- Physical Review Journals
All JournalsPhysics Magazine

Physical Review B

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
  • Editors' Suggestion
  • Open Access

Variational scarring in open two-dimensional quantum dots

Fartash Chalangari1, Joonas Keski-Rahkonen1,2,3, Simo Selinummi1, and Esa Räsänen1

  • 1Computational Physics Laboratory, Tampere University, P.O. Box 600, FI-33014 Tampere, Finland
  • 2Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
  • 3Department of Chemistry and Chemical Biology, Harvard University, Cambridge, Massachusetts 02138, USA

PDF

Phys. Rev. B 112, 115137 – Published 17 September, 2025

DOI: https://doi.org/10.1103/37qq-bd15

Abstract

Quantum scars have recently been directly visualized in graphene quantum dots [Nature (London) 635, 841 (2024)], revealing their resilience and influence on electron dynamics in mesoscopic systems. Here, we examine variational scarring in two-dimensional quantum dots and demonstrate that these states remain robust even in an open system. We show that controlled perturbations enable modulation of electronic transmission via scarred states, presenting a viable approach to tuning quantum transport. These findings provide insights into the role of scarring in mesoscopic transport and open pathways for experimental realization in quantum devices.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (89)

  1. M. V. Berry, The Bakerian Lecture, 1987. Quantum chaology, Proc. R. Soc. London A 413, 183 (1987).
  2. M. Berry, Quantum chaology, not quantum chaos, Phys. Scr. 40, 335 (1989).
  3. A. Einstein, Zum Quantensatz von Sommerfeld und Epstein, Verh. Dtsch. Phys. Ges. 19, 82 (1917).
  4. A. D. Stone, Einstein's unknown insight and the problem of quantizing chaos, Phys. Today 58(8), 37 (2005).
  5. H.-J. Stöckmann, Quantum Chaos: An Introduction (Cambridge University Press, Cambridge, 1999).
  6. M. Gutzwiller, Chaos in Classical and Quantum Mechanics, Interdisciplinary Applied Mathematics (Springer, New York, 1991).
  7. E. J. Heller, The Semiclassical Way to Dynamics and Spectroscopy (Princeton University Press, Princeton, 2018).
  8. G. Casati and T. Prosen, Quantum chaos, in Statistical and Nonlinear Physics, edited by B. Chakraborty (Springer US, New York, 2022), pp. 561–573.
  9. S. Habib, K. Shizume, and W. H. Zurek, Decoherence, chaos, and the correspondence principle, Phys. Rev. Lett. 80, 4361 (1998).
  10. E. Joos and H. D. Zeh, The emergence of classical properties through interaction with the environment, Z. Phys. B 59, 223 (1985).
  11. W. H. Zurek, Decoherence, einselection, and the quantum origins of the classical, Rev. Mod. Phys. 75, 715 (2003).
  12. W. Zurek, Classical selection and quantum darwinism, Phys. Today 68(5), 9 (2015).
  13. W. Zurek, Quantum darwinism, Nat. Phys. 5, 181 (2009).
  14. M. C. Gutzwiller, Periodic orbits and classical quantization conditions, J. Math. Phys 12, 343 (1971).
  15. E. J. Heller, Bound-state eigenfunctions of classically chaotic hamiltonian systems: Scars of periodic orbits, Phys. Rev. Lett. 53, 1515 (1984).
  16. Y. Colin de Verdiére, Ergodicité et fonctions propres du laplacien, Commun. Math. Phys. 102, 497 (1985).
  17. S. Zelditch, Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. J. 55, 919 (1987).
  18. A. I. Shnirel'man, Ergodic properties of eigenfunctions, Uspekhi Mat. Nauk 29, 181 (1974).
  19. L. Kaplan and E. J. Heller, Linear and nonlinear theory of eigenfunction scars, Ann. Phys. 264, 171 (1998).
  20. L. Kaplan, Scars in quantum chaotic wavefunctions, Nonlinearity 12, R1 (1999).
  21. E. B. Bogomolny, Smoothed wave functions of chaotic quantum systems, Physica D 31, 169 (1988).
  22. M. V. Berry, Quantum scars of classical closed orbits in phase space, Proc. R. Soc. London A 423, 219 (1989).
  23. M. Kuś, J. Zakrzewski, and K. Życzkowski, Quantum scars on a sphere, Phys. Rev. A 43, 4244 (1991).
  24. G. M. D'Ariano, L. R. Evangelista, and M. Saraceno, Classical and quantum structures in the kicked-top model, Phys. Rev. A 45, 3646 (1992).
  25. S. Tomsovic and E. J. Heller, Semiclassical construction of chaotic eigenstates, Phys. Rev. Lett. 70, 1405 (1993).
  26. F. Revuelta, E. Vergini, R. M. Benito, and F. Borondo, Short-periodic-orbit method for excited chaotic eigenfunctions, Phys. Rev. E 102, 042210 (2020).
  27. O. Agam and S. Fishman, Quantum eigenfunctions in terms of periodic orbits of chaotic systems, J. Phys. A: Math. Gen. 26, 2113 (1993).
  28. O. Bohigas, S. Tomsovic, and D. Ullmo, Manifestations of classical phase space structures in quantum mechanics, Phys. Rep. 223, 43 (1993).
  29. D. A. Wisniacki, E. Vergini, R. M. Benito, and F. Borondo, Scarring by homoclinic and heteroclinic orbits, Phys. Rev. Lett. 97, 094101 (2006).
  30. T. M. Fromhold, P. B. Wilkinson, F. W. Sheard, L. Eaves, J. Miao, and G. Edwards, Manifestations of classical chaos in the energy level spectrum of a quantum well, Phys. Rev. Lett. 75, 1142 (1995).
  31. P. B. Wilkinson, T. M. Fromhold, L. Eaves, F. W. Sheard, N. Miura, and T. Takamasu, Observation of ‘scarred' wavefunctions in a quantum well with chaotic electron dynamics, Nature (London) 380, 608 (1996).
  32. E. E. Narimanov and A. D. Stone, Origin of strong scarring of wave functions in quantum wells in a tilted magnetic field, Phys. Rev. Lett. 80, 49 (1998).
  33. A. Hönig and D. Wintgen, Spectral properties of strongly perturbed coulomb systems: Fluctuation properties, Phys. Rev. A 39, 5642 (1989).
  34. E. Bogomolny, B. Dietz, T. Friedrich, M. Miski-Oglu, A. Richter, F. Schäfer, and C. Schmit, First experimental observation of superscars in a pseudointegrable barrier billiard, Phys. Rev. Lett. 97, 254102 (2006).
  35. Y.-H. Kim, M. Barth, H.-J. Stöckmann, and J. P. Bird, Wave function scarring in open quantum dots: A microwave-billiard analog study, Phys. Rev. B 65, 165317 (2002).
  36. H.-J. Stöckmann and J. Stein, “Quantum” chaos in billiards studied by microwave absorption, Phys. Rev. Lett. 64, 2215 (1990).
  37. U. Dörr, H.-J. Stöckmann, M. Barth, and U. Kuhl, Scarred and chaotic field distributions in a three-dimensional Sinai-microwave resonator, Phys. Rev. Lett. 80, 1030 (1998).
  38. S. Sridhar, Experimental observation of scarred eigenfunctions of chaotic microwave cavities, Phys. Rev. Lett. 67, 785 (1991).
  39. J. Stein and H.-J. Stöckmann, Experimental determination of billiard wave functions, Phys. Rev. Lett. 68, 2867 (1992).
  40. J. Nöckel and A. Stone, Ray and wave chaos in asymmetric resonant optical cavities, Nature (London) 385, 45 (1997).
  41. S.-B. Lee, J.-H. Lee, J.-S. Chang, H.-J. Moon, S. W. Kim, and K. An, Observation of scarred modes in asymmetrically deformed microcylinder lasers, Phys. Rev. Lett. 88, 033903 (2002).
  42. T. Harayama, T. Fukushima, P. Davis, P. O. Vaccaro, T. Miyasaka, T. Nishimura, and T. Aida, Lasing on scar modes in fully chaotic microcavities, Phys. Rev. E 67, 015207(R) (2003).
  43. P. A. Chinnery and V. F. Humphrey, Experimental visualization of acoustic resonances within a stadium-shaped cavity, Phys. Rev. E 53, 272 (1996).
  44. Z. Ge, A. Graf, J. Keski-Rahkonen, S. Slizovskiy, P. Polizogopoulos, T. Taniguchi, K. Watanabe, R. Van Haren, D. Lederman, V. Fal'ko, E. Heller, and J. Velasco, Direct visualization of relativistic quantum scars in graphene quantum dots, Nature (London) 635, 841 (2024).
  45. D. Wisniacki and G. G. Carlo, Scarring in open quantum systems, Phys. Rev. E 77, 045201(R) (2008).
  46. D. Cabosart, A. Felten, N. Reckinger, A. Iordanescu, S. Toussaint, S. Faniel, and B. Hackens, Recurrent quantum scars in a mesoscopic graphene ring, Nano Lett. 17, 1344 (2017).
  47. R. Ketzmerick, Fractal conductance fluctuations in generic chaotic cavities, Phys. Rev. B 54, 10841 (1996).
  48. A. S. Sachrajda, R. Ketzmerick, C. Gould, Y. Feng, P. J. Kelly, A. Delage, and Z. Wasilewski, Fractal conductance fluctuations in a soft-wall stadium and a Sinai billiard, Phys. Rev. Lett. 80, 1948 (1998).
  49. P. J. J. Luukko, B. Drury, A. Klales, L. Kaplan, E. J. Heller, and E. Räsänen, Strong quantum scarring by local impurities, Sci. Rep. 6, 37656 (2016).
  50. J. Keski-Rahkonen, P. J. J. Luukko, L. Kaplan, E. J. Heller, and E. Räsänen, Controllable quantum scars in semiconductor quantum dots, Phys. Rev. B 96, 094204 (2017).
  51. J. Keski-Rahkonen, P. J. J. Luukko, S. Åberg, and E. Räsänen, Effects of scarring on quantum chaos in disordered quantum wells, J. Phys.: Condens. Matter 31, 105301 (2019).
  52. J. Keski-Rahkonen, A. Ruhanen, E. J. Heller, and E. Räsänen, Quantum Lissajous scars, Phys. Rev. Lett. 123, 214101 (2019).
  53. P. J. J. Luukko and J.-M. Rost, Polyatomic trilobite Rydberg molecules in a dense random gas, Phys. Rev. Lett. 119, 203001 (2017).
  54. J. Keski-Rahkonen, A. Graf, and E. Heller, Antiscarring in chaotic quantum wells, arXiv:2403.18081.
  55. S. Selinummi, J. Keski-Rahkonen, F. Chalangari, and E. Räsänen, Formation, prevalence, and stability of bouncing-ball quantum scars, Phys. Rev. B 110, 235420 (2024).
  56. M. Di Ventra, Electrical Transport in Nanoscale Systems (Cambridge University Press, Cambridge, 2008).
  57. M. A. Kastner, The single-electron transistor, Rev. Mod. Phys. 64, 849 (1992).
  58. S. M. Reimann and M. Manninen, Electronic structure of quantum dots, Rev. Mod. Phys. 74, 1283 (2002).
  59. L. Kouwenhoven, Few-electron quantum dots, Rep. Prog. Phys. 64, 701 (2001).
  60. R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha, and L. M. K. Vandersypen, Spins in few-electron quantum dots, Rev. Mod. Phys. 79, 1217 (2007).
  61. W. Fan, H. Zhang, and Z. Li, Unify the effect of anharmonicity in double-wells and anharmonic oscillators, Int. J. Theor. Phys. 63, 266 (2024).
  62. M. Troppenz, S. Sala, P.-I. Schneider, and A. Saenz, Inelastic confinement-induced resonances in quantum dots, arXiv:1509.01159.
  63. M. Dineykhan, G. Efimov, G. Ganbold, and S. Nedelko, Anharmonic potentials, in Oscillator Representation in Quantum Physics (Springer, Berlin, 1995), pp. 235–245.
  64. S. Schröter, P.-A. Hervieux, G. Manfredi, J. Eiglsperger, and J. Madroñero, Exact treatment of planar two-electron quantum dots: Effects of anharmonicity on the complexity, Phys. Rev. B 87, 155413 (2013).
  65. V. Halonen, P. Hyvönen, P. Pietiläinen, and T. Chakraborty, Effects of scattering centers on the energy spectrum of a quantum dot, Phys. Rev. B 53, 6971 (1996).
  66. E. Räsänen, J. Könemann, R. J. Haug, M. J. Puska, and R. M. Nieminen, Impurity effects in quantum dots: Toward quantitative modeling, Phys. Rev. B 70, 115308 (2004).
  67. A. D. Güçlü, J.-S. Wang, and H. Guo, Disordered quantum dots: A diffusion quantum Monte Carlo study, Phys. Rev. B 68, 035304 (2003).
  68. S. Girvin and K. Yang, Modern Condensed Matter Physics (Cambridge University Press, Cambridge, 2019).
  69. M. Lei, N. Zhu, and H. Guo, Quantum hall effect in the presence of an antidot potential, Phys. Rev. B 52, 16784 (1995).
  70. S. Baer and K. Ensslin, Quantum dots in the quantum Hall regime, in Transport Spectroscopy of Confined Fractional Quantum Hall Systems (Springer International Publishing, Cham, 2015), pp. 233–246.
  71. H. Sari, E. Kasapoglu, S. Sakiroglu, I. Sökmen, and C. A. Duque, Effect of position-dependent effective mass on donor impurity- and exciton-related electronic and optical properties of 2D Gaussian quantum dots, Eur. Phys. J. Plus 137, 341 (2022).
  72. R. En-nadir, H. El-ghazi, M. Tihtih, W. Belaid, S. E. Zaki, I. Maouhoubi, and I. Zorkani, Analyzing the combined influences of external electric field, impurity-location, in-content, and QW's number on donor-impurity binding energy in multiple quantum wells with finite squared potential, Opt. Quantum Electron. 55, 597 (2023).
  73. K. Hirose and N. S. Wingreen, Ground-state energy and spin in disordered quantum dots, Phys. Rev. B 65, 193305 (2002).
  74. K. Hirose, F. Zhou, and N. S. Wingreen, Density-functional theory of spin-polarized disordered quantum dots, Phys. Rev. B 63, 075301 (2001).
  75. A. C. Bleszynski, F. A. Zwanenburg, R. M. Westervelt, A. L. Roest, E. P. A. M. Bakkers, and L. P. Kouwenhoven, Scanned probe imaging of quantum dots inside inas nanowires, Nano Lett. 7, 2559 (2007).
  76. E. E. Boyd, K. Storm, L. Samuelson, and R. M. Westervelt, Scanning gate imaging of quantum dots in 1d ultra-thin InAs/InP nanowires, Nanotechnology 22, 185201 (2011).
  77. T. Blasi, M. F. Borunda, E. Räsänen, and E. J. Heller, Optimal local control of coherent dynamics in custom-made nanostructures, Phys. Rev. B 87, 241303(R) (2013).
  78. P. Luukko and E. Räsänen, Imaginary time propagation code for large-scale two-dimensional eigenvalue problems in magnetic fields, Comput. Phys. Commun. 184, 769 (2013).
  79. P. W. O'Connor and E. J. Heller, Quantum localization for a strongly classically chaotic system, Phys. Rev. Lett. 61, 2288 (1988).
  80. L. Kaplan and E. Heller, Weak quantum ergodicity, Physica D 121, 1 (1998).
  81. R. Duda, J. Keski-Rahkonen, J. Solanpää, and E. Räsänen, Tinie – A software package for electronic transport through two-dimensional cavities in a magnetic field, Comput. Phys. Commun. 270, 108141 (2022).
  82. G. Stefanucci and R. van Leeuwen, Nonequilibrium Many-Body Theory of Quantum Systems (Cambridge University Press, Cambridge, 2013).
  83. S. Datta, Quantum Transport - Atom to Transistor (Cambridge University Press, Cambridge, 2005).
  84. U. Fano, Effects of configuration interaction on intensities and phase shifts, Phys. Rev. 124, 1866 (1961).
  85. A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, Fano resonances in nanoscale structures, Rev. Mod. Phys. 82, 2257 (2010).
  86. L. Kaplan, Scar and antiscar quantum effects in open chaotic systems, Phys. Rev. E 59, 5325 (1999).
  87. J. Keski-Rahkonen, C. Zou, A. Graf, Q. Yao, T. Zhu, J. Velasco, Jr., and E. Heller, Variational scarring in graphene quantum dots, Phys. Rev. E 112, L012201 (2025).
  88. L. Ousmane, Scanning gate microscopy response for local tip potentials beyond perturbation theory, Phys. Rev. B 108, 195415 (2023).
  89. H. Sellier, B. Hackens, M. G. Pala, F. Martins, S. Baltazar, X. Wallart, L. Desplanque, V. Bayot, and S. Huant, On the imaging of electron transport in semiconductor quantum structures by scanning-gate microscopy: successes and limitations, Semicond. Sci. Technol. 26, 064008 (2011).

Outline

Information

Phys. Rev. B 112, 115137– Published 17 September, 2025
  • Received 6 February 2025
  • Revised 17 July 2025
  • Accepted 22 July 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