- Physical Review Journals
All JournalsPhysics Magazine

Physical Review A

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Bilayer crystals in a polar-molecule system

Vinicius Zampronio1,2,*, Matteo Ciardi3,†, and Fabio Cinti1,4,‡

  • *Deceased.
  • †Contact author: matteo.ciardi@tuwien.ac.at
  • ‡Contact author: fabio.cinti@unifi.it
PDF

Phys. Rev. A 114, 033325 – Published 29 September, 2026

DOI: https://doi.org/10.1103/64fx-rnc1

Abstract

We investigate the finite-temperature phase diagram of polar molecules confined in a quasi-two-dimensional geometry by a harmonic potential along the polarization axis. We employ quantum Monte Carlo simulations to explore the strongly correlated regime accessible with current experimental setups. By tuning temperature and confinement strength, we identify a rich set of phases, including normal fluid, superfluid, supersolid, cluster crystal, and bilayer crystal states. Our results reveal the emergence of crystallization upon increasing temperature, highlighting the nontrivial role of thermal fluctuations in dipolar systems. In particular, we show that a bilayer crystal with one molecule per lattice site can be stabilized by varying the confinement strength at fixed interaction. Moreover, we show evidence of layering of superfluid states with phase coherence between the two layers. These findings provide insight into the interplay between interactions, confinement, and temperature in low-dimensional dipolar systems, and suggest new directions for engineering quantum phases with ultracold polar molecules.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (68)

  1. I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics with ultracold gases, Rev. Mod. Phys. 80, 885 (2008).
  2. F. Schäfer, T. Fukuhara, S. Sugawa, Y. Takasu, and Y. Takahashi, Tools for quantum simulation with ultracold atoms in optical lattices, Nat. Rev. Phys. 2, 411 (2020).
  3. M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Observation of Bose-Einstein condensation in a dilute atomic vapor, Science 269, 198 (1995).
  4. K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, Bose-Einstein condensation in a gas of sodium atoms, Phys. Rev. Lett. 75, 3969 (1995).
  5. N. P. Proukakis, A century of Bose-Einstein condensation, Commun. Phys. 8, 264 (2025).
  6. M. Schmitt, M. Wenzel, F. Böttcher, I. Ferrier-Barbut, and T. Pfau, Self-bound droplets of a dilute magnetic quantum liquid, Nature (London) 539, 259 (2016).
  7. L. Chomaz, S. Baier, D. Petter, M. J. Mark, F. Wächtler, L. Santos, and F. Ferlaino, Quantum-fluctuation-driven crossover from a dilute Bose-Einstein condensate to a macrodroplet in a dipolar quantum fluid, Phys. Rev. X 6, 041039 (2016).
  8. F. Böttcher, J. N. Schmidt, M. Wenzel, J. Hertkorn, M. Guo, T. Langen, and T. Pfau, Transient supersolid properties in an array of dipolar quantum droplets, Phys. Rev. X 9, 011051 (2019).
  9. L. Tanzi, E. Lucioni, F. Famà, J. Catani, A. Fioretti, C. Gabbanini, R. N. Bisset, L. Santos, and G. Modugno, Observation of a dipolar quantum gas with metastable supersolid properties, Phys. Rev. Lett. 122, 130405 (2019).
  10. L. Chomaz, D. Petter, P. Ilzhöfer, G. Natale, A. Trautmann, C. Politi, G. Durastante, R. M. W. van Bijnen, A. Patscheider, M. Sohmen, M. J. Mark, and F. Ferlaino, Long-lived and transient supersolid behaviors in dipolar quantum gases, Phys. Rev. X 9, 021012 (2019).
  11. Y.-C. Zhang, F. Maucher, and T. Pohl, Supersolidity around a critical point in dipolar Bose-Einstein condensates, Phys. Rev. Lett. 123, 015301 (2019).
  12. M. Schmidt, L. Lassablière, G. Quéméner, and T. Langen, Self-bound dipolar droplets and supersolids in molecular Bose-Einstein condensates, Phys. Rev. Res. 4, 013235 (2022).
  13. B. T. E. Ripley, D. Baillie, and P. B. Blakie, Two-dimensional supersolidity in a planar dipolar Bose gas, Phys. Rev. A 108, 053321 (2023).
  14. D. Lima, M. Grossklags, V. Zampronio, F. Cinti, and A. Mendoza-Coto, Supersolid dipolar phases in planar geometry: Effects of tilted polarization, Phys. Rev. A 111, 063311 (2025).
  15. R. N. Bisset, R. M. Wilson, D. Baillie, and P. B. Blakie, Ground-state phase diagram of a dipolar condensate with quantum fluctuations, Phys. Rev. A 94, 033619 (2016).
  16. F. Wächtler and L. Santos, Quantum filaments in dipolar Bose-Einstein condensates, Phys. Rev. A 93, 061603(R) (2016).
  17. F. Wächtler and L. Santos, Ground-state properties and elementary excitations of quantum droplets in dipolar Bose-Einstein condensates, Phys. Rev. A 94, 043618 (2016).
  18. T. D. Lee, K. Huang, and C. N. Yang, Eigenvalues and eigenfunctions of a Bose system of hard spheres and its low-temperature properties, Phys. Rev. 106, 1135 (1957).
  19. Aristeu R. P. Lima and A. Pelster, Quantum fluctuations in dipolar Bose gases, Phys. Rev. A 84, 041604(R) (2011).
  20. P. Jain, F. Cinti, and M. Boninsegni, Structure, Bose-Einstein condensation, and superfluidity of two-dimensional confined dipolar assemblies, Phys. Rev. B 84, 014534 (2011).
  21. F. Cinti and M. Boninsegni, Classical and quantum filaments in the ground state of trapped dipolar Bose gases, Phys. Rev. A 96, 013627 (2017).
  22. F. Cinti, A. Cappellaro, L. Salasnich, and T. Macrì, Superfluid filaments of dipolar bosons in free space, Phys. Rev. Lett. 119, 215302 (2017).
  23. H. Saito, Path-integral Monte Carlo study on a droplet of a dipolar Bose–Einstein condensate stabilized by quantum fluctuation, J. Phys. Soc. Jpn. 85, 053001 (2016).
  24. A. Macia, J. Sánchez-Baena, J. Boronat, and F. Mazzanti, Droplets of trapped quantum dipolar bosons, Phys. Rev. Lett. 117, 205301 (2016).
