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Quantum-gas microscopy and Talbot interferometry of the Bose-glass phase
Phys. Rev. A 113, 043303 – Published 3 April, 2026
DOI: https://doi.org/10.1103/xb42-j6px
Abstract
Disordered potentials fundamentally affect transport and coherence in quantum systems, giving rise to a Bose-glass phase in interacting bosonic systems—an insulating yet compressible phase lacking long-range coherence. Directly measuring a reduced coherence length of the Bose glass has been a outstanding challenge. We address this by employing Talbot interferometry combined with single-atom-resolved detection in a quantum-gas microscope. Using ultracold bosonic atoms in a two-dimensional lattice with site-resolved, reproducible disorder, we identify the Bose-glass phase through in situ density distributions and particle-number fluctuations, quantified via the Edwards-Anderson parameter, and through the visibility of interference patterns after time of flight. By driving the system across the Bose-glass phase, we further observe signatures of nonergodic dynamics. Our studies provide a starting point to further explore disordered systems in and out of equilibrium, and are relevant for understanding the dynamics and stability of disordered and glasslike quantum states in solid-state systems.
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References (71)
- L. Krusin-Elbaum, L. Civale, G. Blatter, A. D. Marwick, F. Holtzberg, and C. Feild, Bose-glass melting in YBaCuO crystals with correlated disorder, Phys. Rev. Lett. 72, 1914 (1994).
- L. Berthier and G. Biroli, Theoretical perspective on the glass transition and amorphous materials, Rev. Mod. Phys. 83, 587 (2011).
- F. Ducry, J. Aeschlimann, and M. Luisier, Electro-thermal transport in disordered nanostructures: A modeling perspective, Nanoscale Adv. 2, 2648 (2020).
- L. Sanchez-Palencia and M. Lewenstein, Disordered quantum gases under control, Nat. Phys. 6, 87 (2010).
- B. Shapiro, Cold atoms in the presence of disorder, J. Phys. A: Math. Theor. 45, 143001 (2012).
- L. Pollet, A review of Monte Carlo simulations for the Bose-Hubbard model with diagonal disorder, C.R. Phys. 14, 712 (2013).
- D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Colloquium: Many-body localization, thermalization, and entanglement, Rev. Mod. Phys. 91, 021001 (2019).
- P. W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev. 109, 1492 (1958).
- T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Transport and Anderson localization in disordered two-dimensional photonic lattices, Nature (London) 446, 52 (2007).
- Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Anderson localization and nonlinearity in one-dimensional disordered photonic lattices, Phys. Rev. Lett. 100, 013906 (2008).
- G. Roati, C. D'Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, Anderson localization of a non-interacting Bose–Einstein condensate, Nature (London) 453, 895 (2008).
- M. Pasienski, D. McKay, M. White, and B. DeMarco, A disordered insulator in an optical lattice, Nat. Phys. 6, 677 (2010).
- S. S. Kondov, W. R. McGehee, J. J. Zirbel, and B. DeMarco, Three-dimensional Anderson localization of ultracold matter, Science 334, 66 (2011).
- F. Jendrzejewski, A. Bernard, K. Mueller, P. Cheinet, V. Josse, M. Piraud, L. Pezzé, L. Sanchez-Palencia, A. Aspect, and P. Bouyer, Three-dimensional localization of ultracold atoms in an optical disordered potential, Nat. Phys. 8, 398 (2012).
- D. H. White, T. A. Haase, D. J. Brown, M. D. Hoogerland, M. S. Najafabadi, J. L. Helm, C. Gies, D. Schumayer, and D. A. Hutchinson, Observation of two-dimensional Anderson localisation of ultracold atoms, Nat. Commun. 11, 4942 (2020).
- M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, Boson localization and the superfluid-insulator transition, Phys. Rev. B 40, 546 (1989).
- J.-W. Lee, M.-C. Cha, and D. Kim, Phase diagram of a disordered Boson Hubbard model in two dimensions, Phys. Rev. Lett. 87, 247006 (2001).
- 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).
- I. Aleiner, B. Altshuler, and G. Shlyapnikov, A finite-temperature phase transition for disordered weakly interacting bosons in one dimension, Nat. Phys. 6, 900 (2010).
- K. Binder and A. P. Young, Spin glasses: Experimental facts, theoretical concepts, and open questions, Rev. Mod. Phys. 58, 801 (1986).
- F. Lin, E. S. Sørensen, and D. M. Ceperley, Superfluid-insulator transition in the disordered two-dimensional Bose-Hubbard model, Phys. Rev. B 84, 094507 (2011).
- Ş. G. Söyler, M. Kiselev, N. V. Prokof'ev, and B. V. Svistunov, Phase diagram of the commensurate two-dimensional disordered Bose-Hubbard model, Phys. Rev. Lett. 107, 185301 (2011).
- P. B. Weichman, Dirty bosons: Twenty years later, Mod. Phys. Lett. B 22, 2623 (2008).
- L. Pollet, N. V. Prokof'ev, B. V. Svistunov, and M. Troyer, Absence of a direct Superfluid to Mott insulator transition in disordered Bose systems, Phys. Rev. Lett. 103, 140402 (2009).
- G. Bertoli, V. P. Michal, B. L. Altshuler, and G. V. Shlyapnikov, Finite-temperature disordered bosons in two dimensions, Phys. Rev. Lett. 121, 030403 (2018).
- Z. Zhu, H. Yao, and L. Sanchez-Palencia, Thermodynamic phase diagram of two-dimensional bosons in a quasicrystal potential, Phys. Rev. Lett. 130, 220402 (2023).
- C. Meldgin, U. Ray, P. Russ, D. Chen, D. Ceperley, and B. DeMarco, Probing the Bose glass–superfluid transition using quantum quenches of disorder, Nat. Phys. 12, 646 (2016).
- B. Nagler, S. Barbosa, J. Koch, G. Orso, and A. Widera, Observing the loss and revival of long-range phase coherence through disorder quenches, Proc. Natl. Acad. Sci. USA 119, e2111078118 (2022).
- M. Schreiber, S. S. Hodgman, P. Bordia, H. P. Lüschen, M. H. Fischer, R. Vosk, E. Altman, U. Schneider, and I. Bloch, Observation of many-body localization of interacting fermions in a quasirandom optical lattice, Science 349, 842 (2015).
- J. Choi, S. Hild, J. Zeiher, P. Schauß, A. Rubio-Abadal, T. Yefsah, V. Khemani, D. A. Huse, I. Bloch, and C. Gross, Exploring the many-body localization transition in two dimensions, Science 352, 1547 (2016).
- M. Rispoli, A. Lukin, R. Schittko, S. Kim, M. E. Tai, J. Léonard, and M. Greiner, Quantum critical behaviour at the many-body localization transition, Nature (London) 573, 385 (2019).
- 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).
- S. F. Edwards and P. W. Anderson, Theory of spin glasses, J. Phys. F 5, 965 (1975).
- S. Morrison, A. Kantian, A. J. Daley, H. G. Katzgraber, M. Lewenstein, H. P. Büchler, and P. Zoller, Physical replicas and the Bose glass in cold atomic gases, New J. Phys. 10, 073032 (2008).
- S. J. Thomson, L. S. Walker, T. L. Harte, and G. D. Bruce, Measuring the Edwards-Anderson order parameter of the Bose glass: A quantum gas microscope approach, Phys. Rev. A 94, 051601(R) (2016).
- A. R. Abadal, Probing quantum thermalization and localization in Bose-Hubbard systems, Ph.D. thesis, Ludwig-Maximilians-Universität München, 2020.
- A. Di Carli, C. Parsonage, A. La Rooij, L. Koehn, C. Ulm, C. W. Duncan, A. J. Daley, E. Haller, and S. Kuhr, Commensurate and incommensurate 1D interacting quantum systems, Nat. Commun. 15, 474 (2024).
- J. F. Sherson, C. Weitenberg, M. Endres, M. Cheneau, I. Bloch, and S. Kuhr, Single-atom-resolved fluorescence imaging of an atomic Mott insulator, Nature (London) 467, 68 (2010).
- J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht, P. Lugan, D. Clément, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, Direct observation of Anderson localization of matter waves in a controlled disorder, Nature (London) 453, 891 (2008).
- M. Sbroscia, K. Viebahn, E. Carter, J.-C. Yu, A. Gaunt, and U. Schneider, Observing localization in a 2D quasicrystalline optical lattice, Phys. Rev. Lett. 125, 200604 (2020).
- 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).
- See Supplemental Material at http://link.aps.org/supplemental/10.1103/xb42-j6px for additional information, which also includes Refs. [45, 46, 47, 48, 49, 50].
- F. Gerbier, A. Widera, S. Fölling, O. Mandel, T. Gericke, and I. Bloch, Phase coherence of an atomic Mott insulator, Phys. Rev. Lett. 95, 050404 (2005).
- B. Capogrosso-Sansone, Ş. G. Söyler, N. Prokof'ev, and B. Svistunov, Monte Carlo study of the two-dimensional Bose-Hubbard model, Phys. Rev. A 77, 015602 (2008).
- F. Alet, P. Dayal, A. Grzesik, A. Honecker, M. Körner, A. Läuchli, S. R. Manmana, I. P. McCulloch, F. Michel, R. M. Noack, et al., The ALPS project: Open source software for strongly correlated systems, J. Phys. Soc. Jpn. 74, 30 (2005).
- A. F. Albuquerque, F. Alet, P. Corboz, P. Dayal, A. Feiguin, S. Fuchs, L. Gamper, E. Gull, S. Gürtler, A. Honecker, et al., The ALPS project release 1.3: Open-source software for strongly correlated systems, J. Magn. Magn. Mater. 310, 1187 (2007).
- B. Bauer, L. Carr, H. G. Evertz, A. Feiguin, J. Freire, S. Fuchs, L. Gamper, J. Gukelberger, E. Gull, S. Guertler, et al., The ALPS project release 2.0: Open source software for strongly correlated systems, J. Stat. Mech. P05001 (2011).
- A. W. Sandvik, Stochastic series expansion method with operator-loop update, Phys. Rev. B 59, R14157(R) (1999).
- L. Pollet, S. M. A. Rombouts, K. Van Houcke, and K. Heyde, Optimal Monte Carlo updating, Phys. Rev. E 70, 056705 (2004).
- F. Alet, S. Wessel, and M. Troyer, Generalized directed loop method for quantum Monte Carlo simulations, Phys. Rev. E 71, 036706 (2005).
- S. Pal, R. Bai, S. Bandyopadhyay, K. Suthar, and D. Angom, Enhancement of the Bose glass phase in the presence of an artificial gauge field, Phys. Rev. A 99, 053610 (2019).
- B. Santra, C. Baals, R. Labouvie, A. B. Bhattacherjee, A. Pelster, and H. Ott, Measuring finite-range phase coherence in an optical lattice using Talbot interferometry, Nat. Commun. 8, 15601 (2017).
- V. Gurarie, L. Pollet, N. V. Prokof'ev, B. V. Svistunov, and M. Troyer, Phase diagram of the disordered Bose-Hubbard model, Phys. Rev. B 80, 214519 (2009).
- H. J. Manetsch, G. Nomura, E. Bataille, K. H. Leung, X. Lv, and M. Endres, A tweezer array with 6100 highly coherent atomic qubits, Nature (London) 647, 60 (2025).
- G. Malpuech, D. D. Solnyshkov, H. Ouerdane, M. M. Glazov, and I. Shelykh, Bose glass and superfluid phases of cavity polaritons, Phys. Rev. Lett. 98, 206402 (2007).
- R. Yu, L. Yin, N. S. Sullivan, J. Xia, C. Huan, A. Paduan-Filho, N. F. Oliveira Jr, S. Haas, A. Steppke, C. F. Miclea, et al., Bose glass and Mott glass of quasiparticles in a doped quantum magnet, Nature (London) 489, 379 (2012).
- W. O. Tromp, T. Benschop, J.-F. Ge, I. Battisti, K. M. Bastiaans, D. Chatzopoulos, A. H. Vervloet, S. Smit, E. van Heumen, M. S. Golden, et al., Puddle formation and persistent gaps across the non-mean-field breakdown of superconductivity in overdoped , Nat. Mater. 22, 703 (2023).
- L. A. González-García, S. F. Caballero-Benítez, and R. Paredes, Localisation of weakly interacting bosons in two dimensions: Disorder vs lattice geometry effects, Sci. Rep. 9, 11049 (2019).
- X. Mao, J. Liu, J. Zhong, and R. A. Römer, Disorder effects in the two-dimensional Lieb lattice and its extensions, Physica E 124, 114340 (2020).
- J. Carrasquilla, F. Becca, A. Trombettoni, and M. Fabrizio, Characterization of the Bose-glass phase in low-dimensional lattices, Phys. Rev. B 81, 195129 (2010).
- Z. Ristivojevic, A. Petković, P. Le Doussal, and T. Giamarchi, Superfluid/Bose-glass transition in one dimension, Phys. Rev. B 90, 125144 (2014).
- J. Léonard, S. Kim, M. Rispoli, A. Lukin, R. Schittko, J. Kwan, E. Demler, D. Sels, and M. Greiner, Probing the onset of quantum avalanches in a many-body localized system, Nat. Phys. 19, 481 (2023).
- M. A. Tusch and D. E. Logan, Interplay between disorder and electron interactions in a site-disordered Anderson-Hubbard model: A numerical mean-field study, Phys. Rev. B 48, 14843 (1993).
- A. Sanpera, A. Kantian, L. Sanchez-Palencia, J. Zakrzewski, and M. Lewenstein, Atomic Fermi-Bose mixtures in inhomogeneous and random lattices: From Fermi glass to quantum spin glass and quantum percolation, Phys. Rev. Lett. 93, 040401 (2004).
- V. Ahufinger, L. Sanchez-Palencia, A. Kantian, A. Sanpera, and M. Lewenstein, Disordered ultracold atomic gases in optical lattices: A case study of Fermi-Bose mixtures, Phys. Rev. A 72, 063616 (2005).
- B. Paredes, C. Tejedor, and J. I. Cirac, Fermionic atoms in optical superlattices, Phys. Rev. A 71, 063608 (2005).
- A. Szabó and U. Schneider, Mixed spectra and partially extended states in a two-dimensional quasiperiodic model, Phys. Rev. B 101, 014205 (2020).
- C. W. Duncan, Critical states and anomalous mobility edges in two-dimensional diagonal quasicrystals, Phys. Rev. B 109, 014210 (2024).
- N. Ticea, et al., Observation of disorder-induced superfluidity, arXiv:2512.21416.
- The data used in this publication are openly available at the University of Strathclyde KnowledgeBase, https://doi.org/10.15129/af5cd941-2496-4a50-a10e-14df02036546.
- A. La Rooij, C. Ulm, E. Haller, and S. Kuhr, A comparative study of deconvolution techniques for quantum-gas microscope images, New J. Phys. 25, 083036 (2023).