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Quantum Phases for Finite-Temperature Gases of Bosonic Polar Molecules Shielded by Dual Microwaves

Wei Zhang1,2, Kun Chen1,2,*, Su Yi3,4,†, and Tao Shi1,2,‡

  • *Contact author: chenkun@itp.ac.cn
  • †Contact author: yisu@nbu.edu.cn
  • ‡Contact author: tshi@itp.ac.cn
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PRX Quantum 6, 040307 – Published 9 October, 2025

DOI: https://doi.org/10.1103/9cxl-d9zg

Abstract

We investigate the finite-temperature phase diagram of polar molecules shielded by dual microwave fields using the path integral Monte Carlo method combined with the worm algorithm. We determine the critical temperature Tc for Bose-Einstein condensations (BECs) and identify two distinct phases below Tc: the expanding gas (EG) phase and the self-bound gas (SBG) phase. We further analyze the temperature and interaction-strength dependence of the condensate and superfluid fractions. Notably, in contrast to dilute atomic BECs, the SBG phase displays a low condensate fraction and a high superfluid fraction, resembling the behavior of strongly correlated 4He superfluids. These significant many-body correlations arise from the interplay between long-range dipole-dipole interactions and the short-range shielding potential. Furthermore, we demonstrate that the aspect ratio of the gas provides a characteristic geometric signature to accurately determine the EG-to-SBG transition, robust against external trapping potentials. Our findings provide unbiased and numerically exact results to guide upcoming experiments with polar molecules.

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Popular Summary

In this study, we present a numerically exact finite-temperature phase diagram for polar molecules under dual microwave shielding. Focusing on Bose-Einstein condensates (BECs) of such molecules—recently realized experimentally—we employ advanced computational methods to uncover two distinct quantum phases: an expanding gas and a self-bound gas, both exhibiting strong many-body correlations.

These findings open up a new chapter in the study of ultracold molecules, demonstrating that microwave-shielded polar molecules (MSPMs) offer a promising new platform that bridges the weakly interacting regime of conventional atomic BECs and the strongly correlated behavior in systems like superfluid helium-4 (4He). MSPMs are not merely atomic BECs; they constitute a new class of quantum matter. Dual microwave shielding creates highly tunable long-range dipolar interactions and short-range repulsion, enabling unprecedented strongly correlated phases. These exhibit antibunching and a low condensate fraction alongside high superfluidity, closely resembling superfluid 4He.

Our results provide clear guidance for future experiments with polar molecules such as NaRb and CsAg, supporting efforts to refine microwave shielding and molecular interaction models. They also open pathways to exotic phases, including high-temperature self-bound gases. Ultimately, our work sets the stage for future exploration of strongly correlated quantum systems, with the potential to explore both supersolids and droplet arrays, opening up new possibilities in quantum simulation, quantum computing, and ultracold chemistry. This research not only provides insight into MSPMs but also serves as a guide for future theoretical and experimental work that could expand the boundaries of quantum matter.

Article Text

References (73)

  1. Tijs Karman and Jeremy M. Hutson, Microwave shielding of ultracold polar molecules, Phys. Rev. Lett. 121, 163401 (2018).
  2. Lucas Lassablière and Goulven Quéméner, Controlling the scattering length of ultracold dipolar molecules, Phys. Rev. Lett. 121, 163402 (2018).
  3. N. R. Cooper and G. V. Shlyapnikov, Stable topological superfluid phase of ultracold polar fermionic molecules, Phys. Rev. Lett. 103, 155302 (2009).
  4. Loïc Anderegg, Sean Burchesky, Yicheng Bao, Scarlett S. Yu, Tijs Karman, Eunmi Chae, Kang-Kuen Ni, Wolfgang Ketterle, and John M. Doyle, Observation of microwave shielding of ultracold molecules, Science 373, 779 (2021).
  5. Junyu Lin, Guanghua Chen, Mucan Jin, Zhaopeng Shi, Fulin Deng, Wenxian Zhang, Goulven Quéméner, Tao Shi, Su Yi, and Dajun Wang, Microwave shielding of bosonic NaRb molecules, Phys. Rev. X 13, 031032 (2023).
  6. Niccolò Bigagli, Claire Warner, Weijun Yuan, Siwei Zhang, Ian Stevenson, Tijs Karman, and Sebastian Will, Collisionally stable gas of bosonic dipolar ground state molecules, Nat. Phys. 19, 1579 (2023).
  7. Andreas Schindewolf, Roman Bause, Xing-Yan Chen, Marcel Duda, Tijs Karman, Immanuel Bloch, and Xin-Yu Luo, Evaporation of microwave-shielded polar molecules to quantum degeneracy, Nature 607, 677 (2022).
  8. Xing-Yan Chen, Andreas Schindewolf, Sebastian Eppelt, Roman Bause, Marcel Duda, Shrestha Biswas, Tijs Karman, Timon Hilker, Immanuel Bloch, and Xin-Yu Luo, Field-linked resonances of polar molecules, Nature 614, 59 (2022).
  9. Xing-Yan Chen, Shrestha Biswas, Sebastian Eppelt, Andreas Schindewolf, Fulin Deng, Tao Shi, Su Yi, Timon A. Hilker, Immanuel Bloch, and Xin-Yu Luo, Ultracold field-linked tetratomic molecules, Nature 626, 283 (2024).
  10. Fulin Deng, Xingyan Chen, Xinyu Luo, Wenxian Zhang, Su Yi, and Tao Shi, Formation and dissociation of field-linked tetramers, arXiv:2405.13645.
  11. Niccolò Bigagli, Weijun Yuan, Siwei Zhang, Boris Bulatovic, Tijs Karman, Ian Stevenson, and Sebastian Will, Observation of Bose-Einstein condensation of dipolar molecules, Nature 631, 289 (2024).
  12. L. You and M. Marinescu, Prospects for p-wave paired Bardeen-Cooper-Schrieffer states of fermionic atoms, Phys. Rev. A 60, 2324 (1999).
  13. M. A. Baranov, M. S. Mar’enko, Val. S. Rychkov, and G. V. Shlyapnikov, Superfluid pairing in a polarized dipolar Fermi gas, Phys. Rev. A 66, 013606 (2002).
  14. M. A. Baranov, Ł. Dobrek, and M. Lewenstein, Superfluidity of trapped dipolar Fermi gases, Phys. Rev. Lett. 92, 250403 (2004).
  15. K. V. Samokhin and M. S. Mar’enko, Nonuniform mixed-parity superfluid state in Fermi gases, Phys. Rev. Lett. 97, 197003 (2006).
  16. T. Shi, J.-N. Zhang, C.-P. Sun, and S. Yi, Singlet and triplet Bardeen-Cooper-Schrieffer pairs in a gas of two-species fermionic polar molecules, Phys. Rev. A 82, 033623 (2010).
  17. Congjun Wu and J. E. Hirsch, Mixed triplet and singlet pairing in ultracold multicomponent fermion systems with dipolar interactions, Phys. Rev. B 81, 020508 (2010).
  18. Cheng Zhao, Lei Jiang, Xunxu Liu, W. M. Liu, Xubo Zou, and Han Pu, Hartree-Fock-Bogoliubov theory of dipolar Fermi gases, Phys. Rev. A 81, 063642 (2010).
  19. J. Levinsen, N. R. Cooper, and G. V. Shlyapnikov, Topological px+ipy superfluid phase of fermionic polar molecules, Phys. Rev. A 84, 013603 (2011).
