

E. Temprana, E. Myslivets, B.-P. Kuo, L. Liu, V. Ataie, N. Alic, and S. Radic, “Overcoming Kerr-induced capacity limit in optical fiber transmission,” Science 348, 1445–1448 (2015).
[Crossref]

Lvovsky, A. I.

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

Mandarino, A.

A. Mandarino, M. Bina, C. Porto, S. Cialdi, S. Olivares, and M. G. A. Paris, “Assessing the significance of fidelity as a figure of merit in quantum state reconstruction of discrete and continuous-variable systems,” Phys. Rev. A 93, 062118 (2016).
[Crossref]

M. Bina, A. Mandarino, S. Olivares, and M. G. A. Paris, “Drawbacks of the use of fidelity to assess quantum resources,” Phys. Rev. A 89, 012305 (2014).
[Crossref]

Marquardt, C.

U. L. Andersen, T. Gehring, C. Marquardt, and G. Leuchs, “30 years of squeezed light generation,” Phys. Scripta 91, 053001 (2016).
[Crossref]

Martinez, A.

Milburn, G. J.

D. J. Daniel and G. J. Milburn, “Destruction of quantum coherence in a nonlinear oscillator via attenuation and amplification,” Phys. Rev. A 39, 4628–4640 (1989).
[Crossref]

G. J. Milburn and C. A. Holmes, “Dissipative quantum and classical Liouville mechanics of the anharmonic oscillator,” Phys. Rev. Lett. 56, 2237–2240 (1986).
[Crossref]

Mitra, P.

L. Wegener, M. Povinelli, A. Green, P. Mitra, J. Stark, and P. Littlewood, “The effect of propagation nonlinearities on the information capacity of WDM optical fiber systems: cross-phase modulation and four-wave mixing,” Physica D 189, 81–99 (2004).
[Crossref]

Mitra, P. P.

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature 411, 1027–1030 (2001).
[Crossref]

Mollenauer, L. F.

Mostowski, J.

M. G. Raymer and J. Mostowski, “Stimulated Raman scattering: unified treatment of spontaneous initiation and spatial propagation,” Phys. Rev. A 24, 1980–1993 (1981).
[Crossref]

Myslivets, E.

E. Temprana, E. Myslivets, B.-P. Kuo, L. Liu, V. Ataie, N. Alic, and S. Radic, “Overcoming Kerr-induced capacity limit in optical fiber transmission,” Science 348, 1445–1448 (2015).
[Crossref]

Olivares, S.

A. Mandarino, M. Bina, C. Porto, S. Cialdi, S. Olivares, and M. G. A. Paris, “Assessing the significance of fidelity as a figure of merit in quantum state reconstruction of discrete and continuous-variable systems,” Phys. Rev. A 93, 062118 (2016).
[Crossref]

M. Bina, A. Mandarino, S. Olivares, and M. G. A. Paris, “Drawbacks of the use of fidelity to assess quantum resources,” Phys. Rev. A 89, 012305 (2014).
[Crossref]

Paris, M. G. A.

A. Mandarino, M. Bina, C. Porto, S. Cialdi, S. Olivares, and M. G. A. Paris, “Assessing the significance of fidelity as a figure of merit in quantum state reconstruction of discrete and continuous-variable systems,” Phys. Rev. A 93, 062118 (2016).
[Crossref]

M. Bina, A. Mandarino, S. Olivares, and M. G. A. Paris, “Drawbacks of the use of fidelity to assess quantum resources,” Phys. Rev. A 89, 012305 (2014).
[Crossref]

C. Benedetti, A. P. Shurupov, M. G. A. Paris, G. Brida, and M. Genovese, “Experimental estimation of quantum discord for a polarization qubit and the use of fidelity to assess quantum correlations,” Phys. Rev. A 87, 052136 (2013).
[Crossref]

M. G. A. Paris, “Generation of mesoscopic quantum superpositions through Kerr-stimulated degenerate downconversion,” J. Opt. B 1, 662–667 (1999).
[Crossref]

Paul, H.

U. Leonhardt and H. Paul, “Realistic optical homodyne measurements and quasiprobability distributions,” Phys. Rev. A 48, 4598–4604 (1993).
[Crossref]

Paz, J. P.

