Stokes-space analysis of modal dispersion in fibers with multiple mode transmission

Cristian Antonelli; Antonio Mecozzi; Mark Shtaif; Peter J. Winzer

doi:10.1364/OE.20.011718

Stokes-space analysis of modal dispersion in fibers with multiple mode transmission

Cristian Antonelli, Antonio Mecozzi, Mark Shtaif, and Peter J. Winzer

Optics Express
Vol. 20,
Issue 11,
pp. 11718-11733
(2012)
•https://doi.org/10.1364/OE.20.011718

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Cristian Antonelli, Antonio Mecozzi, Mark Shtaif, and Peter J. Winzer, "Stokes-space analysis of modal dispersion in fibers with multiple mode transmission," Opt. Express 20, 11718-11733 (2012)

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Related Topics
Table of Contents Category
- Fiber Optics and Optical Communications
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Fiber optics communications (060.2330)
- Multiplexing (060.4230)
- Optical communications (060.4510)

About this Article
History
- Original Manuscript: February 2, 2012
- Revised Manuscript: March 28, 2012
- Manuscript Accepted: March 29, 2012
- Published: May 9, 2012

Accessible

Open Access

Abstract

Modal dispersion (MD) in a multimode fiber may be considered as a generalized form of polarization mode dispersion (PMD) in single mode fibers. Using this analogy, we extend the formalism developed for PMD to characterize MD in fibers with multiple spatial modes. We introduce a MD vector defined in a D-dimensional extended Stokes space whose square length is the sum of the square group delays of the generalized principal states. For strong mode coupling, the MD vector undertakes a D-dimensional isotropic random walk, so that the distribution of its length is a chi distribution with D degrees of freedom. We also characterize the largest differential group delay, that is the difference between the delays of the fastest and the slowest principal states, and show that it too is very well approximated by a chi distribution, although in general with a smaller number of degrees of freedom. Finally, we study the spectral properties of MD in terms of the frequency autocorrelation functions of the MD vector, of the square modulus of the MD vector, and of the largest differential group delay. The analytical results are supported by extensive numerical simulations.

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References

View by:
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|
Year
|
Author
|
Publication

