Abstract

A synthetic photonic lattice (SPL) is a re-configurable test-bed for studying the dynamics of one-dimensional mesh lattices including the photonic implementations of discrete time quantum walks. Unlike other realizations of photonic lattices, SPL possesses easy and fast control of lattice parameters. Here we consider disordered SPL where the coupling ratio between the two fiber loops realizing the lattice is random but does not change between the round trips. We obtain a new analytical result for the localization length (inverse Lyapunov exponent) for a practical case of weak coupling disorder. We also numerically study the dynamics of the pulse train circulating within the loops and observe that despite delocalization transition at the band center the pulse spreading is arrested even at small values of the disorder.

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

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References

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2018 (3)

S. H. Choi, S.-W. Kim, Z. Ku, M. A. Visbal-Onufrak, S.-R. Kim, K.-H. Choi, H. Ko, W. Choi, A. M. Urbas, and T.-W. Goo, “Anderson light localization in biological nanostructures of native silk,” Nat. Commun. 9, 452 (2018).
[Crossref] [PubMed]

S. Wang, C. Qin, B. Wang, and P. Lu, “Discrete temporal Talbot effect in synthetic mesh lattices,” Opt. Express 26, 19235 (2018).
[Crossref] [PubMed]

S. Derevyanko, “Anderson localization of a one-dimensional quantum walker,” Sci. Rep. 8, 1795 (2018).
[Crossref] [PubMed]

2017 (4)

I. D. Vatnik, A. Tikan, G. Onishchukov, D. V. Churkin, and A. A. Sukhorukov, “Anderson localization in synthetic photonic lattices,” Sci. Rep. 7, 4301 (2017).
[Crossref] [PubMed]

H. E. Kondakci, A. Szameit, A. F. Abouraddy, D. N. Christodoulides, and B. E. Saleh, “Interferometric control of the photon-number distribution,” APL Photonics 2, 071301 (2017).
[Crossref]

H. E. Kondakci, A. F. Abouraddy, and B. E. Saleh, “Lattice topology dictates photon statistics,” Sci. Rep. 7, 8948 (2017).
[Crossref] [PubMed]

I. Vakulchyk, M. Fistul, P. Qin, and S. Flach, “Anderson localization in generalized discrete-time quantum walks,” Phys. Rev. B 96, 144204 (2017).
[Crossref]

2016 (1)

2015 (5)

M. Wimmer, M.-A. Miri, D. Christodoulides, and U. Peschel, “Observation of bloch oscillations in complex PT-symmetric photonic lattices,” Sci. Rep. 5, 17760 (2015).
[Crossref] [PubMed]

T. Rakovszky and J. K. Asboth, “Localization, delocalization, and topological phase transitions in the one-dimensional split-step quantum walk,” Phys. Rev. A 92, 052311 (2015).
[Crossref]

J. M. Edge and J. K. Asboth, “Localization, delocalization, and topological transitions in disordered two-dimensional quantum walks,” Phys. Rev. B 91, 104202 (2015).
[Crossref]

H. E. Kondakci, A. F. Abouraddy, and B. E. Saleh, “A photonic thermalization gap in disordered lattices,” Nat. Phys. 11, 930 (2015).
[Crossref]

M. Wimmer, A. Regensburger, M.-A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6, 7782 (2015).
[Crossref] [PubMed]

2013 (2)

M. Wimmer, A. Regensburger, C. Bersch, M.-A. Miri, S. Batz, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Optical diametric drive acceleration through action–reaction symmetry breaking,” Nat. Phys. 9, 780 (2013).
[Crossref]

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in p t-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
[Crossref]

2012 (2)

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity–time synthetic photonic lattices,” Nature 488, 167 (2012).
[Crossref] [PubMed]

M.-A. Miri, A. Regensburger, U. Peschel, and D. N. Christodoulides, “Optical mesh lattices with PT symmetry,” Phys. Rev. A 86, 023807 (2012).
[Crossref]

2011 (3)

A. Schreiber, K. Cassemiro, V. Potoček, A. Gábris, I. Jex, and C. Silberhorn, “Decoherence and disorder in quantum walks: from ballistic spread to localization,” Phys. Rev. Lett. 106, 180403 (2011).
[Crossref] [PubMed]

L. Martin, G. Di Giuseppe, A. Perez-Leija, R. Keil, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, A. F. Abouraddy, and D. N. Christodoulides, “Anderson localization in optical waveguide arrays with off-diagonal coupling disorder,” Opt. Express 19, 13636–13646 (2011).
[Crossref] [PubMed]

H. Obuse and N. Kawakami, “Topological phases and delocalization of quantum walks in random environments,” Phys. Rev. B 84, 195139 (2011).
[Crossref]

2010 (2)

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: a quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104, 050502 (2010).
[Crossref] [PubMed]

T. Kitagawa, M. S. Rudner, E. Berg, and E. Demler, “Exploring topological phases with quantum walks,” Phys. Rev. A 82, 033429 (2010).
[Crossref]

2008 (1)

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[Crossref] [PubMed]

2007 (1)

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52 (2007).
[Crossref] [PubMed]

2003 (1)

J. Kempe, “Quantum random walks: an introductory overview,” Contemp. Phys. 44, 307–327 (2003).
[Crossref]

2000 (1)

L. Tessieri and F. Izrailev, “Anderson localization as a parametric instability of the linear kicked oscillator,” Phys. Rev. E 62, 3090 (2000).
[Crossref]

1993 (2)

Y. Aharonov, L. Davidovich, and N. Zagury, “Quantum random walks,” Phys. Rev. A 48, 1687 (1993).
[Crossref] [PubMed]

B. Kramer and A. MacKinnon, “Localization: theory and experiment,” Rep.Prog.Phys. 56, 1469 (1993).

1984 (1)

B. Derrida and E. Gardner, “Lyapounov exponent of the one dimensional anderson model: weak disorder expansions,” Eur. Phys. J. 45, 1283–1295 (1984).

