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  • Letter
  • Open Access

Tunable intertwining via collective excitations

András Szabó1,2,* and R. Chitra1

  • *Contact author: a.szabo@fkf.mpg.de
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Phys. Rev. Research 8, L022053 – Published 18 June, 2026

DOI: https://doi.org/10.1103/tq9z-xg7z

Abstract

The intertwining of multiple order parameters is a widespread phenomenon in equilibrium condensed matter systems, yet its exploration is often hindered by the complexity of real materials. Here, we present a controlled study of intertwined orders in a minimal and versatile driven-dissipative quantum-engineered platform. We consider a Bose-Einstein condensate at the intersection of two optical cavities, realizing two competing copies of a Z2 symmetry-breaking superradiant phase transition characterized by density-wave orders. Using periodic drives that exploit dynamical symmetry reduction, we show that collective excitations can be harnessed to stabilize a variety of out-of-equilibrium intertwined orders. Going beyond the conventional phenomenology involving Landau orders, we show the emergence of a larger class of out-of-equilibrium intertwined phases, including intertwining of purely time-crystalline orders, as well as between Landau and time crystal orders. These results should be observable in state-of-the-art experimental setups.

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References (41)

  1. B. V. Svistunov, E. S. Babaev, and N. V. Prokof'ev, Superfluid States of Matter (CRC Press, Boca Raton, FL, 2015).
  2. E. Fradkin, S. A. Kivelson, and J. M. Tranquada, Colloquium: Theory of intertwined orders in high temperature superconductors, Rev. Mod. Phys. 87, 457 (2015).
  3. R. M. Fernandes, P. P. Orth, and J. Schmalian, Intertwined vestigial order in quantum materials: Nematicity and beyond, Annu. Rev. Condens. Matter Phys. 10, 133 (2019).
  4. E. Berg, E. Fradkin, S. A. Kivelson, and J. M. Tranquada, Striped superconductors: How spin, charge and superconducting orders intertwine in the cuprates, New J. Phys. 11, 115004 (2009).
  5. T. Neupert, M. M. Denner, J.-X. Yin, R. Thomale, and M. Z. Hasan, Charge order and superconductivity in kagome materials, Nat. Phys. 18, 137 (2022).
  6. Y. Wang, H. Wu, G. T. McCandless, J. Y. Chan, and M. N. Ali, Quantum states and intertwining phases in kagome materials, Nat. Rev. Phys. 5, 635 (2023).
  7. D. Y. Kim, S.-Z. Lin, F. Weickert, M. Kenzelmann, E. D. Bauer, F. Ronning, J. D. Thompson, and R. Movshovich, Intertwined orders in heavy-fermion superconductor CeCoIn5, Phys. Rev. X 6, 041059 (2016).
  8. Q. Gu, J. P. Carroll, S. Wang, S. Ran, C. Broyles, H. Siddiquee, N. P. Butch, S. R. Saha, J. Paglione, J. S. Davis, et al., Detection of a pair density wave state in UTe2, Nature (London) 618, 921 (2023).
  9. X. Liu, Y. X. Chong, R. Sharma, and J. C. S. Davis, Discovery of a Cooper-pair density wave state in a transition-metal dichalcogenide, Science 372, 1447 (2021).
  10. C. Candelora, M. Xu, S. Cheng, A. De Vita, D. Romanin, C. Bigi, M. B. Petersen, A. LaFleur, W. A. Castro, F. Motti, P. Orgiani, G. Vinai, M. Calandra, J. Miwa, Y. Hwang, Z. Wang, F. Mazzola, and I. Zeljkovic, Discovery of magnetic-field-tunable density modulations and spin tilting in a layered altermagnet, Commun. Mater. 7, 74 (2026).
  11. A. Cavalleri, Photo-induced superconductivity, Contemp. Phys. 59, 31 (2018).
  12. M. Budden, T. Gebert, M. Buzzi, G. Jotzu, E. Wang, T. Matsuyama, G. Meier, Y. Laplace, D. Pontiroli, M. Riccò, et al., Evidence for metastable photo-induced superconductivity in K3C60, Nat. Phys. 17, 611 (2021).
  13. A. de la Torre, D. M. Kennes, M. Claassen, S. Gerber, J. W. McIver, and M. A. Sentef, Colloquium: Nonthermal pathways to ultrafast control in quantum materials, Rev. Mod. Phys. 93, 041002 (2021).
  14. T. Oka and S. Kitamura, Floquet engineering of quantum materials, Annu. Rev. Condens. Matter Phys. 10, 387 (2019).
  15. S. Kitamura and H. Aoki, Floquet topological superconductivity induced by chiral many-body interaction, Commun. Phys. 5, 174 (2022).
  16. G. Kipp, H. M. Bretscher, B. Schulte, D. Herrmann, K. Kusyak, M. W. Day, S. Kesavan, T. Matsuyama, X. Li, S. M. Langner, et al., Cavity electrodynamics of van der Waals heterostructures, Nat. Phys. 21, 1926 (2025).
  17. M. Kiffner, J. R. Coulthard, F. Schlawin, A. Ardavan, and D. Jaksch, Manipulating quantum materials with quantum light, Phys. Rev. B 99, 085116 (2019).
  18. I.-T. Lu, D. Shin, M. Kamper Svendsen, S. Latini, H. Hübener, M. Ruggenthaler, and A. Rubio, Cavity engineering of solid-state materials without external driving, Adv. Opt. Photon. 17, 441 (2025).
  19. F. Schlawin, D. M. Kennes, and M. A. Sentef, Cavity quantum materials, Appl. Phys. Rev. 9, 011312 (2022).
  20. J. Li and M. Eckstein, Manipulating intertwined orders in solids with quantum light, Phys. Rev. Lett. 125, 217402 (2020).
  21. D. Basov, A. Asenjo-Garcia, P. J. Schuck, X. Zhu, A. Rubio, A. Cavalleri, M. Delor, M. M. Fogler, and M. Liu, Polaritonic quantum matter, Nanophotonics 14, 3723 (2025).
  22. S. Kitamura and H. Aoki, η-pairing superfluid in periodically-driven fermionic Hubbard model with strong attraction, Phys. Rev. B 94, 174503 (2016).
  23. J. Li, D. Golez, P. Werner, and M. Eckstein, η-paired superconducting hidden phase in photodoped Mott insulators, Phys. Rev. B 102, 165136 (2020).
  24. D. F. Agterberg, J. S. Davis, S. D. Edkins, E. Fradkin, D. J. Van Harlingen, S. A. Kivelson, P. A. Lee, L. Radzihovsky, J. M. Tranquada, and Y. Wang, The physics of pair-density waves: Cuprate superconductors and beyond, Annu. Rev. Condens. Matter Phys. 11, 231 (2020).
  25. K. Baumann, C. Guerlin, F. Brennecke, and T. Esslinger, Dicke quantum phase transition with a superfluid gas in an optical cavity, Nature (London) 464, 1301 (2010).
  26. K. Baumann, R. Mottl, F. Brennecke, and T. Esslinger, Exploring symmetry breaking at the Dicke quantum phase transition, Phys. Rev. Lett. 107, 140402 (2011).
  27. V. Helson, T. Zwettler, F. Mivehvar, E. Colella, K. Roux, H. Konishi, H. Ritsch, and J.-P. Brantut, Density-wave ordering in a unitary Fermi gas with photon-mediated interactions, Nature (London) 618, 716 (2023).
  28. R. Landig, L. Hruby, N. Dogra, M. Landini, R. Mottl, T. Donner, and T. Esslinger, Quantum phases from competing short- and long-range interactions in an optical lattice, Nature (London) 532, 476 (2016).
  29. J. Léonard, A. Morales, P. Zupancic, T. Donner, and T. Esslinger, Monitoring and manipulating Higgs and Goldstone modes in a supersolid quantum gas, Science 358, 1415 (2017).
  30. S. Gopalakrishnan, Y. E. Shchadilova, and E. Demler, Intertwined and vestigial order with ultracold atoms in multiple cavity modes, Phys. Rev. A 96, 063828 (2017).
  31. A. Morales, P. Zupancic, J. Léonard, T. Esslinger, and T. Donner, Coupling two order parameters in a quantum gas, Nat. Mater. 17, 686 (2018).
  32. J. Léonard, A. Morales, P. Zupancic, T. Esslinger, and T. Donner, Supersolid formation in a quantum gas breaking a continuous translational symmetry, Nature (London) 543, 87 (2017).
  33. D. V. Else, B. Bauer, and C. Nayak, Floquet time crystals, Phys. Rev. Lett. 117, 090402 (2016).
  34. H. Keßler, P. Kongkhambut, C. Georges, L. Mathey, J. G. Cosme, and A. Hemmerich, Observation of a dissipative time crystal, Phys. Rev. Lett. 127, 043602 (2021).
  35. R. Chitra and O. Zilberberg, Dynamical many-body phases of the parametrically driven, dissipative Dicke model, Phys. Rev. A 92, 023815 (2015).
  36. P. Molignini, L. Papariello, A. U. J. Lode, and R. Chitra, Superlattice switching from parametric instabilities in a driven-dissipative Bose-Einstein condensate in a cavity, Phys. Rev. A 98, 053620 (2018).
  37. Y. Hou, Z. Fu, R. Moessner, M. Bukov, and H. Zhao, Floquet-engineered emergent massive Nambu-Goldstone modes, Phys. Rev. B 112, L020305 (2025).
  38. See Supplemental Material at http://link.aps.org/supplemental/10.1103/tq9z-xg7z for the following: details on the mean-field phase diagram and fluctuation spectra of the cross-cavity system; high-frequency expansion; Heisenberg equations of motion; and extended dataset for dispersively coupled cavity.
  39. C. Zerbe, P. Jung, and P. Hänggi, Brownian parametric oscillators, Phys. Rev. E 49, 3626 (1994).
  40. J. Léonard, A supersolid of matter and light, Ph.D. thesis, ETH Zurich, 2017.
  41. R. Lin, C. Georges, J. Klinder, P. Molignini, M. Büttner, A. U. J. Lode, R. Chitra, A. Hemmerich, and H. Keßler, Mott transition in a cavity-boson system: A quantitative comparison between theory and experiment, SciPost Phys. 11, 030 (2021).

Outline

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Phys. Rev. Research 8, L022053– Published 18 June, 2026
  • Received 13 June 2025
  • Revised 31 October 2025
  • Accepted 15 May 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by Max Planck Society.

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