- Open Access
Symmetry and self-bound droplets in dipolar molecular gases
Phys. Rev. Research 8, 023219 – Published 28 May, 2026
DOI: https://doi.org/10.1103/l9wk-v25v
Abstract
Recent experiments with degenerate molecular gases dressed by elliptically polarized microwave fields have enabled new control of dipolar interactions via engineered anisotropy. We reveal a symmetry structure of the dipolar interaction that generates degeneracies among the interaction parameters, enabling a classification of spatial symmetries and equilibrium shapes of the gases. Exploiting these symmetries, we analyze solutions including beyond-mean-field quantum fluctuations in the dilute gas regime and develop a complementary variational theory. We map out the phase diagram of self-bound droplets and characterize their widths, energies, and densities.
Physics Subject Headings (PhySH)
Article Text
References (63)
- K. K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Pe’er, B. Neyenhuis, J. J. Zirbel, S. Kotochigova, P. S. Julienne, D. S. Jin, and J. Ye, A high phase-space-density gas of polar molecules, Science 322, 231 (2008).
- S. Ospelkaus, K.-K. Ni, D. Wang, M. H. G. de Miranda, B. Neyenhuis, G. Quéméner, P. S. Julienne, J. L. Bohn, D. S. Jin, and J. Ye, Quantum-state controlled chemical reactions of ultracold potassium-rubidium molecules, Science 327, 853 (2010).
- K.-K. Ni, S. Ospelkaus, D. Wang, G. Quemener, B. Neyenhuis, M. H. G. de Miranda, J. L. Bohn, J. Ye, and D. S. Jin, Dipolar collisions of polar molecules in the quantum regime, Nature (London) 464, 1324 (2010).
- L. De Marco, G. Valtolina, K. Matsuda, W. G. Tobias, J. P. Covey, and J. Ye, A degenerate Fermi gas of polar molecules, Science 363, 853 (2019).
- T. Karman and J. M. Hutson, Microwave shielding of ultracold polar molecules, Phys. Rev. Lett. 121, 163401 (2018).
- L. Lassablière and G. Quéméner, Controlling the scattering length of ultracold dipolar molecules, Phys. Rev. Lett. 121, 163402 (2018).
- L. Anderegg, S. Burchesky, Y. Bao, S. S. Yu, T. Karman, E. Chae, K.-K. Ni, W. Ketterle, and J. M. Doyle, Observation of microwave shielding of ultracold molecules, Science 373, 779 (2021).
- N. Bigagli, C. Warner, W. Yuan, S. Zhang, I. Stevenson, T. Karman, and S. Will, Collisionally stable gas of bosonic dipolar ground-state molecules, Nat. Phys. 19, 1579 (2023).
- 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).
- T. Langen, G. Valtolina, D. Wang, and J. Ye, Quantum state manipulation and cooling of ultracold molecules, Nat. Phys. 20, 702 (2024).
- J. Lin, G. Chen, M. Jin, Z. Shi, F. Deng, W. Zhang, G. Quéméner, T. Shi, S. Yi, and D. Wang, Microwave shielding of bosonic NaRb molecules, Phys. Rev. X 13, 031032 (2023).
- 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, arXiv:2508.20518.
- H. Kadau, M. Schmitt, M. Wenzel, C. Wink, T. Maier, I. Ferrier-Barbut, and T. Pfau, Observing the Rosensweig instability of a quantum ferrofluid, Nature (London) 530, 194 (2016).
- 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).
- 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).
- D. S. Petrov, Quantum mechanical stabilization of a collapsing Bose-Bose mixture, Phys. Rev. Lett. 115, 155302 (2015).
- T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, and T. Pfau, The physics of dipolar bosonic quantum gases, Rep. Prog. Phys. 72, 126401 (2009).
- 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).
- L. Chomaz, Quantum-stabilized states in magnetic dipolar quantum gases, Annu. Rev. Condens. Matter Phys. 17, 47 (2026).
- D. Baillie, R. M. Wilson, R. N. Bisset, and P. B. Blakie, Self-bound dipolar droplet: A localized matter wave in free space, Phys. Rev. A 94, 021602(R) (2016).
- 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).
- F. Wächtler and L. Santos, Quantum filaments in dipolar Bose-Einstein condensates, Phys. Rev. A 93, 061603(R) (2016).
- 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).
- C. R. Cabrera, L. Tanzi, J. Sanz, B. Naylor, P. Thomas, P. Cheiney, and L. Tarruell, Quantum liquid droplets in a mixture of Bose-Einstein condensates, Science 359, 301 (2018).
- G. Semeghini, G. Ferioli, L. Masi, C. Mazzinghi, L. Wolswijk, F. Minardi, M. Modugno, G. Modugno, M. Inguscio, and M. Fattori, Self-bound quantum droplets of atomic mixtures in free space, Phys. Rev. Lett. 120, 235301 (2018).
- S. Zhang, W. Yuan, N. Bigagli, H. Kwak, T. Karman, I. Stevenson, and S. Will, Observation of self-bound droplets of ultracold dipolar molecules, Nature (London) 651, 601 (2026).
- S. Giovanazzi, A. Görlitz, and T. Pfau, Tuning the dipolar interaction in quantum gases, Phys. Rev. Lett. 89, 130401 (2002).
- D. Baillie and P. B. Blakie, Rotational tuning of the dipole-dipole interaction in a Bose gas of magnetic atoms, Phys. Rev. A 101, 043606 (2020).
- Y. Tang, W. Kao, K.-Y. Li, and B. L. Lev, Tuning the dipole-dipole interaction in a quantum gas with a rotating magnetic field, Phys. Rev. Lett. 120, 230401 (2018).
- M. Wenzel, Macroscopic states of dipolar quantum gases, Ph.D. thesis, University of Stuttgart, Germany, 2018.
- K. Mukherjee, T. A. Cardinale, L. Chergui, P. Stürmer, and S. M. Reimann, Droplets and supersolids in ultra-cold atomic quantum gases, Eur. Phys. J. Spec. Top. 232, 3417 (2023).
- K. Mukherjee, M. N. Tengstrand, T. A. Cardinale, and S. M. Reimann, Supersolid stacks in antidipolar Bose-Einstein condensates, Phys. Rev. A 108, 023302 (2023).
- W.-J. Jin, F. Deng, S. Yi, and T. Shi, Bose-Einstein condensates of microwave-shielded polar molecules, Phys. Rev. Lett. 134, 233003 (2025).
- W. Zhang, K. Chen, S. Yi, and T. Shi, Quantum phases for finite-temperature gases of bosonic polar molecules shielded by dual microwaves, PRX Quantum 6, 040307 (2025).
- 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).
- 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).
- C. J. Polterauer and R. E. Zillich, Phases of a Bose-Einstein condensate of microwave-shielded dipolar molecules, arXiv:2507.18986.
- T. A. Cardinale, T. Bland, and S. M. Reimann, Exploring molecular supersolidity via exact and mean-field theories: Single microwave shielding, arXiv:2509.18051.
- W. Zhang, H. Liu, F. Deng, K. Chen, S. Yi, and T. Shi, Supersolid phases in ultracold gases of microwave shielded polar molecules, arXiv:2506.23820.
- F. Deng, X.-Y. Chen, X.-Y. Luo, W. Zhang, S. Yi, and T. Shi, Effective potential and superfluidity of microwave-shielded polar molecules, Phys. Rev. Lett. 130, 183001 (2023).
- T. Karman, N. Bigagli, W. Yuan, S. Zhang, I. Stevenson, and S. Will, Double microwave shielding, PRX Quantum 6, 020358 (2025).
- 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).
- A. Schindewolf, J. Hertkorn, I. Stevenson, M. Ciardi, P. Gross, D. Wang, T. Karman, G. Quemener, S. Will, T. Pohl, and T. Langen, Colloquium: Strongly dipolar molecular Bose-Einstein condensates: From few- to many-body physics, Rev. Mod. Phys. (2026), https://doi.org/10.1103/r55l-f93m.
- Using defined in Ref. [26], our for , 2. Using defined in Ref. [39], our .
- Not to be confused with the vacuum permittivity.
- W. Yuan, S. Zhang, N. Bigagli, H. Kwak, C. Warner, T. Karman, I. Stevenson, and S. Will, Extreme loss suppression and wide tunability of dipolar interactions in an ultracold molecular gas, arXiv:2505.08773.
- A. R. P. Lima and A. Pelster, Quantum fluctuations in dipolar Bose gases, Phys. Rev. A 84, 041604(R) (2011).
- A. R. P. Lima and A. Pelster, Beyond mean-field low-lying excitations of dipolar Bose gases, Phys. Rev. A 86, 063609 (2012).
- By taking the Fourier transform, changing variables to , integrating the angular variables, and changing variables back to .
- S. Giovanazzi, P. Pedri, L. Santos, A. Griesmaier, M. Fattori, T. Koch, J. Stuhler, and T. Pfau, Expansion dynamics of a dipolar Bose-Einstein condensate, Phys. Rev. A 74, 013621 (2006).
- K. Glaum and A. Pelster, Bose-Einstein condensation temperature of dipolar gas in anisotropic harmonic trap, Phys. Rev. A 76, 023604 (2007).
- We align our dipoles along , whereas Ref. [50] aligns along , but our function is exactly the same as theirs.
- For , as is reduced changes sharply from for , through zero at , to 1 at .
- L. Santos, G. V. Shlyapnikov, P. Zoller, and M. Lewenstein, Bose-Einstein condensation in trapped dipolar gases, Phys. Rev. Lett. 85, 1791 (2000).
- I. Ferrier-Barbut, H. Kadau, M. Schmitt, M. Wenzel, and T. Pfau, Observation of quantum droplets in a strongly dipolar Bose gas, Phys. Rev. Lett. 116, 215301 (2016).
- J. Dalibard, Magnetic interactions between cold atoms: Quantum droplets and supersolid states, Lecture series, Collège de France, 2024, https://pro.college-de-france.fr/jean.dalibard/CdF/2024/total_en_2024.pdf.
- D. Baillie, R. M. Wilson, and P. B. Blakie, Collective excitations of self-bound droplets of a dipolar quantum fluid, Phys. Rev. Lett. 119, 255302 (2017).
- D. Baillie, Symmetry and self-bound droplets in dipolar molecular gases [data set], Zenodo (2026), https://doi.org/10.5281/zenodo.20078499.
- J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999).
- S. Ronen, D. C. E. Bortolotti, and J. L. Bohn, Bogoliubov modes of a dipolar condensate in a cylindrical trap, Phys. Rev. A 74, 013623 (2006).
- 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).
- The prefactor of Eq. (A10) of Ref. [21] should be , not .
- For larger parameters increases like or [see Eq. (4)], but the imaginary part of will then be significant.