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:

Experimental determination of the finite-temperature phase diagram of a spin–orbit coupled Bose gas

Nature Physics volume 10, pages 314–320 (2014)Cite this article

Subjects

Abstract

Spin–orbit (SO) coupling leads to numerous phenomena in electron systems. Artificial SO coupling in ultracold neutral atoms provides the opportunity to study such phenomena in bosonic systems, which exhibit superfluidity and various symmetry-breaking condensate phases. In general, a richer structure of symmetry breaking results in a nontrivial finite-temperature phase diagram, but the thermodynamics of the SO-coupled Bose gas at finite temperature remains unknown both in theory and experiment. Here we experimentally determine a new finite-temperature phase transition that is consistent with the transition between the stripe ordered phase and the magnetized phase. We also observe that the magnetic phase and the Bose condensate transitions occur simultaneously as temperature decreases. We determine the entire finite-temperature phase diagram of the SO-coupled Bose gas, thus illustrating the power of quantum simulation.

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

Figure 1: Zero-temperature phase diagram and three scenarios for the finite-temperature phase diagram for an SO-coupled Bose gas.
Figure 2: Critical temperature Tc of a spin–orbit coupled Bose gas.
Figure 3: Histogram of condensate magnetization for Ω<Ω1.
Figure 4: Phase transition between the ST condensate and the MG condensate at very low temperatures.
Figure 5: Finite-temperature phase diagram of spin–orbit coupled Bose gas.
Figure 6: Formation of magnetic order for Ω1<Ω<Ω2.

Similar content being viewed by others

References

  1. Hasan, M. Z. & Kane, C. L. Topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).

    Article  ADS  Google Scholar 

  2. Qi, X-L. & Zhang, S-C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011).

    Article  ADS  Google Scholar 

  3. Lin, Y-J., Jiménez-García, K. & Spielman, I. B. Spin–orbit-coupled Bose–Einstein condensates. Nature 471, 83–86 (2011).

    Article  ADS  Google Scholar 

  4. Wang, P. et al. Spin–orbit coupled degenerate Fermi gases. Phys. Rev. Lett. 109, 095301 (2012).

    Article  ADS  Google Scholar 

  5. Cheuk, L. M. et al. Spin-injection spectroscopy of a spin–orbit coupled Fermi gas. Phys. Rev. Lett. 109, 095302 (2012).

    Article  ADS  Google Scholar 

  6. Zhang, J-Y. et al. Collective dipole oscillation of a spin–orbit coupled Bose–Einstein condensate. Phys. Rev. Lett. 109, 115301 (2012).

    Article  ADS  Google Scholar 

  7. Williams, R. A. et al. Synthetic partial waves in ultracold atomic collisions. Science 335, 314–317 (2012).

    Article  ADS  Google Scholar 

  8. Qu, C., Hamner, C., Gong, M., Zhang, C. & Engels, P. Observation of Zitterbewegung in a spin–orbit-coupled Bose–Einstein condensate. Phys. Rev. A 88, 021604(R) (2013).

    Article  ADS  Google Scholar 

  9. Zhang, L. et al. Stability of excited dressed states with spin–orbit coupling. Phys. Rev. A 87, 011601(R) (2013).

    Article  ADS  Google Scholar 

  10. Wang, C., Gao, C., Jian, C-M. & Zhai, H. Spin–orbit coupled spinor Bose–Einstein condensates. Phys. Rev. Lett. 105, 160403 (2010).

    Article  ADS  Google Scholar 

  11. Ho, T-L. & Zhang, S. Bose–Einstein condensates with spin–orbit interaction. Phys. Rev. Lett. 107, 150403 (2011).

    Article  ADS  Google Scholar 

  12. Li, Y., Pitaevskii, L. P. & Stringari, S. Quantum tricriticality and phase transitions in spin–orbit coupled Bose–Einstein condensates. Phys. Rev. Lett. 108, 225301 (2012).

    Article  ADS  Google Scholar 

  13. Wu, C-J., Mondragon-Shem, I. & Zhou, X-F. Unconventional Bose–Einstein condensations from spin–orbit coupling. Chin. Phys. Lett. 28, 097102 (2011).

    Article  ADS  Google Scholar 

  14. Jian, C-M. & Zhai, H. Paired superfluidity and fractionalized vortices in systems of spin–orbit coupled bosons. Phys. Rev. B 84, 060508(R) (2011).

    Article  ADS  Google Scholar 

  15. Gopalakrishnan, S., Lamacraft, A. & Goldbart, P. M. Universal phase structure of dilute Bose gases with Rashba spin–orbit coupling. Phys. Rev. A 84, 061604(R) (2011).

    Article  ADS  Google Scholar 

  16. Ozawa, T. & Baym, G. Stability of ultracold atomic Bose condensates with Rashba spin–orbit coupling against quantum and thermal fluctuations. Phys. Rev. Lett. 109, 025301 (2012).

    Article  ADS  Google Scholar 

  17. Cui, X. & Zhou, Q. Enhancement of condensate depletion due to spin–orbit coupling. Phys. Rev. A 87, 031604(R) (2013).

    Article  ADS  Google Scholar 

  18. Zheng, W., Yu, Z-Q., Cui, X. & Zhai, H. Properties of Bose gases with Raman-induced spin–orbit coupling. J. Phys. B 46, 134007 (2013).

    Article  ADS  Google Scholar 

  19. Sedrakyan, T. A., Kamenev, A. & Glazman, L. I. Composite fermion state of spin–orbit-coupled bosons. Phys. Rev. A 86, 063639 (2012).

    Article  ADS  Google Scholar 

  20. Zhu, Q., Zhang, C. & Wu, B. Exotic superfluidity in spin–orbit coupled Bose–Einstein condensates. Europhys. Lett. 100, 50003 (2012).

    Article  ADS  Google Scholar 

  21. Martone, G. I., Li, Y., Pitaevskii, L. P. & Stringari, S. Anisotropic dynamics of a spin–orbit coupled Bose–Einstein condensate. Phys. Rev. A 86, 063621 (2012).

    Article  ADS  Google Scholar 

  22. Mukerjee, S., Xu, C. & Moore, J. E. Topological defects and the superfluid transition of the s=1 spinor condensate in two dimensions. Phys. Rev. Lett. 97, 120406 (2006).

    Article  ADS  Google Scholar 

  23. James, A. J. A. & Lamacraft, A. Phase diagram of two-dimensional polar condensates in a magnetic field. Phys. Rev. Lett. 106, 140402 (2011).

    Article  ADS  Google Scholar 

  24. Shi, Y., Lamacraft, A. & Fendley, P. Boson pairing and unusual criticality in a generalized XY model. Phys. Rev. Lett. 107, 240601 (2011).

    Article  ADS  Google Scholar 

  25. Spielman, I. B. Raman process and effective gauge potentials. Phys. Rev. A 79, 063613 (2009).

    Article  ADS  Google Scholar 

  26. Wheatley, J. C. Experimental properties of superfluid 3He. Rev. Mod. Phys. 47, 415–470 (1975).

    Article  ADS  Google Scholar 

  27. de la Cruz, C. et al. Magnetic order close to superconductivity in the iron-based layered LaO1−xFxFeAs system. Nature 453, 899–902 (2008).

    Article  ADS  Google Scholar 

  28. Fang, C., Yao, H., Tsai, W-F., Hu, J-P. & Kivelson, S. A. Theory of electron nematic order in LaFeAsO. Phys. Rev. B 77, 224509 (2008).

    Article  ADS  Google Scholar 

  29. Xu, C-K., Müller, M. & Sachdev, S. Ising and spin orders in the iron-based superconductors. Phys. Rev. B 78, 020501(R) (2008).

    Article  ADS  Google Scholar 

  30. Li, Y., Martone, G. I., Pitaevskii, L. P. & Stringari, S. Superstripes and the excitation spectrum of a spin–orbit-coupled Bose–Einstein condensate. Phys. Rev. Lett. 110, 235302 (2013).

    Article  ADS  Google Scholar 

Download references

Acknowledgements

We acknowledge insightful discussions with C. Chin and T-L. Ho. S. C. thanks B. Zhao for his careful reading of the manuscript. This work has been supported by the NNSF of China, the CAS, the National Fundamental Research Program (under Grant No. 2011CB921300, No. 2011CB921500), NSERC and Tsinghua University Initiative Scientific Research Program.

Author information

Author notes

  1. Si-Cong Ji: These authors contributed equally to this work.

Authors and Affiliations

  1. Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, Shanghai Branch, University of Science and Technology of China, Shanghai 201315, China

    Si-Cong Ji, Jin-Yi Zhang, Long Zhang, Zhi-Dong Du, You-Jin Deng, Shuai Chen & Jian-Wei Pan

  2. Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China

    Si-Cong Ji, Jin-Yi Zhang, Long Zhang, Zhi-Dong Du, You-Jin Deng, Shuai Chen & Jian-Wei Pan

  3. Institute for Advanced Study, Tsinghua University, Beijing 100084, China

    Wei Zheng & Hui Zhai

Authors
  1. Si-Cong Ji
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jin-Yi Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Long Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Zhi-Dong Du
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Wei Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. You-Jin Deng
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Hui Zhai
    View author publications

    You can also search for this author in PubMed Google Scholar

  8. Shuai Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  9. Jian-Wei Pan
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

S.C. and J-W.P. planned and supervised the project. S-C.J., J-Y.Z., Z-D.D. and S.C. performed the experiments, L.Z., W.Z., Y-J.D. and H.Z. provided theoretical support, and all the authors contributed to analysis of the data and writing the manuscript.

Corresponding authors

Correspondence to Shuai Chen or Jian-Wei Pan.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Supplementary information

Supplementary Information

Supplementary Information (PDF 483 kb)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ji, SC., Zhang, JY., Zhang, L. et al. Experimental determination of the finite-temperature phase diagram of a spin–orbit coupled Bose gas. Nature Phys 10, 314–320 (2014). https://doi.org/10.1038/nphys2905

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nphys2905

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