1981 (2)

M. Kappus and F. Wegner, “Anomaly in the band centre of the one-dimensional anderson model,” Z. Phys. B Con. Mat. 45, 15–21 (1981).
[Crossref]

C. Soukoulis and E. Economou, “Off-diagonal disorder in one-dimensional systems,” Phys. Rev. B 24, 5698 (1981).
[Crossref]

1958 (1)

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492 (1958).
[Crossref]

Abouraddy, A. F.

H. E. Kondakci, A. Szameit, A. F. Abouraddy, D. N. Christodoulides, and B. E. Saleh, “Interferometric control of the photon-number distribution,” APL Photonics 2, 071301 (2017).
[Crossref]

H. E. Kondakci, A. F. Abouraddy, and B. E. Saleh, “Lattice topology dictates photon statistics,” Sci. Rep. 7, 8948 (2017).
[Crossref] [PubMed]

H. E. Kondakci, A. Szameit, A. F. Abouraddy, D. N. Christodoulides, and B. E. Saleh, “Sub-thermal to super-thermal light statistics from a disordered lattice via deterministic control of excitation symmetry,” Optica 3, 477–482 (2016).
[Crossref]

H. E. Kondakci, A. F. Abouraddy, and B. E. Saleh, “A photonic thermalization gap in disordered lattices,” Nat. Phys. 11, 930 (2015).
[Crossref]

L. Martin, G. Di Giuseppe, A. Perez-Leija, R. Keil, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, A. F. Abouraddy, and D. N. Christodoulides, “Anderson localization in optical waveguide arrays with off-diagonal coupling disorder,” Opt. Express 19, 13636–13646 (2011).
[Crossref] [PubMed]

Aharonov, Y.

Y. Aharonov, L. Davidovich, and N. Zagury, “Quantum random walks,” Phys. Rev. A 48, 1687 (1993).
[Crossref] [PubMed]

Anderson, P. W.

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492 (1958).
[Crossref]

Andersson, E.

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: a quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104, 050502 (2010).
[Crossref] [PubMed]

Asboth, J. K.

T. Rakovszky and J. K. Asboth, “Localization, delocalization, and topological phase transitions in the one-dimensional split-step quantum walk,” Phys. Rev. A 92, 052311 (2015).
[Crossref]

J. M. Edge and J. K. Asboth, “Localization, delocalization, and topological transitions in disordered two-dimensional quantum walks,” Phys. Rev. B 91, 104202 (2015).
[Crossref]

Avidan, A.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[Crossref] [PubMed]

Bartal, G.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52 (2007).
[Crossref] [PubMed]

Batz, S.

M. Wimmer, A. Regensburger, C. Bersch, M.-A. Miri, S. Batz, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Optical diametric drive acceleration through action–reaction symmetry breaking,” Nat. Phys. 9, 780 (2013).
[Crossref]

Berg, E.

T. Kitagawa, M. S. Rudner, E. Berg, and E. Demler, “Exploring topological phases with quantum walks,” Phys. Rev. A 82, 033429 (2010).
[Crossref]

Bersch, C.

M. Wimmer, A. Regensburger, M.-A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6, 7782 (2015).
[Crossref] [PubMed]

M. Wimmer, A. Regensburger, C. Bersch, M.-A. Miri, S. Batz, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Optical diametric drive acceleration through action–reaction symmetry breaking,” Nat. Phys. 9, 780 (2013).
[Crossref]

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in p t-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
[Crossref]

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity–time synthetic photonic lattices,” Nature 488, 167 (2012).
[Crossref] [PubMed]

Cassemiro, K.

A. Schreiber, K. Cassemiro, V. Potoček, A. Gábris, I. Jex, and C. Silberhorn, “Decoherence and disorder in quantum walks: from ballistic spread to localization,” Phys. Rev. Lett. 106, 180403 (2011).
[Crossref] [PubMed]

Cassemiro, K. N.

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: a quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104, 050502 (2010).
[Crossref] [PubMed]

Chen, Y.

Y. Wang, J. Gao, X.-L. Pang, Z.-Q. Jiao, H. Tang, Y. Chen, L.-F. Qiao, Z.-W. Gao, J.-P. Dou, and A.-L. Yang, “Experimental parity-induced thermalization gap in disordered ring lattices,” arXiv preprint arXiv:1803.10838 (2018).

Choi, K.-H.