  25. F. Böttcher, M. Wenzel, J. N. Schmidt, M. Guo, T. Langen, I. Ferrier-Barbut, T. Pfau, R. Bombín, J. Sánchez-Baena, J. Boronat, and F. Mazzanti, Dilute dipolar quantum droplets beyond the extended Gross-Pitaevskii equation, Phys. Rev. Res. 1, 033088 (2019).
  26. M. Boninsegni, Morphology of dipolar Bose droplets, Results Phys. 31, 104935 (2021).
  27. Y. Kora and M. Boninsegni, Patterned supersolids in dipolar Bose systems, J. Low Temp. Phys. 197, 337 (2019).
  28. M. Sohmen, C. Politi, L. Klaus, L. Chomaz, M. J. Mark, M. A. Norcia, and F. Ferlaino, Birth, life, and death of a dipolar supersolid, Phys. Rev. Lett. 126, 233401 (2021).
  29. M. A. Norcia, C. Politi, L. Klaus, E. Poli, M. Sohmen, M. J. Mark, R. N. Bisset, L. Santos, and F. Ferlaino, Two-dimensional supersolidity in a dipolar quantum gas, Nature (London) 596, 357 (2021).
  30. S. Sinha and S. Sinha, Supersolid phases of bosons, J. Phys.: Condens. Matter 37, 333001 (2025).
  31. A. Recati and S. Stringari, Supersolidity in ultracold dipolar gases, Nat. Rev. Phys. 5, 735 (2023).
  32. L. Chomaz, I. Ferrier-Barbut, F. Ferlaino, B. Laburthe-Tolra, B. L. Lev, and T. Pfau, Dipolar physics: A review of experiments with magnetic quantum gases, Rep. Prog. Phys. 86, 026401 (2023).
  33. A. Schindewolf, J. Hertkorn, I. Stevenson, M. Ciardi, P. Groß, D. Wang, T. Karman, G. Quéméner, S. Will, T. Pohl, and T. Langen, Colloquium: Strongly dipolar molecular Bose-Einstein condensates: From few- to many-body physics, Rev. Mod. Phys. 98, 031002 (2026).
  34. T. Karman and J. M. Hutson, Microwave shielding of ultracold polar molecules, Phys. Rev. Lett. 121, 163401 (2018).
  35. L. Anderegg, S. Burchesky, Y. Bao, S. S. Yu, T. Karman, E. Chae, K. Ni, W. Ketterle, and J. M. Doyle, Observation of microwave shielding of ultracold molecules, Science 373, 779 (2021).
  36. A. Schindewolf, R. Bause, X. Chen, M. Duda, T. Karman, I. Bloch, and X. Luo, Evaporation of microwave-shielded polar molecules to quantum degeneracy, Nature (London) 607, 677 (2022).
  37. A. F. Andreev and I. M. Lifshitz, Quantum theory of defects in crystals, Sov. Phys. Usp. 13, 670 (1971).
  38. F. Cinti, T. Macrì, W. Lechner, G. Pupillo, and T. Pohl, Defect-induced supersolidity with soft-core bosons, Nat. Commun. 5, 3235 (2014).
  39. N. Bigagli, W. Yuan, S. Zhang, B. Bulatovic, T. Karman, I. Stevenson, and S. Will, Observation of Bose–Einstein condensation of dipolar molecules, Nature (London) 631, 289 (2024).
  40. Z. Shi, Z. Huang, F. Deng, W.-J. Jin, S. Yi, T. Shi, and D. Wang, Bose-Einstein condensate of ultracold sodium-rubidium molecules with tunable dipolar interactions, Nat. Phys. 22, 1467 (2026).
  41. J. Sánchez-Baena, G. Pascual, R. Bombín, F. Mazzanti, and J. Boronat, Thermal behavior of Bose-Einstein condensates of polar molecules, Phys. Rev. Res. 7, 033080 (2025).
  42. J. Dutta, B. Mukherjee, and J. M. Hutson, Universality in the microwave shielding of ultracold polar molecules, Phys. Rev. Res. 7, 023164 (2025).
  43. H. P. Büchler, E. Demler, M. Lukin, A. Micheli, N. Prokof'ev, G. Pupillo, and P. Zoller, Strongly correlated 2D quantum phases with cold polar molecules: Controlling the shape of the interaction potential, Phys. Rev. Lett. 98, 060404 (2007).
  44. A. Micheli, G. Pupillo, H. P. Büchler, and P. Zoller, Cold polar molecules in two-dimensional traps: Tailoring interactions with external fields for novel quantum phases, Phys. Rev. A 76, 043604 (2007).
  45. L. Lassablière and G. Quéméner, Controlling the scattering length of ultracold dipolar molecules, Phys. Rev. Lett. 121, 163402 (2018).
  46. F. Deng, X. Hu, W.-J. Jin, S. Yi, and T. Shi, Two- and many-body physics of ultracold molecules dressed by dual microwave fields, Nat. Commun. 16, 11219 (2025).
  47. T. Karman, N. Bigagli, W. Yuan, S. Zhang, I. Stevenson, and S. Will, Double microwave shielding, PRX Quantum 6, 020358 (2025).