J. R. Anglin, J. P. Paz, and W. H. Zurek, “Deconstructing decoherence,” Phys. Rev. A 55, 4041–4053 (1997).
[Crossref]

Porto, C.

A. Mandarino, M. Bina, C. Porto, S. Cialdi, S. Olivares, and M. G. A. Paris, “Assessing the significance of fidelity as a figure of merit in quantum state reconstruction of discrete and continuous-variable systems,” Phys. Rev. A 93, 062118 (2016).
[Crossref]

Porzio, A.

Povinelli, M.

L. Wegener, M. Povinelli, A. Green, P. Mitra, J. Stark, and P. Littlewood, “The effect of propagation nonlinearities on the information capacity of WDM optical fiber systems: cross-phase modulation and four-wave mixing,” Physica D 189, 81–99 (2004).
[Crossref]

Radic, S.

E. Temprana, E. Myslivets, B.-P. Kuo, L. Liu, V. Ataie, N. Alic, and S. Radic, “Overcoming Kerr-induced capacity limit in optical fiber transmission,” Science 348, 1445–1448 (2015).
[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, 063819 (2006).
[Crossref]

Raymer, M. G.

C.-C. Cheng and M. G. Raymer, “Propagation of transverse optical coherence in random multiple-scattering media,” Phys. Rev. A 62, 023811 (2000).
[Crossref]

M. G. Raymer and J. Mostowski, “Stimulated Raman scattering: unified treatment of spontaneous initiation and spatial propagation,” Phys. Rev. A 24, 1980–1993 (1981).
[Crossref]

Rosenbluh, M.

M. Rosenbluh and R. M. Shelby, “Squeezed optical solitons,” Phys. Rev. Lett. 66, 153–156 (1991).
[Crossref]

Schmitt, S.

S. Schmitt, J. Ficker, M. Wolff, F. König, A. Sizmann, and G. Leuchs, “Photon-number squeezed solitons from an asymmetric fiber-optic Sagnac interferometer,” Phys. Rev. Lett. 81, 2446–2449 (1998).
[Crossref]

Schumacher, B.

B. Schumacher and M. D. Westmoreland, “Sending classical information via noisy quantum channels,” Phys. Rev. A 56, 131–138 (1997).
[Crossref]

Sharping, J. E.

Shaw, W. T.

Shelby, R. M.

M. Rosenbluh and R. M. Shelby, “Squeezed optical solitons,” Phys. Rev. Lett. 66, 153–156 (1991).
[Crossref]

Shurupov, A. P.

C. Benedetti, A. P. Shurupov, M. G. A. Paris, G. Brida, and M. Genovese, “Experimental estimation of quantum discord for a polarization qubit and the use of fidelity to assess quantum correlations,” Phys. Rev. A 87, 052136 (2013).
[Crossref]

Sizmann, A.

S. Schmitt, J. Ficker, M. Wolff, F. König, A. Sizmann, and G. Leuchs, “Photon-number squeezed solitons from an asymmetric fiber-optic Sagnac interferometer,” Phys. Rev. Lett. 81, 2446–2449 (1998).
[Crossref]

Stark, J.

L. Wegener, M. Povinelli, A. Green, P. Mitra, J. Stark, and P. Littlewood, “The effect of propagation nonlinearities on the information capacity of WDM optical fiber systems: cross-phase modulation and four-wave mixing,” Physica D 189, 81–99 (2004).
[Crossref]

Stark, J. B.

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature 411, 1027–1030 (2001).
[Crossref]

Stoler, D.

B. Yurke and D. Stoler, “Generating quantum mechanical superpositions of macroscopically distinguishable states via amplitude dispersion,” Phys. Rev. Lett. 57, 13–16 (1986).
[Crossref]

Sudarshan, E. C. G.

E. C. G. Sudarshan, “Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams,” Phys. Rev. Lett. 10, 277–279 (1963).
[Crossref]

Temprana, E.

E. Temprana, E. Myslivets, B.-P. Kuo, L. Liu, V. Ataie, N. Alic, and S. Radic, “Overcoming Kerr-induced capacity limit in optical fiber transmission,” Science 348, 1445–1448 (2015).
[Crossref]

Vogel, W.

W. Vogel and J. Grabow, “Statistics of difference events in homodyne detection,” Phys. Rev. A 47, 4227–4235 (1993).
[Crossref]

Wasilewski, W.

R. Chrapkiewicz and W. Wasilewski, “Multimode spontaneous parametric down-conversion in a lossy medium,” J. Mod. Opt. 57, 345–355 (2010).
[Crossref]

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

Wegener, L.

L. Wegener, M. Povinelli, A. Green, P. Mitra, J. Stark, and P. Littlewood, “The effect of propagation nonlinearities on the information capacity of WDM optical fiber systems: cross-phase modulation and four-wave mixing,” Physica D 189, 81–99 (2004).
[Crossref]

Werner, M. J.

M. J. Werner, “Quantum soliton generation using an interferometer,” Phys. Rev. Lett. 81, 4132–4135 (1998).
[Crossref]

Westmoreland, M. D.

B. Schumacher and M. D. Westmoreland, “Sending classical information via noisy quantum channels,” Phys. Rev. A 56, 131–138 (1997).
[Crossref]

Windeler, R. S.

Wódkiewicz, K.

K. Banaszek and K. Wódkiewicz, “Operational theory of homodyne detection,” Phys. Rev. A 55, 3117–3123 (1997).
[Crossref]

Wolff, M.

S. Schmitt, J. Ficker, M. Wolff, F. König, A. Sizmann, and G. Leuchs, “Photon-number squeezed solitons from an asymmetric fiber-optic Sagnac interferometer,” Phys. Rev. Lett. 81, 2446–2449 (1998).
[Crossref]

Yurke, B.

B. Yurke and D. Stoler, “Generating quantum mechanical superpositions of macroscopically distinguishable states via amplitude dispersion,” Phys. Rev. Lett. 57, 13–16 (1986).
[Crossref]

Zhao, J.

Zurek, W. H.

J. R. Anglin, J. P. Paz, and W. H. Zurek, “Deconstructing decoherence,” Phys. Rev. A 55, 4041–4053 (1997).
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IEEE J. Sel. Top. Quantum Electron. (1)

J. M. Kahn and K.-P. Ho, “Spectral efficiency limits and modulation/detection techniques for DWDM systems,” IEEE J. Sel. Top. Quantum Electron. 10, 259–272 (2004).
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IEEE Trans. Inf. Theory (1)

A. S. Holevo, “The capacity of the quantum channel with general signal states,” IEEE Trans. Inf. Theory 44, 269–273 (1998).
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J. Lightwave Technol. (4)

J. Mod. Opt. (1)

R. Chrapkiewicz and W. Wasilewski, “Multimode spontaneous parametric down-conversion in a lossy medium,” J. Mod. Opt. 57, 345–355 (2010).
[Crossref]

J. Opt. B (1)

M. G. A. Paris, “Generation of mesoscopic quantum superpositions through Kerr-stimulated degenerate downconversion,” J. Opt. B 1, 662–667 (1999).
[Crossref]

J. Opt. Soc. Am. B (1)

Nat. Photonics (1)

V. Giovannetti, R. García-Patrón, N. J. Cerf, and A. S. Holevo, “Ultimate classical communication rates of quantum optical channels,” Nat. Photonics 8, 796–800 (2014).
[Crossref]

Nature (1)

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature 411, 1027–1030 (2001).
[Crossref]

Opt. Lett. (2)

Phys. Rev. (1)

R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).
[Crossref]

Phys. Rev. A (13)

J. R. Anglin, J. P. Paz, and W. H. Zurek, “Deconstructing decoherence,” Phys. Rev. A 55, 4041–4053 (1997).
[Crossref]

C.-C. Cheng and M. G. Raymer, “Propagation of transverse optical coherence in random multiple-scattering media,” Phys. Rev. A 62, 023811 (2000).
[Crossref]

W. Vogel and J. Grabow, “Statistics of difference events in homodyne detection,” Phys. Rev. A 47, 4227–4235 (1993).
[Crossref]

U. Leonhardt and H. Paul, “Realistic optical homodyne measurements and quasiprobability distributions,” Phys. Rev. A 48, 4598–4604 (1993).
[Crossref]

K. Banaszek and K. Wódkiewicz, “Operational theory of homodyne detection,” Phys. Rev. A 55, 3117–3123 (1997).
[Crossref]

J. F. Corney, J. Heersink, R. Dong, V. Josse, P. D. Drummond, G. Leuchs, and U. L. Andersen, “Simulations and experiments on polarization squeezing in optical fiber,” Phys. Rev. A 78, 023831 (2008).
[Crossref]

M. G. Raymer and J. Mostowski, “Stimulated Raman scattering: unified treatment of spontaneous initiation and spatial propagation,” Phys. Rev. A 24, 1980–1993 (1981).
[Crossref]

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

C. Benedetti, A. P. Shurupov, M. G. A. Paris, G. Brida, and M. Genovese, “Experimental estimation of quantum discord for a polarization qubit and the use of fidelity to assess quantum correlations,” Phys. Rev. A 87, 052136 (2013).
[Crossref]

M. Bina, A. Mandarino, S. Olivares, and M. G. A. Paris, “Drawbacks of the use of fidelity to assess quantum resources,” Phys. Rev. A 89, 012305 (2014).
[Crossref]

A. Mandarino, M. Bina, C. Porto, S. Cialdi, S. Olivares, and M. G. A. Paris, “Assessing the significance of fidelity as a figure of merit in quantum state reconstruction of discrete and continuous-variable systems,” Phys. Rev. A 93, 062118 (2016).
[Crossref]

D. J. Daniel and G. J. Milburn, “Destruction of quantum coherence in a nonlinear oscillator via attenuation and amplification,” Phys. Rev. A 39, 4628–4640 (1989).
[Crossref]

B. Schumacher and M. D. Westmoreland, “Sending classical information via noisy quantum channels,” Phys. Rev. A 56, 131–138 (1997).
[Crossref]

Phys. Rev. Lett. (6)

G. J. Milburn and C. A. Holmes, “Dissipative quantum and classical Liouville mechanics of the anharmonic oscillator,” Phys. Rev. Lett. 56, 2237–2240 (1986).
[Crossref]

M. Rosenbluh and R. M. Shelby, “Squeezed optical solitons,” Phys. Rev. Lett. 66, 153–156 (1991).
[Crossref]

B. Yurke and D. Stoler, “Generating quantum mechanical superpositions of macroscopically distinguishable states via amplitude dispersion,” Phys. Rev. Lett. 57, 13–16 (1986).
[Crossref]

S. Schmitt, J. Ficker, M. Wolff, F. König, A. Sizmann, and G. Leuchs, “Photon-number squeezed solitons from an asymmetric fiber-optic Sagnac interferometer,” Phys. Rev. Lett. 81, 2446–2449 (1998).
[Crossref]

M. J. Werner, “Quantum soliton generation using an interferometer,” Phys. Rev. Lett. 81, 4132–4135 (1998).
[Crossref]

E. C. G. Sudarshan, “Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams,” Phys. Rev. Lett. 10, 277–279 (1963).
[Crossref]

Phys. Scripta (1)

U. L. Andersen, T. Gehring, C. Marquardt, and G. Leuchs, “30 years of squeezed light generation,” Phys. Scripta 91, 053001 (2016).
[Crossref]

Physica D (1)

L. Wegener, M. Povinelli, A. Green, P. Mitra, J. Stark, and P. Littlewood, “The effect of propagation nonlinearities on the information capacity of WDM optical fiber systems: cross-phase modulation and four-wave mixing,” Physica D 189, 81–99 (2004).
[Crossref]

Problemy Peredachi Informatsii (1)

A. Holevo, “Some estimates of information transmitted through quantum communication channel,” Problemy Peredachi Informatsii 9, 3–11 (1973).

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V. Bužek and P. L. Knight, “Quantum interference, superposition states of light, and nonclassical effects,” Prog. Opt. 34, 1–158 (1995).
[Crossref]

Science (1)

E. Temprana, E. Myslivets, B.-P. Kuo, L. Liu, V. Ataie, N. Alic, and S. Radic, “Overcoming Kerr-induced capacity limit in optical fiber transmission,” Science 348, 1445–1448 (2015).
[Crossref]

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Figures (6)

