- 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

Unified quantum walk model for internal crystal effects in dynamical diffraction

Owen Lailey1,2,*, Dusan Sarenac3, David G. Cory1,4, Michael G. Huber5, and Dmitry A. Pushin1,2,†

  • *Contact author: oalailey@uwaterloo.ca
  • †Contact author: dmitry.pushin@uwaterloo.ca
PDF

Phys. Rev. A 114, 032210 – Published 14 September, 2026

DOI: https://doi.org/10.1103/rwgb-rpjh

Abstract

The theory of dynamical diffraction (DD) in perfect crystals is the backbone of high-precision neutron and x-ray diffraction experiments, enabling accurate determination of crystal-structure factors and the realization of perfect-crystal interferometers. In practice, however, real crystals exhibit deformations and imperfections, including surface roughness, defects, temperature gradients, angled crystal faces, and curvature, that degrade interferometer performance and are difficult to model using conventional DD theory, particularly in complex geometries. To address these challenges, a quantum information (QI) model for DD has been under development, with demonstrated experimental agreement both for ideal crystals and in the presence of some imperfections such as surface roughness and defects. Here, we present a unified quantum walk model that is now suitable for reproducing a broad range of established DD effects. We demonstrate this by incorporating several diverse internal crystal effects influencing DD intensity distributions, including linear temperature gradients, the DD Talbot effect, and angled or miscut crystals. These results establish the QI model as a comprehensive and flexible framework for experimental analysis, as well as for the design of next-generation perfect-crystal neutron interferometers and neutron optical components, such as condensing monochromators.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (63)

  1. U. Bonse and M. Hart, An x-ray interferometer, Appl. Phys. Lett. 6, 155 (1965).
  2. M. Hart, Review lecture: Ten years of x-ray interferometry, Proc. R. Soc. A 346, 1 (1975).
  3. H. Rauch and S. A. Werner, Neutron Interferometry: Lessons in Experimental Quantum Mechanics, Wave-Particle Duality, and Entanglement, 2nd ed. (Oxford University Press, Oxford, 2015).
  4. V. F. Sears, Neutron Optics (Oxford University Press, Oxford, 1989).
  5. T. J. Davis, D. Gao, T. Gureyev, A. Stevenson, and S. Wilkins, Phase-contrast imaging of weakly absorbing materials using hard x-rays, Nature (London) 373, 595 (1995).
  6. A. Momose, T. Takeda, Y. Itai, and K. Hirano, Phase–contrast X–ray computed tomography for observing biological soft tissues, Nat. Med. 2, 473 (1996).
  7. D. Pushin, D. Cory, M. Arif, D. Jacobson, and M. Huber, Reciprocal space approaches to neutron imaging, Appl. Phys. Lett. 90, 224104 (2007).
  8. P. Becker, K. Dorenwendt, G. Ebeling, R. Lauer, W. Lucas, R. Probst, H.-J. Rademacher, G. Reim, P. Seyfried, and H. Siegert, Absolute measurement of the (220) lattice plane spacing in a silicon crystal, Phys. Rev. Lett. 46, 1540 (1981).
  9. L. Ferroglio, G. Mana, and E. Massa, Si lattice parameter measurement by centimeter x-ray interferometry, Opt. Express 16, 16877 (2008).
  10. E. Massa, G. Mana, U. Kuetgens, and L. Ferroglio, Measurement of the {2 2 0} lattice-plane spacing of a Si28 x-ray interferometer, Metrologia 48, S37 (2011).
  11. K. Li, M. Arif, D. G. Cory, R. Haun, B. Heacock, M. G. Huber, J. Nsofini, D. A. Pushin, P. Saggu, D. Sarenac, et al., Neutron limit on the strongly-coupled chameleon field, Phys. Rev. D 93, 062001 (2016).
  12. B. Heacock, T. Fujiie, R. W. Haun, A. Henins, K. Hirota, T. Hosobata, M. G. Huber, M. Kitaguchi, D. A. Pushin, H. Shimizu, et al., Pendellösung interferometry probes the neutron charge radius, lattice dynamics, and fifth forces, Science 373, 1239 (2021).
  13. M. Arif, M. S. Dewey, G. Greene, D. Jacobson, and S. Werner, X-ray determination of the elastic deformation of a perfect crystal neutron interferometer: Implications for gravitational phase shift experiments, Phys. Lett. A 184, 154 (1994).
  14. M. Huber, B. Heacock, I. Taminiau, D. Cory, D. Sarenac, R. Valdillez, and D. Pushin, Achieving a near-ideal silicon crystal neutron interferometer using submicrometer fabrication techniques, Phys. Rev. Res. 6, 043079 (2024).
  15. H. Rauch, Particle and/or wave features in neutron interferometry, J. Phys.: Conf. Ser. 361, 012019 (2012).
  16. G. Mana, G. Mana, and L. Ferroglio, Observation of a bent crystal-lattice by x-ray interferometry, Opt. Express 17, 11172 (2009).
  17. B. Heacock, M. Arif, R. Haun, M. G. Huber, D. A. Pushin, and A. R. Young, Neutron interferometer crystallographic imperfections and gravitationally induced quantum interference measurements, Phys. Rev. A 95, 013840 (2017).
  18. A. Authier, Dynamical theory of x-ray diffraction, in International Tables for Crystallography Volume B: Reciprocal Space, edited by U. Shmueli, International Tables for Crystallography Vol. B (Springer, Dordrecht, 2006), pp. 534–551.
  19. D. A. Pushin, M. Arif, and D. G. Cory, Decoherence-free neutron interferometry, Phys. Rev. A 79, 053635 (2009).
  20. D. Pushin, M. Huber, M. Arif, and D. Cory, Experimental realization of decoherence-free subspace in neutron interferometry, Phys. Rev. Lett. 107, 150401 (2011).
  21. P. Saggu, T. Mineeva, M. Arif, D. G. Cory, R. Haun, B. Heacock, M. G. Huber, K. Li, J. Nsofini, D. Sarenac, et al., Decoupling of a neutron interferometer from temperature gradients, Rev. Sci. Instrum. 87, 123507 (2016).
  22. B. Heacock, M. Arif, D. G. Cory, T. Gnaeupel-Herold, R. Haun, M. G. Huber, M. E. Jamer, J. Nsofini, D. A. Pushin, D. Sarenac, et al., Increased interference fringe visibility from the post-fabrication heat treatment of a perfect crystal silicon neutron interferometer, Rev. Sci. Instrum. 89, 023502 (2018).
  23. B. Heacock, R. Haun, K. Hirota, T. Hosobata, M. G. Huber, M. E. Jamer, M. Kitaguchi, D. A. Pushin, H. Shimizu, I. Taminiau, et al., Measurement and alleviation of subsurface damage in a thick-crystal neutron interferometer, Acta Crystallogr. Sect. A: Found. Adv. 75, 833 (2019).
  24. J. M. Rocha and F. Dahia, Neutron interferometry and tests of short-range modifications of gravity, Phys. Rev. D 103, 124014 (2021).
  25. H. Lemmel, M. Jentschel, H. Abele, F. Lafont, B. Guerard, C. P. Sasso, G. Mana, and E. Massa, Neutron interference from a split-crystal interferometer, J. Appl. Crystallogr. 55, 870 (2022).
  26. A. Zeilinger, M. Horne, and H. Bernstein, Symmetry violations and Schwinger scattering in neutron interferometry, J. Phys. Colloques 45, C3-209 (1984).
  27. T. Dombeck, R. Ringo, D. D. Koetke, H. Kaiser, K. Schoen, S. A. Werner, and D. Dombeck, Measurement of the neutron reflectivity for Bragg reflections off a perfect silicon crystal, Phys. Rev. A 64, 053607 (2001).
  28. V. V. Fedorov, M. Jentschel, I. A. Kuznetsov, E. G. Lapin, E. Lelievre-Berna, V. Nesvizhevsky, A. Petoukhov, S. Y. Semenikhin, T. Soldner, V. V. Voronin, et al., Measurement of the neutron electric dipole moment via spin rotation in a non-centrosymmetric crystal, Phys. Lett. B 694, 22 (2010).
  29. M. V. Lasitsa, Y. P. Braginetz, E. O. Vezhlev, S. Y. Semenikhin, I. A. Kuznetsov, V. V. Fedorov, and V. V. Voronin, Anomalous behavior of neutron refraction index in a perfect crystal near the Bragg reflex, J. Phys.: Conf. Ser. 572, 012008 (2014).
  30. T. R. Gentile, M. G. Huber, D. D. Koetke, M. Peshkin, M. Arif, T. Dombeck, D. S. Hussey, D. L. Jacobson, P. Nord, D. A. Pushin, and R. Smither, Direct observation of neutron spin rotation in Bragg scattering due to the spin-orbit interaction in silicon, Phys. Rev. C 100, 034005 (2019).
  31. O. Lailey, D. Sarenac, D. G. Cory, M. G. Huber, and D. A. Pushin, A perfect crystal neutron loop cavity, arXiv:2604.02541.
  32. J. Nsofini, K. Ghofrani, D. Sarenac, D. G. Cory, and D. A. Pushin, Quantum-information approach to dynamical diffraction theory, Phys. Rev. A 94, 062311 (2016).
  33. J. Nsofini, D. Sarenac, K. Ghofrani, M. G. Huber, M. Arif, D. G. Cory, and D. A. Pushin, Noise refocusing in a five-blade neutron interferometer, J. Appl. Phys. 122, 054501 (2017).
  34. J. Nsofini, D. Sarenac, D. G. Cory, and D. A. Pushin, Coherence optimization in neutron interferometry through defocusing, Phys. Rev. A 99, 043614 (2019).
  35. O. Nahman-Lévesque, D. Sarenac, D. G. Cory, B. Heacock, M. G. Huber, and D. A. Pushin, Generalizing the quantum information model for dynamic diffraction, Phys. Rev. A 105, 022403 (2022).
  36. O. Nahman-Lévesque, D. Sarenac, O. Lailey, D. G. Cory, M. G. Huber, and D. A. Pushin, Quantum information approach to the implementation of a neutron cavity, New J. Phys. 25, 073016 (2023).
  37. M. Hart, Pendellösung fringes in elastically deformed silicon, Z. Phys. 189, 269 (1966).
  38. I. Fankuchen, A condensing monochromator for x-rays, Nature (London) 139, 193 (1937).
  39. M. Balyan, X-ray two-wave dynamical diffraction analogue of Talbot effect, J. Opt. 21, 055603 (2019).
  40. M. Balyan, An x-ray dynamical diffraction analogue of Talbot effect in the transmitted beam, J. Contemp. Phys. (Arm. Acad. Sci.) 54, 253 (2019).
  41. M. K. Balyan, X-ray dynamical diffraction analogues of the integer and fractional Talbot effects, J Synchrotron Radiat. 26, 1650 (2019).
  42. M. K. Balyan, L. V. Levonyan, and K. G. Trouni, Spherical-wave x-ray dynamical diffraction Talbot effect inside a crystal, Acta Crystallogr. Sect. A: Found. Adv. 76, 494 (2020).
  43. M. Balyan, X-ray dynamical diffraction analogue of Talbot effect in case of incident finite front wave, J. Contemp. Phys. (Arm. Acad. Sci.) 55, 70 (2020).
  44. M. Balyan, L. Levonyan, and K. Trouni, X-ray dynamical diffraction Talbot effect behind a crystal in free space, Acta Crystallogr. Sect. A: Found. Adv. 77, 149 (2021).
  45. C. Shull, Observation of Pendellösung fringe structure in neutron diffraction, Phys. Rev. Lett. 21, 1585 (1968).
  46. B. Klar and F. Rustichelli, Dynamical neutron diffraction by ideally curved crystals, Nuovo Cim. B 13, 249 (1973).
  47. G. Albertini, A. Boeuf, G. Cesini, S. Mazkedian, S. Melone, and F. Rustichelli, A simple model for dynamical neutron diffraction by deformed crystals, Acta Cryst. 32, 863 (1976).
  48. G. Albertini, A. Boeuf, B. Klar, S. Lagomarsino, S. Mazkedian, S. Melone, P. Puliti, and F. Rustichelli, Dynamical neutron diffraction by curved crystals in the Laue geometry, Phys. Status Solidi A 44, 127 (1977).
  49. A. Zeilinger and C. Shull, Magnetic field effects on dynamical diffraction of neutrons by perfect crystals, Phys. Rev. B 19, 3957 (1979).
  50. G. E. Sterbinsky and S. M. Heald, A simple method for mitigating error in the fixed-offset of a double-crystal monochromator, Synchrotron Radiat. 28, 1737 (2021).
  51. A. M. Afanas'ev, R. M. Imamov, and E. K. Mukhmedzhanov, Asymmetric x-ray diffraction, Crystallogr. Rev. 3, 157 (1992).
  52. F. Eichhorn, J. Kulda, and P. Mikula, Study of asymmetric neutron diffraction including the transition from Bragg to Laue case, Phys. Status Solidi A 80, 483 (1983).
  53. S. Brauer, G. Stephenson, and M. Sutton, Perfect crystals in the asymmetric Bragg geometry as optical elements for coherent x-ray beams, J. Synchrotron Radiat. 2, 163 (1995).
  54. R. Evans, P. Hirsch, and J. Kellar, A ‘parallel-beam’ concentrating monochromator for x-rays, Acta Crystallogr. 1, 124 (1948).
  55. G. Albertini, A. Boeuf, S. Lagomarsino, S. Mazkedian, S. Melone, and F. Rustichelli, Evidence for the Fankuchen effect in neutron diffraction by curved crystals, Acta Cryst. 33, 360 (1977).
  56. A. G. Wagh, S. Abbas, and W. Treimer, The plain truth about forming a plane wave of neutrons, Nucl. Instrum. Methods Phys. Res. Sect. A 634, S41 (2011).
  57. H. F. Talbot, Facts relating to optical science, Philos. Mag. 9, 401 (1836).
  58. 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).
  59. M. T. Rekveldt and P. Westerhuijs, Bent perfect crystals as neutron reflectors: Test experiments with Si plates, Nucl. Instrum. Methods Phys. Res. Sect. B 28, 583 (1987).
  60. P. Mikula, M. Vrána, J. Šaroun, B. Seong, V. Em, and M. Moon, Observation of multiple Bragg reflections of neutrons in bent perfect crystals, Nucl. Instrum. Methods Phys. Res. Sect. A 634, S108 (2011).
  61. J.-P. Guigay and M. Sanchez del Rio, X-ray focusing by bent crystals: Focal positions as predicted by the crystal lens equation and the dynamical diffraction theory, J. Synchrotron Radiat. 29, 148 (2022).
  62. V. V. Nesvizhevsky, A. Y. Voronin, R. Cubitt, and K. V. Protasov, Neutron whispering gallery, Nat. Phys. 6, 114 (2010).
  63. O. Lailey, Unified QI model code, Zenodo, 2026, https://doi.org/10.5281/zenodo.22048463.

Outline

Information

Phys. Rev. A 114, 032210– Published 14 September, 2026
  • Received 29 April 2026
  • Accepted 21 August 2026
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