Journal of Low Temperature Physics

, Volume 189, Issue 5–6, pp 262–275 | Cite as

Quantum Phases and Collective Excitations of a Spin-Orbit-Coupled Bose–Einstein Condensate in a One-Dimensional Optical Lattice

  • G. I. Martone


The ground state of a spin-orbit-coupled Bose gas in a one-dimensional optical lattice is known to exhibit a mixed regime, where the condensate wave function is given by a superposition of multiple Bloch-wave components, and an unmixed one, in which the atoms occupy a single Bloch state. The unmixed regime features two unpolarized Bloch-wave phases, having quasimomentum at the center or at the edge of the first Brillouin zone, and a polarized Bloch-wave phase at intermediate quasimomenta. By calculating the critical values of the Raman coupling and of the lattice strength at the transitions among the various phases, we show the existence of a tricritical point where the mixed, the polarized and the edge-quasimomentum phases meet, and whose appearance is a consequence of the spin-dependent interaction. Furthermore, we evaluate the excitation spectrum in the unmixed regime and we characterize the behavior of the phonon and the roton modes, pointing out the instabilities occurring when a phase transition is approached.


Bose–Einstein condensation Spin-orbit coupling Optical lattice 



Useful discussions with T. Ozawa, D. Papoular, N. Pavloff, C. Qu, and S. Stringari are acknowledged. The research leading to these results has received funding from the European Research Council under European Community’s Seventh Framework Programme (FP7/2007-2013 Grant Agreement No. 341197).


  1. 1.
    Y.-J. Lin, K. Jimenez-Garcia, I.B. Spielman, Nature 471, 83 (2011). doi: 10.1038/nature09887 ADSCrossRefGoogle Scholar
  2. 2.
    J. Dalibard, F. Gerbier, G. Juzeliūnas, P. Öhberg, Rev. Mod. Phys. 83, 1523 (2011). doi: 10.1103/RevModPhys.83.1523 ADSCrossRefGoogle Scholar
  3. 3.
    V. Galitski, I.B. Spielman, Nature 494, 49 (2013). doi: 10.1038/nature11841 ADSCrossRefGoogle Scholar
  4. 4.
    X. Zhou, Y. Li, Z. Cai, C. Wu, J. Phys. B 46, 134001 (2013). doi: 10.1088/0953-4075/46/13/134001 ADSCrossRefGoogle Scholar
  5. 5.
    N. Goldman, G. Juzeliūnas, P. Öhberg, I.B. Spielman, Rep. Prog. Phys. 77, 126401 (2014). doi: 10.1088/0034-4885/77/12/126401 ADSCrossRefGoogle Scholar
  6. 6.
    H. Zhai, Rep. Prog. Phys. 78, 026001 (2015). doi: 10.1088/0034-4885/78/2/026001 ADSCrossRefGoogle Scholar
  7. 7.
    Y. Li, G.I. Martone, S. Stringari, Spin-orbit-coupled Bose–Einstein condensates, in Annual Review of Cold Atoms and Molecules, vol. 3, chap. 5, ed. by K.W. Madison, K. Bongs, L.D. Carr, A.M. Rey, H. Zhai (World Scientific, Singapore, 2015), pp. 201–250Google Scholar
  8. 8.
    Y. Zhang, M.E. Mossman, T. Busch, P. Engels, C. Zhang, Front. Phys. 11, 118103 (2016). doi: 10.1007/s11467-016-0560-y CrossRefGoogle Scholar
  9. 9.
    J. Larson, J.-P. Martikainen, A. Collin, E. Sjöqvist, Phys. Rev. A 82, 043620 (2010). doi: 10.1103/PhysRevA.82.043620 ADSCrossRefGoogle Scholar
  10. 10.
    H. Sakaguchi, B. Li, Phys. Rev. A 87, 015602 (2013). doi: 10.1103/PhysRevA.87.015602 ADSCrossRefGoogle Scholar
  11. 11.
    Y. Zhang, C. Zhang, Phys. Rev. A 87, 023611 (2013). doi: 10.1103/PhysRevA.87.023611 ADSCrossRefGoogle Scholar
  12. 12.
    Y.V. Kartashov, V.V. Konotop, F.K. Abdullaev, Phys. Rev. Lett. 111, 060402 (2013). doi: 10.1103/PhysRevLett.111.060402 ADSCrossRefGoogle Scholar
  13. 13.
    Y. Cheng, G. Tang, S.K. Adhikari, Phys. Rev. A 89, 063602 (2014). doi: 10.1103/PhysRevA.89.063602 ADSCrossRefGoogle Scholar
  14. 14.
    M. Salerno, F.K. Abdullaev, arXiv:1501.07296
  15. 15.
    W. Li, L. Chen, Z. Chen, Y. Hu, Z. Zhang, Z. Liang, Phys. Rev. A 91, 023629 (2015). doi: 10.1103/PhysRevA.91.023629 ADSCrossRefGoogle Scholar
  16. 16.
    Y. Zhang, Y. Xu, T. Busch, Phys. Rev. A 91, 043629 (2015). doi: 10.1103/PhysRevA.91.043629 ADSCrossRefGoogle Scholar
  17. 17.
    T.F.J. Poon, X.-J. Liu, Phys. Rev. A 93, 063420 (2016). doi: 10.1103/PhysRevA.93.063420 ADSCrossRefGoogle Scholar
  18. 18.
    Z. Chen, Z. Liang, Phys. Rev. A 93, 013601 (2016). doi: 10.1103/PhysRevA.93.013601 ADSCrossRefGoogle Scholar
  19. 19.
    G.I. Martone, T. Ozawa, C. Qu, S. Stringari, Phys. Rev. A 94, 043629 (2016). doi: 10.1103/PhysRevA.94.043629 ADSCrossRefGoogle Scholar
  20. 20.
    H.M. Hurst, J.H. Wilson, J.H. Pixley, I.B. Spielman, S.S. Natu, Phys. Rev. A 94, 063613 (2016). doi: 10.1103/PhysRevA.94.063613 ADSCrossRefGoogle Scholar
  21. 21.
    C. Hamner, Y. Zhang, M.A. Khamehchi, M.J. Davis, P. Engels, Phys. Rev. Lett. 114, 070401 (2015). doi: 10.1103/PhysRevLett.114.070401 ADSCrossRefGoogle Scholar
  22. 22.
    Y.A. Bychkov, E.I. Rashba, J. Phys. C 17, 6039 (1984). doi: 10.1088/0022-3719/17/33/015 ADSCrossRefGoogle Scholar
  23. 23.
    G. Dresselhaus, Phys. Rev. 100, 580 (1955). doi: 10.1103/PhysRev.100.580 ADSCrossRefGoogle Scholar
  24. 24.
    N.W. Ashcroft, N.D. Mermin, Solid State Physics (Saunders College Publishing, Philadelphia, 1976)zbMATHGoogle Scholar
  25. 25.
    C.J. Pethick, H. Smith, Bose–Einstein Condensation in Dilute Gases, 2nd edn. (Cambridge University Press, Cambridge, 2008)CrossRefGoogle Scholar
  26. 26.
    L.P. Pitaevskii, S. Stringari, Bose–Einstein Condensation and Superfluidity (Oxford University Press, Oxford, 2016)CrossRefzbMATHGoogle Scholar
  27. 27.
    T.-L. Ho, S. Zhang, Phys. Rev. Lett. 107, 150403 (2011). doi: 10.1103/PhysRevLett.107.150403 ADSCrossRefGoogle Scholar
  28. 28.
    Y. Li, L.P. Pitaevskii, S. Stringari, Phys. Rev. Lett. 108, 225301 (2012). doi: 10.1103/PhysRevLett.108.225301 ADSCrossRefGoogle Scholar
  29. 29.
    Y. Li, G.I. Martone, L.P. Pitaevskii, S. Stringari, Phys. Rev. Lett. 110, 235302 (2013). doi: 10.1103/PhysRevLett.110.235302 ADSCrossRefGoogle Scholar
  30. 30.
    Y. Li, G.I. Martone, S. Stringari, EPL 99, 56008 (2012). doi: 10.1209/0295-5075/99/56008 ADSCrossRefGoogle Scholar
  31. 31.
    J.-Y. Zhang, S.-C. Ji, Z. Chen, L. Zhang, Z.-D. Du, B. Yan, G.-S. Pan, B. Zhao, Y.-J. Deng, H. Zhai, S. Chen, J.-W. Pan, Phys. Rev. Lett. 109, 115301 (2012). doi: 10.1103/PhysRevLett.109.115301 ADSCrossRefGoogle Scholar
  32. 32.
    G.I. Martone, Y. Li, L.P. Pitaevskii, S. Stringari, Phys. Rev. A 86, 063621 (2012). doi: 10.1103/PhysRevA.86.063621 ADSCrossRefGoogle Scholar
  33. 33.
    S.-C. Ji, L. Zhang, X.-T. Xu, Z. Wu, Y. Deng, S. Chen, J.-W. Pan, Phys. Rev. Lett. 114, 105301 (2015). doi: 10.1103/PhysRevLett.114.105301 ADSCrossRefGoogle Scholar
  34. 34.
    W. Zheng, Z.-Q. Yu, X. Cui, H. Zhai, J. Phys. B 46, 134007 (2013). doi: 10.1088/0953-4075/46/13/134007 ADSCrossRefGoogle Scholar
  35. 35.
    M.A. Khamehchi, Y. Zhang, C. Hamner, T. Busch, P. Engels, Phys. Rev. A 90, 063624 (2014). doi: 10.1103/PhysRevA.90.063624 ADSCrossRefGoogle Scholar
  36. 36.
    D. Toniolo, J. Linder, Phys. Rev. A 89, 061605(R) (2014). doi: 10.1103/PhysRevA.89.061605 ADSCrossRefGoogle Scholar
  37. 37.
    G.I. Martone, Y. Li, S. Stringari, Phys. Rev. A 90, 041604(R) (2014). doi: 10.1103/PhysRevA.90.041604 ADSCrossRefGoogle Scholar
  38. 38.
    J. Li, W. Huang, B. Shteynas, S. Burchesky, F.Ç. Top, E. Su, J. Lee, A.O. Jamison, W. Ketterle, Phys. Rev. Lett. 117, 185301 (2016). doi: 10.1103/PhysRevLett.117.185301 ADSCrossRefGoogle Scholar
  39. 39.
    J. Li, J. Lee, W. Huang, S. Burchesky, B. Shteynas, F.Ç. Top, A.O. Jamison, W. Ketterle, Nature 543, 91 (2017). doi: 10.1038/nature21431 ADSCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.LPTMS, CNRSUniv. Paris-Sud, Université Paris-SaclayOrsayFrance

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