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Coupling two order parameters in a quantum gas


Controlling matter to simultaneously support coupled properties is of fundamental and technological importance1 (for example, in multiferroics2,3,4,5 or high-temperature superconductors6,7,8,9). However, determining the microscopic mechanisms responsible for the simultaneous presence of different orders is difficult, making it hard to predict material phenomenology10,11 or modify properties12,13,14,15,16. Here, using a quantum gas to engineer an adjustable interaction at the microscopic level, we demonstrate scenarios of competition, coexistence and mutual enhancement of two orders. For the enhancement scenario, the presence of one order lowers the critical point of the other. Our system is realized by a Bose–Einstein condensate that can undergo self-organization phase transitions in two optical resonators17, resulting in two distinct crystalline density orders. We characterize the coupling between these orders by measuring the composite order parameter and the elementary excitations and explain our results with a mean-field free-energy model derived from a microscopic Hamiltonian. Our system is ideally suited to explore quantum tricritical points18 and can be extended to study the interplay of spin and density orders19 as a function of temperature20.

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Fig. 1: Phase diagram of two coupled order parameters.
Fig. 2: Observing a mixed-order phase.
Fig. 3: Controlling the coupling of two orders.
Fig. 4: Characterization of coupled orders.


  1. Spaldin, N. A., Cheong, S.-W. & Ramesh, R. Multiferroics: Past, present, and future. Phys. Today 10, 38–43 (October, 2010).

  2. Cheong, A.-W. & Mostovoy, M. Multiferroics: a magnetic twist for ferroelectricity. Nat. Mater. 6, 13–20 (2007).

    Article  Google Scholar 

  3. Hur, N. et al. Electric polarization reversal and memory in a multiferroic material induced by magnetic fields. Nature 429, 392–395 (2004).

    Article  Google Scholar 

  4. Lawes, G. et al. Magnetically driven ferroelectric order in Ni3V2O8. Phys. Rev. Lett. 95, 087205 (2005).

    Article  Google Scholar 

  5. Heyer, O. et al. A new multiferroic material: MnWO4. J. Phys. Condens. Matter 18, L471–L475 (2006).

    Article  Google Scholar 

  6. Tranquada, J. M., Sternlieb, B. J., Axe, J. D., Nakamura, Y. & Uchida, S. Evidence for stripe correlations of spins and holes in copper oxide superconductors. Nature 375, 561–563 (1995).

    Article  Google Scholar 

  7. Carbotte, J. P., Schachinger, E. & Basov, D. N. Coupling strength of charge carriers to spin fluctuations in high-temperature superconductors. Nature 401, 354–356 (1999).

    Article  Google Scholar 

  8. Daou, R. et al. Broken rotational symmetry in the pseudogap phase of a high-T c superconductor. Nature 463, 519–522 (2010).

    Article  Google Scholar 

  9. Hinkov, V. et al. Electronic liquid crystal state in the high-temperature superconductor YBa2Cu3O6.45. Science 319, 597–600 (2008).

    Article  Google Scholar 

  10. Demler, E., Hanke, W. & Zhang, S.-C. SO(5) theory of antiferromagnetism and superconductivity. Rev. Mod. Phys. 76, 909–974 (2004).

    Article  Google Scholar 

  11. Artyukhin, S., Delaney, K. T., Spaldin, N. A. & Mostovoy, M. Landau theory of topological defects in multiferroic hexagonal manganites. Nat. Mater. 13, 42–49 (2014).

    Article  Google Scholar 

  12. Norman, M. R., Randeria, M., Ding, H. & Campuzano, J. C. Phenomenology of the low-energy spectral function in high-T c superconductors. Phys. Rev. B 57, R11093(R) (1998).

    Article  Google Scholar 

  13. Hill, N. A. Why are there so few magnetic ferroelectrics? J. Phys. Chem. B 104, 6694–6709 (2000).

    Article  Google Scholar 

  14. Aschauer, U. & Spaldin, N. A. Competition and cooperation between antiferrodistortive and ferroelectric instabilities in the model perovskite SrTiO3. J. Phys. Condens. Matter 26, 122203 (2014).

    Article  Google Scholar 

  15. Tsvelik, A. M. & Chubukov, A. V. Phenomenological theory of the underdoped phase of a high-T c superconductor. Phys. Rev. Lett. 98, 237001 (2007).

    Article  Google Scholar 

  16. Nyéki, J. et al. Intertwined superfluid and density wave order in two-dimensional 4He. Nat. Phys. 13, 455–459 (2017).

    Article  Google Scholar 

  17. Léonard, J., Morales, A., Zupancic, P., Esslinger, T. & Donner, T. Supersolid formation in a quantum gas breaking a continuous translational symmetry. Nature 543, 87–90 (2016).

    Article  Google Scholar 

  18. Friedemann, S. et al. Quantum tricritical points in NbFe2. Nat. Phys. 14, 62–67 (2017).

    Article  Google Scholar 

  19. Mivehvar, F., Piazza, F. & Ritsch, H. Disorder-driven density and spin self-ordering of a Bose–Einstein condensate in a cavity. Phys. Rev. Lett. 119, 063602 (2017).

    Article  Google Scholar 

  20. Piazza, F., Strack, P. & Zwerger, W. Bose–Einstein condensation versus Dicke–Hepp–Lieb transition in an optical cavity. Ann. Phys. (NY) 339, 135–159 (2013).

    Article  Google Scholar 

  21. Sachdev, S. Quantum Phase Transitions (Cambridge Univ. Press, Cambridge, 1999).

    Google Scholar 

  22. Fradkin, E., Kivelson, S. A. & Tranquada, J. M. Colloquium: Theory of intertwined orders in high temperature superconductors. Rev. Mod. Phys. 87, 457–482 (2015).

    Article  Google Scholar 

  23. Baumann, K., Guerlin, C., Brennecke, F. & Esslinger, T. Dicke quantum phase transition with a superfluid gas in an optical cavity. Nature 464, 1301–1306 (2010).

    Article  Google Scholar 

  24. Baumann, K., Mottl, R., Brennecke, F. & Esslinger, T. Exploring symmetry breaking at the Dicke quantum phase transition. Phys. Rev. Lett. 107, 140402 (2011).

    Article  Google Scholar 

  25. Ketterle, W., Durfee, D. S. & Stamper-Kurn, D. M. Making, Probing and Understanding BEC Course CXL (eds Inguscio, M. et al.) 67 (IOS, Amsterdam, 1999).

  26. Léonard, J., Morales, A., Zupancic, P., Donner, T. & Esslinger, T. Monitoring and manipulating Higgs and Goldstone modes in a supersolid quantum gas. Science 358, 1415–1418 (2017).

    Article  Google Scholar 

  27. Lang, J., Piazza, F. & Zwerger, W. Collective excitations and supersolid behavior of bosonic atoms inside two crossed optical cavities. New J. Phys. 19, 123027 (2017).

    Article  Google Scholar 

  28. Bornholdt, S., Tetradis, N. & Wetterich, C. Coleman–Weinberg phase transition in two–scalar models. Phys. Rev. B 348, 89–99 (1995).

    Google Scholar 

  29. Gopalakrishnan, S., Shchadilova, Y. E. & Demler, E. Intertwined and vestigial order with ultracold atoms in multiple cavity modes. Phys. Rev. A 96, 063828 (2017).

    Article  Google Scholar 

  30. Awadhesh, N., Balatsky, A. V. & Spaldin, N. A. Multiferroic quantum criticality. Preprint at (2017).

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We thank E. Demler, S. Gopalakrishnan, A. Narayan, Y. E. Shchadilova and N. Spaldin for insightful discussions. We thank D. Dreon for careful reading of the manuscript and X. Li for experimental assistance. We acknowledge funding for the SBFI Horizon2020 project QUIC (grant agreement 641122) and the Horizon2020 European Training Network ColOpt (grant agreement 721465), and SNF support for the NCCR QSIT and the DACH project ‘Quantum Crystals of Matter and Light’.

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Correspondence to Tilman Esslinger.

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Morales, A., Zupancic, P., Léonard, J. et al. Coupling two order parameters in a quantum gas. Nature Mater 17, 686–690 (2018).

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