- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

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

Laser-dressed states on Riemannian manifolds: A generalization of the Kramers-Henneberger transformation

Hannah Bendin*, Benjamin Schwager*,†, and Jamal Berakdar

  • *These authors contributed equally to this work.
  • †Contact author: benjamin.schwager@physik.uni-halle.de
PDF

Phys. Rev. Research 7, 043183 – Published 19 November, 2025

DOI: https://doi.org/10.1103/fc67-47zv

Abstract

Quantum particles under geometric constraints are sensitive to the geometry and topology of the underlying space. We analytically study the laser-driven nonlinear dynamics of a quantum particle whose motion is constrained to a two-dimensional Riemannian manifold embedded in a three-dimensional hyperspace. The geometry of space may result in a potential-like term that supports bound states on the manifold. In the presence of a laser field, we derive expressions for a generalized Kramers-Henneberger-type unitary transformation, which is shown to be generally space- and time-dependent, and deduce a Schrödinger-like equation in the Kramers-Henneberger frame. Compared to a flat (geometrically trivial) space, new time-averaged coefficients of differential operators and operator-valued perturbation terms appear, which determine the geometry-dependent laser-dressed states on Riemannian manifolds.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (51)

  1. A. Di Piazza, C. Müller, K. Z. Hatsagortsyan, and C. H. Keitel, Extremely high-intensity laser interactions with fundamental quantum systems, Rev. Mod. Phys. 84, 1177 (2012).
  2. A. M. Popov, O. V. Tikhonova, and E. A. Volkova, Applicability of the Kramers-Henneberger approximation in the theory of strong-field ionization, J. Phys. B: At. Mol. Opt. Phys. 32, 3331 (1999).
  3. A. M. Popov, O. V. Tikhonova, and E. A. Volkova, Strong-field atomic stabilization: Numerical simulation and analytical modelling, J. Phys. B: At. Mol. Opt. Phys. 36, R125 (2003).
  4. H. A. Kramers, Collected Scientific Papers, edited by N. G. van Kampen (North-Holland Publishing Company, Amsterdam, 1956).
  5. W. C. Henneberger, Perturbation method for atoms in intense light beams, Phys. Rev. Lett. 21, 838 (1968).
  6. M. Pont, N. R. Walet, M. Gavrila, and C. W. McCurdy, Dichotomy of the hydrogen atom in superintense, high-frequency laser fields, Phys. Rev. Lett. 61, 939 (1988).
  7. V. C. Reed, P. L. Knight, and K. Burnett, Suppression of ionization in superintense fields without dichotomy, Phys. Rev. Lett. 67, 1415 (1991).
  8. E. A. Volkova, A. M. Popov, O. V. Smirnova, and O. V. Tikhonova, Emergence of a stabilization regime in quantum systems subject to a strong laser field and the Kramers-Henneberger approximation, J. Exp. Theor. Phys. 84, 658 (1997).
  9. J. Liang, Y. Zhou, Y. Liao, W.-C. Jiang, M. Li, and P. Lu, Direct visualization of deforming atomic wavefunction in ultraintense high-frequency laser pulses, Ultrafast Sci. 2022, 9842716 (2022).
  10. Q. Su and J. H. Eberly, Stabilization of a model atom in superintense field ionization, J. Opt. Soc. Am. B 7, 564 (1990).
  11. R. M. A. Vivirito and P. L. Knight, Adiabatic excitation and stabilization in short-range potentials, J. Phys. B: At. Mol. Opt. Phys. 28, 4357 (1995).
  12. F. Morales, M. Richter, S. Patchkovskii, and O. Smirnova, Imaging the Kramers–Henneberger atom, Proc. Natl. Acad. Sci. USA 108, 16906 (2011).
  13. K. Yamanouchi, K. Midorikawa, and L. Roso, Progress in Ultrafast Intense Laser Science XVI (Springer International Publishing AG, Cham, 2021), p. 222.
  14. M. Førre, S. Selstø, J. P. Hansen, and L. B. Madsen, Exact nondipole Kramers-Henneberger form of the light-atom Hamiltonian: An application to atomic stabilization and photoelectron energy spectra, Phys. Rev. Lett. 95, 043601 (2005).
  15. A. D. Bandrauk, F. Fillion-Gourdeau, and E. Lorin, Atoms and molecules in intense laser fields: Gauge invariance of theory and models, J. Phys. B: At. Mol. Opt. Phys. 46, 153001 (2013).
  16. J. Wätzel and J. Berakdar, Electrons in intense laser fields with local phase, polarization, and skyrmionic textures, Phys. Rev. A 102, 063105 (2020).
  17. H. Jensen and H. Koppe, Quantum mechanics with constraints, Ann. Phys. 63, 586 (1971).
  18. R. C. T. da Costa, Quantum mechanics of a constrained particle, Phys. Rev. A 23, 1982 (1981).
  19. F. T. Brandt and J. A. Sánchez-Monroy, Quantum dynamics of spinless particles on a brane coupled to a bulk gauge field, Class. Quantum Grav. 34, 075010 (2017).
  20. J. Jost, Riemannian Geometry and Geometric Analysis (Springer, Cham, 2011).
  21. V. Atanasov and R. Dandoloff, Curvature induced quantum potential on deformed surfaces, Phys. Lett. A 371, 118 (2007).
  22. M. Førre and S. Selstø, Schrödinger formulation of the nondipole light-matter interaction consistent with relativity, Phys. Rev. A 101, 063416 (2020).
  23. A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, Geometric potential and transport in photonic topological crystals, Phys. Rev. Lett. 104, 150403 (2010).
  24. J. Onoe, T. Ito, H. Shima, H. Yoshioka, and S.-I. Kimura, Observation of Riemannian geometric effects on electronic states, Europhys. Lett. 98, 27001 (2012).
  25. M. G. Harats, J. N. Kirchhof, M. Qiao, K. Greben, and K. I. Bolotin, Dynamics and efficient conversion of excitons to trions in non-uniformly strained monolayer WS2, Nat. Photon. 14, 324 (2020).
  26. P. Maraner, A complete perturbative expansion for quantum mechanics with constraints, J. Phys. A: Math. Gen. 28, 2939 (1995).
  27. P. C. Schuster and R. L. Jaffe, Quantum mechanics on manifolds embedded in Euclidean space, Ann. Phys. 307, 132 (2003).
  28. S. Jalalzadeh and H. R. Sepangi, Classical and quantum dynamics of confined test particles in brane gravity, Class. Quantum Grav. 22, 2035 (2005).
  29. P. Maraner and J. K. Pachos, Universal features of dimensional reduction schemes from general covariance breaking, Ann. Phys. 323, 2044 (2008).
  30. G. Ferrari and G. Cuoghi, Schrödinger equation for a particle on a curved surface in an electric and magnetic field, Phys. Rev. Lett. 100, 230403 (2008).
  31. B. Schwager, L. Meschede, and J. Berakdar, Wigner time delay and Hartman effect in quantum motion along deformed Riemannian manifolds, Can. J. Phys. 103, 51 (2025).
  32. In this case, the resulting effective flat space Schrödinger equation carries the signature of the metric tensor through the geometric potential and the position-dependent mass. Both of these aspects are decisive for the obtained spectrum. Yet, we want to point out that our geometry-induced quantum phenomenon is fundamentally different from what is called the “quantum metric” [51], which refers to analyzing in geometric terms the spectrum of a quantum particle in flat space, where, e.g., the geometric potential would vanish.
  33. F. T. Brandt and J. A. Sánchez-Monroy, Absence of effects of an in-plane magnetic field in a quasi-two-dimensional electron system, Int. J. Mod. Phys. B 32, 1850083 (2018).
  34. T. Schürmann, On momentum operators given by killing vectors whose integral curves are geodesics Physics 4, 1440 (2022).
  35. R. M. Wilcox, Exponential operators and parameter differentiation in quantum physics, J. Math. Phys. 8, 962 (1967).
  36. E. A. Volkova, A. M. Popov, and O. V. Tikhonova, Study of the structure of the energy spectrum of the system “atom+strong external electromagnetic field”, Zh. Éksp. Teor. Fiz. 109, 1586 (1996) [Sov. Phys. JETP 82, 853 (1996)].
  37. Q. Su and J. H. Eberly, Suppression of ionization and atomic electron localization by short intense laser pulses, Phys. Rev. A 43, 2474 (1991).
  38. The Lie algebra of endomorphisms gl(H) contains (probably) unbounded operators so that the convergence of the operator exponentials and hence the existence of a corresponding Lie group are not guaranteed. Furthermore, the bounded operators would be a necessary prerequisite for the BCH formula and the Hadamard lemma to secure their convergence. Therefore, additional assumptions concerning the operator domain may be necessary to fully rely on these expansions. Nevertheless, we can either compute all commutators explicitly in this work or only apply the formulas to find approximations.
  39. B. A. Dubrovin, A. T. Fomenko, and S. P. Novikov, Modern Geometry (Springer-Verlag, New York, 1992), p. 483.
  40. B. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction (Springer, Cham, 2015), p. 453.
  41. C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (Princeton University Press, Princeton, 2017), p. 1280.
  42. S. Helgason, Geometric Analysis on Symmetric Spaces (American Mathematical Society, Providence, 2008).
  43. F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (Cambridge University Press, New York, 2010), p. 951.
  44. Y.-L. Wang, H. Jiang, and H.-S. Zong, Geometric influences of a particle confined to a curved surface embedded in three-dimensional Euclidean space, Phys. Rev. A 96, 022116 (2017).
  45. L. Zhang, Y. Chao, K. Yang, D. Xue, and S. Zhou, Recent Advances in Metal/Alloy Nano Coatings for Carbon Nanotubes Based on Electroless Plating, Front. Chem. 9 (2022).
  46. Y. Zhang, N. W. Franklin, R. J. Chen, and H. Dai, Metal coating on suspended carbon nanotubes and its implication to metal–tube interaction, Chem. Phys. Lett. 331, 35 (2000).
  47. G. L. Goh, S. Agarwala, and W. Y. Yeong, Directed and on-demand alignment of carbon nanotube: A review toward 3D printing of electronics, Adv. Mater. Interfaces 6, 1801318 (2019),.
  48. K. Wang, Y. Wu, H. Wu, Y. Luo, D. Wang, K. Jiang, Q. Li, Y. Li, S. Fan, and J. Wang, Super-aligned carbon nanotube films with a thin metal coating as highly conductive and ultralight current collectors for lithium-ion batteries, J. Power Sources 351, 160 (2017).
  49. M. Hamermesh, Group Theory and Its Application to Physical Problems (Dover Publications, Mineola, 1989), p. 509.
  50. J. Aczél, Lectures on Functional Equations and Their Applications (Academic Press, New York, 1966), p. 510.
  51. N. Wang, D. Kaplan, Z. Zhang, T. Holder, N. Cao, A. Wang, X. Zhou, F. Zhou, Z. Jiang, C. Zhang, S. Ru, H. Cai, K. Watanabe, T. Taniguchi, B. Yan, and W. Gao, Quantum-metric-induced nonlinear transport in a topological antiferromagnet, Nature (London) 621, 487 (2023).

Outline

Information

Phys. Rev. Research 7, 043183– Published 19 November, 2025
  • Received 15 February 2024
  • Accepted 16 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