S. H. Choi, S.-W. Kim, Z. Ku, M. A. Visbal-Onufrak, S.-R. Kim, K.-H. Choi, H. Ko, W. Choi, A. M. Urbas, and T.-W. Goo, “Anderson light localization in biological nanostructures of native silk,” Nat. Commun. 9, 452 (2018).
[Crossref] [PubMed]

Choi, S. H.

S. H. Choi, S.-W. Kim, Z. Ku, M. A. Visbal-Onufrak, S.-R. Kim, K.-H. Choi, H. Ko, W. Choi, A. M. Urbas, and T.-W. Goo, “Anderson light localization in biological nanostructures of native silk,” Nat. Commun. 9, 452 (2018).
[Crossref] [PubMed]

Choi, W.

S. H. Choi, S.-W. Kim, Z. Ku, M. A. Visbal-Onufrak, S.-R. Kim, K.-H. Choi, H. Ko, W. Choi, A. M. Urbas, and T.-W. Goo, “Anderson light localization in biological nanostructures of native silk,” Nat. Commun. 9, 452 (2018).
[Crossref] [PubMed]

Christodoulides, D.

M. Wimmer, M.-A. Miri, D. Christodoulides, and U. Peschel, “Observation of bloch oscillations in complex PT-symmetric photonic lattices,” Sci. Rep. 5, 17760 (2015).
[Crossref] [PubMed]

Christodoulides, D. N.

H. E. Kondakci, A. Szameit, A. F. Abouraddy, D. N. Christodoulides, and B. E. Saleh, “Interferometric control of the photon-number distribution,” APL Photonics 2, 071301 (2017).
[Crossref]

H. E. Kondakci, A. Szameit, A. F. Abouraddy, D. N. Christodoulides, and B. E. Saleh, “Sub-thermal to super-thermal light statistics from a disordered lattice via deterministic control of excitation symmetry,” Optica 3, 477–482 (2016).
[Crossref]

M. Wimmer, A. Regensburger, M.-A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6, 7782 (2015).
[Crossref] [PubMed]

M. Wimmer, A. Regensburger, C. Bersch, M.-A. Miri, S. Batz, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Optical diametric drive acceleration through action–reaction symmetry breaking,” Nat. Phys. 9, 780 (2013).
[Crossref]

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in p t-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
[Crossref]

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity–time synthetic photonic lattices,” Nature 488, 167 (2012).
[Crossref] [PubMed]

M.-A. Miri, A. Regensburger, U. Peschel, and D. N. Christodoulides, “Optical mesh lattices with PT symmetry,” Phys. Rev. A 86, 023807 (2012).
[Crossref]

L. Martin, G. Di Giuseppe, A. Perez-Leija, R. Keil, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, A. F. Abouraddy, and D. N. Christodoulides, “Anderson localization in optical waveguide arrays with off-diagonal coupling disorder,” Opt. Express 19, 13636–13646 (2011).
[Crossref] [PubMed]

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[Crossref] [PubMed]

Churkin, D. V.

I. D. Vatnik, A. Tikan, G. Onishchukov, D. V. Churkin, and A. A. Sukhorukov, “Anderson localization in synthetic photonic lattices,” Sci. Rep. 7, 4301 (2017).
[Crossref] [PubMed]

Davidovich, L.

Y. Aharonov, L. Davidovich, and N. Zagury, “Quantum random walks,” Phys. Rev. A 48, 1687 (1993).
[Crossref] [PubMed]

Demler, E.

T. Kitagawa, M. S. Rudner, E. Berg, and E. Demler, “Exploring topological phases with quantum walks,” Phys. Rev. A 82, 033429 (2010).
[Crossref]

Derevyanko, S.

S. Derevyanko, “Anderson localization of a one-dimensional quantum walker,” Sci. Rep. 8, 1795 (2018).
[Crossref] [PubMed]

Derrida, B.

B. Derrida and E. Gardner, “Lyapounov exponent of the one dimensional anderson model: weak disorder expansions,” Eur. Phys. J. 45, 1283–1295 (1984).

Di Giuseppe, G.

Dou, J.-P.

Y. Wang, J. Gao, X.-L. Pang, Z.-Q. Jiao, H. Tang, Y. Chen, L.-F. Qiao, Z.-W. Gao, J.-P. Dou, and A.-L. Yang, “Experimental parity-induced thermalization gap in disordered ring lattices,” arXiv preprint arXiv:1803.10838 (2018).

Dreisow, F.

Economou, E.

C. Soukoulis and E. Economou, “Off-diagonal disorder in one-dimensional systems,” Phys. Rev. B 24, 5698 (1981).
[Crossref]

Edge, J. M.

J. M. Edge and J. K. Asboth, “Localization, delocalization, and topological transitions in disordered two-dimensional quantum walks,” Phys. Rev. B 91, 104202 (2015).
[Crossref]

Fishman, S.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52 (2007).
[Crossref] [PubMed]

Fistul, M.

I. Vakulchyk, M. Fistul, P. Qin, and S. Flach, “Anderson localization in generalized discrete-time quantum walks,” Phys. Rev. B 96, 144204 (2017).
[Crossref]

Flach, S.

I. Vakulchyk, M. Fistul, P. Qin, and S. Flach, “Anderson localization in generalized discrete-time quantum walks,” Phys. Rev. B 96, 144204 (2017).
[Crossref]

Gábris, A.

A. Schreiber, K. Cassemiro, V. Potoček, A. Gábris, I. Jex, and C. Silberhorn, “Decoherence and disorder in quantum walks: from ballistic spread to localization,” Phys. Rev. Lett. 106, 180403 (2011).
[Crossref] [PubMed]

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: a quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104, 050502 (2010).
[Crossref] [PubMed]

Gao, J.

Y. Wang, J. Gao, X.-L. Pang, Z.-Q. Jiao, H. Tang, Y. Chen, L.-F. Qiao, Z.-W. Gao, J.-P. Dou, and A.-L. Yang, “Experimental parity-induced thermalization gap in disordered ring lattices,” arXiv preprint arXiv:1803.10838 (2018).

Gao, Z.-W.

Y. Wang, J. Gao, X.-L. Pang, Z.-Q. Jiao, H. Tang, Y. Chen, L.-F. Qiao, Z.-W. Gao, J.-P. Dou, and A.-L. Yang, “Experimental parity-induced thermalization gap in disordered ring lattices,” arXiv preprint arXiv:1803.10838 (2018).

Gardner, E.

B. Derrida and E. Gardner, “Lyapounov exponent of the one dimensional anderson model: weak disorder expansions,” Eur. Phys. J. 45, 1283–1295 (1984).

Goo, T.-W.

S. H. Choi, S.-W. Kim, Z. Ku, M. A. Visbal-Onufrak, S.-R. Kim, K.-H. Choi, H. Ko, W. Choi, A. M. Urbas, and T.-W. Goo, “Anderson light localization in biological nanostructures of native silk,” Nat. Commun. 9, 452 (2018).
[Crossref] [PubMed]

Heinrich, M.

Izrailev, F.

L. Tessieri and F. Izrailev, “Anderson localization as a parametric instability of the linear kicked oscillator,” Phys. Rev. E 62, 3090 (2000).
[Crossref]

Jex, I.

A. Schreiber, K. Cassemiro, V. Potoček, A. Gábris, I. Jex, and C. Silberhorn, “Decoherence and disorder in quantum walks: from ballistic spread to localization,” Phys. Rev. Lett. 106, 180403 (2011).
[Crossref] [PubMed]

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: a quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104, 050502 (2010).
[Crossref] [PubMed]

Jiao, Z.-Q.

Y. Wang, J. Gao, X.-L. Pang, Z.-Q. Jiao, H. Tang, Y. Chen, L.-F. Qiao, Z.-W. Gao, J.-P. Dou, and A.-L. Yang, “Experimental parity-induced thermalization gap in disordered ring lattices,” arXiv preprint arXiv:1803.10838 (2018).

Kappus, M.

M. Kappus and F. Wegner, “Anomaly in the band centre of the one-dimensional anderson model,” Z. Phys. B Con. Mat. 45, 15–21 (1981).
[Crossref]

Kawakami, N.

H. Obuse and N. Kawakami, “Topological phases and delocalization of quantum walks in random environments,” Phys. Rev. B 84, 195139 (2011).
[Crossref]

Keil, R.

Kempe, J.

J. Kempe, “Quantum random walks: an introductory overview,” Contemp. Phys. 44, 307–327 (2003).
[Crossref]

Kim, S.-R.

S. H. Choi, S.-W. Kim, Z. Ku, M. A. Visbal-Onufrak, S.-R. Kim, K.-H. Choi, H. Ko, W. Choi, A. M. Urbas, and T.-W. Goo, “Anderson light localization in biological nanostructures of native silk,” Nat. Commun. 9, 452 (2018).
[Crossref] [PubMed]

Kim, S.-W.

S. H. Choi, S.-W. Kim, Z. Ku, M. A. Visbal-Onufrak, S.-R. Kim, K.-H. Choi, H. Ko, W. Choi, A. M. Urbas, and T.-W. Goo, “Anderson light localization in biological nanostructures of native silk,” Nat. Commun. 9, 452 (2018).
[Crossref] [PubMed]

Kitagawa, T.

T. Kitagawa, M. S. Rudner, E. Berg, and E. Demler, “Exploring topological phases with quantum walks,” Phys. Rev. A 82, 033429 (2010).
[Crossref]

Ko, H.

S. H. Choi, S.-W. Kim, Z. Ku, M. A. Visbal-Onufrak, S.-R. Kim, K.-H. Choi, H. Ko, W. Choi, A. M. Urbas, and T.-W. Goo, “Anderson light localization in biological nanostructures of native silk,” Nat. Commun. 9, 452 (2018).
[Crossref] [PubMed]

Kondakci, H. E.

H. E. Kondakci, A. F. Abouraddy, and B. E. Saleh, “Lattice topology dictates photon statistics,” Sci. Rep. 7, 8948 (2017).
[Crossref] [PubMed]

H. E. Kondakci, A. Szameit, A. F. Abouraddy, D. N. Christodoulides, and B. E. Saleh, “Interferometric control of the photon-number distribution,” APL Photonics 2, 071301 (2017).
[Crossref]

H. E. Kondakci, A. Szameit, A. F. Abouraddy, D. N. Christodoulides, and B. E. Saleh, “Sub-thermal to super-thermal light statistics from a disordered lattice via deterministic control of excitation symmetry,” Optica 3, 477–482 (2016).
[Crossref]

H. E. Kondakci, A. F. Abouraddy, and B. E. Saleh, “A photonic thermalization gap in disordered lattices,” Nat. Phys. 11, 930 (2015).
[Crossref]

Kramer, B.