Fig. 1.
Fig. 1. Decomposition of noisy propagation of a coherent light pulse in a lossy Kerr medium into a sequence of three processes. Process 1 is linear attenuation of the input complex amplitude ζ 0 by a factor τ , where τ is the power transmission of the medium. Process 2 is nonlinear phase noise, which shifts the phase by ϕ 0 and introduces Gaussian phase diffusion characterized by variance σ 2 . Finally, process 3 is unitary Kerr evolution, which would have occurred in the absence of attenuation, applied to the output of process 2.
Fig. 2.
Fig. 2. Husimi Q function of the output state emerging from a lossy nonlinear medium with transmission τ = 10 8 and nonlinearity ϰ = 5 × 10 6 , shown for the output mean photon number (a)  τ n ¯ = 1 , (b)  τ n ¯ = 3 , (c)  τ n ¯ = 15 , and (d)  τ n ¯ = 60 . A coherent state with a real amplitude n ¯ has been assumed at the input. Nonlinear phase noise is clearly seen to increase with the pulse intensity.
Fig. 3.
Fig. 3. Real and the imaginary parts of the function f τ ( ϰ ) defined in Eq. (7) for a transmission of (a)  τ = 10 8 and (b)  τ = 0.8 .
Fig. 4.
Fig. 4. Infidelity 1 F between the actual density matrix ϱ ^ and its approximation ϱ ^ G based on the Gaussian phase diffusion model for the transmission τ = 0.8 as a function of the output mean photon number τ n ¯ and the dimensionless nonlinearity ϰ .
Fig. 5.
Fig. 5. (a) Continuous phase shift keying. Information is encoded in the phase ϕ of a constellation of coherent states with constant intensity uniformly distributed on a circle. (b) Holevo quantity for the continuous phase shift keying constellation as a function of the output mean photon number τ n ¯ for several values of the nonlinearity ϰ .
Fig. 6.
Fig. 6. Attainable squeezing ( Δ x ^ θ ) 2 for the input mean photon number n ¯ = 10 8 and several different values of the transmission τ as a function of the dimensionless nonlinear coefficient ϰ . For convenience, the abscissa is parameterized with sinh    r = 2 ϰ ( log τ ) τ n ¯ . Dashed horizontal lines correspond to the estimate ( 1 τ ) / 3 .

Tables (1)

Tables Icon

Table 1. Exemplary Combinations of Optical Fiber and Pulse Parameters that Give the Value of the Dimensionless Nonlinearity Parameter ϰ = 5 × 10 6

Equations (36)

Equations on this page are rendered with MathJax. Learn more.

d ϱ ^ d z = i μ [ n ^ 2 , ϱ ^ ( z ) ] + α 2 ( 2 a ^ ϱ ^ ( z ) a ^ n ^ ϱ ^ ( z ) ϱ ^ ( z ) n ^ ) ,
ϱ ^ ( z ) = exp ( i μ z n ^ 2 ) ϱ ^ ( z ) exp ( i μ z n ^ 2 ) .
d ϱ ^ d z = α 2 ( 2 a ^ ( z ) ϱ ^ ( z ) [ a ^ ( z ) ] n ^ ϱ ^ ( z ) ϱ ^ ( z ) n ^ ) ,
| ζ 0 = e n ¯ / 2 n = 0 ζ 0 n n ! | n .
ϱ mn ( z ) = ζ 0 m ( ζ 0 * ) n m ! n ! e ( m + n ) α z / 2 c m n ( z ) .
ϱ m n ( z ) = m | τ ζ 0 τ ζ 0 | n exp [ n ¯ f τ ( ( m n ) ϰ ) ] .
f τ ( ϰ ) = 1 τ 1 τ 1 2 i ϰ 1 2 i ϰ .
f τ ( ϰ ) 2 i ( 1 τ + τ log τ ) ϰ + [ 4 2 τ 2 τ ( 1 log τ ) 2 ] ϰ 2 .
m , n = 0 | m n | m | ζ ζ | n exp [ i ( m n ) ϕ 0 ( m n ) 2 σ 2 2 ] = d ϕ 1 2 π σ 2 e ( ϕ ϕ 0 ) 2 / 2 σ 2 | e i ϕ ζ e i ϕ ζ | .
ϕ 0 = 2 ϰ n ¯ ( 1 τ + τ log τ ) ,
σ 2 = 4 ϰ 2 n ¯ [ 2 τ τ ( 1 log τ ) 2 ] .
χ = S ( 0 2 π d ϕ p ϕ ϱ ^ ϕ ) 0 2 π d ϕ p ϕ S ( ϱ ^ ϕ ) .
ϱ ^ G = d ϕ 1 2 π σ 2 e ϕ 2 / 2 σ 2 | e i ϕ ζ e i ϕ ζ | .
d a ^ d z = i μ [ n ^ 2 , a ^ ( z ) ] = i μ [ 2 a ^ ( z ) a ^ ( z ) + 1 ] a ^ ( z ) .
a ^ ( z ) = e i μ z ( 2 ζ 2 + 1 ) e i ϕ [ ζ + b ^ ( z ) ] ,
d b ^ d z = 2 i μ ζ 2 [ b ^ ( z ) + b ^ ( z ) ] ,
b ^ ( z ) = ( 1 + 2 i μ z ζ 2 ) b ^ ( 0 ) + 2 i μ z ζ 2 b ^ ( 0 ) .
q ^ = e i μ z ( 2 ζ 2 + 1 ) a ^ ( z ) + h.c. = e i ϕ [ ζ + b ^ ( z ) ] + h.c. ,
p ^ = i e i μ z ( 2 ζ 2 + 1 ) a ^ ( z ) + h.c. = i e i ϕ [ ζ + b ^ ( z ) ] + h.c. ,
q ^ = 2 ζ cos ϕ , p ^ = 2 ζ sin ϕ
q ^ 2 = 1 + 4 ζ 2 cos 2 ϕ 4 μ z ζ 2 sin 2 ϕ + ( 4 μ z ζ 2 ) 2 sin 2 ϕ ,
p ^ 2 = 1 + 4 ζ 2 sin 2 ϕ + 4 μ z ζ 2 sin 2 ϕ + ( 4 μ z ζ 2 ) 2 cos 2 ϕ
1 2 ( q ^ p ^ + p ^ q ^ ) = 2 ζ 2 sin 2 ϕ + 4 μ z ζ 2 cos 2 ϕ 8 ( μ t ζ 2 ) 2 sin 2 ϕ .
cos ϕ = e σ 2 / 2 , cos 2 ϕ = e 2 σ 2 .
( Δ q ^ ) 2 = 1 + 2 ( 1 e σ 2 ) 2 ζ 2 + 2 ( 1 e 2 σ 2 ) sinh 2 r ,
( Δ p ^ ) 2 = 1 + 2 ( 1 e 2 σ 2 ) ζ 2 + 2 ( 1 + e 2 σ 2 ) sinh 2 r ,
1 2 ( Δ q ^ Δ p ^ + Δ p ^ Δ q ^ ) = 2 e 2 σ 2 sinh    r .
tan 2 θ = sinh    r sinh 2 r + ζ 2 ( e σ 2 1 ) ,
( Δ x ^ θ ) 2 = 1 + 2 ζ 2 ( 1 e σ 2 ) + 2 sinh 2 r 2 e 2 σ 2 sinh 2 r + [ sinh 2 r + ζ 2 ( e σ 2 1 ) ] 2 .
g ( τ ) = 2 τ τ ( 1 log τ ) 2 τ ( log τ ) 2 1 3 ( 1 τ ) ,
( Δ x ^ θ 0 ) 2 = e 2 r + ( Δ p ) exc 2 sin 2 θ 0 = e 2 r + g ( τ ) ( 1 e 2 r ) tanh r .
d ϱ ^ d z = α 2 ( 2 a ^ ( z ) ϱ ^ ( z ) [ a ^ ( z ) ] n ^ ϱ ^ ( z ) ϱ ^ ( z ) n ^ ) ,
d ϱ m , n d z = α 2 [ 2 ρ m + 1 , n + 1 ( z ) ( m + 1 ) ( n + 1 ) e 2 i μ z ( m n ) ρ m , n ( z ) ( m + n ) ] .
d c j d z = α n ¯ e α z e 2 i μ z j c j ( z ) .
c j ( z ) = exp ( n ¯ α α 2 i μ j e α z e 2 i μ z j + A 0 ) = exp ( n ¯ τ 1 2 i ϰ j τ 2 i ϰ j + A 0 ) ,
ϱ m n ( z ) = ( τ ζ 0 ) m ( τ ζ 0 * ) n m ! n ! exp ( n ¯ τ 1 2 i ϰ ( m n ) 1 2 i ϰ ( m n ) + A 0 ) .

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