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Quantum transport in fractal networks


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.

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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.

Data availability

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


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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




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.

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The authors declare no competing interests.

Additional information

Peer review informationNature 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.

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Xu, XY., Wang, XW., Chen, DY. et al. Quantum transport in fractal networks. Nat. Photon. (2021).

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