  20. T. Shi, S.-H. Zou, H. Hu, C.-P. Sun, and S. Yi, Ultracold Fermi gases with resonant dipole-dipole interaction, Phys. Rev. Lett. 110, 045301 (2013).
  21. Wei-Jian Jin, Fulin Deng, Su Yi, and Tao Shi, Bose Einstein condensates of microwave-shielded polar molecules, Phys. Rev. Lett. 134, 233003 (2025).
  22. Tim Langen, Jordi Boronat, Juan Sánchez-Baena, Raúl Bombín, Tijs Karman, and Ferran Mazzanti, Dipolar droplets of strongly interacting molecules, Phys. Rev. Lett. 134, 053001 (2025).
  23. A. Micheli, G. K. Brennen, and P. Zoller, A toolbox for lattice-spin models with polar molecules, Nat. Phys. 2, 341 (2006).
  24. T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, and T. Pfau, The physics of dipolar bosonic quantum gases, Rep. Prog. Phys. 72, 126401 (2009).
  25. M. A. Baranov, M. Dalmonte, G. Pupillo, and P. Zoller, Condensed matter theory of dipolar quantum gases, Chem. Rev. 112, 5012 (2012).
  26. Ehud Altman et al., Quantum simulators: Architectures and opportunities, PRX Quantum 2, 017003 (2021).
  27. D. DeMille, Quantum computation with trapped polar molecules, Phys. Rev. Lett. 88, 067901 (2002).
  28. Rahul Sawant, Jacob A. Blackmore, Philip D. Gregory, Jordi Mur-Petit, Dieter Jaksch, Jesús Aldegunde, Jeremy M. Hutson, M. R. Tarbutt, and Simon L. Cornish, Ultracold polar molecules as qudits, New J. Phys. 22, 013027 (2020).
  29. P. Rabl, D. DeMille, J. M. Doyle, M. D. Lukin, R. J. Schoelkopf, and P. Zoller, Hybrid quantum processors: Molecular ensembles as quantum memory for solid state circuits, Phys. Rev. Lett. 97, 033003 (2006).
  30. Carmen M. Tesch and Regina de Vivie-Riedle, Quantum computation with vibrationally excited molecules, Phys. Rev. Lett. 89, 157901 (2002).
  31. M. L. Wall, K. Maeda, and Lincoln D. Carr, Realizing unconventional quantum magnetism with symmetric top molecules, New J. Phys. 17, 025001 (2015).
  32. Victor V. Albert, Jacob P. Covey, and John Preskill, Robust encoding of a qubit in a molecule, Phys. Rev. X 10, 031050 (2020).
  33. R. V. Krems, Cold controlled chemistry, Phys. Chem. Chem. Phys. 10, 4079 (2008).
  34. M.-G. Hu, Y. Liu, D. D. Grimes, Y.-W. Lin, A. H. Gheorghe, R. Vexiau, N. Bouloufa-Maafa, O. Dulieu, T. Rosenband, and K.-K. Ni, Direct observation of bimolecular reactions of ultracold KRb molecules, Science 366, 1111 (2019).
  35. Yu Liu and Kang-Kuen Ni, Bimolecular chemistry in the ultracold regime, Annu. Rev. Phys. Chem. 73, 73 (2022).
  36. V. V. Flambaum and M. G. Kozlov, Enhanced sensitivity to the time variation of the fine-structure constant and mp/me in diatomic molecules, Phys. Rev. Lett. 99, 150801 (2007).
  37. T. A. Isaev, S. Hoekstra, and R. Berger, Laser-cooled RaF as a promising candidate to measure molecular parity violation, Phys. Rev. A 82, 052521 (2010).
  38. J. J. Hudson, D. M. Kara, I. J. Smallman, B. E. Sauer, M. R. Tarbutt, and E. A. Hinds, Improved measurement of the shape of the electron, Nature 473, 493 (2011).
  39. The ACME Collaboration, J. Baron, W. C. Campbell, D. DeMille, J. M. Doyle, G. Gabrielse, Y. V. Gurevich, P. W. Hess, N. R. Hutzler, E. Kirilov, I. Kozyryev, B. R. O’Leary, C. D. Panda, M. F. Parsons, E. S. Petrik, B. Spaun, A. C. Vutha, and A. D. West, Order of magnitude smaller limit on the electric dipole moment of the electron, Science 343, 269 (2014).
  40. V. Andreev, D. G. Ang, D. DeMille, J. M. Doyle, G. Gabrielse, J. Haefner, N. R. Hutzler, Z. Lasner, C. Meisenhelder, B. R. O’Leary, C. D. Panda, A. D. West, E. P. West, and X. Wu, ACME Collaboration, Improved limit on the electric dipole moment of the electron, Nature 562, 355 (2018).
  41. Nicholas R. Hutzler, Polyatomic molecules as quantum sensors for fundamental physics, Quantum Sci. Technol. 5, 044011 (2020).
  42. Fulin Deng, Xing-Yan Chen, Xin-Yu Luo, Wenxian Zhang, Su Yi, and Tao Shi, Effective potential and superfluidity of microwave-shielded polar molecules, Phys. Rev. Lett. 130, 183001 (2023).
  43. Fulin Deng, Xinyuan Hu, Wei-Jian Jin, Su Yi, and Tao Shi, Two- and many-body physics of ultracold molecules dressed by dual microwave fields, arXiv:2501.05210.
  44. Tijs Karman, Niccolò Bigagli, Weijun Yuan, Siwei Zhang, Ian Stevenson, and Sebastian Will, Double microwave shielding, PRX Quantum 6, 020358 (2025).
  45. Matthias Schmitt, Matthias Wenzel, Fabian Böttcher, Igor Ferrier-Barbut, and Tilman Pfau, Self-bound droplets of a dilute magnetic quantum liquid, Nature 539, 259 (2016).
  46. Fabian Böttcher, Matthias Wenzel, Jan-Niklas Schmidt, Mingyang Guo, Tim Langen, Igor Ferrier-Barbut, Tilman Pfau, Raúl Bombín, Joan Sánchez-Baena, Jordi Boronat, and Ferran Mazzanti, Dilute dipolar quantum droplets beyond the extended Gross-Pitaevskii equation, Phys. Rev. Res. 1, 033088 (2019).
  47. Oliver Penrose and Lars Onsager, Bose-Einstein condensation and liquid helium, Phys. Rev. 104, 576 (1956).
  48. V. F. Sears, E. C. Svensson, P. Martel, and A. D. B. Woods, Neutron-scattering determination of the momentum distribution and the condensate fraction in liquid 4He, Phys. Rev. Lett. 49, 279 (1982).
  49. V. F. Sears, Kinetic energy and condensate fraction of superfluid 4He, Phys. Rev. B 28, 5109 (1983).
  50. T. R. Sosnick, W. M. Snow, and P. E. Sokol, Deep-inelastic neutron scattering from liquid 4He, Phys. Rev. B 41, 11185 (1990).
  51. T. R. Sosnick, W. M. Snow, R. N. Silver, and P. E. Sokol, Deviations from the impulse approximation in liquid 4He: An experimental test at q=23 å −1, Phys. Rev. B 43, 216 (1991).
  52. N. V. Prokof’Ev, B. V. Svistunov, and I. S. Tupitsyn, Exact, complete, and universal continuous-time worldline Monte Carlo approach to the statistics of discrete quantum systems, J. Exp. Theor. Phys. 87, 310 (1998).
  53. Massimo Boninsegni, Nikolay Prokof’ev, and Boris Svistunov, Worm algorithm for continuous-space path integral Monte Carlo simulations, Phys. Rev. Lett. 96, 070601 (2006).
  54. M. Boninsegni, N. V. Prokof’ev, and B. V. Svistunov, Worm algorithm and diagrammatic Monte Carlo: A new approach to continuous-space path integral Monte Carlo simulations, Phys. Rev. E 74, 036701 (2006).
  55. Kun Chen, Longxiang Liu, Youjin Deng, Lode Pollet, and Nikolay Prokof’ev, Universal conductivity in a two-dimensional superfluid-to-insulator quantum critical system, Phys. Rev. Lett. 112, 030402 (2014).
  56. Kun Chen, Yuan Huang, Youjin Deng, A. B. Kuklov, N. V. Prokof’ev, and B. V. Svistunov, Deconfined criticality flow in the Heisenberg model with ring-exchange interactions, Phys. Rev. Lett. 110, 185701 (2013).
  57. Massimo Boninsegni, Nikolay Prokof’ev, and Boris Svistunov, Superglass phase of 4He, Phys. Rev. Lett. 96, 105301 (2006).
  58. Anatoly Kuklov, Lode Pollet, Nikolay Prokof’ev, and Boris Svistunov, Transverse quantum superfluids, Annu. Rev. Condens. Matter Phys. 16, 209 (2024).
  59. Chao Zhang, Barbara Capogrosso-Sansone, Massimo Boninsegni, Nikolay V. Prokof’ev, and Boris V. Svistunov, Superconducting transition temperature of the Bose one-component plasma, Phys. Rev. Lett. 130, 236001 (2023).
  60. M. Aymar and O. Dulieu, Calculation of accurate permanent dipole moments of the lowest sigma+1,3 states of heteronuclear alkali dimers using extended basis sets, J. Chem. Phys. 122, 204302 (2005).
  61. Ian Stevenson, Shayamal Singh, Ahmed Elkamshishy, Niccoló Bigagli, Weijun Yuan, Siwei Zhang, Chris H. Greene, and Sebastian Will, Three-body recombination of ultracold microwave-shielded polar molecules, Phys. Rev. Lett. 133, 263402 (2024).
  62. Wolfgang Ketterle and N. J. van Druten, Bose-Einstein condensation of a finite number of particles trapped in one or three dimensions, Phys. Rev. A 54, 656 (1996).
  63. Franco Dalfovo, Stefano Giorgini, Lev P. Pitaevskii, and Sandro Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys. 71, 463 (1999).
  64. J. R. Ensher, D. S. Jin, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Bose-Einstein condensation in a dilute gas: Measurement of energy and ground-state occupation, Phys. Rev. Lett. 77, 4984 (1996).
  65. F. Gerbier, J. H. Thywissen, S. Richard, M. Hugbart, P. Bouyer, and A. Aspect, Critical temperature of a trapped, weakly interacting Bose gas, Phys. Rev. Lett. 92, 030405 (2004).
  66. R. Meppelink, R. A. Rozendaal, S. B. Koller, J. M. Vogels, and P. van der Straten, Thermodynamics of Bose-Einstein-condensed clouds using phase-contrast imaging, Phys. Rev. A 81, 053632 (2010).
  67. Robert P. Smith, Robert L. D. Campbell, Naaman Tammuz, and Zoran Hadzibabic, Effects of interactions on the critical temperature of a trapped Bose gas, Phys. Rev. Lett. 106, 250403 (2011).
  68. Milan Krstajić, Jiří Kučera, Lucas R. Hofer, Gavin Lamb, Péter Juhász, and Robert P. Smith, Interaction shift of the Bose-Einstein condensation temperature in a dipolar gas, Phys. Rev. A 111, L051303 (2025).
  69. Philippe Sindzingre, Michael L. Klein, and David M. Ceperley, Path-integral Monte Carlo study of low-temperature 4He clusters, Phys. Rev. Lett. 63, 1601 (1989).
  70. V. L. Berezinskii, Destruction of long range order in one-dimensional and two-dimensional systems having a continuous symmetry group. I. Classical systems, Sov. Phys. JETP 32, 493 (1971).
  71. J. M. Kosterlitz and D. J. Thouless, Ordering, metastability and phase transitions in two-dimensional systems, J. Phys. C 6, 1181 (1973).
  72. Here, the axial trap frequency is kept fixed. Lowering this frequency shifts the EG-to-SBG transition boundary toward smaller Ωπ and to the lower-temperature side.
  73. Mingyang Guo, Fabian Boettcher, Jens Hertkorn, Jan-Niklas Schmidt, Matthias Wenzel, Hans Peter Buechler, Tim Langen, and Tilman Pfau, The low-energy Goldstone mode in a trapped dipolar supersolid, Nature 574, 386 (2019).

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PRX Quantum 6, 040307– Published 9 October, 2025
  • Received 7 April 2025
  • Accepted 26 August 2025
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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.

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