B. Kramer and A. MacKinnon, “Localization: theory and experiment,” Rep.Prog.Phys. 56, 1469 (1993).

Ku, Z.

S. H. Choi, S.-W. Kim, Z. Ku, M. A. Visbal-Onufrak, S.-R. Kim, K.-H. Choi, H. Ko, W. Choi, A. M. Urbas, and T.-W. Goo, “Anderson light localization in biological nanostructures of native silk,” Nat. Commun. 9, 452 (2018).
[Crossref] [PubMed]

Lahini, Y.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[Crossref] [PubMed]

Lu, P.

MacKinnon, A.

B. Kramer and A. MacKinnon, “Localization: theory and experiment,” Rep.Prog.Phys. 56, 1469 (1993).

Martin, L.

Miri, M.-A.

M. Wimmer, M.-A. Miri, D. Christodoulides, and U. Peschel, “Observation of bloch oscillations in complex PT-symmetric photonic lattices,” Sci. Rep. 5, 17760 (2015).
[Crossref] [PubMed]

M. Wimmer, A. Regensburger, M.-A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6, 7782 (2015).
[Crossref] [PubMed]

M. Wimmer, A. Regensburger, C. Bersch, M.-A. Miri, S. Batz, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Optical diametric drive acceleration through action–reaction symmetry breaking,” Nat. Phys. 9, 780 (2013).
[Crossref]

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in p t-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
[Crossref]

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity–time synthetic photonic lattices,” Nature 488, 167 (2012).
[Crossref] [PubMed]

M.-A. Miri, A. Regensburger, U. Peschel, and D. N. Christodoulides, “Optical mesh lattices with PT symmetry,” Phys. Rev. A 86, 023807 (2012).
[Crossref]

Morandotti, R.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[Crossref] [PubMed]

Mosley, P. J.

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: a quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104, 050502 (2010).
[Crossref] [PubMed]

Näger, J.

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in p t-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
[Crossref]

Nolte, S.

Obuse, H.

H. Obuse and N. Kawakami, “Topological phases and delocalization of quantum walks in random environments,” Phys. Rev. B 84, 195139 (2011).
[Crossref]

Onishchukov, G.

I. D. Vatnik, A. Tikan, G. Onishchukov, D. V. Churkin, and A. A. Sukhorukov, “Anderson localization in synthetic photonic lattices,” Sci. Rep. 7, 4301 (2017).
[Crossref] [PubMed]

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in p t-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
[Crossref]

M. Wimmer, A. Regensburger, C. Bersch, M.-A. Miri, S. Batz, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Optical diametric drive acceleration through action–reaction symmetry breaking,” Nat. Phys. 9, 780 (2013).
[Crossref]

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity–time synthetic photonic lattices,” Nature 488, 167 (2012).
[Crossref] [PubMed]

Pang, X.-L.

Y. Wang, J. Gao, X.-L. Pang, Z.-Q. Jiao, H. Tang, Y. Chen, L.-F. Qiao, Z.-W. Gao, J.-P. Dou, and A.-L. Yang, “Experimental parity-induced thermalization gap in disordered ring lattices,” arXiv preprint arXiv:1803.10838 (2018).

Perez-Leija, A.

Peschel, U.

M. Wimmer, M.-A. Miri, D. Christodoulides, and U. Peschel, “Observation of bloch oscillations in complex PT-symmetric photonic lattices,” Sci. Rep. 5, 17760 (2015).
[Crossref] [PubMed]

M. Wimmer, A. Regensburger, M.-A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6, 7782 (2015).
[Crossref] [PubMed]

M. Wimmer, A. Regensburger, C. Bersch, M.-A. Miri, S. Batz, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Optical diametric drive acceleration through action–reaction symmetry breaking,” Nat. Phys. 9, 780 (2013).
[Crossref]

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in p t-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
[Crossref]

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity–time synthetic photonic lattices,” Nature 488, 167 (2012).
[Crossref] [PubMed]

M.-A. Miri, A. Regensburger, U. Peschel, and D. N. Christodoulides, “Optical mesh lattices with PT symmetry,” Phys. Rev. A 86, 023807 (2012).
[Crossref]

Potocek, V.

A. Schreiber, K. Cassemiro, V. Potoček, A. Gábris, I. Jex, and C. Silberhorn, “Decoherence and disorder in quantum walks: from ballistic spread to localization,” Phys. Rev. Lett. 106, 180403 (2011).
[Crossref] [PubMed]

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: a quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104, 050502 (2010).
[Crossref] [PubMed]

Pozzi, F.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[Crossref] [PubMed]

Qiao, L.-F.

Y. Wang, J. Gao, X.-L. Pang, Z.-Q. Jiao, H. Tang, Y. Chen, L.-F. Qiao, Z.-W. Gao, J.-P. Dou, and A.-L. Yang, “Experimental parity-induced thermalization gap in disordered ring lattices,” arXiv preprint arXiv:1803.10838 (2018).

Qin, C.

Qin, P.

I. Vakulchyk, M. Fistul, P. Qin, and S. Flach, “Anderson localization in generalized discrete-time quantum walks,” Phys. Rev. B 96, 144204 (2017).
[Crossref]

Rakovszky, T.

T. Rakovszky and J. K. Asboth, “Localization, delocalization, and topological phase transitions in the one-dimensional split-step quantum walk,” Phys. Rev. A 92, 052311 (2015).
[Crossref]

Regensburger, A.

M. Wimmer, A. Regensburger, M.-A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6, 7782 (2015).
[Crossref] [PubMed]

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in p t-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
[Crossref]

M. Wimmer, A. Regensburger, C. Bersch, M.-A. Miri, S. Batz, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Optical diametric drive acceleration through action–reaction symmetry breaking,” Nat. Phys. 9, 780 (2013).
[Crossref]

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity–time synthetic photonic lattices,” Nature 488, 167 (2012).
[Crossref] [PubMed]

M.-A. Miri, A. Regensburger, U. Peschel, and D. N. Christodoulides, “Optical mesh lattices with PT symmetry,” Phys. Rev. A 86, 023807 (2012).
[Crossref]

Rudner, M. S.

T. Kitagawa, M. S. Rudner, E. Berg, and E. Demler, “Exploring topological phases with quantum walks,” Phys. Rev. A 82, 033429 (2010).
[Crossref]

Saleh, B. E.

H. E. Kondakci, A. F. Abouraddy, and B. E. Saleh, “Lattice topology dictates photon statistics,” Sci. Rep. 7, 8948 (2017).
[Crossref] [PubMed]

H. E. Kondakci, A. Szameit, A. F. Abouraddy, D. N. Christodoulides, and B. E. Saleh, “Interferometric control of the photon-number distribution,” APL Photonics 2, 071301 (2017).
[Crossref]

H. E. Kondakci, A. Szameit, A. F. Abouraddy, D. N. Christodoulides, and B. E. Saleh, “Sub-thermal to super-thermal light statistics from a disordered lattice via deterministic control of excitation symmetry,” Optica 3, 477–482 (2016).
[Crossref]

H. E. Kondakci, A. F. Abouraddy, and B. E. Saleh, “A photonic thermalization gap in disordered lattices,” Nat. Phys. 11, 930 (2015).
[Crossref]

Schreiber, A.

A. Schreiber, K. Cassemiro, V. Potoček, A. Gábris, I. Jex, and C. Silberhorn, “Decoherence and disorder in quantum walks: from ballistic spread to localization,” Phys. Rev. Lett. 106, 180403 (2011).
[Crossref] [PubMed]

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: a quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104, 050502 (2010).
[Crossref] [PubMed]

Schwartz, T.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52 (2007).
[Crossref] [PubMed]

Segev, M.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52 (2007).
[Crossref] [PubMed]

Silberberg, Y.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[Crossref] [PubMed]

Silberhorn, C.

A. Schreiber, K. Cassemiro, V. Potoček, A. Gábris, I. Jex, and C. Silberhorn, “Decoherence and disorder in quantum walks: from ballistic spread to localization,” Phys. Rev. Lett. 106, 180403 (2011).
[Crossref] [PubMed]

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: a quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104, 050502 (2010).
[Crossref] [PubMed]

Sorel, M.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[Crossref] [PubMed]

Soukoulis, C.

C. Soukoulis and E. Economou, “Off-diagonal disorder in one-dimensional systems,” Phys. Rev. B 24, 5698 (1981).
[Crossref]

Sukhorukov, A. A.

I. D. Vatnik, A. Tikan, G. Onishchukov, D. V. Churkin, and A. A. Sukhorukov, “Anderson localization in synthetic photonic lattices,” Sci. Rep. 7, 4301 (2017).
[Crossref] [PubMed]

Szameit, A.

Tang, H.

Y. Wang, J. Gao, X.-L. Pang, Z.-Q. Jiao, H. Tang, Y. Chen, L.-F. Qiao, Z.-W. Gao, J.-P. Dou, and A.-L. Yang, “Experimental parity-induced thermalization gap in disordered ring lattices,” arXiv preprint arXiv:1803.10838 (2018).

Tessieri, L.

L. Tessieri and F. Izrailev, “Anderson localization as a parametric instability of the linear kicked oscillator,” Phys. Rev. E 62, 3090 (2000).
[Crossref]

Tikan, A.

I. D. Vatnik, A. Tikan, G. Onishchukov, D. V. Churkin, and A. A. Sukhorukov, “Anderson localization in synthetic photonic lattices,” Sci. Rep. 7, 4301 (2017).
[Crossref] [PubMed]

Urbas, A. M.

S. H. Choi, S.-W. Kim, Z. Ku, M. A. Visbal-Onufrak, S.-R. Kim, K.-H. Choi, H. Ko, W. Choi, A. M. Urbas, and T.-W. Goo, “Anderson light localization in biological nanostructures of native silk,” Nat. Commun. 9, 452 (2018).
[Crossref] [PubMed]

Vakulchyk, I.

