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.

  • Letter
  • Published:

Pattern formation in a driven Bose–Einstein condensate

Nature Physics volume 16, pages 652–656 (2020)Cite this article

Subjects

Abstract

Pattern formation is ubiquitous in nature at all scales, from morphogenesis and cloud formation to galaxy filamentation. How patterns emerge in a homogeneous system is a fundamental question across interdisciplinary research including hydrodynamics1, condensed matter physics2, nonlinear optics3, cosmology4 and bio-chemistry5,6. Paradigmatic examples, such as Rayleigh–Bénard convection rolls and Faraday waves7,8, have been studied extensively and found numerous applications9,10,11. How such knowledge applies to quantum systems and whether the patterns in a quantum system can be controlled remain intriguing questions. Here we show that the density patterns with two- (D2), four- (D4) and six-fold (D6) symmetries can emerge in Bose–Einstein condensates on demand when the atomic interactions are modulated at multiple frequencies. The D6 pattern, in particular, arises from a resonant wave-mixing process that establishes phase coherence of the excitations that respect the symmetry. Our experiments explore a novel class of non-equilibrium phenomena in quantum gases, as well as a new route to prepare quantum states with desired correlations.

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

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

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

Fig. 1: Pattern formation in a BEC with interaction modulation at two frequencies.
Fig. 2: Formation of density waves with D6 symmetry.
Fig. 3: Density wave patterns in real space.
Fig. 4: Coherent properties of D4 and D6 density waves.

Similar content being viewed by others

Data availability

The data represented in Figs. 2d, 3c, 4b,f are available as Source Data Figs. 2, 3 and 4. All other data that support the plots within this paper are available from the corresponding author upon reasonable request.

References

  1. Cross, M. C. & Hohenberg, P. C. Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–1112 (1993).

    Article  ADS  Google Scholar 

  2. Mouritsen, O. G. Pattern formation in condensed matter. Int. J. Modern Phys. B 04, 1925–1954 (1990).

    Article  ADS  Google Scholar 

  3. Arecchi, F., Boccaletti, S. & Ramazza, P. Pattern formation and competition in nonlinear optics. Phys. Rep. 318, 1–83 (1999).

    Article  ADS  Google Scholar 

  4. Liddle, A. R. & Lyth, D. H. Cosmological Inflation and Large-Scale Structure (Cambridge Univ. Press, 2000).

  5. Turing, A. M. The chemical basis of morphogenesis. Phil. Trans. R. Soc. Lond. B Biol. Sci. 237, 37–72 (1952).

    Article  ADS  MathSciNet  Google Scholar 

  6. Maini, P. K., Painter, K. J. & Chau, H. N. P. Spatial pattern formation in chemical and biological systems. J. Chem. Soc. Faraday Trans. 93, 3601–3610 (1997).

    Article  Google Scholar 

  7. Bodenschatz, E., Pesch, W. & Ahlers, G. Recent developments in Rayleigh–Bénard convection. Annu. Rev. Fluid Mech. 32, 709–778 (2000).

    Article  ADS  Google Scholar 

  8. Miles, J. On Faraday waves. J. Fluid Mech. 248, 671–683 (1993).

    Article  ADS  MathSciNet  Google Scholar 

  9. Edwards, W. S. & Fauve, S. Patterns and quasi-patterns in the Faraday experiment. J. Fluid Mech. 278, 123–148 (1994).

    Article  ADS  MathSciNet  Google Scholar 

  10. Lifshitz, R. & Petrich, D. M. Theoretical model for Faraday waves with multiple-frequency forcing. Phys. Rev. Lett. 79, 1261–1264 (1997).

    Article  ADS  Google Scholar 

  11. Arbell, H. & Fineberg, J. Pattern formation in two-frequency forced parametric waves. Phys. Rev. E 65, 036224 (2002).

    Article  ADS  Google Scholar 

  12. Swift, J. & Hohenberg, P. C. Hydrodynamic fluctuations at the convective instability. Phys. Rev. A 15, 319–328 (1977).

    Article  ADS  Google Scholar 

  13. Pomeau, Y. & Manneville, P. Intermittent transition to turbulence in dissipative dynamical systems. Commun. Math. Phys. 74, 189–197 (1980).

    Article  ADS  MathSciNet  Google Scholar 

  14. Sakaguchi, H. & Brand, H. R. Localized patterns for the quintic complex Swift–Hohenberg equation. Physica D 117, 95–105 (1998).

    Article  ADS  Google Scholar 

  15. Hoyle, R. Pattern Formation: An Introduction to Methods (Cambridge Univ. Press, 2006).

  16. Landau, L. D. & Lifshitz, E. M. Fluid Mechanics Vol. 6 (Course of Theoretical Physics, Pergamon Press, 1959).

  17. Temam, R. & Chorin, A. Navier Stokes Equations: Theory and Numerical Analysis (AMS Chelsea Publishing, 1978).

  18. Ardizzone, V. et al. Formation and control of Turing patterns in a coherent quantum fluid. Sci. Rep. 3, 3016 (2013).

    Article  Google Scholar 

  19. Engels, P., Atherton, C. & Hoefer, M. A. Observation of Faraday waves in a Bose–Einstein condensate. Phys. Rev. Lett. 98, 095301 (2007).

    Article  ADS  Google Scholar 

  20. Groot, A. Excitations in Hydrodynamic Ultra-Cold Bose Gases. PhD thesis, Utrecht Univ. (2015).

  21. Nguyen, J. H. V. et al. Parametric excitation of a Bose–Einstein condensate: from Faraday waves to granulation. Phys. Rev. X 9, 011052 (2019).

    Google Scholar 

  22. Pollack, S. E. et al. Collective excitation of a Bose–Einstein condensate by modulation of the atomic scattering length. Phys. Rev. A 81, 053627 (2010).

    Article  ADS  Google Scholar 

  23. Kronjäger, J., Becker, C., Soltan-Panahi, P., Bongs, K. & Sengstock, K. Spontaneous pattern formation in an antiferromagnetic quantum gas. Phys. Rev. Lett. 105, 090402 (2010).

    Article  ADS  Google Scholar 

  24. Hung, C.-L., Gurarie, V. & Chin, C. From cosmology to cold atoms: observation of Sakharov oscillations in a quenched atomic superfluid. Science 341, 1213–1215 (2013).

    Article  ADS  Google Scholar 

  25. Kadau, H. et al. Observing the Rosensweig instability of a quantum ferrofluid. Nature 530, 194–197 (2016).

    Article  ADS  Google Scholar 

  26. Li, J.-R. et al. A stripe phase with supersolid properties in spin–orbit-coupled Bose–Einstein condensates. Nature 543, 91–94 (2017).

    Article  Google Scholar 

  27. Böttcher, F. et al. Transient supersolid properties in an array of dipolar quantum droplets. Phys. Rev. X 9, 011051 (2019).

    Google Scholar 

  28. Tanzi, L. et al. Observation of a dipolar quantum gas with metastable supersolid properties. Phys. Rev. Lett. 122, 130405 (2019).

    Article  ADS  Google Scholar 

  29. Chomaz, L. et al. Long-lived and transient supersolid behaviors in dipolar quantum gases. Phys. Rev. X 9, 021012 (2019).

    Google Scholar 

  30. Clark, L. W., Gaj, A., Feng, L. & Chin, C. Collective emission of matter-wave jets from driven Bose–Einstein condensates. Nature 551, 356–359 (2017).

    Article  Google Scholar 

  31. Fu, H. et al. Density waves and jet emission asymmetry in Bose fireworks. Phys. Rev. Lett. 121, 243001 (2018).

    Article  ADS  Google Scholar 

  32. Chin, C., Grimm, R., Julienne, P. & Tiesinga, E. Feshbach resonances in ultracold gases. Rev. Mod. Phys. 82, 1225–1286 (2010).

    Article  ADS  Google Scholar 

  33. Feng, L., Hu, J., Clark, L. W. & Chin, C. Correlations in high-harmonic generation of matter-wave jets revealed by pattern recognition. Science 363, 521–524 (2019).

    Article  ADS  Google Scholar 

  34. Abe, H. et al. Faraday instability of superfluid surface. Phys. Rev. E 76, 046305 (2007).

    Article  ADS  Google Scholar 

  35. Hung, C.-L. et al. Extracting density–density correlations from in situ images of atomic quantum gases. New J. Phys. 13, 075019 (2011).

    Article  ADS  Google Scholar 

  36. Hu, J., Feng, L., Zhang, Z. & Chin, C. Quantum simulation of Unruh radiation. Nat. Phys. 15, 785–789 (2019).

    Article  Google Scholar 

  37. Petrov, D. S., Holzmann, M. & Shlyapnikov, G. V. Bose–Einstein condensation in quasi-2D trapped gases. Phys. Rev. Lett. 84, 2551–2555 (2000).

    Google Scholar 

Download references

Acknowledgements

We thank K. Patel for careful reading of the manuscript. This work is supported by National Science Foundation (NSF) grant no. PHY-1511696, the Army Research Office Multidisciplinary Research Initiative under grant no. W911NF-14-1-0003 and the University of Chicago Materials Research Science and Engineering Center, which is funded by the NSF under grant no. DMR-1420709. J.H. acknowledges financial support from the National Natural Science Foundation of China under grant no. 11974202.

Author information

Authors and Affiliations

  1. James Franck Institute, University of Chicago, Chicago, IL, USA

    Zhendong Zhang, Kai-Xuan Yao, Lei Feng & Cheng Chin

  2. Enrico Fermi Institute, University of Chicago, Chicago, IL, USA

    Zhendong Zhang, Kai-Xuan Yao, Lei Feng & Cheng Chin

  3. Department of Physics, University of Chicago, Chicago, IL, USA

    Zhendong Zhang, Kai-Xuan Yao, Lei Feng & Cheng Chin

  4. Department of Physics, Tsinghua University, Beijing, China

    Jiazhong Hu

  5. State Key Laboratory, Tsinghua University, Beijing, China

    Jiazhong Hu

  6. Frontier Science Center for Quantum Information, Beijing, China

    Jiazhong Hu

Authors
  1. Zhendong Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kai-Xuan Yao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Lei Feng
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jiazhong Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Cheng Chin
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Z.Z. and K.-X.Y. performed the experiments, built the theoretical model and analysed the data. L.F. contributed to development of the modulation hardware and pattern recognition scheme. J.H. contributed to the discussion of the results. C.C. supervised the work. All authors contributed to writing the manuscript.

Corresponding author

Correspondence to Cheng Chin.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature Physics thanks Simon Cornish and other, anonymous, reviewer(s) 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

Supplementary Information

Supplementary Information.

Source data

Source Data Fig. 2

Correlation of D6 pattern formation.

Source Data Fig. 3

Spatial correlation for different schemes.

Source Data Fig. 4

Angular correlation and phase histogram.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, Z., Yao, KX., Feng, L. et al. Pattern formation in a driven Bose–Einstein condensate. Nat. Phys. 16, 652–656 (2020). https://doi.org/10.1038/s41567-020-0839-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41567-020-0839-3

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