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Stability and mixed phases of three-component droplets in one dimension

I. A. Englezos1,*, E. G. Charalampidis2,†, P. Schmelcher1, and S. I. Mistakidis3

  • *Contact author: ilias.englezos@uni-hamburg.de
  • †Contact author: echaralampidis@sdsu.edu
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Phys. Rev. A 114, 053304 – Published 3 November, 2026

DOI: https://doi.org/10.1103/ppf6-ccsb

Abstract

We explore the ground state properties and excitation spectra of one-dimensional, three-component bosonic mixtures, where two majority components form a self-bound droplet coupled to a third, minority component. Within a self-consistent Lee-Huang-Yang framework, we identify distinct mixed self-bound droplet phases and characterize their phase transitions through variations of either the particle number of the majority components or the intercomponent coupling. The ensuing phases demonstrate that the minority component is untrapped, partially trapped, or fully trapped by the majority droplet species. These states are characterized by their binding energies captured by the chemical potentials and their low-amplitude excitation spectrum, including mode crossings at the particle-emission threshold. We further derive effective reduced models in the high-imbalance limit, which accurately reproduce the numerically computed ground states, while providing analytical insights into the role of quantum fluctuations. Our results establish the stability and structure of mixed droplet phases and provide experimentally relevant signatures for their identification.

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References (76)

  1. F. Böttcher, J.-N. Schmidt, J. Hertkorn, K. S. H. Ng, S. D. Graham, M. Guo, T. Langen, and T. Pfau, New states of matter with fine-tuned interactions: Quantum droplets and dipolar supersolids, Rep. Prog. Phys. 84, 012403 (2021).
  2. B. A. Malomed, The family of quantum droplets keeps expanding, Front. Phys. 16, 22504 (2021).
  3. Z.-H. Luo, W. Pang, B. Liu, Y.-Y. Li, and B. A. Malomed, A new form of liquid matter: Quantum droplets, Front. Phys. 16, 32201 (2021).
  4. S. I. Mistakidis, A. G. Volosniev, R. E. Barfknecht, T. Fogarty, T. Busch, A. Foerster, P. Schmelcher, and N. T. Zinner, Few-body Bose gases in low dimensions—A laboratory for quantum dynamics, Phys. Rep. 1042, 1 (2023).
  5. 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).
  6. D. S. Petrov, Quantum mechanical stabilization of a collapsing Bose-Bose mixture, Phys. Rev. Lett. 115, 155302 (2015).
  7. D. S. Petrov and G. E. Astrakharchik, Ultradilute low-dimensional liquids, Phys. Rev. Lett. 117, 100401 (2016).
  8. I. Englezos, P. Schmelcher, and S. I. Mistakidis, Multicomponent one-dimensional quantum droplets across the mean-field stability regime, SciPost Phys. 19, 133 (2025).
  9. 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).
  10. 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).
  11. 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).
  12. 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).
  13. P. Cheiney, C. R. Cabrera, J. Sanz, B. Naylor, L. Tanzi, and L. Tarruell, Bright soliton to quantum droplet transition in a mixture of Bose-Einstein condensates, Phys. Rev. Lett. 120, 135301 (2018).
  14. 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).
  15. C. D'Errico, A. Burchianti, M. Prevedelli, L. Salasnich, F. Ancilotto, M. Modugno, F. Minardi, and C. Fort, Observation of quantum droplets in a heteronuclear bosonic mixture, Phys. Rev. Res. 1, 033155 (2019).
  16. Z. Guo, F. Jia, L. Li, Y. Ma, J. M. Hutson, X. Cui, and D. Wang, Lee-Huang-Yang effects in the ultracold mixture of Na23 and Rb87 with attractive interspecies interactions, Phys. Rev. Res. 3, 033247 (2021).
  17. L. Cavicchioli, C. Fort, F. Ancilotto, M. Modugno, F. Minardi, and A. Burchianti, Dynamical formation of multiple quantum droplets in a Bose-Bose mixture, Phys. Rev. Lett. 134, 093401 (2025).
  18. T. A. Flynn, L. Parisi, T. P. Billam, and N. G. Parker, Quantum droplets in imbalanced atomic mixtures, Phys. Rev. Res. 5, 033167 (2023).
  19. T. A. Flynn, N. A. Keepfer, N. G. Parker, and T. P. Billam, Harmonically trapped imbalanced quantum droplets, Phys. Rev. Res. 6, 013209 (2024).
  20. L. He, H. Li, W. Yi, and Z.-Q. Yu, Quantum criticality of liquid-gas transition in a binary Bose mixture, Phys. Rev. Lett. 130, 193001 (2023).
  21. J. C. Pelayo, G. A. Bougas, T. Fogarty, T. Busch, and S. I. Mistakidis, Droplet-gas phases and their dynamical formation in particle imbalanced mixtures, Quantum Sci. Technol. 10, 045074 (2025).
  22. G. E. Astrakharchik and B. A. Malomed, Dynamics of one-dimensional quantum droplets, Phys. Rev. A 98, 013631 (2018).
  23. M. Tylutki, G. E. Astrakharchik, B. A. Malomed, and D. S. Petrov, Collective excitations of a one-dimensional quantum droplet, Phys. Rev. A 101, 051601(R) (2020).
  24. S. I. Mistakidis, T. Mithun, P. G. Kevrekidis, H. R. Sadeghpour, and P. Schmelcher, Formation and quench of homonuclear and heteronuclear quantum droplets in one dimension, Phys. Rev. Res. 3, 043128 (2021).
  25. I. A. Englezos, S. I. Mistakidis, and P. Schmelcher, Correlated dynamics of collective droplet excitations in a one-dimensional harmonic trap, Phys. Rev. A 107, 023320 (2023).
  26. Z. Zhao, G. Li, Z. Chen, H.-B. Luo, B. Liu, B. A. Malomed, and Y. Li, Bulk modulus of three-dimensional quantum droplets, Phys. Rev. A 113, 043315 (2026).
  27. L. Dong, D. Liu, and B. A. Malomed, Robust quantum-droplet necklace clusters in three dimensions, Chaos, Solitons & Fractals 203, 117676 (2026).
  28. G. Ferioli, G. Semeghini, L. Masi, G. Giusti, G. Modugno, M. Inguscio, A. Gallemí, A. Recati, and M. Fattori, Collisions of self-bound quantum droplets, Phys. Rev. Lett. 122, 090401 (2019).
  29. G. C. Katsimiga, S. I. Mistakidis, G. N. Koutsokostas, D. J. Frantzeskakis, R. Carretero-González, and P. G. Kevrekidis, Solitary waves in a quantum droplet-bearing system, Phys. Rev. A 107, 063308 (2023).
  30. M. Edmonds, Dark quantum droplets and solitary waves in beyond-mean-field Bose-Einstein condensate mixtures, Phys. Rev. Res. 5, 023175 (2023).
  31. Y. Li, Z. Chen, Z. Luo, C. Huang, H. Tan, W. Pang, and B. A. Malomed, Two-dimensional vortex quantum droplets, Phys. Rev. A 98, 063602 (2018).
  32. G. Bougas, G. C. Katsimiga, P. G. Kevrekidis, and S. I. Mistakidis, Stability and dynamics of nonlinear excitations in a two-dimensional droplet-bearing environment, Phys. Rev. A 110, 033317 (2024).
  33. M. N. Tengstrand, P. Stürmer, E. O. Karabulut, and S. M. Reimann, Rotating binary Bose-Einstein condensates and vortex clusters in quantum droplets, Phys. Rev. Lett. 123, 160405 (2019).
  34. T. A. Yoğurt, U. Tanyeri, A. Keleş, and M. Ö. Oktel, Vortex lattices in strongly confined quantum droplets, Phys. Rev. A 108, 033315 (2023).
  35. S. Chandramouli, S. I. Mistakidis, G. C. Katsimiga, and P. G. Kevrekidis, Dispersive shock waves in a one-dimensional droplet-bearing environment, Phys. Rev. A 110, 023304 (2024).
  36. Y. V. Kartashov, V. M. Lashkin, M. Modugno, and L. Torner, Spinor-induced instability of kinks, holes and quantum droplets, New J. Phys. 24, 073012 (2022).
  37. G. C. Katsimiga, S. I. Mistakidis, B. A. Malomed, D. J. Frantzeskakis, R. Carretero-Gonzalez, and P. G. Kevrekidis, Interactions and dynamics of one-dimensional droplets, bubbles and kinks, Condens. Matter 8, 67 (2023).
  38. S. I. Mistakidis, G. Bougas, G. C. Katsimiga, and P. G. Kevrekidis, Generic transverse stability of kink structures in atomic and optical nonlinear media with competing attractive and repulsive interactions, Phys. Rev. Lett. 134, 123402 (2025).
  39. S. Chandramouli, S. I. Mistakidis, G. C. Katsimiga, D. J. Ratliff, D. J. Frantzeskakis, and P. G. Kevrekidis, Rogue waves in extended Gross-Pitaevskii models with a Lee-Huang-Yang correction, Phys. Rev. A 113, 013308 (2026).
  40. I. A. Englezos, P. Schmelcher, and S. I. Mistakidis, Particle-imbalanced weakly interacting quantum droplets in one dimension, Phys. Rev. A 110, 023324 (2024).
  41. S. Gangwar, R. Ravisankar, S. Mistakidis, P. Muruganandam, and P. K. Mishra, Interaction imbalanced spin-orbit coupled quantum droplets, Phys. Lett. A 542, 130426 (2025).
  42. Y. V. Kartashov and D. A. Zezyulin, Multipole quantum droplets in quasi-one-dimensional asymmetric mixtures, Phys. Rev. A 110, L021304 (2024).
  43. J. Vallès-Muns, I. Morera, G. E. Astrakharchik, and B. Juliá-Díaz, Quantum droplets with particle imbalance in one-dimensional optical lattices, SciPost Phys. 16, 074 (2024).
  44. E. G. Charalampidis and S. I. Mistakidis, Two-component droplet phases and their spectral stability in one dimension, Phys. Rev. A 111, 013318 (2025).
  45. A. Boudjemâa and K. Abbas, Quantum droplets in one-dimensional Bose-Bose mixtures: Beyond the Lee-Huang-Yang description, New J. Phys. 25, 093052 (2023).
  46. L. Parisi and S. Giorgini, Quantum droplets in one-dimensional Bose mixtures: A quantum Monte Carlo study, Phys. Rev. A 102, 023318 (2020).
  47. L. Parisi, G. E. Astrakharchik, and S. Giorgini, Liquid state of one-dimensional Bose mixtures: A quantum Monte Carlo study, Phys. Rev. Lett. 122, 105302 (2019).
  48. A. Debnath, A. Khan, and B. Malomed, Interaction of one-dimensional quantum droplets with potential wells and barriers, Commun. Nonlinear Sci. Numer. Simul. 126, 107457 (2023).
  49. F. Bristy, G. A. Bougas, G. C. Katsimiga, and S. I. Mistakidis, Localization and splitting of a quantum droplet with a potential defect, Chaos, Solitons & Fractals 201, 117383 (2025).
  50. A. Boudjemâa, N. Guebli, M. Sekmane, and S. K.-Karfa, Breathing modes of repulsive polarons in Bose–Bose mixtures, J. Phys.: Condens. Matter 32, 415401 (2020).
  51. D. Diplaris, I. A. Englezos, F. Theel, P. Schmelcher, and S. I. Mistakidis, Phases and dynamics of an impurity immersed in one-dimensional quantum droplets, Entropy 28, 626 (2026).
  52. J. Jogania, S. Das, A. Nath, and J. Bera, Oscillation-induced frequency generation in 1D quantum droplets under harmonic-Gaussian confinements, Phys. Scr. 100, 085212 (2025).
  53. Y. Ma, C. Peng, and X. Cui, Borromean droplet in three-component ultracold Bose gases, Phys. Rev. Lett. 127, 043002 (2021).
  54. D. Edler, Luis A. Pena Ardila, C. R. Cabrera, and L. Santos, Anomalous buoyancy of quantum bubbles in immiscible Bose mixtures, Phys. Rev. Res. 4, 033017 (2022).
  55. Y. Ma and X. Cui, Shell-shaped quantum droplet in a three-component ultracold Bose gas, Phys. Rev. Lett. 134, 043402 (2025).
  56. G. Bighin, A. Burchianti, F. Minardi, and T. Macrì, Impurity in a heteronuclear two-component Bose mixture, Phys. Rev. A 106, 023301 (2022).
  57. J. C. Pelayo, T. Fogarty, T. Busch, and S. I. Mistakidis, Phases and dynamics of few fermionic impurities immersed in two-dimensional boson droplets, Phys. Rev. Res. 6, 033219 (2024).
  58. S. Sinha, S. Biswas, L. Santos, and S. Sinha, Impurities in quasi-one-dimensional droplets of binary Bose mixtures, Phys. Rev. A 108, 023311 (2023).
  59. F. Grusdt and E. Demler, in Quantum Matter at Ultralow Temperatures (IOS Press, Varenna, 2016), pp. 325–411.
  60. F. Scazza, M. Zaccanti, P. Massignan, M. M. Parish, and J. Levinsen, Repulsive Fermi and Bose polarons in quantum gases, Atoms 10, 55 (2022).
  61. R. Schmidt, M. Knap, D. A. Ivanov, J.-S. You, M. Cetina, and E. Demler, Universal many-body response of heavy impurities coupled to a Fermi sea: A review of recent progress, Rep. Prog. Phys. 81, 024401 (2018).
  62. L. Lavoine and T. Bourdel, Beyond-mean-field crossover from one dimension to three dimensions in quantum droplets of binary mixtures, Phys. Rev. A 103, 033312 (2021).
  63. C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Feshbach resonances in ultracold gases, Rev. Mod. Phys. 82, 1225 (2010).
  64. F. K. Abdullaev and R. Galimzyanov, Bosonic impurity in a one-dimensional quantum droplet in the Bose–Bose mixture, J. Phys. B: At. Mol. Opt. Phys. 53, 165301 (2020).
  65. C. T. Kelley, Solving Nonlinear Equations with Newton's Method (SIAM, Philadelphia, 2003).
  66. H. Uecker, Numerical Continuation and Bifurcation in Nonlinear PDEs (SIAM, Philadelphia, 2021).
  67. T. Kapitula and K. Promislow, Spectral and Dynamical Stability of Nonlinear Waves (Springer-Verlag, New York, 2013).
  68. N. J. Willis, A. K. Didier, and K. M. Sonnanburg, How to compute a Puiseux expansion, arXiv:0807.4674.
  69. S. I. Mistakidis, G. C. Katsimiga, G. M. Koutentakis, T. Busch, and P. Schmelcher, Quench dynamics and orthogonality catastrophe of Bose polarons, Phys. Rev. Lett. 122, 183001 (2019).
  70. A. Romero-Ros, G. C. Katsimiga, S. I. Mistakidis, S. Mossman, G. Biondini, P. Schmelcher, P. Engels, and P. G. Kevrekidis, Experimental realization of the Peregrine soliton in repulsive two-component Bose-Einstein condensates, Phys. Rev. Lett. 132, 033402 (2024).
  71. G. A. Bougas, G. C. Katsimiga, S. Mossman, P. Engels, P. G. Kevrekidis, and S. I. Mistakidis, Observation of vector rogue waves in repulsive threecomponent atomic mixtures, Commun. Phys. (2026), doi:10.1038/s42005-026-02818-x.
  72. B. Bakkali-Hassani, C. Maury, Y.-Q. Zou, E. Le Cerf, R. Saint-Jalm, P. C. M. Castilho, S. Nascimbene, J. Dalibard, and J. Beugnon, Realization of a Townes soliton in a two-component planar Bose gas, Phys. Rev. Lett. 127, 023603 (2021).
  73. B. Bakkali-Hassani and J. Dalibard, Townes soliton and beyond: Non-miscible Bose mixtures in 2D, arXiv:2210.14045.
  74. L. Cao, V. Bolsinger, S. I. Mistakidis, G. M. Koutentakis, S. Krönke, J. M. Schurer, and P. Schmelcher, A unified ab initio approach to the correlated quantum dynamics of ultracold fermionic and bosonic mixtures, J. Chem. Phys. 147, 044106 (2017).
  75. R. J. Walker, Algebraic Curves (Springer, Princeton, 1950).
  76. C. J. Pethick and H. Smith, Bose–Einstein Condensation in Dilute Gases, 2nd ed. (Cambridge University Press, Cambridge, 2008).

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Phys. Rev. A 114, 053304– Published 3 November, 2026
  • Received 9 January 2026
  • Accepted 1 October 2026
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