  48. T. Langen, J. Boronat, J. Sánchez-Baena, R. Bombín, T. Karman, and F. Mazzanti, Dipolar droplets of strongly interacting molecules, Phys. Rev. Lett. 134, 053001 (2025).
  49. M. Ciardi, K. R. Pedersen, T. Langen, and T. Pohl, Self-bound superfluid membranes and monolayer crystals of ultracold polar molecules, Phys. Rev. Lett. 135, 153401 (2025).
  50. D.-W. Wang, Quantum phase transitions of polar molecules in bilayer systems, Phys. Rev. Lett. 98, 060403 (2007).
  51. E. Rydow, V. P. Singh, A. Beregi, E. Chang, L. Mathey, C. J. Foot, and S. Sunami, Observation of a bilayer superfluid with interlayer coherence, Nat. Commun. 16, 7201 (2025).
  52. F. Cinti, M. Ciardi, S. Prestipino, and G. Pellicane, Layering and superfluidity of soft-core bosons in shallow spherical traps, Phys. Rev. A 114, 013313 (2026).
  53. J. P. Eisenstein, Experimental studies of multicomponent quantum Hall systems, in Perspectives in Quantum Hall Effects, edited by S. Das Sarma and A. Pinczuk (Wiley, New York, 1996), Chap. 2, pp. 37–70.
  54. S. M. Girvin and A. H. MacDonald, Multicomponent quantum Hall systems The sum of their parts and more, in Perspectives in Quantum Hall Effects Novel Quantum Liquids in Low-Dimensional Semiconductor Structures, edited by S. Das Sarma and A. Pinczuk (Wiley, New York, 1996), Chap. 5, pp. 161–224.
  55. A. Perali, D. Neilson, and A. R. Hamilton, High-temperature superfluidity in double-bilayer graphene, Phys. Rev. Lett. 110, 146803 (2013).
  56. Q. Gao, Y.-h. Chan, Y. Wang, H. Zhang, P. Jinxu, S. Cui, Y. Yang, Z. Liu, D. Shen, Z. Sun, J. Jiang, T. C. Chiang, and P. Chen, Evidence of high-temperature exciton condensation in a two-dimensional semimetal, Nat. Commun. 14, 994 (2023).
  57. D. M. Ceperley, Path integrals in the theory of condensed helium, Rev. Mod. Phys. 67, 279 (1995).
  58. M. Boninsegni, N. Prokof'ev, and B. Svistunov, Worm algorithm for continuous-space path integral Monte Carlo simulations, Phys. Rev. Lett. 96, 070601 (2006).
  59. E. L. Pollock and D. M. Ceperley, Path-integral computation of superfluid densities, Phys. Rev. B 36, 8343 (1987).
  60. M. Ciardi, F. Cinti, G. Pellicane, and S. Prestipino, Effects of gravity on supersolid order in bubble-trapped bosons, Phys. Rev. B 111, 024512 (2025).
  61. X. Lu, C.-Q. Wu, A. Micheli, and G. Pupillo, Structure and melting behavior of classical bilayer crystals of dipoles, Phys. Rev. B 78, 024108 (2008).
  62. F. Cinti, D.-W. Wang, and M. Boninsegni, Phases of dipolar bosons in a bilayer geometry, Phys. Rev. A 95, 023622 (2017).
  63. J. Sánchez-Baena, C. Politi, F. Maucher, F. Ferlaino, and T. Pohl, Heating a dipolar quantum fluid into a solid, Nat. Commun. 14, 1868 (2023).
  64. J. Jung, L. Futamura, M. Ciardi, J. Rosenberg, Y. A. Alaoui, Y. Lu, T. Pohl, and W. S. Bakr, Static-field shielding of bosonic molecules evaporation to degeneracy and self-bound droplets, arXiv:2608.31116.
  65. V. Zampronio, A. Mendoza-Coto, T. Macrì, and F. Cinti, Exploring quantum phases of dipolar gases through quasicrystalline confinement, Phys. Rev. Lett. 133, 196001 (2024).
  66. J.-C. Yu, S. Bhave, L. Reeve, B. Song, and U. Schneider, Observing the two-dimensional Bose glass in an optical quasicrystal, Nature (London) 633, 338 (2024).
  67. S. N. Kempkes, M. R. Slot, S. E. Freeney, S. J. M. Zevenhuizen, D. Vanmaekelbergh, I. Swart, and C. M. Smith, Design and characterization of electrons in a fractal geometry, Nat. Phys. 15, 127 (2019).
  68. R. C. Verstraten, I. H. A. Knottnerus, Y. C. Tseng, A. Urech, T. S. d. E. Santo, V. Zampronio, F. Schreck, R. J. C. Spreeuw, and C. M. Smith, Control of single spin-flips in a Rydberg atomic fractal arXiv:2509.03514.

Outline

Information

Phys. Rev. A 114, 033325– Published 29 September, 2026
  • Received 18 May 2026
  • Accepted 8 September 2026
Crossmark - Check for updates button
Creative Commons Logo

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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation