Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Quantum transport in fractal networks

Nature Photonics volume 15, pages 703–710 (2021)Cite this article

Subjects

Abstract

Fractals are fascinating, not only for their aesthetic appeal but also for allowing the investigation of physical properties in non-integer dimensions. In these unconventional systems, many intrinsic features might come into play, including the fractal dimension and the fractal geometry. Despite abundant theoretical studies, experiments in fractal networks remain elusive. Here we experimentally investigate quantum transport in fractal networks by performing continuous-time quantum walks in fractal photonic lattices. We unveil the transport properties through the photon evolution patterns, the mean square displacement and the Pólya number. Contrarily to classical fractals, we observe anomalous transport governed solely by the fractal dimension. In addition, the critical point at which there is a transition from normal to anomalous transport depends on the fractal geometry. Our experiment allows the verification of physical laws in a quantitative manner and reveals the transport dynamics in great detail, thus opening a path to the understanding of more complex quantum phenomena governed by fractality.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on SpringerLink
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Geometry and connectivity analysis of the fractals, and implementation of quantum walks.
Fig. 2: Quantum transport in the Sierpiński gasket.
Fig. 3: Quantum transport in the Sierpiński carpet.
Fig. 4: Quantum transport in the dual Sierpiński carpet.

Similar content being viewed by others

Data availability

The data that support the findings of this study are available from the corresponding authors on reasonable request.

References

  1. Mandelbrot, B. B. Fractals: Form, Chance and Dimension (W. H. Freeman, 1977).

  2. ben-Avraham, D. & Havlin, S. Diffusion and Reactions in Fractals and Disordered Systems (Cambridge Univ. Press, 2000).

  3. Mandelbrot, B. B. The Fractal Geometry of Nature (W. H. Freeman, 1983).

  4. Gouyet, J.-F. Physics and Fractal Structures (Springer, 1996).

  5. Ivanov, P. C. et al. Multifractality in human heartbeat dynamics. Nature 399, 461–465 (1999).

    Article  ADS  Google Scholar 

  6. Bandres, M. A., Rechtsman, M. C. & Segev, M. Topological photonic quasicrystals: fractal topological spectrum and protected transport. Phys. Rev. X 6, 011016 (2016).

    Google Scholar 

  7. Tanese, D. et al. Fractal energy spectrum of a polariton gas in a Fibonacci quasiperiodic potential. Phys. Rev. Lett. 112, 146404 (2014).

    Article  ADS  Google Scholar 

  8. Hofstadter, D. R. Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Phys. Rev. B 14, 2239–2249 (1976).

    Article  ADS  Google Scholar 

  9. Brady, R. & Ball, R. Fractal growth of copper electrodeposits. Nature 309, 225–229 (1984).

    Article  ADS  Google Scholar 

  10. Goldberger, A. L. et al. Fractal dynamics in physiology: alternations with disease and aging. Proc. Natl Acad. Sci. USA 99, 2466–2472 (2002).

    Article  ADS  Google Scholar 

  11. Bassingthwaighte, J. B., Liebovitch, L. S. & West, B. J. Fractal Physiology (Springer, 2013).

  12. Peters, E. E. Fractal structure in the capital markets. Financ. Anal. J. 45, 32–37 (1989).

    Article  Google Scholar 

  13. Dean, C. R. et al. Hofstadteras butterfly and the fractal quantum Hall effect in moiré superlattices. Nature 497, 598–602 (2013).

    Article  ADS  Google Scholar 

  14. Wang, L. et al. Evidence for a fractional fractal quantum Hall effect in graphene superlattices. Science 350, 1231–1234 (2015).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Benedetti, D. Fractal properties of quantum spacetime. Phys. Rev. Lett. 102, 111303 (2009).

    Article  ADS  MathSciNet  Google Scholar 

  16. Karman, G., McDonald, G., New, G. & Woerdman, J. Fractal modes in unstable resonators. Nature 402, 138 (1999).

    Article  ADS  Google Scholar 

  17. Dudley, J. M., Finot, C., Richardson, D. J. & Millot, G. Self-similarity in ultrafast nonlinear optics. Nat. Phys. 3, 597–603 (2007).