A. R. Chraplyvy, “The coming capacity crunch,” European Conference on Optical Communication 2009 (ECOC09), plenary talk (2009).
R. W. Tkach, “Scaling optical communications for the next decade and beyond,” Bell Labs Tech. J. 14, 3–9 (2010).
[Crossref]
Peter J. Winzer and Gerard J. Foschini, “MIMO capacities and outage probabilities in spatially multiplexed optical transport systems,” Opt. Express 19, 16680–16696 (2011).
[Crossref] [PubMed]
S. Chandrasekhar, A. H. Gnauck, X. Liu, P. J. Winzer, Y. Pan, E. C. Burrows, B. Zhu, T. F. Taunay, M. Fishteyn, M. F. Yan, J. M. Fini, E. M. Monberg, and F. V. Dimarcello, “WDM/SDM transmission of 10 × 128-Gb/s PDM-QPSK over 2688-km 7-core fiber with a per-fiber net aggregate spectral-efficiency distance product of 40,320 km×b/s/Hz,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Th.13.C4 (2011).
S. Randel, R. Ryf, A. Sierra, P.J. Winzer, A.H. Gnauck, C.A. Bolle, R-J. Essiambre, D.W. Peckham, A. McCurdy, and R. Lingle “6×56-Gb/s mode-division multiplexed transmission over 33-km few-mode fiber enabled by 6×6 MIMO equalization,” Opt”. Express 19, 16697–16707 (2011).
[Crossref] [PubMed]
E. Ip, N. Bai, Y.K. Huang, E. Mateo, F. Yaman, S. Bickham, H.Y. Tam, C. Lu, M.J. Li, S. Ten, A.P. Tao Lau, V. Tse, G.D. Peng, C. Montero, X. Prieto, and G. Li, “88 × 3 × 112-Gb/s WDM transmission over 50 km of three-mode fiber with inline few-mode fiber amplifier,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Th.13.C2 (2011).
S. Fan and J. M. Kahn, “Principal modes in multi-mode waveguides”, Opt. Lett. 30, 135–137 (2005).
[Crossref] [PubMed]
K-P. Ho and J. M Kahn, “Statistics of group delays in multi-mode fibers with strong mode coupling,” J. Lightwave Technol. 29, 3119–3128 (2011).
[Crossref]
K-P. Ho and J. M Kahn, “Mode-dependent loss and gain: statistics and effect on mode-division multiplexing,” Opt. Express 19, 16612–16635 (2011).
[Crossref] [PubMed]
J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA 97, 4541–4550 (2000).
[Crossref] [PubMed]
N. Gisin, “Statistics of polarization dependent losses,” Opt. Commun. 114, 399–405 (1995).
[Crossref]
D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]
M. Nazarathy and E. Simony, “Generalized Stokes parameters shift keying approach to multichip differential phase encoded optical modulation formats,” Opt. Lett. 31, 435–437 (2006).
[Crossref] [PubMed]
M. Salsi, C. Koebele, D. Sperti, P. Tran, H. Mardoyan, P. Brindel, S. Bigo, A. Boutin, F. Verluise, P. Sillard, M. Astruc, L. Provost, and G. Charlet, “Mode division multiplexing of 2 × 100Gb/s channels using an LCOS based spatial modulator,” J. Lightwave Technol. 30, 618–623 (2012).
[Crossref]
S. Randel, M. Magarini, R. Ryf, R-J. Essiambre, A. H. Gnauck, P. J. Winzer, T. Hayashi, T. Taru, and T. Sasaki “MIMO-based signal processing of spatially multiplexed 112-Gb/s PDM-QPSK signals using strongly-coupled 3-core fiber,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Tu.5.B1 (2011).
D. Cassioli, M. Z. Win, and A. F. Molisch, “The ultra-wide bandwidth indoor channel: from statistical model to simulations,” IEEE J. Sel. Areas Commun. 20, 1247–1257 (2002).
[Crossref]
In the literature the shape parameter KN is typically referred to as the number of degrees of freedom of the chi distribution. We refrain from this terminology so as to avoid confusion with the number of degrees of freedom Deff representing the dimension of the manyfold spanned by the MD vector in the generalized Stokes space.
M. Karlsson and J. Brentel, “The autocorrelation function of the polarization-mode dispersion vector,” Opt. Lett. 24, 939–941 (1999).
[Crossref]
M. Shtaif and A. Mecozzi, “Study of the frequency autocorrelation of the differential group delay in fibers with polarization mode dispersion,” Opt. Lett. 25, 707–709 (2000).
[Crossref]
R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R-J. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96km of few-mode fiber using coherent 6 × 6 MIMO processing,” J. Lightwave Technol. 30, 521–531 (2012).
[Crossref]
M. Gell-Mann, “Symmetries of baryons and mesons,” Phys. Rev. 125, 1067–1084 (1962).
[Crossref]
A. Galtarossa, L. Palmieri, M. Schiano, and T. Tambosso, “Measurement of birefringence correlation length in long, single-mode fibers,” Opt. Lett. 26, 962–964 (2001).
[Crossref]
F. Curti, B. Daino, G. De Marchis, and F. Matera, “Statistical treatment of the evolution of the principal states of polarization in single-mode fibers,” J. Lightwave Technol. 8, 1162–1166 (1990).
[Crossref]
The more common terminology for the vector operation defined in Eq. (28) is Lie bracket. We prefer the term generalized cross-product for emphasizing the analogy with the case of PMD.

2012 (2)

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R-J. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96km of few-mode fiber using coherent 6 × 6 MIMO processing,” J. Lightwave Technol. 30, 521–531 (2012).
[Crossref]

M. Salsi, C. Koebele, D. Sperti, P. Tran, H. Mardoyan, P. Brindel, S. Bigo, A. Boutin, F. Verluise, P. Sillard, M. Astruc, L. Provost, and G. Charlet, “Mode division multiplexing of 2 × 100Gb/s channels using an LCOS based spatial modulator,” J. Lightwave Technol. 30, 618–623 (2012).
[Crossref]

2011 (4)

K-P. Ho and J. M Kahn, “Mode-dependent loss and gain: statistics and effect on mode-division multiplexing,” Opt. Express 19, 16612–16635 (2011).
[Crossref] [PubMed]

Peter J. Winzer and Gerard J. Foschini, “MIMO capacities and outage probabilities in spatially multiplexed optical transport systems,” Opt. Express 19, 16680–16696 (2011).
[Crossref] [PubMed]

K-P. Ho and J. M Kahn, “Statistics of group delays in multi-mode fibers with strong mode coupling,” J. Lightwave Technol. 29, 3119–3128 (2011).
[Crossref]

S. Randel, R. Ryf, A. Sierra, P.J. Winzer, A.H. Gnauck, C.A. Bolle, R-J. Essiambre, D.W. Peckham, A. McCurdy, and R. Lingle “6×56-Gb/s mode-division multiplexed transmission over 33-km few-mode fiber enabled by 6×6 MIMO equalization,” Opt”. Express 19, 16697–16707 (2011).
[Crossref] [PubMed]

2010 (1)

R. W. Tkach, “Scaling optical communications for the next decade and beyond,” Bell Labs Tech. J. 14, 3–9 (2010).
[Crossref]

2006 (1)

M. Nazarathy and E. Simony, “Generalized Stokes parameters shift keying approach to multichip differential phase encoded optical modulation formats,” Opt. Lett. 31, 435–437 (2006).
[Crossref] [PubMed]

2005 (1)

S. Fan and J. M. Kahn, “Principal modes in multi-mode waveguides”, Opt. Lett. 30, 135–137 (2005).
[Crossref] [PubMed]

2002 (1)

D. Cassioli, M. Z. Win, and A. F. Molisch, “The ultra-wide bandwidth indoor channel: from statistical model to simulations,” IEEE J. Sel. Areas Commun. 20, 1247–1257 (2002).
[Crossref]

2001 (2)

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]

A. Galtarossa, L. Palmieri, M. Schiano, and T. Tambosso, “Measurement of birefringence correlation length in long, single-mode fibers,” Opt. Lett. 26, 962–964 (2001).
[Crossref]

2000 (2)

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA 97, 4541–4550 (2000).
[Crossref] [PubMed]

M. Shtaif and A. Mecozzi, “Study of the frequency autocorrelation of the differential group delay in fibers with polarization mode dispersion,” Opt. Lett. 25, 707–709 (2000).
[Crossref]

1999 (1)

M. Karlsson and J. Brentel, “The autocorrelation function of the polarization-mode dispersion vector,” Opt. Lett. 24, 939–941 (1999).
[Crossref]

1995 (1)

N. Gisin, “Statistics of polarization dependent losses,” Opt. Commun. 114, 399–405 (1995).
[Crossref]

1990 (1)

F. Curti, B. Daino, G. De Marchis, and F. Matera, “Statistical treatment of the evolution of the principal states of polarization in single-mode fibers,” J. Lightwave Technol. 8, 1162–1166 (1990).
[Crossref]

1962 (1)

M. Gell-Mann, “Symmetries of baryons and mesons,” Phys. Rev. 125, 1067–1084 (1962).
[Crossref]

Astruc, M.

Bai, N.

E. Ip, N. Bai, Y.K. Huang, E. Mateo, F. Yaman, S. Bickham, H.Y. Tam, C. Lu, M.J. Li, S. Ten, A.P. Tao Lau, V. Tse, G.D. Peng, C. Montero, X. Prieto, and G. Li, “88 × 3 × 112-Gb/s WDM transmission over 50 km of three-mode fiber with inline few-mode fiber amplifier,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Th.13.C2 (2011).

Bickham, S.

Bigo, S.

Bolle, C.

Bolle, C.A.

Boutin, A.

Brentel, J.

M. Karlsson and J. Brentel, “The autocorrelation function of the polarization-mode dispersion vector,” Opt. Lett. 24, 939–941 (1999).
[Crossref]

Brindel, P.

Burrows, E. C.

S. Chandrasekhar, A. H. Gnauck, X. Liu, P. J. Winzer, Y. Pan, E. C. Burrows, B. Zhu, T. F. Taunay, M. Fishteyn, M. F. Yan, J. M. Fini, E. M. Monberg, and F. V. Dimarcello, “WDM/SDM transmission of 10 × 128-Gb/s PDM-QPSK over 2688-km 7-core fiber with a per-fiber net aggregate spectral-efficiency distance product of 40,320 km×b/s/Hz,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Th.13.C4 (2011).

Cassioli, D.

D. Cassioli, M. Z. Win, and A. F. Molisch, “The ultra-wide bandwidth indoor channel: from statistical model to simulations,” IEEE J. Sel. Areas Commun. 20, 1247–1257 (2002).
[Crossref]

Chandrasekhar, S.

Charlet, G.

Chraplyvy, A. R.

A. R. Chraplyvy, “The coming capacity crunch,” European Conference on Optical Communication 2009 (ECOC09), plenary talk (2009).

Curti, F.

Daino, B.

De Marchis, G.

Dimarcello, F. V.

Esmaeelpour, M.

Essiambre, R-J.

S. Randel, M. Magarini, R. Ryf, R-J. Essiambre, A. H. Gnauck, P. J. Winzer, T. Hayashi, T. Taru, and T. Sasaki “MIMO-based signal processing of spatially multiplexed 112-Gb/s PDM-QPSK signals using strongly-coupled 3-core fiber,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Tu.5.B1 (2011).

Fan, S.

S. Fan and J. M. Kahn, “Principal modes in multi-mode waveguides”, Opt. Lett. 30, 135–137 (2005).
[Crossref] [PubMed]

Fini, J. M.

Fishteyn, M.

Foschini, Gerard J.

Peter J. Winzer and Gerard J. Foschini, “MIMO capacities and outage probabilities in spatially multiplexed optical transport systems,” Opt. Express 19, 16680–16696 (2011).
[Crossref] [PubMed]

Galtarossa, A.

A. Galtarossa, L. Palmieri, M. Schiano, and T. Tambosso, “Measurement of birefringence correlation length in long, single-mode fibers,” Opt. Lett. 26, 962–964 (2001).
[Crossref]

Gell-Mann, M.

M. Gell-Mann, “Symmetries of baryons and mesons,” Phys. Rev. 125, 1067–1084 (1962).
[Crossref]

Gisin, N.

N. Gisin, “Statistics of polarization dependent losses,” Opt. Commun. 114, 399–405 (1995).
[Crossref]

Gnauck, A. H.

Gnauck, A.H.

Gordon, J. P.

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA 97, 4541–4550 (2000).
[Crossref] [PubMed]

Hayashi, T.

Ho, K-P.

K-P. Ho and J. M Kahn, “Mode-dependent loss and gain: statistics and effect on mode-division multiplexing,” Opt. Express 19, 16612–16635 (2011).
[Crossref] [PubMed]

K-P. Ho and J. M Kahn, “Statistics of group delays in multi-mode fibers with strong mode coupling,” J. Lightwave Technol. 29, 3119–3128 (2011).
[Crossref]

Huang, Y.K.

Ip, E.

James, D. F. V.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]

Koebele, C.

Kogelnik, H.

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA 97, 4541–4550 (2000).
[Crossref] [PubMed]

Kwiat, P. G.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]

Li, G.

Li, M.J.

Lingle, R.

Liu, X.

Lu, C.

Magarini, M.

Mardoyan, H.

Mateo, E.

Matera, F.

McCurdy, A.

McCurdy, A. H.

Mecozzi, A.

M. Shtaif and A. Mecozzi, “Study of the frequency autocorrelation of the differential group delay in fibers with polarization mode dispersion,” Opt. Lett. 25, 707–709 (2000).
[Crossref]

Molisch, A. F.

D. Cassioli, M. Z. Win, and A. F. Molisch, “The ultra-wide bandwidth indoor channel: from statistical model to simulations,” IEEE J. Sel. Areas Commun. 20, 1247–1257 (2002).
[Crossref]

Monberg, E. M.

Montero, C.

Mumtaz, S.

Munro, W. J.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]

Nazarathy, M.

Palmieri, L.

A. Galtarossa, L. Palmieri, M. Schiano, and T. Tambosso, “Measurement of birefringence correlation length in long, single-mode fibers,” Opt. Lett. 26, 962–964 (2001).
[Crossref]

Pan, Y.

Peckham, D. W.

Peckham, D.W.

Peng, G.D.

Prieto, X.

Provost, L.

Randel, S.

Ryf, R.

Salsi, M.

Sasaki, T.

Schiano, M.

A. Galtarossa, L. Palmieri, M. Schiano, and T. Tambosso, “Measurement of birefringence correlation length in long, single-mode fibers,” Opt. Lett. 26, 962–964 (2001).
[Crossref]

Shtaif, M.

M. Shtaif and A. Mecozzi, “Study of the frequency autocorrelation of the differential group delay in fibers with polarization mode dispersion,” Opt. Lett. 25, 707–709 (2000).
[Crossref]

Sierra, A.

Sillard, P.

