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Quantum scattering of spinless particles in Riemannian manifolds

Lars Meschede, Benjamin Schwager*, Dominik Schulz, and Jamal Berakdar

  • Institut für Physik, Martin-Luther-Universität Halle-Wittenberg, 06120 Halle (Saale), Germany

  • *Corresponding autor: benjamin.schwager@physik.uni-halle.de
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Phys. Rev. A 107, 062806 – Published 6 June, 2023

DOI: https://doi.org/10.1103/PhysRevA.107.062806

Abstract

Quantum mechanics is sensitive to the geometry of the underlying space. Here we present a framework for quantum scattering of a nonrelativistic particle confined to a two-dimensional space. When the motion manifold hosts localized curvature modulations, scattering occurs from an emergent geometric potential and the metric tensor field. Analytical and full numerical simulations identify the geometric potential as the primary source for low-energy scattering, while the metric tensor field of the curved space governs high-energy diffraction. Compared to flat spaces, important differences in the validity range of perturbation approaches are found and demonstrated by full numerical simulations using combined finite element and boundary element methods. As an illustration, we consider a Gaussian-shaped dent leading to effects known as gravitational lensing. Experimentally, the considered setup is realizable based on geometrically engineered two-dimensional materials.

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References (51)

  1. B. S. DeWitt, Point transformations in quantum mechanics, Phys. Rev. 85, 653 (1952).
  2. B. S. DeWitt, Dynamical theory in curved spaces. I. A review of the classical and quantum action principles, Rev. Mod. Phys. 29, 377 (1957).
  3. M. S. Marinov, Path integrals in quantum theory: An outlook of basic concepts, Phys. Rep. 60, 1 (1980).
  4. A. Mostafazadeh, Supersymmetry and the Atiyah-Singer index theorem. I. Peierls brackets, Green's functions, and a proof of the index theorem via Gaussian superdeterminants, J. Math. Phys. 35, 1095 (1994).
  5. A. Mostafazadeh, Supersymmetry and the Atiyah-Singer index theorem. II. The scalar curvature factor in the Schrödinger equation, J. Math. Phys. 35, 1125 (1994).
  6. H. Jensen and H. Koppe, Quantum mechanics with constraints, Ann. Phys. (NY) 63, 586 (1971).
  7. R. C. T. da Costa, Quantum mechanics of a constrained particle, Phys. Rev. A 23, 1982 (1981).
  8. L. Kaplan, N. T. Maitra, and E. J. Heller, Quantizing constrained systems, Phys. Rev. A 56, 2592 (1997).
  9. G.-H. Liang and M.-Y. Lai, Effective quantum dynamics in curved thin-layer system with inhomogeneous confinement, Phys. Rev. A 107, 022213 (2023).
  10. 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).
  11. V. Atanasov and R. Dandoloff, Curvature induced quantum potential on deformed surfaces, Phys. Lett. A 371, 118 (2007).
  12. K. V. R. A. Silva, C. F. de Freitas, and C. Filgueiras, Geometry-induced quantum dots on surfaces with Gaussian bumps, Eur. Phys. J. B 86, 147 (2013).
  13. F. Serafim, F. A. N. Santos, J. R. F. Lima, S. Fumeron, B. Berche, and F. Moraes, Magnetic and geometric effects on the electronic transport of metallic nanotubes, J. Appl. Phys. 129, 044301 (2021).
  14. R. Spittel, P. Uebel, H. Bartelt, and M. A. Schmidt, Curvature-induced geometric momenta: The origin of waveguide dispersion of surface plasmons on metallic wires, Opt. Express 23, 12174 (2015).
  15. C. Ortix, S. Kiravittaya, O. G. Schmidt, and J. van den Brink, Curvature-induced geometric potential in strain-driven nanostructures, Phys. Rev. B 84, 045438 (2011).
  16. H. Shima, H. Yoshioka, and J. Onoe, Geometry-driven shift in the Tomonaga-Luttinger exponent of deformed cylinders, Phys. Rev. B 79, 201401(R) (2009).
  17. 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).
  18. B. Jensen and R. Dandoloff, Quantum mechanics of a constrained electrically charged particle in the presence of electric currents, Phys. Rev. A 80, 052109 (2009); Erratum: Quantum mechanics of a constrained electrically charged particle in the presence of electric currents [Phys. Rev. A 80, 052109 (2009)], Phys. Rev. A 81, 049905 (2010).
  19. M. V. Entin and L. I. Magarill, Spin-orbit interaction of electrons on a curved surface, Phys. Rev. B 64, 085330 (2001).
  20. J.-Y. Chang, J.-S. Wu, and C.-R. Chang, Exact Hamiltonians with Rashba and cubic Dresselhaus spin-orbit couplings on a curved surface, Phys. Rev. B 87, 174413 (2013).
  21. P. Gentile, M. Cuoco, and C. Ortix, Edge States and Topological Insulating Phases Generated by Curving a Nanowire with Rashba Spin-Orbit Coupling, Phys. Rev. Lett. 115, 256801 (2015).
  22. G.-H. Liang, Y.-L. Wang, M.-Y. Lai, H. Liu, H.-S. Zong, and S.-N. Zhu, Pseudo-magnetic-field and effective spin-orbit interaction for a spin-1/2 particle confined to a curved surface, Phys. Rev. A 98, 062112 (2018).
  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. Kimura, Observation of Riemannian geometric effects on electronic states, Europhys. Lett. 98, 27001 (2012).
  25. T. Ando, A. B. Fowler, and F. Stern, Electronic properties of two-dimensional systems, Rev. Mod. Phys. 54, 437 (1982).
  26. V. Vadakkumbatt, E. Joseph, A. Pal, and A. Ghosh, Studying electrons on curved surfaces by trapping and manipulating multielectron bubbles in liquid helium, Nat. Commun. 5, 4571 (2014).
  27. J. Zhang and T.-L. Ho, Potential scattering on a spherical surface, J. Phys. B 51, 115301 (2018).
  28. R. A. Carollo, D. C. Aveline, B. Rhyno, S. Vishveshwara, C. Lannert, J. D. Murphree, E. R. Elliott, J. R. Williams, R. J. Thompson, and N. Lundblad, Observation of ultracold atomic bubbles in orbital microgravity, Nature (London) 606, 281 (2022).
  29. C. Ticknor, Two-dimensional dipolar scattering with a tilt, Phys. Rev. A 84, 032702 (2011).
  30. E. A. Koval, O. A. Koval, and V. S. Melezhik, Anisotropic quantum scattering in two dimensions, Phys. Rev. A 89, 052710 (2014).
  31. A. Mostafazadeh, Scalar curvature factor in the Schrödinger equation and scattering on a curved surface, Phys. Rev. A 54, 1165 (1996).
  32. N. Oflaz, A. Mostafazadeh, and M. Ahmady, Scattering due to geometry: Case of a spinless particle moving on an asymptotically flat embedded surface, Phys. Rev. A 98, 022126 (2018).
  33. H. V. Bui and A. Mostafazadeh, Geometric scattering of a scalar particle moving on a curved surface in the presence of point defects, Ann. Phys. (NY) 407, 228 (2019).
  34. J. R. Anglin and E. Wamba, Sphere on a plane: Two-dimensional scattering from a finite curved region, Phys. Rev. A 106, 042216 (2022).
  35. P. Maraner, A complete perturbative expansion for quantum mechanics with constraints, J. Phys. A: Math. Gen. 28, 2939 (1995).
  36. P. C. Schuster and R. L. Jaffe, Quantum mechanics on manifolds embedded in Euclidean space, Ann. Phys. (NY) 307, 132 (2003).
  37. S. Jalalzadeh and H. R. Sepangi, Classical and quantum dynamics of confined test particles in brane gravity, Class. Quantum Grav. 22, 2035 (2005).
  38. J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics (Cambridge University Press, Cambridge, 2020).
  39. F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (Cambridge University Press, Cambridge, 2010).
  40. I. R. Lapidus, Quantum-mechanical scattering in two dimensions, Am. J. Phys. 50, 45 (1982).
  41. M. Papoular, Some properties of low-energy resonance scattering on surfaces, J. Phys. B 18, L821 (1985).
  42. I. R. Lapidus, Scattering by two-dimensional circular barrier, hard circle, and delta function ring potentials, Am. J. Phys. 54, 459 (1986).
  43. P. G. Burke, R-Matrix Theory of Atomic Collisions (Springer, Berlin, 2011).
  44. M. W. Scroggs, J. S. Dokken, C. N. Richardson, and G. N. Wells, Construction of arbitrary order finite element degree-of-freedom maps on polygonal and polyhedral cell meshes, ACM Trans. Math. Softw. 48, 1 (2022).
  45. M. W. Scroggs, I. A. Baratta, C. N. Richardson, and G. N. Wells, Basix: A runtime finite element basis evaluation library, J. Open Source Softw. 7, 3982 (2022).
  46. M. S. Alnæs, A. Logg, K. B. Ølgaard, M. E. Rognes, and G. N. Wells, Unified form language: A domain-specific language for weak formulations of partial differential equations, ACM Trans. Math. Softw. 40, 1 (2014).
  47. S. Refsdal and H. Bondi, The Gravitational lens effect, Mon. Not. R. Astron. Soc. 128, 295 (1964).
  48. M. Bartelmann and P. Schneider, Weak gravitational lensing, Phys. Rep. 340, 291 (2001).
  49. S. Deng and V. Berry, Wrinkled, rippled and crumpled graphene: An overview of formation mechanism, electronic properties, and applications, Mater. Today 19, 197 (2016).
  50. 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).
  51. Z.-Y. Gu and S.-W. Qian, Some remarks on the two-dimensional optical theorem and Aharonov-Bohm scattering, Phys. Lett. A 136, 6 (1989).

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Phys. Rev. A 107, 062806– Published 6 June, 2023
  • Received 24 January 2023
  • Accepted 4 May 2023
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