    Article  Google Scholar 

  18. Rivera, J. A., Galvin, T. C., Steinforth, A. W. & Eden, J. G. Fractal modes and multi-beam generation from hybrid microlaser resonators. Nat. Commun. 9, 2594 (2018).

    Article  ADS  Google Scholar 

  19. Ding, J., Fan, L., Zhang, S.-Y., Zhang, H. & Yu, W.-W. Simultaneous realization of slow and fast acoustic waves using a fractal structure of Koch curve. Sci. Rep. 8, 1481 (2018).

    Article  ADS  Google Scholar 

  20. Fan, J. A. et al. Fractal design concepts for stretchable electronics. Nat. Commun. 5, 3266 (2014).

    Article  ADS  Google Scholar 

  21. Fazio, B. et al. Strongly enhanced light trapping in a two-dimensional silicon nanowire random fractal array. Light Sci. Appl. 5, e16062 (2016).

    Article  Google Scholar 

  22. Gottheim, S., Zhang, H., Govorov, A. O. & Halas, N. J. Fractal nanoparticle plasmonics: the Cayley tree. ACS Nano 9, 3284–3292 (2015).

    Article  Google Scholar 

  23. De Nicola, F. et al. Multiband plasmonic Sierpinski carpet fractal antennas. ACS Photonics 5, 2418–2425 (2018).

    Article  Google Scholar 

  24. Zhu, L.-H. et al. Broadband absorption and efficiency enhancement of an ultra-thin silicon solar cell with a plasmonic fractal. Opt. Express 21, A313–A323 (2013).

    Article  Google Scholar 

  25. Havlin, S. & Ben-Avraham, D. Diffusion in disordered media. Adv. Phys. 36, 695–798 (1987).

    Article  ADS  Google Scholar 

  26. Blumen, A., Klafter, J., White, B. & Zumofen, G. Continuous-time random walks on fractals. Phys. Rev. Lett. 53, 1301 (1984).

    Article  ADS  Google Scholar 

  27. Alexander, S. & Orbach, R. Density of states on fractals: fractons. J. Phys. Lett. 43, 625–631 (1982).

    Article  Google Scholar 

  28. Orbach, R. Dynamics of fractal networks. Science 231, 814–819 (1986).

    Article  ADS  Google Scholar 

  29. Ben-Avraham, D. & Havlin, S. Diffusion on percolation clusters at criticality. J. Phys. A Math. Gen. 15, L691 (1982).

    Article  ADS  Google Scholar 

  30. Sokolov, I. M. What is the alternative to the Alexander–Orbach relation? J. Phys. A Math. Theor. 49, 095003 (2016).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Reis, F. D. A. & Voller, V. R. Models of infiltration into homogeneous and fractal porous media with localized sources. Phys. Rev. E 99, 042111 (2019).

    Article  ADS  Google Scholar 

  32. Mülken, O. & Blumen, A. Continuous-time quantum walks: models for coherent transport on complex networks. Phys. Rep. 502, 37–87 (2011).

    Article  ADS  MathSciNet  Google Scholar 

  33. Agliari, E., Blumen, A. & Mülken, O. Dynamics of continuous-time quantum walks in restricted geometries. J. Phys. A Math. Theor. 41, 445301 (2008).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. Darázs, Z., Anishchenko, A., Kiss, T., Blumen, A. & Mülken, O. Transport properties of continuous-time quantum walks on Sierpinski fractals. Phys. Rev. E 90, 032113 (2014).

    Article  ADS  Google Scholar 

  35. Volta, A. Quantum walks and trapping on regular hyperbranched fractals. J. Phys. A Math. Theor. 42, 225003 (2009).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  36. van Veen, E., Yuan, S., Katsnelson, M. I., Polini, M. & Tomadin, A. Quantum transport in Sierpinski carpets. Phys. Rev. B 93, 115428 (2016).

