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Stark Many-Body Localization in One-Dimensional Bose Gas under the Continuous Limit

Zihan Zhao1,2, Zhongchi Zhang2, Huaichuan Wang2, Ken Deng2, Wenlan Chen2,*, and Jiazhong Hu1,†

  • *Contact author: cwlaser@ultracold.cn
  • †Contact author: hujiazhong01@ultracold.cn
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PRX Quantum 7, 010307 – Published 13 January, 2026

DOI: https://doi.org/10.1103/ygdy-hrgj

Abstract

Many-body localization, an emergent quantum phenomenon, challenges the fundamental principles of thermodynamics and has attracted significant attention. In this study, we demonstrate Stark many-body localization in a one-dimensional Bose gas under the continuous limit, a scenario that extends well beyond the conventional tight-binding model commonly employed in most discussions. This localization occurs within a background harmonic trap without any random disorder, where the potential gradient effectively halts thermalization. Contrary to prior studies on Stark localization, our findings reveal that an increasing gradient does not necessarily lead to greater localization. Instead, we observe a complex transition within this continuous interacting system. Another surprising result is that a higher initial density, which typically correlates with a higher collisional rate and thermalization rate, actually induces stronger localization in this one-dimensional model. Additionally, we observe a density plateau with zero compressibility and low-density fluctuations, which may signal an insulating behavior associated with prethermalization during nonequilibrium dynamics.

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Popular Summary

Many-body localization is a fascinating quantum phenomenon where interacting particles avoid thermalization, defying the conventional laws of thermodynamics. While typically studied in tight-binding models, such as optical lattices with deep potentials, it remains unclear whether localization persists in systems beyond this approximation—where particles behave more like a continuous gas. Here, we demonstrate Stark many-body localization in a one-dimensional Bose gas under the continuous limit, using a very shallow optical lattice and a large number of atoms. This setup allows us to explore quantum dynamics in a regime where traditional models break down.

Our experiment employs rubidium-85 atoms confined in a weak one-dimensional lattice with a background harmonic trap. By suddenly releasing the atoms and monitoring their evolution, we observe a complex transition between oscillation and stationary behaviors for the case of short-time evolution, and localized behavior is observed after a long-term evolution with relatively small scattering length. Surprisingly, increasing the atomic density enhances localization—contrary to expectations that higher collisional rates promote thermalization. We also observe a density plateau with zero compressibility and low-density fluctuations, which may signal an insulating behavior associated with prethermalization during nonequilibrium dynamics.

Looking ahead, our work reveals that localization behavior is present even in systems under the continuous limit, which will spur further theoretical research in this field. Our work will also inspire researchers to further study transport or thermodynamic properties in systems beyond the tight-binding approximation.

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References (54)

  1. Rahul Nandkishore and David A. Huse, Many-body localization and thermalization in quantum statistical mechanics, Ann. Rev. Condens. Matter Phys. 6, 15 (2015).
  2. Dmitry A. Abanin, Ehud Altman, Immanuel Bloch, and Maksym Serbyn, Colloquium: Many-body localization, thermalization, and entanglement, Rev. Mod. Phys. 91, 021001 (2019).
  3. L. Fleishman and P. W. Anderson, Interactions and the Anderson transition, Phys. Rev. B 21, 2366 (1980).
  4. L. Fallani, J. E. Lye, V. Guarrera, C. Fort, and M. Inguscio, Ultracold atoms in a disordered crystal of light: Towards a Bose glass, Phys. Rev. Lett. 98, 130404 (2007).
  5. S. S. Kondov, W. R. McGehee, W. Xu, and B. DeMarco, Disorder-induced localization in a strongly correlated atomic Hubbard gas, Phys. Rev. Lett. 114, 083002 (2015).
  6. Michael Schreiber, Sean S. Hodgman, Pranjal Bordia, Henrik P. Lüschen, Mark H. Fischer, Ronen Vosk, Ehud Altman, Ulrich Schneider, and Immanuel Bloch, Observation of many-body localization of interacting fermions in a quasirandom optical lattice, Science 349, 842 (2015).
  7. J. Y. Choi, Sebastian Hild, Johannes Zeiher, Peter Schauß, Antonio Rubio-Abadal, Tarik Yefsah, Vedika Khemani, David A. Huse, Immanuel Bloch, and Christian Gross, Exploring the many-body localization transition in two dimensions, Science 352, 1547 (2016).
  8. Alexander Lukin, Matthew Rispoli, Robert Schittko, M. Eric Tai, Adam M. Kaufman, Soonwon Choi, Vedika Khemani, Julian Léonard, and Markus Greiner, Probing entanglement in a many-body–localized system, Science 364, 256 (2019).
  9. J. Smith, A. Lee, P. Richerme, B. Neyenhuis, P. W. Hess, P. Hauke, M. Heyl, D. A. Huse, and C. Monroe, Many-body localization in a quantum simulator with programmable random disorder, Nat. Phys. 12, 907 (2016).
  10. W. Morong, F. Liu, P. Becker, K. S. Collins, L. Feng, A. Kyprianidis, G. Pagano, T. You, A. V. Gorshkov, and C. Monroe, Observation of Stark many-body localization without disorder, Nature 599, 393 (2021).
  11. Qiujiang Guo, Chen Cheng, Hekang Li, Shibo Xu, Pengfei Zhang, Zhen Wang, Chao Song, Wuxin Liu, Wenhui Ren, Hang Dong, Rubem Mondaini, and H. Wang, Stark many-body localization on a superconducting quantum processor, Phys. Rev. Lett. 127, 240502 (2021).
  12. Sebastian Scherg, Thomas Kohlert, Pablo Sala, Frank Pollmann, Bharath Hebbe Madhusudhana, Immanuel Bloch, and Monika Aidelsburger, Observing non-ergodicity due to kinetic constraints in tilted Fermi-Hubbard chains, Nat. Commun. 12, 4490 (2021).
  13. M. Schulz, C. A. Hooley, R. Moessner, and F. Pollmann, Stark many-body localization, Phys. Rev. Lett. 122, 040606 (2019).
  14. Evert van Nieuwenburg, Yuval Baum, and Gil Refael, From Bloch oscillations to many-body localization in clean interacting systems, Proc. Natl. Acad. Sci. U.S.A. 116, 9269 (2019).
  15. Hristiana Atanasova, André Erpenbeck, Emanuel Gull, Yevgeny Bar Lev, and Guy Cohen, Stark many-body localization in interacting infinite dimensional systems, Phys. Rev. Lett. 132, 166301 (2024).
  16. Haoyuan Li, Jia Wang, Xia-Ji Liu, and Hui Hu, Many-body localization in Ising models with random long-range interactions, Phys. Rev. A 94, 063625 (2016).
  17. M. Serbyn, M. Knap, S. Gopalakrishnan, Z. Papić, N. Y. Yao, C. R. Laumann, D. A. Abanin, M. D. Lukin, and E. A. Demler, Interferometric probes of many-body localization, Phys. Rev. Lett. 113, 147204 (2014).
  18. Shuo Liu, Shi-Xin Zhang, Chang-Yu Hsieh, Shengyu Zhang, and Hong Yao, Discrete time crystal enabled by Stark many-body localization, Phys. Rev. Lett. 130, 120403 (2023).
  19. Hajime Nakanotani, Taro Furukawa, Kei Morimoto, and Chihaya Adachi, Long-range coupling of electron-hole pairs in spatially separated organic donor-acceptor layers, Sci. Adv. 2, e1501470 (2016).
  20. Gyunghun Kim, Joseph Suh, Dayeong Lee, Namkyoo Park, and Sunkyu Yu, Long-range-interacting topological photonic lattices breaking channel-bandwidth limit, Light: Sci. Appl. 13, 189 (2024).
  21. Haijun Tang, Can Huang, Yuhan Wang, Xiong Jiang, Ruiheng Jin, Yue Cui, Shumin Xiao, and Qinghai Song, Dynamically tunable long-range coupling enabled by bound state in the continuum, Light: Sci. Appl. 14, 278 (2025).
  22. N. S. Israel and L. Ramunno, in 2023 Photonics North (PN) (IEEE, New York City, USA, 2023), pp. 1–2.
  23. Changqing Xu, Gang Wang, Zhi Hong Hang, Jie Luo, C. T. Chan, and Yun Lai, Design of full-k-space flat bands in photonic crystals beyond the tight-binding picture, Sci. Rep. 5, 18181 (2015).
  24. Stefano Longhi, Non-Hermitian skin effect beyond the tight-binding models, Phys. Rev. B 104, 125109 (2021).
  25. Zhoufei Liu, Peng Jin, Min Lei, Chengmeng Wang, Fabio Marchesoni, Jian-Hua Jiang, and Jiping Huang, Topological thermal transport, Nat. Rev. Phys. 6, 554 (2024).
  26. Holger Fehske, Ralf Schneider, and Alexander Weisse, Computational Many-Particle Physics (Springer, Berlin, Germany, 2007), Vol. 739.
  27. Oliver Morsch and Markus Oberthaler, Dynamics of Bose-Einstein condensates in optical lattices, Rev. Mod. Phys. 78, 179 (2006).
  28. Hui Zhai, Ultracold Atomic Physics (Cambridge University, Cambridge, United Kingdom, 2021).
  29. Nathan Gemelke, Xibo Zhang, Chen-Lung Hung, and Cheng Chin, In situ observation of incompressible Mott-insulating domains in ultracold atomic gases, Nature 460, 995 (2009).
  30. Jacob F. Sherson, Christof Weitenberg, Manuel Endres, Marc Cheneau, Immanuel Bloch, and Stefan Kuhr, Single-atom-resolved fluorescence imaging of an atomic Mott insulator, Nature 467, 68 (2010).
  31. Markus Greiner, Olaf Mandel, Tilman Esslinger, Theodor W. Hnsch, and Immanuel Bloch, Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms, Nature 415, 39 (2002).
  32. Thilo Stöferle, Henning Moritz, Christian Schori, Michael Köhl, and Tilman Esslinger, Transition from a strongly interacting 1D superfluid to a Mott insulator, Phys. Rev. Lett. 92, 130403 (2004).
  33. W. S. Bakr, A. Peng, M. E. Tai, R. Ma, J. Simon, J. I. Gillen, S. Fölling, L. Pollet, and M. Greiner, Probing the superfluid–to–Mott insulator transition at the single-atom level, Science 329, 547 (2010).
  34. Chen-Lung Hung, Xibo Zhang, Nathan Gemelke, and Cheng Chin, Accelerating evaporative cooling of atoms into Bose-Einstein condensation in optical traps, Phys. Rev. A 78, 011604 (2008).
  35. N. R. Claussen, S. J. J. M. F. Kokkelmans, S. T. Thompson, E. A. Donley, E. Hodby, and C. E. Wieman, Very-high-precision bound-state spectroscopy near a 85Rb Feshbach resonance, Phys. Rev. A 67, 060701 (2003).
  36. Cheng Chin, Rudolf Grimm, Paul Julienne, and Eite Tiesinga, Feshbach resonances in ultracold gases, Rev. Mod. Phys. 82, 1225 (2010).
  37. See Supplemental Material at http://link.aps.org/supplemental/10.1103/ygdy-hrgj for the additional experimental measurements and the details of the relevant theory, which includes Refs. [52, 53, 54].
  38. M. M. Cerimele, M. L. Chiofalo, F. Pistella, S. Succi, and M. P. Tosi, Numerical solution of the Gross-Pitaevskii equation using an explicit finite-difference scheme: An application to trapped Bose-Einstein condensates, Phys. Rev. E 62, 1382 (2000).
  39. Weizhu Bao, Dieter Jaksch, and Peter A. Markowich, Numerical solution of the Gross–Pitaevskii equation for Bose–Einstein condensation, J. Comput. Phys. 187, 318 (2003).
  40. Xavier Antoine, Weizhu Bao, and Christophe Besse, Computational methods for the dynamics of the nonlinear Schrödinger/Gross–Pitaevskii equations, Comput. Phys. Commun. 184, 2621 (2013).
  41. C. Sias, A. Zenesini, H. Lignier, S. Wimberger, D. Ciampini, O. Morsch, and E. Arimondo, Resonantly enhanced tunneling of Bose-Einstein condensates in periodic potentials, Phys. Rev. Lett. 98, 120403 (2007).
  42. Alessandro Zenesini, Carlo Sias, Hans Lignier, Yeshpal Singh, Donatella Ciampini, Oliver Morsch, Riccardo Mannella, Ennio Arimondo, Andrea Tomadin, and Sandro Wimberger, Resonant tunneling of Bose-Einstein condensates in optical lattices, New J. Phys. 10, 053038 (2008).
  43. Florian Meinert, Manfred J. Mark, Emil Kirilov, Katharina Lauber, Philipp Weinmann, Michael Gröbner, Andrew J. Daley, and Hanns-Christoph Nägerl, Observation of many-body dynamics in long-range tunneling after a quantum quench, Science 344, 1259 (2014).
  44. P. O. Fedichev, M. W. Reynolds, and G. V. Shlyapnikov, Three-body recombination of ultracold atoms to a weakly bound s level, Phys. Rev. Lett. 77, 2921 (1996).
  45. B. D. Esry, Chris H. Greene, and James P. Burke, Recombination of three atoms in the ultracold limit, Phys. Rev. Lett. 83, 1751 (1999).
  46. Tino Weber, Jens Herbig, Michael Mark, Hanns-Christoph Nägerl, and Rudolf Grimm, Three-body recombination at large scattering lengths in an ultracold atomic gas, Phys. Rev. Lett. 91, 123201 (2003).
  47. P. A. Altin, G. R. Dennis, G. D. McDonald, D. Döring, J. E. Debs, J. D. Close, C. M. Savage, and N. P. Robins, Collapse and three-body loss in a 85Rb Bose-Einstein condensate, Phys. Rev. A 84, 033632 (2011).
  48. Zhao-Di Liu, Henri Lyyra, Yong-Nan Sun, Bi-Heng Liu, Chuan-Feng Li, Guang-Can Guo, Sabrina Maniscalco, and Jyrki Piilo, Experimental implementation of fully controlled dephasing dynamics and synthetic spectral densities, Nat. Commun. 9, 3453 (2018).
  49. M. Kolesik and J. V. Moloney, Numerical discreteness and dephasing in high-harmonic calculations in solids, Phys. Rev. B 108, 115433 (2023).
  50. Nathan Gemelke, Xibo Zhang, Chen-Lung Hung, and Cheng Chin, In situ observation of incompressible Mott-insulating domains in ultracold atomic gases, Nature 460, 995 (2009).
  51. Zihan Zhao, Stark many-body localization in one-dimensional Bose gas under the continuous limit, 2025. https://doi.org/10.5281/zenodo.17966182.
  52. G. M. Muslu and H. A. Erbay, A split-step Fourier method for the complex modified Korteweg-de Vries equation, Comput. Math. Appl. 45, 503 (2003).
  53. J. A. C. Weideman and B. M. Herbst, Split-step methods for the solution of the nonlinear Schrodinger equation, SIAM J. Numer. Anal. 23, 485 (1986).
  54. Clarence Zener and Ralph Howard Fowler, Non-adiabatic crossing of energy levels, Proc. R. Soc. London, Ser. A 137, 696 (1932).

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PRX Quantum 7, 010307– Published 13 January, 2026
  • Received 22 July 2025
  • Accepted 15 December 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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