- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Letters

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Quantum Carpets of Higgs Quasiparticles in a Supersolid

K. Mukherjee1,*, M. Schubert1, R. Klemt2, T. Bland1, T. Pfau2, and S. M. Reimann1

  • 1Mathematical Physics and NanoLund, LTH, Lund University, Box 118, 22100 Lund, Sweden
  • 25. Physikalisches Institut and Center for Integrated Quantum Science and Technology, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany

  • *Contact author: koushikphysics21@gmail.com
PDF

Phys. Rev. Lett. 135, 223402 – Published 24 November, 2025

DOI: https://doi.org/10.1103/6d1g-671p

Abstract

Supersolids formed from dipolar Bose Einstein condensates (BECs) exhibit spontaneous density modulation while maintaining global phase coherence. This state of matter supports gapped amplitude (Higgs) excitations featuring a quadratic dispersion relation. While Higgs modes are typically strongly damped due to coupling with other amplitude and phase modes, imposing an experimentally realistic toroidal geometry allows us to numerically study the time evolution and dispersion of a localized Higgs quasiparticle excitation, with minimal residual coupling to sound modes. Strikingly, the quadratic dispersion leads to the occurrence of (fractional) revivals, similar to those seen in the optical Talbot effect or the so-called quantum carpets. The revival times provide a novel method for determining the effective mass of the Higgs quasiparticle through a nonspectroscopic approach. These results pave the way for further studies of coherent Higgs dynamics and mutual interactions between Higgs quasiparticle.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (94)

  1. J. Goldstone, A. Salam, and S. Weinberg, Broken symmetries, Phys. Rev. 127, 965 (1962).
  2. P. W. Anderson, Plasmons, gauge invariance, and mass, Phys. Rev. 130, 439 (1963).
  3. P. W. Higgs, Broken symmetries and the masses of gauge bosons, Phys. Rev. Lett. 13, 508 (1964).
  4. D. Pekker and C. Varma, Amplitude/Higgs modes in condensed matter physics, Annu. Rev. Condens. Matter Phys. 6, 269 (2015).
  5. R. Sooryakumar and M. V. Klein, Raman scattering from superconducting gap excitations in the presence of a magnetic field, Phys. Rev. B 23, 3213 (1981).
  6. R. Matsunaga, Y. I. Hamada, K. Makise, Y. Uzawa, H. Terai, Z. Wang, and R. Shimano, Higgs amplitude mode in the BCS superconductors Nb1−xTixN induced by terahertz pulse excitation, Phys. Rev. Lett. 111, 057002 (2013).
  7. D. Sherman, U. S. Pracht, B. Gorshunov, S. Poran, J. Jesudasan, M. Chand, P. Raychaudhuri, M. Swanson, N. Trivedi, A. Auerbach, M. Scheffler, A. Frydman, and M. Dressel, The Higgs mode in disordered superconductors close to a quantum phase transition, Nat. Phys. 11, 188 (2015).
  8. T. Cea, C. Castellani, and L. Benfatto, Nonlinear optical effects and third-harmonic generation in superconductors: Cooper pairs versus Higgs mode contribution, Phys. Rev. B 93, 180507(R) (2016).
  9. R. Shimano and N. Tsuji, Higgs mode in superconductors, Annu. Rev. Condens. Matter Phys. 11, 103 (2020).
  10. C. Rüegg, B. Normand, M. Matsumoto, A. Furrer, D. F. McMorrow, K. W. Krämer, H. U. Güdel, S. N. Gvasaliya, H. Mutka, and M. Boehm, Quantum magnets under pressure: Controlling elementary excitations in TlCuCl3, Phys. Rev. Lett. 100, 205701 (2008).
  11. A. Behrle, T. Harrison, J. Kombe, K. Gao, M. Link, J. S. Bernier, C. Kollath, and M. Köhl, Higgs mode in a strongly interacting fermionic superfluid, Nat. Phys. 14, 781 (2018).
  12. P. Dyke, S. Musolino, H. Kurkjian, D. J. M. Ahmed-Braun, A. Pennings, I. Herrera, S. Hoinka, S. J. J. M. F. Kokkelmans, V. E. Colussi, and C. J. Vale, Higgs oscillations in a unitary fermi superfluid, Phys. Rev. Lett. 132, 223402 (2024).
  13. A. Kell, M. Breyer, D. Eberz, and M. Köhl, Exciting the Higgs mode in a strongly interacting fermi gas by interaction modulation, Phys. Rev. Lett. 133, 150403 (2024).
  14. C. R. Cabrera, R. Henke, L. Broers, J. Skulte, H. P. Ojeda Collado, H. Biss, L. Mathey, and H. Moritz, Hybridization of the amplitude mode in a confined fermionic superfluid, Phys. Rev. Res. 7, L032018 (2025).
  15. U. Bissbort, S. Götze, Y. Li, J. Heinze, J. S. Krauser, M. Weinberg, C. Becker, K. Sengstock, and W. Hofstetter, Detecting the amplitude mode of strongly interacting lattice bosons by Bragg scattering, Phys. Rev. Lett. 106, 205303 (2011).
  16. M. Endres, T. Fukuhara, D. Pekker, M. Cheneau, P. Schauß, C. Gross, E. Demler, S. Kuhr, and I. Bloch, The ’Higgs’ amplitude mode at the two-dimensional superfluid/Mott insulator transition, Nature (London) 487, 454 (2012).
  17. T. M. Hoang, H. M. Bharath, M. J. Boguslawski, M. Anquez, B. A. Robbins, and M. S. Chapman, Adiabatic quenches and characterization of amplitude excitations in a continuous quantum phase transition, Proc. Natl. Acad. Sci. U.S.A. 113, 9475 (2016).
  18. J. Lonard, A. Morales, P. Zupancic, T. Donner, and T. Esslinger, Monitoring and manipulating Higgs and goldstone modes in a supersolid quantum gas, Science 358, 1415 (2017).
  19. J. Bjerlin, S. M. Reimann, and G. M. Bruun, Few-body precursor of the Higgs mode in a fermi gas, Phys. Rev. Lett. 116, 155302 (2016).
  20. L. Bayha, M. Holten, R. Klemt, K. Subramanian, J. Bjerlin, S. M. Reimann, G. M. Bruun, P. M. Preiss, and S. Jochim, Observing the emergence of a quantum phase transition shell by shell, Nature (London) 587, 583 (2020).
  21. S. D. Huber, B. Theiler, E. Altman, and G. Blatter, Amplitude mode in the quantum phase model, Phys. Rev. Lett. 100, 050404 (2008).
  22. L. Pollet and N. Prokof’ev, Higgs mode in a two-dimensional superfluid, Phys. Rev. Lett. 109, 010401 (2012).
  23. K. Chen, L. Liu, Y. Deng, L. Pollet, and N. Prokof’ev, Universal properties of the Higgs resonance in (2+1)-dimensional U(1) critical systems, Phys. Rev. Lett. 110, 170403 (2013).
  24. L. Liu, K. Chen, Y. Deng, M. Endres, L. Pollet, and N. Prokof’ev, Massive goldstone (Higgs) mode in two-dimensional ultracold atomic lattice systems, Phys. Rev. B 92, 174521 (2015).
  25. M. Boninsegni and N. V. Prokof’ev, Colloquium: Supersolids: What and where are they?, Rev. Mod. Phys. 84, 759 (2012).
  26. J.-R. Li, J. Lee, W. Huang, S. Burchesky, B. Shteynas, F. C. Top, A. O. Jamison, and W. Ketterle, A stripe phase with supersolid properties in spin-orbit-coupled Bose-Einstein condensates, Nature (London) 543, 9194 (2017).
  27. J. Léonard, A. Morales, P. Zupancic, T. Esslinger, and T. Donner, Supersolid formation in a quantum gas breaking a continuous translational symmetry, Nature (London) 543, 87 (2017).
  28. C. Chisholm, S. Hirthe, V. Makhalov, R. Ramos, R. Vatré, J. Cabedo, A. Celi, and L. Tarruell, Probing supersolidity through excitations in a spin-orbit-coupled Bose-Einstein condensate, arXiv:2412.13861.
  29. L. Tanzi, E. Lucioni, F. Famà, J. Catani, A. Fioretti, C. Gabbanini, R. N. Bisset, L. Santos, and G. Modugno, Observation of a dipolar quantum gas with metastable supersolid properties, Phys. Rev. Lett. 122, 130405 (2019).
  30. F. Bttcher, M. Wenzel, J.-N. Schmidt, M. Guo, T. Langen, I. Ferrier-Barbut, T. Pfau, R. Bombín, J. Sánchez-Baena, J. Boronat, and F. Mazzanti, Dilute dipolar quantum droplets beyond the extended Gross-Pitaevskii equation, Phys. Rev. Res. 1, 033088 (2019).
  31. L. Chomaz, D. Petter, P. Ilzhöfer, G. Natale, A. Trautmann, C. Politi, G. Durastante, R. M. W. van Bijnen, A. Patscheider, M. Sohmen, M. J. Mark, and F. Ferlaino, Long-lived and transient supersolid behaviors in dipolar quantum gases, Phys. Rev. X 9, 021012 (2019).
  32. M. A. Norcia, C. Politi, L. Klaus, E. Poli, M. Sohmen, M. J. Mark, R. N. Bisset, L. Santos, and F. Ferlaino, Two-dimensional supersolidity in a dipolar quantum gas, Nature (London) 596, 357 (2021).
  33. F. Bttcher, 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 (2020).
  34. 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).
  35. A. Recati and S. Stringari, Supersolidity in ultracold dipolar gases, Nat. Rev. Phys. 5, 735 (2023).
  36. G. Natale, R. M. W. van Bijnen, A. Patscheider, D. Petter, M. J. Mark, L. Chomaz, and F. Ferlaino, Excitation spectrum of a trapped dipolar supersolid and its experimental evidence, Phys. Rev. Lett. 123, 050402 (2019).
  37. M. Guo, F. Böttcher, J. Hertkorn, J.-N. Schmidt, M. Wenzel, H. P. Büchler, T. Langen, and T. Pfau, The low-energy Goldstone mode in a trapped dipolar supersolid, Nature (London) 574, 386 (2019).
  38. D. Petter, G. Natale, R. M. W. van Bijnen, A. Patscheider, M. J. Mark, L. Chomaz, and F. Ferlaino, Probing the roton excitation spectrum of a stable dipolar Bose gas, Phys. Rev. Lett. 122, 183401 (2019).
  39. J.-N. Schmidt, J. Hertkorn, M. Guo, F. Böttcher, M. Schmidt, K. S. H. Ng, S. D. Graham, T. Langen, M. Zwierlein, and T. Pfau, Roton excitations in an oblate dipolar quantum gas, Phys. Rev. Lett. 126, 193002 (2021).
  40. J. Hertkorn, J.-N. Schmidt, M. Guo, F. Böttcher, K. S. H. Ng, S. D. Graham, P. Uerlings, H. P. Büchler, T. Langen, M. Zwierlein, and T. Pfau, Supersolidity in two-dimensional trapped dipolar droplet arrays, Phys. Rev. Lett. 127, 155301 (2021).
  41. J. Hertkorn, F. Böttcher, M. Guo, J. N. Schmidt, T. Langen, H. P. Büchler, and T. Pfau, Fate of the amplitude mode in a trapped dipolar supersolid, Phys. Rev. Lett. 123, 193002 (2019).
  42. J. Hofmann and W. Zwerger, Hydrodynamics of a superfluid smectic, J. Stat. Mech. (2021) P033104.
  43. P. B. Blakie, L. Chomaz, D. Baillie, and F. Ferlaino, Compressibility and speeds of sound across the superfluid-to-supersolid phase transition of an elongated dipolar gas, Phys. Rev. Res. 5, 033161 (2023).
  44. M. Šindik, T. Zawiślak, A. Recati, and S. Stringari, Sound, superfluidity, and layer compressibility in a ring dipolar supersolid, Phys. Rev. Lett. 132, 146001 (2024).
  45. L. M Platt, D. Baillie, and P. B. Blakie, Sound waves and fluctuations in one-dimensional supersolids, Phys. Rev. A 110, 023320 (2024).
  46. J. Hertkorn, P. Stürmer, K. Mukherjee, K. S. H. Ng, P. Uerlings, F. Hellstern, L. Lavoine, S. M. Reimann, T. Pfau, and R. Klemt, Decoupled sound and amplitude modes in trapped dipolar supersolids, Phys. Rev. Res. 6, L042056 (2024).
  47. M. N. Tengstrand, D. Boholm, R. Sachdeva, J. Bengtsson, and S. M. Reimann, Persistent currents in toroidal dipolar supersolids, Phys. Rev. A 103, 013313 (2021).
  48. M. Nilsson Tengstrand and P. Stürmer, and J. Ribbing, and S. M. Reimann, Toroidal dipolar supersolid with a rotating weak link, Phys. Rev. A 107, 063316 (2023).
  49. J. Azaña and M. A. Muriel, Temporal talbot effect in fiber gratings and its applications, Appl. Opt. 38, 6700 (1999).
  50. J. Wen, Y. Zhang, and M. Xiao, The talbot effect: Recent advances in classical optics, nonlinear optics, and quantum optics, Adv. Opt. Photonics 5, 83 (2013).
  51. H. F. Talbot, Facts relating to optical science. No. IV, London Edinburgh Dublin Phil. Mag. J. Sci. 9, 401 (1836).
  52. L. Rayleigh, On copying diffraction-gratings, and on some phenomena connected therewith, London Edinburgh Dublin Phil. Mag. J. Sci. 11, 196 (1881).
  53. M. V. Berry and S. Klein, Integer, fractional and fractal talbot effects, J. Mod. Opt. 43, 2139 (1996).
  54. M. R. Dennis, N. I. Zheludev, and F. J. G. de Abajo, The plasmon talbot effect, Opt. Express 15, 9692 (2007).
  55. L. Li, Y. Fu, H. Wu, L. Zheng, H. Zhang, Z. Lu, Q. Sun, and W. Yu, The talbot effect of plasmonic nanolenses, Opt. Express 19, 19365 (2011).
  56. Y. Zhang, M. R. Belić, H. Zheng, H. Chen, C. Li, J. Song, and Y. Zhang, Nonlinear talbot effect of rogue waves, Phys. Rev. E 89, 032902 (2014).
  57. G. G. Rozenman, W. P. Schleich, L. Shemer, and A. Arie, Periodic wave trains in nonlinear media: Talbot revivals, akhmediev breathers, and asymmetry breaking, Phys. Rev. Lett. 128, 214101 (2022).
  58. H. Kaijun, J. Jahns, and A. W. Lohmann, Talbot interferometry with a vibrating phase object, Opt. Commun. 45, 295 (1983).
  59. S. Prakash, S. Upadhyay, and C. Shakher, Real-time out-of-plane vibration measurement and monitoring using talbot interferometry, Opt. Lasers Eng. 33, 251 (2000).
  60. J. F. Clauser and M. W. Reinsch, New theoretical and experimental results in fresnel optics with applications to matter-wave and x-ray interferometry, Appl. Phys. B 54, 380 (1992).
  61. M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, Near-field imaging of atom diffraction gratings: The atomic talbot effect, Phys. Rev. A 51, R14 (1995).
  62. S. Nowak, C. Kurtsiefer, T. Pfau, and C. David, High-order talbot fringes for atomic matter waves, Opt. Lett. 22, 1430 (1997).
  63. S. B. Cahn, A. Kumarakrishnan, U. Shim, T. Sleator, P. R. Berman, and B. Dubetsky, Time-domain de broglie wave interferometry, Phys. Rev. Lett. 79, 784 (1997).
  64. M. J. Mark, E. Haller, J. G. Danzl, K. Lauber, M. Gustavsson, and H.-C. Ngerl, Demonstration of the temporal matter-wave talbot effect for trapped matter waves, New J. Phys. 13, 085008 (2011).
  65. 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).
  66. P. Höllmer, J.-S. Bernier, C. Kollath, C. Baals, B. Santra, and H. Ott, Talbot effect in the presence of interactions, Phys. Rev. A 100, 063613 (2019).
  67. F. Wei, Z. Zhang, Y. Chen, H. Shui, Y. Liang, C. Li, and X. Zhou, Temporal talbot interferometer of a strongly interacting molecular Bose-Einstein condensate, Phys. Rev. A 109, 043313 (2024).
  68. C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Feshbach resonances in ultracold gases, Rev. Mod. Phys. 82, 1225 (2010).
  69. Y. Tang, A. Sykes, N. Q. Burdick, J. L. Bohn, and B. L. Lev, s-wave scattering lengths of the strongly dipolar bosons Dy162 and Dy164, Phys. Rev. A 92, 022703 (2015).
  70. 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).
  71. 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).
  72. L. Chomaz, S. Baier, D. Petter, M. J. Mark, F. Wächtler, L. Santos, and F. Ferlaino, Quantum-fuctuation-driven crossover from a dilute Bose-Einstein condensate to a macrodroplet in a dipolar quantum fluid, Phys. Rev. X 6, 041039 (2016).
  73. 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).
  74. See Supplemental Material http://link.aps.org/supplemental/10.1103/6d1g-671p for further details, which includes information on (i) construction of a Higgs wave packet, (ii) analytical insights, (iii) quantum carpets for varying number of localized density sites, (iv) details of the numerical simulations, and additional Refs. [75–79].
  75. P. B. Blakie, D. Baillie, L. Chomaz, and F. Ferlaino, Supersolidity in an elongated dipolar condensate, Phys. Rev. Res. 2, 043318 (2020).
  76. J. C. Smith, D. Baillie, and P. B. Blakie, Supersolidity and crystallization of a dipolar Bose gas in an infinite tube, Phys. Rev. A 107, 033301 (2023).
  77. 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).
  78. R. Schtzhold, M. Uhlmann, Y. Xu, and U. R. Fischer, Mean-field expansion in BoseEinstein condensates with finite-range interactions, Int. J. Mod. Phys. B 20, 3555 (2006).
  79. A. R. P. Lima and A. Pelster, Quantum fluctuations in dipolar Bose gases, Phys. Rev. A 84, 041604(R) (2011).
  80. G. Biagioni, N. Antolini, B. Donelli, L. Pezzè, A. Smerzi, M. Fattori, A. Fioretti, C. Gabbanini, M. Inguscio, L. Tanzi et al., Measurement of the superfluid fraction of a supersolid by Josephson effect, Nature (London) 629, 773 (2024).
  81. G. Del Pace, K. Xhani, A. Muzi Falconi, M. Fedrizzi, N. Grani, D. Hernandez Rajkov, M. Inguscio, F. Scazza, W. J. Kwon, and G. Roati, Imprinting persistent currents in tunable fermionic rings, Phys. Rev. X 12, 041037 (2022).
  82. A. E. Kaplan, I. Marzoli, W. E. Lamb, and W. P. Schleich, Multimode interference: Highly regular pattern formation in quantum wave-packet evolution, Phys. Rev. A 61, 032101 (2000).
  83. J. Ruostekoski, B. Kneer, W. P. Schleich, and G. Rempe, Interference of a Bose-Einstein condensate in a hard-wall trap: From the nonlinear talbot effect to the formation of vorticity, Phys. Rev. A 63, 043613 (2001).
  84. P. T. Grochowski, T. Karpiuk, M. Brewczyk, and K. Rzażewski, Fermionic quantum carpets: From canals and ridges to solitonlike structures, Phys. Rev. Res. 2, 013119 (2020).
  85. M. Łebek, P. T. Grochowski, and K. Rzażewski, Single- to many-body crossover of a quantum carpet, Phys. Rev. Res. 3, 023009 (2021).
  86. D. Podolsky, A. Auerbach, and D. P. Arovas, Visibility of the amplitude (Higgs) mode in condensed matter, Phys. Rev. B 84, 174522 (2011).
  87. S. Gazit, D. Podolsky, and A. Auerbach, Fate of the Higgs mode near quantum criticality, Phys. Rev. Lett. 110, 140401 (2013).
  88. F. Bttcher, J.-N. Schmidt, M. Wenzel, J. Hertkorn, M. Guo, T. Langen, and T. Pfau, Transient supersolid properties in an array of dipolar quantum droplets, Phys. Rev. X 9, 011051 (2019).
  89. J. Leonard, A. Morales, P. Zupancic, T. Esslinger, and T. Donner, A stripe phase with supersolid properties in spin-orbit-coupled Bose-Einstein condensates, Nature (London) 543, 87 (2017).
  90. 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.
  91. S. Zhang, W. Yuan, N. Bigagli, H. Kwak, T. Karman, I. Stevenson, and S. Will, Observation of self-bound droplets of ultracold dipolar molecules, arXiv:2507.15208.
  92. T. Arnone Cardinale, T. Bland, and S. M. Reimann, Exploring molecular supersolidity via exact and mean-field theories: single microwave shielding, arXiv:2509.18051.
  93. F. Manni, K. G. Lagoudakis, T. C. H. Liew, R. André, and B. Deveaud-Plédran, Spontaneous pattern formation in a polariton condensate, Phys. Rev. Lett. 107, 106401 (2011).
  94. D. Trypogeorgos, A. Gianfrate, M. Landini, D. Nigro, D. Gerace, I. Carusotto, F. Riminucci, K. W. Baldwin, L. N. Pfeiffer, G. I. Martone, M. De Giorgi, D. Ballarini, and D. Sanvitto, Emerging supersolidity in photonic-crystal polariton condensates, Nature (London) 639, 337 (2025).

Outline

Information

Phys. Rev. Lett. 135, 223402– Published 24 November, 2025
  • Received 1 July 2025
  • Revised 1 October 2025
  • Accepted 28 October 2025
Crossmark - Check for updates button
Creative Commons Logo

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. Funded by Bibsam.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation