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Identifying quantum phase transitions using artificial neural networks on experimental data


Machine-learning techniques such as artificial neural networks are currently revolutionizing many technological areas and have also proven successful in quantum physics applications1,2,3,4. Here, we employ an artificial neural network and deep-learning techniques to identify quantum phase transitions from single-shot experimental momentum-space density images of ultracold quantum gases and obtain results that were not feasible with conventional methods. We map out the complete two-dimensional topological phase diagram of the Haldane model5,6,7 and provide an improved characterization of the superfluid-to-Mott-insulator transition in an inhomogeneous Bose–Hubbard system8,9,10. Our work points the way to unravel complex phase diagrams of general experimental systems, where the Hamiltonian and the order parameters might not be known.

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Data availability

Source data for Figs. 2 and 3 are available in the Supplementary information. All data files including onnx files of the trained networks are available from the corresponding author on request.


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We thank M. Lewenstein for stimulating our interest in machine learning of quantum phase transitions and A. Dauphin and J. Thywissen for useful discussions. The computational resources were provided by the PHYSnet-Rechenzentrum of Universität Hamburg and we thank B. Krause-Kyora and M. Stieben for technical support. We acknowledge financial support from the Deutsche Forschungsgemeinschaft via the Research Unit FOR 2414 and the Collaborative Research Center SFB 925. B.S.R. acknowledges financial support from the European Commission (Marie Skłodowska Curie Fellowship ISOTOP, grant number 652837).

Author information

B.S.R., M.T. and L.A. took the experimental data on the Haldane system. C.B. took the experimental data on the Hubbard system. N.K., B.S.R., M.T., L.A. and C.B. evaluated and analysed the data. M.T., N.K., N.F. and L.A. performed numerical calculations including the Floquet phase diagram. K.S. and C.W. conceived and supervised the project. All authors substantially contributed to the interpretation of the results and the writing of the manuscript.

Correspondence to Klaus Sengstock.

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The authors declare no competing interests.

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Supplementary information

Supplementary Information

Supplementary text, Figs. 1–7 and references.

Supplementary Data 1

Source data for Fig. 2b.

Supplementary Data 2

Source data for Fig. 2c.

Supplementary Data 3

Source data for Fig. 2d.

Supplementary Data 4

Source data for Fig. 3b.

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Further reading

Fig. 1: Using a neural network to identify physical phases from experimental images.
Fig. 2: Mapping out a topological phase diagram using a neural network.
Fig. 3: Characterizing the superfluid-to-Mott-insulator transition.