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Tying quantum knots

Nature Physics volume 12, pages 478–483 (2016)Cite this article

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Abstract

As topologically stable objects in field theories, knots have been put forward to explain various persistent phenomena in systems ranging from atoms and molecules to cosmic textures in the universe. Recent experiments have reported the observation of knots in different classical contexts. However, no experimental observation of knots has yet been reported in quantum matter. Here we demonstrate the experimental creation and detection of knot solitons in the order parameter of a spinor Bose–Einstein condensate. The observed texture corresponds to a topologically nontrivial element of the third homotopy group and exhibits the celebrated Hopf fibration, which unites many seemingly unrelated physical phenomena. Our work calls for future studies of the stability and dynamics knot solitons in the quantum regime.

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Figure 1: Structure of the knot soliton and the method of its creation.
Figure 2: Tying the knot soliton by winding the nematic vector.
Figure 3: Comparison of experiment with theory.
Figure 4: Numerical simulation of the knot creation before expansion.
Figure 5: Linked preimages.
Figure 6: Comparison of experiment with theory for x and y projections.

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Acknowledgements

We acknowledge funding by the National Science Foundation (grant PHY-1205822), by the Academy of Finland through its Centres of Excellence Program (grant nos 251748 and 284621) and grants (nos 135794 and 272806), Finnish Doctoral Programme in Computational Sciences, and the Magnus Ehrnrooth Foundation. CSC—IT Center for Science Ltd. (Project No. ay2090) and Aalto Science-IT project are acknowledged for computational resources. We thank N. Johnson for making public his Hopf fibration code, A. Li for assistance with figures, and W. Lee and S. J. Vickery for experimental assistance.

Author information

Author notes

  1. M. W. Ray & A. H. Gheorghe

    Present address: Present addresses: Department of Physics and Astronomy, California State University Sacramento, Sacramento, California 95819–6041, USA (M.W.R.); Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA (A.H.G.).,

Authors and Affiliations

  1. Department of Physics and Astronomy, Amherst College, Amherst, Massachusetts 01002–5000, USA

    D. S. Hall, M. W. Ray & A. H. Gheorghe

  2. Department of Applied Physics, QCD Labs, COMP Centre of Excellence, Aalto University, PO Box 13500, FI–00076 Aalto, Finland

    K. Tiurev, E. Ruokokoski & M. Möttönen

  3. Department of Mathematical Information Technology, University of Jyväskylä, PO Box 35, FI–40014 University of Jyväskylä, Finland

    M. Möttönen

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  1. D. S. Hall
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Contributions

M.W.R., A.H.G. and D.S.H. developed and conducted the experiments and analysed the data. K.T. and E.R. perfomed the numerical simulations under the guidance of M.M., who provided the initial suggestions for the experiment. M.M. and D.S.H. developed the analytical interpretation of the m = 0 data as preimages. All authors discussed both experimental and theoretical results and commented on the manuscript.

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Correspondence to D. S. Hall.

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Hall, D., Ray, M., Tiurev, K. et al. Tying quantum knots. Nature Phys 12, 478–483 (2016). https://doi.org/10.1038/nphys3624

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