I. Vakulchyk, M. Fistul, P. Qin, and S. Flach, “Anderson localization in generalized discrete-time quantum walks,” Phys. Rev. B 96, 144204 (2017).
[Crossref]

Vatnik, I. D.

I. D. Vatnik, A. Tikan, G. Onishchukov, D. V. Churkin, and A. A. Sukhorukov, “Anderson localization in synthetic photonic lattices,” Sci. Rep. 7, 4301 (2017).
[Crossref] [PubMed]

Visbal-Onufrak, M. A.

S. H. Choi, S.-W. Kim, Z. Ku, M. A. Visbal-Onufrak, S.-R. Kim, K.-H. Choi, H. Ko, W. Choi, A. M. Urbas, and T.-W. Goo, “Anderson light localization in biological nanostructures of native silk,” Nat. Commun. 9, 452 (2018).
[Crossref] [PubMed]

Wang, B.

Wang, S.

Wang, Y.

Y. Wang, J. Gao, X.-L. Pang, Z.-Q. Jiao, H. Tang, Y. Chen, L.-F. Qiao, Z.-W. Gao, J.-P. Dou, and A.-L. Yang, “Experimental parity-induced thermalization gap in disordered ring lattices,” arXiv preprint arXiv:1803.10838 (2018).

Wegner, F.

M. Kappus and F. Wegner, “Anomaly in the band centre of the one-dimensional anderson model,” Z. Phys. B Con. Mat. 45, 15–21 (1981).
[Crossref]

Wimmer, M.

M. Wimmer, M.-A. Miri, D. Christodoulides, and U. Peschel, “Observation of bloch oscillations in complex PT-symmetric photonic lattices,” Sci. Rep. 5, 17760 (2015).
[Crossref] [PubMed]

M. Wimmer, A. Regensburger, M.-A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6, 7782 (2015).
[Crossref] [PubMed]

M. Wimmer, A. Regensburger, C. Bersch, M.-A. Miri, S. Batz, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Optical diametric drive acceleration through action–reaction symmetry breaking,” Nat. Phys. 9, 780 (2013).
[Crossref]

Yang, A.-L.

Y. Wang, J. Gao, X.-L. Pang, Z.-Q. Jiao, H. Tang, Y. Chen, L.-F. Qiao, Z.-W. Gao, J.-P. Dou, and A.-L. Yang, “Experimental parity-induced thermalization gap in disordered ring lattices,” arXiv preprint arXiv:1803.10838 (2018).

Zagury, N.

Y. Aharonov, L. Davidovich, and N. Zagury, “Quantum random walks,” Phys. Rev. A 48, 1687 (1993).
[Crossref] [PubMed]

APL Photonics (1)

H. E. Kondakci, A. Szameit, A. F. Abouraddy, D. N. Christodoulides, and B. E. Saleh, “Interferometric control of the photon-number distribution,” APL Photonics 2, 071301 (2017).
[Crossref]

Contemp. Phys. (1)

J. Kempe, “Quantum random walks: an introductory overview,” Contemp. Phys. 44, 307–327 (2003).
[Crossref]

Eur. Phys. J. (1)

B. Derrida and E. Gardner, “Lyapounov exponent of the one dimensional anderson model: weak disorder expansions,” Eur. Phys. J. 45, 1283–1295 (1984).

Nat. Commun. (2)

S. H. Choi, S.-W. Kim, Z. Ku, M. A. Visbal-Onufrak, S.-R. Kim, K.-H. Choi, H. Ko, W. Choi, A. M. Urbas, and T.-W. Goo, “Anderson light localization in biological nanostructures of native silk,” Nat. Commun. 9, 452 (2018).
[Crossref] [PubMed]

M. Wimmer, A. Regensburger, M.-A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6, 7782 (2015).
[Crossref] [PubMed]

Nat. Phys. (2)

M. Wimmer, A. Regensburger, C. Bersch, M.-A. Miri, S. Batz, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Optical diametric drive acceleration through action–reaction symmetry breaking,” Nat. Phys. 9, 780 (2013).
[Crossref]

H. E. Kondakci, A. F. Abouraddy, and B. E. Saleh, “A photonic thermalization gap in disordered lattices,” Nat. Phys. 11, 930 (2015).
[Crossref]

Nature (2)

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52 (2007).
[Crossref] [PubMed]

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity–time synthetic photonic lattices,” Nature 488, 167 (2012).
[Crossref] [PubMed]

Opt. Express (2)

Optica (1)

Phys. Rev. (1)

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492 (1958).
[Crossref]

Phys. Rev. A (4)

M.-A. Miri, A. Regensburger, U. Peschel, and D. N. Christodoulides, “Optical mesh lattices with PT symmetry,” Phys. Rev. A 86, 023807 (2012).
[Crossref]

T. Kitagawa, M. S. Rudner, E. Berg, and E. Demler, “Exploring topological phases with quantum walks,” Phys. Rev. A 82, 033429 (2010).
[Crossref]

T. Rakovszky and J. K. Asboth, “Localization, delocalization, and topological phase transitions in the one-dimensional split-step quantum walk,” Phys. Rev. A 92, 052311 (2015).
[Crossref]

Y. Aharonov, L. Davidovich, and N. Zagury, “Quantum random walks,” Phys. Rev. A 48, 1687 (1993).
[Crossref] [PubMed]

Phys. Rev. B (4)

I. Vakulchyk, M. Fistul, P. Qin, and S. Flach, “Anderson localization in generalized discrete-time quantum walks,” Phys. Rev. B 96, 144204 (2017).
[Crossref]

J. M. Edge and J. K. Asboth, “Localization, delocalization, and topological transitions in disordered two-dimensional quantum walks,” Phys. Rev. B 91, 104202 (2015).
[Crossref]

C. Soukoulis and E. Economou, “Off-diagonal disorder in one-dimensional systems,” Phys. Rev. B 24, 5698 (1981).
[Crossref]

H. Obuse and N. Kawakami, “Topological phases and delocalization of quantum walks in random environments,” Phys. Rev. B 84, 195139 (2011).
[Crossref]

Phys. Rev. E (1)

L. Tessieri and F. Izrailev, “Anderson localization as a parametric instability of the linear kicked oscillator,” Phys. Rev. E 62, 3090 (2000).
[Crossref]

Phys. Rev. Lett. (4)

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Figures (5)

Fig. 1
Fig. 1 a) Synthetic photonic lattice b) Schematic representation of pulse evolution in mesh lattice, equivalent to the SPL. Both lattices are equivalent in terms of pulse evolution after applying the appropriate transformations [5].
Fig. 2
Fig. 2 (a) The Lyapunov exponent obtained by iterating the Riccati map (Eq. (11)) with Δθmax from 0 to π/4. (b) Density of states (DOS) obtained for Δθmax from 0 to π/4.
Fig. 3
Fig. 3 The normalized Lyapunov exponent obtained for the DTQW with Δθmax = 0.1. Comparison of theoretical result (Eq. (20)) with the full numerics.
Fig. 4
Fig. 4 Anderson localization in the SPL due to the coupling coefficient random distribution with a) Δθmax = 0.1π, b) Δθmax = 0.15π and c) Δθmax = 0.25π.
Fig. 5
Fig. 5 a) Participation ratio dependence on the roundtrip number m for different Δθmax b) Inverse participation ratio at the fixed roundtrip m = 1000 dependence on Δθmax with fluctuations, averaged over an ensemble of Navg = 2000 realizations of the random coupling coefficient distribution. Number of slots N=1000 for both pictures.

Equations (20)

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S ^ SPL = n { | n 1 , n , | + | n + 1 , n , | }
R ^ SPL = n e i θ n σ x | n n |
{ u n m + 1 = [ cos ( θ n + 1 ) u n + 1 m + i sin ( θ n + 1 ) v n + 1 m ] v n m + 1 = [ cos ( θ n 1 ) v n 1 m + i sin ( θ n 1 ) u n 1 m ]
H ^ SPL = π π d k β ( k ) n ( k ) σ | k k |
| k = 1 2 π n e i k n | n
n ( k ) = [ cos θ cos k , sin θ sin k , cos θ sin k ] sin β ( k )
Γ = e i ( π / 2 ) n A n σ | n n | = n A n σ | n n | , A n = A n ( θ n ) .
u n + 1 1 cos ( θ n + 1 ) [ λ + λ 1 sin θ n + 1 sin θ n ] u n + tan θ n + 1 tan θ n u n 1 = 0
v n + 1 1 cos ( θ n ) [ sin θ n sin θ n 1 λ + λ 1 ] v n + tan θ n tan θ n 1 v n 1 = 0
γ ( β ) = 1 N log | u N | = 1 N n = 1 log | r n U | = E [ log | r n U | ]
r n + 1 U = 1 cos θ n + 1 [ e i β + e i β sin θ n + 1 sin θ n ] tan θ n + 1 tan θ n 1 r n U
x n + 1 = [ λ tan 2 θ n sin θ n 1 + tan θ n λ cos θ n ] tan 2 θ n x n , λ = e i β .
γ ( β ) = lim N E [ ln | v N v N 1 | ] = lim N E [ ln | x N | ]
ε 0 : A = 2 2 cos β 1 A ,
ε 1 : A B n + 1 = 1 A B n + 2 e i β ( 4 V n V n 1 ) + 3 2 e i β V n 4 V n A
ε 2 : A [ C n + 1 + 1 2 B n + 1 2 ] = 1 A [ C n + 1 2 B n 2 ] + 4 B n A V n + + 2 ( 8 V n 2 + 3 2 V n 1 2 4 V n V n 1 ) e i β + 11 2 V n 2 e i β 8 V n 2 A
B n + 1 = B n = 0 ,
B n + 1 2 = B n 2 = A ± 2 A ± 4 1 [ 8 2 A 3 e i β + A 2 ( 4 16 e 2 i β ) 8 2 A ( 7 cos ( β ) + i sin ( β ) ) + 16 i ( sin 2 β ) + 52 cos ( 2 β ) + 48 ]
C n + 1 = C n = A 2 + 1 2 ( A 2 1 ) B n 2 + 1 A 2 1 [ 19 A 2 e i β + 11 A 2 e i β 8 ]
γ ( β ) = ε 2 sin 2 β 1 2 cos 2 β , 0 < | cos β | < 1 / 2 .

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