    Article  ADS  Google Scholar 

  37. Feng, Z. et al. Photonic Newton’s cradle for remote energy transport. Phys. Rev. Appl. 11, 044009 (2019).

    Article  ADS  Google Scholar 

  38. Tang, H. et al. Experimental quantum fast hitting on hexagonal graphs. Nat. Photonics 12, 754–758 (2018).

    Article  ADS  Google Scholar 

  39. Perets, H. B. et al. Realization of quantum walks with negligible decoherence in waveguide lattices. Phys. Rev. Lett. 100, 170506 (2008).

    Article  ADS  Google Scholar 

  40. Peruzzo, A. et al. Quantum walks of correlated photons. Science 329, 1500–1503 (2010).

    Article  ADS  Google Scholar 

  41. Tang, H. et al. Experimental two-dimensional quantum walk on a photonic chip. Sci. Adv. 4, eaat3174 (2018).

    Article  ADS  Google Scholar 

  42. Schwartz, T., Bartal, G., Fishman, S. & Segev, M. Transport and Anderson localization in disordered two-dimensional photonic lattices. Nature 446, 52–55 (2007).

    Article  ADS  Google Scholar 

  43. Lahini, Y. et al. Anderson localization and nonlinearity in one-dimensional disordered photonic lattices. Phys. Rev. Lett. 100, 013906 (2008).

    Article  ADS  Google Scholar 

  44. Segev, M., Silberberg, Y. & Christodoulides, D. N. Anderson localization of light. Nat. Photonics 7, 197–204 (2013).

    Article  ADS  Google Scholar 

  45. Naether, U. et al. Experimental observation of superdiffusive transport in random dimer lattices. New J. Phys. 15, 013045 (2013).

    Article  ADS  Google Scholar 

  46. Eichelkraut, T. et al. Mobility transition from ballistic to diffusive transport in non-Hermitian lattices. Nat. Commun. 4, 2533 (2013).

    Article  ADS  Google Scholar 

  47. Shang, J. et al. Assembling molecular Sierpiński triangle fractals. Nat. Chem. 7, 389–393 (2015).

    Article  Google Scholar 

  48. Newkome, G. R. et al. Nanoassembly of a fractal polymer: a molecular ‘Sierpinski hexagonal gasket’. Science 312, 1782–1785 (2006).

    Article  ADS  MathSciNet  Google Scholar 

  49. Rothemund, P. W., Papadakis, N. & Winfree, E. Algorithmic self-assembly of DNA Sierpinski triangles. PLoS Biol. 2, e424 (2004).

    Article  Google Scholar 

  50. Kempkes, S. N. et al. Design and characterization of electrons in a fractal geometry. Nat. Phys. 15, 127–131 (2019).

    Article  Google Scholar 

  51. Jia, S. & Fleischer, J. W. Nonlinear light propagation in fractal waveguide arrays. Opt. Express 18, 14409–14415 (2010).

    Article  ADS  Google Scholar 

  52. Osellame, R., Cerullo, G. & Ramponi, R. (eds) Femtosecond Laser Micromachining: Photonic and Microfluidic Devices in Transparent Materials Vol. 123 (Springer, 2012).

  53. Xu, X.-Y. et al. A scalable photonic computer solving the subset sum problem. Sci. Adv. 6, eaay5853 (2020).

    Article  ADS  Google Scholar 

  54. Darázs, Z. & Kiss, T. Pólya number of the continuous-time quantum walks. Phys. Rev. A 81, 062319 (2010).

    Article  ADS  Google Scholar 

  55. Fleischmann, R., Geisel, T., Ketzmerick, R. & Petschel, G. Quantum diffusion, fractal spectra, and chaos in semiconductor microstructures. Physica D 86, 171–181 (1995).

    Article  ADS  MATH  Google Scholar 

  56. Hameroff, S. & Penrose, R. Consciousness in the universe: a review of the ‘Orch OR’ theory. Phys. Life Rev. 11, 39–78 (2014).

    Article  ADS  Google Scholar 

  57. Hameroff, S. & Penrose, R. Orchestrated reduction of quantum coherence in brain microtubules: a model for consciousness. Math. Comput. Simul. 40, 453–480 (1996).

    Article  Google Scholar 

  58. Gefen, Y., Aharony, A., Mandelbrot, B. B. & Kirkpatrick, S. Solvable fractal family, and its possible relation to the backbone at percolation. Phys. Rev. Lett. 47, 1771 (1981).

    Article  ADS  MathSciNet  Google Scholar 

  59. Agliari, E., Blumen, A. & Mülken, O. Quantum-walk approach to searching on fractal structures. Phys. Rev. A 82, 012305 (2010).

    Article  ADS  Google Scholar 

Download references

Acknowledgements

We thank J.-W. Pan for helpful discussions, X.-L. Huang and Z.-M. Li for helping in the experiments, J. Gao, R.-J. Ren and S. Freeney for proof reading, and W.-H. Zhou for assistance in formatting the figures. This research is supported by the National Key R&D Program of China (2019YFA0308700, 2019YFA0706302 and 2017YFA0303700); National Natural Science Foundation of China (NSFC) (11904229, 61734005, 11761141014, 11690033); Science and Technology Commission of Shanghai Municipality (STCSM) (20JC1416300, 2019SHZDZX01); Shanghai Municipal Education Commission (SMEC) (2017-01-07-00-02-E00049); China Postdoctoral Science Foundation (2021M692094, 2020M671091). X.-M.J. acknowledges additional support from a Shanghai talent program and support from Zhiyuan Innovative Research Center of Shanghai Jiao Tong University.

Author information

Authors and Affiliations

  1. Center for Integrated Quantum Information Technologies (IQIT), School of Physics and Astronomy and State Key Laboratory of Advanced Optical Communication Systems and Networks, Shanghai Jiao Tong University, Shanghai, China

    Xiao-Yun Xu, Xiao-Wei Wang, Dan-Yang Chen & Xian-Min Jin

  2. CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui, China

    Xiao-Yun Xu, Xiao-Wei Wang & Xian-Min Jin

  3. Institute for Theoretical Physics, Utrecht University, Utrecht, the Netherlands

    C. Morais Smith

  4. Tsung-Dao Lee Institute, Shanghai, China

    C. Morais Smith

  5. Wilczek Quantum Center, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai, China

    C. Morais Smith

  6. TuringQ Co., Ltd., Shanghai, China

    Xian-Min Jin

Authors
  1. Xiao-Yun Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiao-Wei Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dan-Yang Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. C. Morais Smith
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Xian-Min Jin
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

X.-M.J. conceived and supervised the project. X.-Y.X. performed the simulations and fabricated the photonic chips. X.-Y.X., X.-W.W., D.-Y.C. and X.-M.J. performed the experiments and analysed the data. X.-Y.X., C.M.S. and X.-M.J. interpreted the data and wrote the paper, with input from all the other authors.

Corresponding authors

Correspondence to C. Morais Smith or Xian-Min Jin.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature Photonics thanks Eric Heller and Shengjun Yuan for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplemental Information

Supplementary Figs. 1–27 and Discussion.

Supplementary Video 1

The dynamical evolution of the MSD for the Sierpiński gasket and the corresponding evolution patterns of photons.

Supplementary Video 2

The dynamical evolution of the MSD for the Sierpiński carpet and the corresponding evolution patterns of photons.

Supplementary Video 3

The dynamical evolution of the MSD for the dual Sierpiński carpet and the corresponding evolution patterns of photons.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xu, XY., Wang, XW., Chen, DY. et al. Quantum transport in fractal networks. Nat. Photon. 15, 703–710 (2021). https://doi.org/10.1038/s41566-021-00845-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41566-021-00845-4

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing