- Physical Review Journals
All JournalsPhysics Magazine

Physical Review A

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

Long-wavelength optical lattices from optical beatnotes: Theory and applications

T. Petrucciani1, A. Santoni2,3, C. Mazzinghi1,3, D. Trypogeorgos4, F. Minardi5,3, M. Fattori1,3,6,*, and M. Modugno7,8,9

  • *Contact author: fattori@lens.unifi.it
PDF

Phys. Rev. A 112, 043323 – Published 27 October, 2025

DOI: https://doi.org/10.1103/zpxp-btt5

Abstract

We present a theoretical analysis of BeatNote Superlattices (BNSLs), a recently demonstrated technique for generating periodic trapping potentials for ultracold atomic clouds, with arbitrarily large lattice spacings while maintaining interferometric stability. By combining two optical lattices with slightly different wavelengths, a beatnote intensity pattern is formed, generating, for low depths, an effective lattice potential with a periodicity equal to the wavelength associated to the difference between the wave vectors of the two lattices. We study the range of lattice depths and wavelengths under which this approximation is valid and investigate its robustness against perturbations. We present a few examples where the use of BNSLs could offer significant advantages in comparison to well-established techniques for the manipulation of ultracold atomic gases. Our results highlight the potential of BNSLs for quantum simulation, atom interferometry, and other applications in quantum technologies.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (59)

  1. A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. Schmidt, Optical atomic clocks, Rev. Mod. Phys. 87, 637 (2015).
  2. I. Bloch, J. Dalibard, and S. Nascimbène, Quantum simulations with ultracold quantum gases, Nat. Phys. 8, 267 (2012).
  3. A. D. Cronin, J. Schmiedmayer, and D. E. Pritchard, Optics and interferometry with atoms and molecules, Rev. Mod. Phys. 81, 1051 (2009).
  4. M. Saffman, T. G. Walker, and K. Mølmer, Quantum information with Rydberg atoms, Rev. Mod. Phys. 82, 2313 (2010).
  5. O. Morsch and M. Oberthaler, Dynamics of Bose-Einstein condensates in optical lattices, Rev. Mod. Phys. 78, 179 (2006).
  6. M. Kasevich and S. Chu, Atomic interferometry using stimulated Raman transitions, Phys. Rev. Lett. 67, 181 (1991).
  7. I. Bloch, Ultracold quantum gases in optical lattices, Nat. Phys. 1, 23 (2005).
  8. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, Laser phase and frequency stabilization using an optical resonator, Appl. Phys. B 31, 97 (1983).
  9. R. Scheunemann, F. S. Cataliotti, T. W. Hänsch, and M. Weitz, An optical lattice with single lattice site optical control for quantum engineering, J. Opt. B: Quantum Semiclassical Opt. 2, 645 (2000).
  10. K. D. Nelson, X. Li, and D. S. Weiss, Imaging single atoms in a three-dimensional array, Nat. Phys. 3, 556 (2007).
  11. M. Albiez, R. Gati, J. Fölling, S. Hunsmann, M. Cristiani, and M. K. Oberthaler, Direct observation of tunneling and nonlinear self-trapping in a single bosonic Josephson junction, Phys. Rev. Lett. 95, 010402 (2005).
  12. T. C. Li, H. Kelkar, D. Medellin, and M. G. Raizen, Real-time control of the periodicity of a standing wave: an optical accordion, Opt. Express 16, 5465 (2008).
  13. G. Valtolina, K. Matsuda, W. G. Tobias, J.-R. Li, L. De Marco, and J. Ye, Dipolar evaporation of reactive molecules to below the Fermi temperature, Nature (London) 588, 239 (2020).
  14. S. Hirthe, T. Chalopin, D. Bourgund, P. Bojović, A. Bohrdt, E. Demler, F. Grusdt, I. Bloch, and T. A. Hilker, Magnetically mediated hole pairing in fermionic ladders of ultracold atoms, Nature (London) 613, 463 (2023).
  15. L.-C. Ha, L. W. Clark, C. V. Parker, B. M. Anderson, and C. Chin, Roton-maxon excitation spectrum of Bose condensates in a shaken optical lattice, Phys. Rev. Lett. 114, 055301 (2015).
  16. D. Barredo, S. de Léséleuc, V. Lienhard, T. Lahaye, and A. Browaeys, An atom-by-atom assembler of defect-free arbitrary two-dimensional atomic arrays, Science 354, 1021 (2016).
  17. L. Masi, T. Petrucciani, G. Ferioli, G. Semeghini, G. Modugno, M. Inguscio, and M. Fattori, Spatial Bloch oscillations of a quantum gas in a “beat-note” superlattice, Phys. Rev. Lett. 127, 020601 (2021).
  18. L. Masi, T. Petrucciani, A. Burchianti, C. Fort, M. Inguscio, L. Marconi, G. Modugno, N. Preti, D. Trypogeorgos, M. Fattori, and F. Minardi, Multimode trapped interferometer with noninteracting Bose-Einstein condensates, Phys. Rev. Res. 3, 043188 (2021).
  19. J. Sebby-Strabley, M. Anderlini, P. S. Jessen, and J. V. Porto, Lattice of double wells for manipulating pairs of cold atoms, Phys. Rev. A 73, 033605 (2006).
  20. V. Novičenko, J. Ruseckas, and E. Anisimovas, Quantum dynamics in potentials with fast spatial oscillations, Phys. Rev. A 99, 043608 (2019).
  21. O. Morandi and M. Modugno, Multiband envelope function model for quantum transport in a tunneling diode, Phys. Rev. B 71, 235331 (2005).
  22. N. Ashcroft and D. Mermin, Solid State Physics (Saunders College, Philadelphia, 1976).
  23. The derivation of this equation makes use of Eq. (9.13) from Ref. [22], setting K1=0, ɛ≃ɛk0+∑K≠0|UK|2ɛk0−ɛk−K0,where ɛk0 corresponds to the free-particle dispersion, and UK denotes the Fourier components of the potential, which is fixed by an arbitrary offset such that U0=0. For a potential of the form U(x)=−V0αcos(k+x), the only nonzero components are U±K=−V0α/2, with K=k+. After some algebra, and upon reinstating the constant term, this yields Eq. (5).
  24. I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics with ultracold gases, Rev. Mod. Phys. 80, 885 (2008).
  25. This approximation works reasonably well for n≫1.
  26. F. Gerbier, A. Widera, S. Fölling, O. Mandel, T. Gericke, and I. Bloch, Interference pattern and visibility of a Mott insulator, Phys. Rev. A 72, 053606 (2005).
  27. I. Nałcz, L. Masi, G. Ferioli, T. Petrucciani, M. Fattori, and J. Chwedeńczuk, Sensitivity bounds of a spatial Bloch-oscillation atom interferometer, Phys. Rev. A 102, 033318 (2020).
  28. Note that the dependence of the bandwidth on the ratio λ2/λ1 for low amplitudes, V0/EB+≪1, does not contradict the validity of the effective potential. It simply reflects the fact that the value of k−, which defines the first Brillouin zone, increases with λ2/λ1, and consequently, so does the bandwidth (think of the free-particle dispersion relation, in the limit V0→0).
  29. Z. Hadzibabic, P. Krüger, M. Cheneau, B. Battelier, and J. Dalibard, Berezinskii–Kosterlitz–Thouless crossover in a trapped atomic gas, Nature (London) 441, 1118 (2006).
  30. For large n, this occurs when δ≈ℏω, leading to V0≈2[(2n+1)/π]4EB+.
  31. M. Boll, T. A. Hilker, G. Salomon, A. Omran, J. Nespolo, L. Pollet, I. Bloch, and C. Gross, Spin- and density-resolved microscopy of antiferromagnetic correlations in Fermi-Hubbard chains, Science 353, 1257 (2016).
  32. G. Salomon, J. Koepsell, J. Vijayan, T. Hilker, J. Nespolo, L. Pollet, I. Bloch, and C. Gross, Direct observation of incommensurate magnetism in Hubbard chains, Nature (London) 565, 56 (2019).
  33. M. Gall, N. Wurz, J. Samland, C. F. Chan, and M. Köhl, Competing magnetic orders in a bilayer Hubbard model with ultracold atoms, Nature (London) 589, 40 (2021).
  34. T. Hartke, B. Oreg, N. Jia, and M. Zwierlein, Doublon-hole correlations and fluctuation thermometry in a Fermi-Hubbard gas, Phys. Rev. Lett. 125, 113601 (2020).
  35. Z. Hadzibabic, S. Stock, B. Battelier, V. Bretin, and J. B. Dalibard, Interference of an array of independent Bose-Einstein condensates, Phys. Rev. Lett. 93, 180403 (2004).
  36. W. Lechner and P. Zoller, From classical to quantum glasses with ultracold polar molecules, Phys. Rev. Lett. 111, 185306 (2013).
  37. F. Cinti, D.-W. Wang, and M. Boninsegni, Phases of dipolar bosons in a bilayer geometry, Phys. Rev. A 95, 023622 (2017).
  38. G. Anich, N. Höllrigl, M. Kreyer, R. Grimm, and E. Kirilov, Comprehensive characterization of an apparatus for cold electromagnetic dysprosium dipoles, Phys. Rev. A 110, 023311 (2024).
  39. 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).
  40. J. L. Ville, T. Bienaimé, R. Saint-Jalm, L. Corman, M. Aidelsburger, L. Chomaz, K. Kleinlein, D. Perconte, S. Nascimbène, J. Dalibard, and J. Beugnon, Loading and compression of a single two-dimensional Bose gas in an optical accordion, Phys. Rev. A 95, 013632 (2017).
  41. The longitudinal size σ is determined by the effective trapping frequency provided by the lattice potential at each lattice site, which is related to the first energy gap by the relation Δ1≃ℏωeff.
  42. Y. Shin, M. Saba, T. A. Pasquini, W. Ketterle, D. E. Pritchard, and A. E. Leanhardt, Atom interferometry with Bose-Einstein condensates in a double-well potential, Phys. Rev. Lett. 92, 050405 (2004).
  43. T. Berrada, S. van Frank, R. Bücker, T. Schumm, J.-F. Schaff, and J. Schmiedmayer, Integrated Mach–Zehnder interferometer for Bose–Einstein condensates, Nat. Commun. 4, 2077 (2013).
  44. J. Estève, C. Gross, A. Weller, S. Giovanazzi, and M. K. Oberthaler, Squeezing and entanglement in a Bose–Einstein condensate, Nature (London) 455, 1216 (2008).
  45. A. Trenkwalder, G. Spagnolli, G. Semeghini, S. Coop, M. Landini, P. Castilho, L. Pezzè, G. Modugno, M. Inguscio, A. Smerzi, and M. Fattori, Quantum phase transitions with parity-symmetry breaking and hysteresis, Nat. Phys. 12, 826 (2016).
  46. G. Spagnolli, G. Semeghini, L. Masi, G. Ferioli, A. Trenkwalder, S. Coop, M. Landini, L. Pezzè, G. Modugno, M. Inguscio, A. Smerzi, and M. Fattori, Crossing over from attractive to repulsive interactions in a tunneling bosonic Josephson junction, Phys. Rev. Lett. 118, 230403 (2017).
  47. T. Petrucciani, A. Santoni, C. Mazzinghi, D. Trypogeorgos, F. S. Cataliotti, M. Inguscio, G. Modugno, A. Smerzi, L. Pezzé, and M. Fattori, Mach-Zehnder atom interferometry with non-interacting trapped Bose Einstein condensates, arXiv:2504.17391.
  48. T. Chalopin, P. Bojović, D. Bourgund, S. Wang, T. Franz, I. Bloch, and T. Hilker, Optical superlattice for engineering Hubbard couplings in quantum simulation, Phys. Rev. Lett. 134, 053402 (2025).
  49. W. D. Li, T. He, and A. Smerzi, Multimode Kapitza-Dirac interferometry with trapped cold atoms, Phys. Rev. Lett. 113, 023003 (2014).
  50. J. Stenger, S. Inouye, A. P. Chikkatur, D. M. Stamper-Kurn, D. E. Pritchard, and W. Ketterle, Bragg spectroscopy of a Bose-Einstein condensate, Phys. Rev. Lett. 82, 4569 (1999).
  51. S. Stringari and L. Pitaevskii, Bose-Einstein Condensation (Clarendon, Oxford, 2003).
  52. R. Bracewell, The Hilbert transform, The Fourier Transform and its Applications, 3rd ed. (McGraw-Hill, Singapore, 2000), Ch. 13, pp. 359–367.
  53. E. W. Weisstein, Hilbert transform, from MathWorld—A Wolfram Web Resource, https://mathworld.wolfram.com/HilbertTransform.html.
  54. D. Trypogeorgos and C. J. Foot, Cotrapping different species in ion traps using multiple radio frequencies, Phys. Rev. A 94, 023609 (2016).
  55. T. L. Harte, E. Bentine, K. Luksch, A. J. Barker, D. Trypogeorgos, B. Yuen, and C. J. Foot, Ultracold atoms in multiple radio-frequency dressed adiabatic potentials, Phys. Rev. A 97, 013616 (2018).
  56. C. J. Foot, D. Trypogeorgos, E. Bentine, A. Gardner, and M. Keller, Two-frequency operation of a Paul trap to optimise confinement of two species of ions, Int. J. Mass Spectrom. 430, 117 (2018).
  57. D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, 4th ed. (Oxford University Press, Oxford, 2007).
  58. N. V. Konenkov, M. Sudakov, and D. J. Douglas, Matrix methods for the calculation of stability diagrams in quadrupole mass spectrometry, J. Am. Soc. Mass Spectrom. 13, 597 (2002).
  59. H. Broer and C. Simó, Hill's equation with quasi-periodic forcing: resonance tongues, instability pockets and global phenomena, Bull. Braz. Math. Soc. 29, 253 (1998).

Outline

Information

Phys. Rev. A 112, 043323– Published 27 October, 2025
  • Received 29 April 2025
  • Accepted 17 September 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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation