Topological pumping of light governed by Fibonacci numbers

Ruihan Peng; Kai Yang; Qidong Fu; Yanli Chen; Peng Wang; Yaroslav V. Kartashov; Vladimir V. Konotop; Fangwei Ye

doi:10.1186/s43593-025-00095-9

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Topological pumping of light governed by Fibonacci numbers

eLight 2025年5卷

Research Article | Updated：2025-12-17

- Topological pumping of light governed by Fibonacci numbers
- Topological pumping of light governed by Fibonacci numbers
- Affiliations：
- Author bio：
- Funds：
  
  Shanghai Outstanding Academic Leaders Plan;Shanghai Leading Talent Program 1 of Eastern Talent Plan (The 16th Shanghai Leading Talent Program);National Natural Science Foundation of China;National Natural Science Foundation of China;China Postdoctoral Science Foundation;China Postdoctoral Science Foundation;Portuguese Foundation for Science and Technology (FCT);Institute of Spectroscopy of the Russian Academy of Sciences
- DOI：10.1186/s43593-025-00095-9
  中图分类号：
- 收稿：2025-02-06，
  
  修回：2025-06-02，
  
  录用：2025-06-14，
  
  网络出版：2025-08-05，
  
  纸质出版：2025-12
- Accepted：
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Ruihan Peng, Kai Yang, Qidong Fu, 等. Topological pumping of light governed by Fibonacci numbers[J]. eLight, 2025,5.

Ruihan Peng, Kai Yang, Qidong Fu, et al. Topological pumping of light governed by Fibonacci numbers[J]. eLight, 2025, 5.
Ruihan Peng, Kai Yang, Qidong Fu, 等. Topological pumping of light governed by Fibonacci numbers[J]. eLight, 2025,5. DOI： 10.1186/s43593-025-00095-9.

Ruihan Peng, Kai Yang, Qidong Fu, et al. Topological pumping of light governed by Fibonacci numbers[J]. eLight, 2025, 5. DOI： 10.1186/s43593-025-00095-9.

摘要

Abstract

Topological pumping refers to transfer of a physical quantity governed by the system topology

resulting in quantized amounts of the transferred quantities. It is a ubiquitous wave phenomenon typically considered subject to exactly periodic adiabatic variation of the system parameters. Recently

proposals for generalizing quasi-periodic topological pumping and identifying possible physical settings for its implementation have emerged. In a strict sense

pumping with incommensurate frequencies can only manifest over infinite evolution distances

raising a fundamental question about its observability in real-world finite-dimensional systems. Here we demonstrate that bi-chromatic topological pumping with two frequencies

whose ratio is an irrational number

can be viewed as the convergence limit of pumping with two commensurate frequencies representing the best rational approximations of that irrational number. In our experiment

this phenomenon is observed as the displacement of a light beam center in photorefractive crystals induced by two optical lattices. The longitudinal periods of the lattices

that in the paraxial approximation emulate two pumping frequencies

are related as Fibonacci numbers

successively approaching the golden ratio. We observed that a one-cycle displacement of the beam center at each successive approximation is determined by the relation between successive Fibonacci numbers

while the average direction of propagation (emulating average pumping velocity) of the beam is determined by the golden ratio.

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Keywords

references

D.J. Thouless , Quantization of particle transport . Phys. Rev. B 27 , 6083 - 6087 ( 1983 ). http://doi.org/10.1103/PhysRevB.27.6083 http://doi.org/10.1103/PhysRevB.27.6083

R. Citro , M. Aidelsburger , Thouless pumping and topology . Nat. Rev. Phys. 5 , 87 ( 2023 ). http://doi.org/10.1038/s42254-022-00545-0 http://doi.org/10.1038/s42254-022-00545-0

O. Zilberberg et al. , Photonic topological boundary pumping as a probe of 4D quantum Hall physics . Nature 553 , 59 - 62 ( 2018 ). http://doi.org/10.1038/nature25011 http://doi.org/10.1038/nature25011

P. Wang , Q. Fu , R. Peng , Y.V. Kartashov , L. Torner , V.V. Konotop , F. Ye , Two-dimensional Thouless pumping of light in photonic moiré lattices . Nat. Commun. 13 , 6738 ( 2022 ). http://doi.org/10.1038/s41467-022-34394-3 http://doi.org/10.1038/s41467-022-34394-3

W.A. Benalcazar , J. Noh , M. Wang , S. Huang , K.P. Chen , M.C. Rechtsman , Higher-order topological pumping and its observation in photonic lattices . Phys. Rev. B 105 ,( 2022 ). http://doi.org/10.1103/PhysRevB.105.195129 http://doi.org/10.1103/PhysRevB.105.195129

Q. Cheng , H. Wang , Y. Ke , T. Chen , Y. Yu , Y.S. Kivshar , C. Lee , Y. Pan , Asymmetric topological pumping in nonparaxial photonics . Nat. Commun. 13 , 249 ( 2022 ). http://doi.org/10.1038/s41467-021-27773-9 http://doi.org/10.1038/s41467-021-27773-9

Y. Ke , X. Qin , F. Mei , H. Zhong , Y.S. Kivshar , C. Lee , Topological phase transitions and Thouless pumping of light in photonic waveguide arrays . Laser Photon. Rev. 10 , 995 - 1001 ( 2016 ). http://doi.org/10.1002/lpor.201600119 http://doi.org/10.1002/lpor.201600119

M. Lohse , C. Schweizer , O. Zilberberg , M. Aidelsburger , I. Bloch , A Thouless quantum pump with ultracold bosonic atoms in an optical superlattice . Nat. Phys. 12 , 350 - 354 ( 2016 ). http://doi.org/10.1038/nphys3584 http://doi.org/10.1038/nphys3584

S. Nakajima et al. , Topological Thouless pumping of ultracold fermions . Nat. Phys. 12 , 296 - 300 ( 2016 ). http://doi.org/10.1038/nphys3622 http://doi.org/10.1038/nphys3622

L. Taddia , E. Cornfeld , D. Rossini , L. Mazza , E. Sela , R. Fazio , Topological fractional pumping with Alkaline-earth-like atoms in synthetic lattices . Phys. Rev. Lett. 118 ,( 2017 ). http://doi.org/10.1103/PhysRevLett.118.230402 http://doi.org/10.1103/PhysRevLett.118.230402

M. Lohse , C. Schweizer , H.M. Price , O. Zilberberg , I. Bloch , Exploring 4D quantum Hall physics with a 2D topological charge pump . Nature 553 , 55 - 58 ( 2018 ). http://doi.org/10.1038/nature25000 http://doi.org/10.1038/nature25000

W. Ma , L. Zhou , Q. Zhang , M. Li , C. Cheng , J. Geng , X. Rong , F. Shi , J. Gong , J. Du , Experimental observation of a generalized Thouless pump with a single spin . Phys. Rev. Lett. 120 ,( 2018 ). http://doi.org/10.1103/PhysRevLett.120.120501 http://doi.org/10.1103/PhysRevLett.120.120501

M.I.N. Rosa , R.K. Pal , J.R.F. Arruda , M. Ruzzene , Edge states and topological pumping in spatially modulated elastic lattices . Phys. Rev. Lett. 123 ,( 2019 ). http://doi.org/10.1103/PhysRevLett.123.034301 http://doi.org/10.1103/PhysRevLett.123.034301

W. Cheng , E. Prodan , C. Prodan , Experimental demonstration of dynamic topological pumping across incommensurate bilayered acoustic metamaterials . Phys. Rev. Lett. 125 ,( 2020 ). http://doi.org/10.1103/PhysRevLett.125.224301 http://doi.org/10.1103/PhysRevLett.125.224301

H. Chen , H. Zhang , Q. Wu , Y. Huang , H. Nguyen , E. Prodan , X. Zhou , G. Huang , Creating synthetic spaces for higher-order topological sound transport . Nat. Commun. 12 , 5028 ( 2021 ). http://doi.org/10.1038/s41467-021-25305-z http://doi.org/10.1038/s41467-021-25305-z

Y.E. Kraus , Y. Lahini , Z. Ringel , M. Verbin , O. Zilberberg , Topological states and adiabatic pumping in quasicrystals . Phys. Rev. Lett. 109 ,( 2012 ). http://doi.org/10.1103/PhysRevLett.109.106402 http://doi.org/10.1103/PhysRevLett.109.106402

M. Verbin , O. Zilberberg , Y. Lahini , Y.E. Kraus , Y. Silberberg , Topological pumping over a photonic Fibonacci quasicrystal . Phys. Rev. B 91 ,( 2015 ). http://doi.org/10.1103/PhysRevB.91.064201 http://doi.org/10.1103/PhysRevB.91.064201

Y.E. Kraus , Z. Ringel , O. Zilberberg , Four-dimensional quantum hall effect in a two-dimensional quasicrystal . Phys. Rev. Lett. 111 ,( 2013 ). http://doi.org/10.1103/PhysRevLett.111.226401 http://doi.org/10.1103/PhysRevLett.111.226401

K. Yang , Q. Fu , C. Prates , P. Huang , Y.V. Wang , V.V. Kartashov , F. Ye Konotop , Observation of Thouless pumping of light in quasi-periodic photonic crystals . Proc. Natl Acad. Sci. USA 121 ,( 2024 ). http://doi.org/10.1073/pnas.2411793121 http://doi.org/10.1073/pnas.2411793121

A. Cerjan , M. Wang , S. Huang , K.P. Chen , M.C. Rechtsman , Thouless pumping in disordered photonic systems . Light. Sci. Appl. 9 , 178 ( 2020 ). http://doi.org/10.1038/s41377-020-00408-2 http://doi.org/10.1038/s41377-020-00408-2

S. Nakajima et al. , Competition and interplay between topology and quasi-periodic disorder in Thouless pumping of ultracold atoms . Nat. Phys. 17 , 844 - 849 ( 2021 ). http://doi.org/10.1038/s41567-021-01229-9 http://doi.org/10.1038/s41567-021-01229-9

D.J. Thouless , M. Kohmoto , M.P. Nightingale , M. den Nijs , Quantized Hall conductance in a two-dimensional periodic potential . Phys. Rev. Lett. 49 , 405 ( 1982 ). http://doi.org/10.1103/PhysRevLett.49.405 http://doi.org/10.1103/PhysRevLett.49.405

J.E. Avron , R. Seiler , L.G. Yaffe , Adiabatic theorems and applications to the quantum Hall effect . Com. Math. Phys. 110 , 33 - 49 ( 1987 ). http://doi.org/10.1007/BF01209015 http://doi.org/10.1007/BF01209015

Q. Fu , P. Wang , Y.V. Kartashov , V.V. Konotop , F. Ye , Two-dimensional nonlinear Thouless pumping of matter waves . Phys. Rev. Lett. 129 ,( 2022 ). http://doi.org/10.1103/PhysRevLett.129.183901 http://doi.org/10.1103/PhysRevLett.129.183901

M. Schiavoni , L. Sanchez-Palencia , F. Renzoni , G. Grynberg , Phase control of directed diffusion in a symmetric optical lattice . Phys. Rev. Lett. 90 ,( 2003 ). http://doi.org/10.1103/PhysRevLett.90.094101 http://doi.org/10.1103/PhysRevLett.90.094101

R. Gommers , S. Bergamini , F. Renzoni , Dissipation induced symmetry breaking in a driven optical lattice . Phys. Rev. Lett. 95 ,( 2005 ). http://doi.org/10.1103/PhysRevLett.95.073003 http://doi.org/10.1103/PhysRevLett.95.073003

C. Grossert , M. Leder , S. Denisov , P. Hänggi , M. Weitz , Experimental control of transport resonances in a coherent quantum rocking ratchet . Nat. Commun. 7 , 10440 ( 2016 ). http://doi.org/10.1038/ncomms10440 http://doi.org/10.1038/ncomms10440

F. Görg , K. Sandholzer , J. Minguzzi , R. Desbuquois , M. Messer , T. Esslinger , Realization of density-dependent Peierls phases to engineer quantized gauge fields coupled to ultracold matter . Nat. Phys. 15 , 1161 ( 2019 ). http://doi.org/10.1038/s41567-019-0615-4 http://doi.org/10.1038/s41567-019-0615-4

J. Minguzzi , Z. Zhu , K. Sandholzer , A.S. Walter , K. Viebahn , T. Esslinger , Topological pumping in a floquet-bloch band . Phys. Rev. Lett. 129 ,( 2022 ). http://doi.org/10.1103/PhysRevLett.129.053201 http://doi.org/10.1103/PhysRevLett.129.053201

H. Hatami , C. Danieli , J.D. Bodyfelt , S. Flach , Quasiperiodic driving of Anderson localized waves in one dimension . Phys. Rev. E 93 ,( 2016 ). http://doi.org/10.1103/PhysRevE.93.062205 http://doi.org/10.1103/PhysRevE.93.062205

H. Zhao , F. Mintert , J. Knolle , R. Moessner , Localization persisting under aperiodic driving . Phys. Rev. B 105 , L220202 ( 2022 ). http://doi.org/10.1103/PhysRevB.105.L220202 http://doi.org/10.1103/PhysRevB.105.L220202

D.V. Else , W.W. Ho , P.T. Dumitrescu , Long-Lived interacting phases of matter protected by multiple time-translation symmetries in quasiperiodically driven systems . Phys. Rev. X 10 ,( 2020 ).

F. Nathan , R. Ge , S. Gazit , M. Rudner , M. Kolodrubetz , Quasiperiodic Floquet-Thouless energy pump . Phys. Rev. Lett. 127 ,( 2021 ). http://doi.org/10.1103/PhysRevLett.127.166804 http://doi.org/10.1103/PhysRevLett.127.166804

D.M. Long , P.J.D. Crowley , A. Chandran , Nonadiabatic topological energy pumps with quasiperiodic driving . Phys. Rev. Lett. 126 ,( 2021 ). http://doi.org/10.1103/PhysRevLett.126.106805 http://doi.org/10.1103/PhysRevLett.126.106805

Z. Qi , G. Refael , Y. Peng , Universal nonadiabatic energy pumping in a quasiperiodically driven extended system . Phys. Rev. B 104 ,( 2021 ). http://doi.org/10.1103/PhysRevB.104.224301 http://doi.org/10.1103/PhysRevB.104.224301

M.H. Kolodrubetz , F. Nathan , S. Gazit , T. Morimoto , J.E. Moore , Topological Floquet-Thouless energy pump . Phys. Rev. Lett. 120 ,( 2018 ). http://doi.org/10.1103/PhysRevLett.120.150601 http://doi.org/10.1103/PhysRevLett.120.150601

P. Hu , H.W. Wu , P. Xie , Y. Zhuo , W. Sun , Z. Sheng , Y. Pan , Hearing the dynamical Floquet-Thouless pump of a sound pulse . Phys. Rev. B 110 ,( 2024 ). http://doi.org/10.1103/PhysRevB.110.085137 http://doi.org/10.1103/PhysRevB.110.085137

A.Y. Khinchin , T. Teichmann , Continued fractions . Phys. Today 17 , 70 - 71 ( 1964 ). http://doi.org/10.1063/1.3051235 http://doi.org/10.1063/1.3051235

S. Ostlund , R. Pandit , D. Rand , H.J. Schellnhuber , E.D. Siggia , One-dimensional Schrödinger equation with an almost periodic potential . Phys. Rev. Lett. 50 , 1873 ( 1983 ). http://doi.org/10.1103/PhysRevLett.50.1873 http://doi.org/10.1103/PhysRevLett.50.1873

R.B. Diener , G.A. Georgakis , J. Zhong , M. Raizen , Q. Niu , Transitions between extended and localized states in a one-dimensional incommensurate optical lattice . Phys. Rev. A 64 ,( 2001 ). http://doi.org/10.1103/PhysRevA.64.033416 http://doi.org/10.1103/PhysRevA.64.033416

M. Modugno , Exponential localization in one-dimensional quasi-periodic optical lattices . New J. Phys. 11 ,( 2009 ). http://doi.org/10.1088/1367-2630/11/3/033023 http://doi.org/10.1088/1367-2630/11/3/033023

P. Marra , M. Nitta , Topologically quantized current in quasiperiodic Thouless pumps . Phys. Rev. Res. 2 , 042035(R) ( 2020 ). http://doi.org/10.1103/PhysRevResearch.2.042035 http://doi.org/10.1103/PhysRevResearch.2.042035

D.A. Zezyulin , V.V. Konotop , Localization of ultracold atoms in Zeeman lattices with incommensurate spin-orbit coupling . Phys. Rev. A 105 ,( 2022 ). http://doi.org/10.1103/PhysRevA.105.063323 http://doi.org/10.1103/PhysRevA.105.063323

N.K. Efremidis , S. Sears , D.N. Christodoulides , J.W. Fleischer , M. Segev , Discrete solitons in photorefractive optically induced photonic lattices . Phys. Rev. E 66 ( 4 ),( 2002 ). http://doi.org/10.1103/PhysRevE.66.046602 http://doi.org/10.1103/PhysRevE.66.046602

J.W. Fleischer , M. Segev , N.K. Efremidis , D.N. Christodoulides , Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices . Nature 422 , 147 - 150 ( 2003 ). http://doi.org/10.1038/nature01452 http://doi.org/10.1038/nature01452

Z. Fedorova , H. Qiu , S. Linden , J. Kroha , Observation of topological transport quantization by dissipation in fast Thouless pumps . Nat. Commun. 11 , 3758 ( 2020 ). http://doi.org/10.1038/s41467-020-17510-z http://doi.org/10.1038/s41467-020-17510-z

Q. Fu , P. Wang , Y.V. Kartashov , V.V. Konotop , F. Ye , Nonlinear Thouless pumping: solitons and transport breakdown . Phys. Rev. Lett. 128 ,( 2022 ). http://doi.org/10.1103/PhysRevLett.128.154101 http://doi.org/10.1103/PhysRevLett.128.154101

P. Wang , Y. Zheng , X. Chen , C. Huang , Y.V. Kartashov , L. Torner , V.V. Konotop , F. Ye , Localization and delocalization of light in photonic moiré lattices . Nature 577 , 42 - 46 ( 2020 ). http://doi.org/10.1038/s41586-019-1851-6 http://doi.org/10.1038/s41586-019-1851-6

F. Lederer , G.I. Stegeman , D.N. Christodoulides , G. Assanto , M. Segev , Y. Silberberg , Discrete solitons in optics . Phys. Rep. 463 , 1 - 126 ( 2008 ). http://doi.org/10.1016/j.physrep.2008.04.004 http://doi.org/10.1016/j.physrep.2008.04.004

G.L. Alfimov , P.G. Kevrekidis , V.V. Konotop , M. Salerno , Wannier functions analysis of the nonlinear Schrödinger equation with a periodic potential . Phys. Rev. E 66 ,( 2002 ). http://doi.org/10.1103/PhysRevE.66.046608 http://doi.org/10.1103/PhysRevE.66.046608

W. Kohn , Analytic properties of Bloch waves and Wannier functions . Phys. Rev. 115 , 809 - 821 ( 1959 ). http://doi.org/10.1103/PhysRev.115.809 http://doi.org/10.1103/PhysRev.115.809

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