Simony, E.

Sperti, D.

Tam, H.Y.

Tambosso, T.

A. Galtarossa, L. Palmieri, M. Schiano, and T. Tambosso, “Measurement of birefringence correlation length in long, single-mode fibers,” Opt. Lett. 26, 962–964 (2001).
[Crossref]

Tao Lau, A.P.

Taru, T.

Taunay, T. F.

Ten, S.

Tkach, R. W.

R. W. Tkach, “Scaling optical communications for the next decade and beyond,” Bell Labs Tech. J. 14, 3–9 (2010).
[Crossref]

Tran, P.

Tse, V.

Verluise, F.

White, A. G.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]

Win, M. Z.

D. Cassioli, M. Z. Win, and A. F. Molisch, “The ultra-wide bandwidth indoor channel: from statistical model to simulations,” IEEE J. Sel. Areas Commun. 20, 1247–1257 (2002).
[Crossref]

Winzer, P. J.

Winzer, P.J.

Winzer, Peter J.

Peter J. Winzer and Gerard J. Foschini, “MIMO capacities and outage probabilities in spatially multiplexed optical transport systems,” Opt. Express 19, 16680–16696 (2011).
[Crossref] [PubMed]

Yaman, F.

Yan, M. F.

Zhu, B.

Bell Labs Tech. J. (1)

R. W. Tkach, “Scaling optical communications for the next decade and beyond,” Bell Labs Tech. J. 14, 3–9 (2010).
[Crossref]

IEEE J. Sel. Areas Commun. (1)

D. Cassioli, M. Z. Win, and A. F. Molisch, “The ultra-wide bandwidth indoor channel: from statistical model to simulations,” IEEE J. Sel. Areas Commun. 20, 1247–1257 (2002).
[Crossref]

J. Lightwave Technol. (4)

K-P. Ho and J. M Kahn, “Statistics of group delays in multi-mode fibers with strong mode coupling,” J. Lightwave Technol. 29, 3119–3128 (2011).
[Crossref]

Opt. Commun. (1)

N. Gisin, “Statistics of polarization dependent losses,” Opt. Commun. 114, 399–405 (1995).
[Crossref]

Opt. Express (2)

K-P. Ho and J. M Kahn, “Mode-dependent loss and gain: statistics and effect on mode-division multiplexing,” Opt. Express 19, 16612–16635 (2011).
[Crossref] [PubMed]

Peter J. Winzer and Gerard J. Foschini, “MIMO capacities and outage probabilities in spatially multiplexed optical transport systems,” Opt. Express 19, 16680–16696 (2011).
[Crossref] [PubMed]

Opt. Lett. (5)

M. Shtaif and A. Mecozzi, “Study of the frequency autocorrelation of the differential group delay in fibers with polarization mode dispersion,” Opt. Lett. 25, 707–709 (2000).
[Crossref]

M. Karlsson and J. Brentel, “The autocorrelation function of the polarization-mode dispersion vector,” Opt. Lett. 24, 939–941 (1999).
[Crossref]

A. Galtarossa, L. Palmieri, M. Schiano, and T. Tambosso, “Measurement of birefringence correlation length in long, single-mode fibers,” Opt. Lett. 26, 962–964 (2001).
[Crossref]

S. Fan and J. M. Kahn, “Principal modes in multi-mode waveguides”, Opt. Lett. 30, 135–137 (2005).
[Crossref] [PubMed]

Opt”. Express (1)

Phys. Rev. (1)

M. Gell-Mann, “Symmetries of baryons and mesons,” Phys. Rev. 125, 1067–1084 (1962).
[Crossref]

Phys. Rev. A (1)

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]

Proc. Natl. Acad. Sci. USA (1)

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA 97, 4541–4550 (2000).
[Crossref] [PubMed]

Other (6)

In the literature the shape parameter KN is typically referred to as the number of degrees of freedom of the chi distribution. We refrain from this terminology so as to avoid confusion with the number of degrees of freedom Deff representing the dimension of the manyfold spanned by the MD vector in the generalized Stokes space.

A. R. Chraplyvy, “The coming capacity crunch,” European Conference on Optical Communication 2009 (ECOC09), plenary talk (2009).

The more common terminology for the vector operation defined in Eq. (28) is Lie bracket. We prefer the term generalized cross-product for emphasizing the analogy with the case of PMD.

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Fig. 1 Probability density function of τ for N = 2, normalized to its root-mean square value. The coupling magnitude increases from no coupling in (a) to full coupling in (d). A similar behavior has been verified for a number of larger values of N.

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Fig. 2 Top panel. The solid line represents the shape parameter K_N of the chi PDF that provides the best approximation of the distribution of the largest DGD T as a function of N; the dashed line shows the mean square value of T normalized to 〈τ²〉/N². Bottom panels. Probability density function of T for (A) N = 3, (B) N = 50 and (C) N = 100. Dots are the results of Monte-Carlo simulations, solid lines are the plot of chi PDFs with parameters taken from the top panel.

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Fig. 3 Top panels: normalized autocorrelation function of τ⃗ according to theory (solid curve) and simulations (stars). Bottom panels: the normalized autocorrelation function of τ² according to theory (solid curve) and simulations (stars) as well as the autocorrelation function of T (circles). Dot-dashed curves represent the re-scaled ACF of τ² Eq. (5), modified by replacing D with K_N as from Fig. 2.

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Fig. 4 (a) Probability density function of the length of the MD vector τ⃗_LP₁₁ that represents the four LP₁₁ modes (subspace dimension D_LP₁₁ = 15) and of the length of the MD vector τ⃗_LP₀₁ that represents the two LP₀₁ modes (subspace dimension D_LP₁₁ = 3). The degree of coupling between the two groups of modes is discussed in the text. Figures (b) and (c) show the ACFs of the the two MD vectors. Symbols refer to numerical simulations, solid lines to the theory.

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Equations (33)

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

τ^{2} = 2 N \sum_{i = 1}^{2 N} t_{i}^{2} .

K_{N} ≃ 3 + ⌈ 10.39 \times {(N - 1)}^{1.36} ⌉

〈 T^{2} 〉 ≃ \frac{{(N - 1)}^{2} + 24.7 \times (N - 1) + 16.14}{0.2532 \times {(N - 1)}^{2} + 7.401 \times (N - 1) + 16.14} 〈 τ^{2} 〉 / N^{2},

R_{\vec{τ}} (ω) = 〈 \vec{τ} (Ω + ω) \cdot \vec{τ} (Ω) 〉 = \frac{D}{ω^{2}} [1 - exp (- \frac{ω^{2} 〈 τ^{2} 〉}{D})],

R_{τ^{2}} (ω) = 〈 τ^{2} (Ω + ω) τ^{2} (Ω) 〉 = {〈 τ^{2} 〉}^{2} + \frac{4 〈 τ^{2} 〉}{ω^{2}} - \frac{4 D}{ω^{4}} [1 - exp (- \frac{ω^{2} 〈 τ^{2} 〉}{D})] .

| ψ (z) 〉 = U (z, z_{0}) | ψ (z_{0}) 〉,

\frac{d U (z, z_{0})}{d z} = i B (z) U (z, z_{0})

\frac{d U (z, z_{0})}{d ω} = i T (z, z_{0}) U (z, z_{0}),

\frac{1}{2 N} Trace {Λ_{i} Λ_{j}} = δ_{i, j},

H = \frac{1}{2 N} (η_{0} 1 + \vec{η} \cdot \vec{Λ}),

\vec{η} \cdot \vec{Λ} = \sum_{j = 1}^{4 N^{2} - 1} η_{j} Λ_{j} .

\begin{matrix} η_{i} = trace {Λ_{i} H} & and & η_{0} = trace {H} . \end{matrix}

| ψ 〉 〈 ψ | = \frac{1}{2 N} (1 + \vec{ψ} \cdot \vec{Λ}) .

ψ_{j} = trace {Λ_{j} | ψ 〉 〈 ψ |} = 〈 ψ | Λ_{j} | ψ 〉,

{| 〈 ψ | ϕ 〉 |}^{2} = \frac{1}{2 N} (1 + \vec{ψ} \cdot \vec{ϕ}),

\frac{1}{2 N} trace {(\vec{A} \cdot \vec{Λ}) (\vec{B} \cdot \vec{Λ})} = \vec{A} \cdot \vec{B},

\vec{S} \cdot {\vec{ψ}}_{i} = θ_{i},

2 N {| \vec{S} |}^{2} = \sum θ_{i}^{2} .

\frac{d U (z, z_{0})}{d z} = \frac{i}{2 N} (β_{0} 1 + \vec{β} \cdot \vec{Λ}) U (z, z_{0})

\frac{d U (z, z_{0})}{d ω} = \frac{i}{2 N} (τ_{0} 1 + \vec{τ} \cdot \vec{Λ}) U (z, z_{0}),

\vec{τ} \cdot \vec{Λ} = - 2 N i \frac{d U}{d ω} U^{†} = - 2 N i \frac{d U_{2}}{d ω} U_{2}^{†} + U_{2} (- 2 N i \frac{d U_{1}}{d ω} U_{1}^{†}) U_{2}^{†} = {\vec{τ}}_{2} \cdot \vec{Λ} + U_{2} ({\vec{τ}}_{1} \cdot \vec{Λ}) U_{2}^{†},

{| {\vec{τ}}_{1}^{(R)} |}^{2} = \frac{1}{2 N} trace {{[U_{2} ({\vec{τ}}_{1} \cdot \vec{Λ}) U_{2}^{†}]}^{2}} = {| {\vec{τ}}_{1} |}^{2},

\frac{\partial}{\partial z} (\vec{τ} \cdot \vec{Λ}) = {\vec{β}}_{ω} \cdot \vec{Λ} + \frac{i}{2 N} [(\vec{β} \cdot \vec{Λ}) (\vec{τ} \cdot \vec{Λ}) - (\vec{τ} \cdot \vec{Λ}) (\vec{β} \cdot \vec{Λ})],

\vec{τ} (L) = \int_{0}^{L} R (L, z) {\vec{β}}_{ω} d z

U (L, z) ({\vec{β}}_{ω} \cdot \vec{Λ}) U^{†} (L, z) = (R (L, z) {\vec{β}}_{ω}) \cdot \vec{Λ} .

τ^{2} = 2 N \sum_{i = 1}^{2 N} t_{i}^{2},

f_{i, j, k} = \frac{i}{{(2 N)}^{2}} trace {Λ_{k} (Λ_{i} Λ_{j} - Λ_{j} Λ_{i})} .

\vec{A} \times \vec{β} = \sum_{i, j, k} f_{i, j, k} A_{i} B_{j} {\vec{e}}_{k}

\frac{\partial \vec{τ}}{\partial z} = {\vec{β}}_{ω} + \vec{β} \times \vec{τ},

\begin{matrix} \frac{\partial \vec{S}}{\partial z} = \vec{β} \times \vec{S}, & \frac{\partial \vec{S}}{\partial ω} = \vec{τ} \times \vec{S}, \end{matrix}

d \vec{τ} = d \vec{W} + ω d \vec{W} \times \vec{τ},

\begin{matrix} Λ_{1} = σ_{1} = (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}), & Λ_{2} = σ_{2} = (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}), & Λ_{3} = σ_{3} = (\begin{matrix} 0 & - i \\ i & 0 \end{matrix}) . \end{matrix}

\begin{array}{l} Λ_{1} = \sqrt{2} (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}), & Λ_{2} = \sqrt{2} (\begin{matrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}), & Λ_{3} = \sqrt{2} (\begin{matrix} 0 & - i & 0 & 0 \\ i & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}), \\ Λ_{4} = \sqrt{2} (\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & - 1 \end{matrix}), & Λ_{5} = \sqrt{2} (\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{matrix}), & Λ_{6} = \sqrt{2} (\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & - i \\ 0 & 0 & i & 0 \end{matrix}), \\ Λ_{7} = \sqrt{2} (\begin{matrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}), & Λ_{8} = \sqrt{2} (\begin{matrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix}), & Λ_{9} = \sqrt{2} (\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}), \\ Λ_{10} = \sqrt{2} (\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix}), & Λ_{11} = \sqrt{2} (\begin{matrix} 0 & 0 & i & 0 \\ 0 & 0 & 0 & 0 \\ - i & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}), & Λ_{12} = \sqrt{2} (\begin{matrix} 0 & 0 & 0 & i \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ - i & 0 & 0 & 0 \end{matrix}), \\ Λ_{13} = \sqrt{2} (\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & i & 0 \\ 0 & - i & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}), & Λ_{14} = \sqrt{2} (\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & i \\ 0 & 0 & 0 & 0 \\ 0 & - i & 0 & 0 \end{matrix}), & Λ_{15} = (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & - 1 \end{matrix}) . \end{array}

Abstract

References

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