Direct visualization of relativistic quantum scars in graphene quantum dots – Nature
Heller, E. J. Bound-state eigenfunctions of classically chaotic Hamiltonian systems: scars of periodic orbits. Phys. Rev. Lett. 53, 1515–1518 (1984).
Google Scholar
Stöckmann, H.-J. Quantum Chaos: An Introduction (American Association of Physics Teachers, 2000).
Gutzwiller, M. C. Chaos in Classical and Quantum Mechanics Vol. 1 (Springer Science & Business Media, 2013).
Heller, E., Crommie, M., Lutz, C. & Eigler, D. Scattering and absorption of surface electron waves in quantum corrals. Nature 369, 464–466 (1994).
Google Scholar
Crook, R. et al. Imaging fractal conductance fluctuations and scarred wave functions in a quantum billiard. Phys. Rev. Lett. 91, 246803 (2003).
Google Scholar
Martins, F. et al. Imaging electron wave functions inside open quantum rings. Phys. Rev. Lett. 99, 136807 (2007).
Google Scholar
Burke, A. et al. Periodic scarred states in open quantum dots as evidence of quantum Darwinism. Phys. Rev. Lett. 104, 176801 (2010).
Google Scholar
Aoki, N. et al. Direct imaging of electron states in open quantum dots. Phys. Rev. Lett. 108, 136804 (2012).
Google Scholar
Cabosart, D. et al. Recurrent quantum scars in a mesoscopic graphene ring. Nano Lett. 17, 1344–1349 (2017).
Google Scholar
Ge, Z. et al. Imaging quantum interference in stadium-shaped monolayer and bilayer graphene quantum dots. Nano Lett. 21, 8993–8998 (2021).
Google Scholar
Lee, J. et al. Imaging electrostatically confined Dirac fermions in graphene quantum dots. Nat. Phys. 12, 1032–1036 (2016).
Google Scholar
Ge, Z. et al. Visualization and manipulation of bilayer graphene quantum dots with broken rotational symmetry and nontrivial topology. Nano Lett. 20, 8682–8688 (2020).
Google Scholar
Huang, L., Lai, Y.-C., Ferry, D. K., Goodnick, S. M. & Akis, R. Relativistic quantum scars. Phys. Rev. Lett. 103, 054101 (2009).
Google Scholar
Huang, L., Xu, H.-Y., Grebogi, C. & Lai, Y.-C. Relativistic quantum chaos. Phys. Rep. 753, 1–128 (2018).
Google Scholar
Luukko, P. J. et al. Strong quantum scarring by local impurities. Sci. Rep. 6, 37656 (2016).
Google Scholar
Keski-Rahkonen, J., Luukko, P. J., Kaplan, L., Heller, E. & Räsänen, E. Controllable quantum scars in semiconductor quantum dots. Phys. Rev. B 96, 094204 (2017).
Google Scholar
Keski-Rahkonen, J., Ruhanen, A., Heller, E. & Räsänen, E. Quantum Lissajous scars. Phys. Rev. Lett. 123, 214101 (2019).
Google Scholar
Xu, H., Huang, L., Lai, Y.-C. & Grebogi, C. Chiral scars in chaotic dirac fermion systems. Phys. Rev. Lett. 110, 064102 (2013).
Google Scholar
Song, M.-Y., Li, Z.-Y., Xu, H.-Y., Huang, L. & Lai, Y.-C. Quantization of massive Dirac billiards and unification of nonrelativistic and relativistic chiral quantum scars. Phys. Rev. Res. 1, 033008 (2019).
Google Scholar
Keski-Rahkonen, J., Graf, A. & Heller, E. Antiscarring in chaotic quantum wells. Preprint at https://arxiv.org/abs/2403.18081 (2024).
Berry, M. Quantum chaology, not quantum chaos. Phys. Scr. 40, 335 (1989).
Google Scholar
Einstein, A. Zum quantensatz von Sommerfeld und Epstein. Verh. Dtsch. Phys. Ges. 19, 82–92 (1917).
Stone, A. D. Einstein’s unknown insight and the problem of quantizing chaos. Phys. Today 58, 37 (2005).
Google Scholar
Pilatowsky-Cameo, S. et al. Ubiquitous quantum scarring does not prevent ergodicity. Nat. Commun. 12, 852 (2021).
Google Scholar
Hummel, Q., Richter, K. & Schlagheck, P. Genuine many-body quantum scars along unstable modes in Bose–Hubbard systems. Phys. Rev. Lett. 130, 250402 (2023).
Google Scholar
Evrard, B., Pizzi, A., Mistakidis, S. I. & Dag, C. B. Quantum scars and regular eigenstates in a chaotic spinor condensate. Phys. Rev. Lett. 132, 020401 (2024).
Google Scholar
Bernien, H. et al. Probing many-body dynamics on a 51-atom quantum simulator. Nature 551, 579–584 (2017).
Google Scholar
Heller, E. J. The Semiclassical Way to Dynamics and Spectroscopy (Princeton Univ. Press, 2018).
Zelditch, S. Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Math. J. 55, 919–941 (1987).
Google Scholar
Bohigas, O., Giannoni, M.-J. & Schmit, C. Characterization of chaotic quantum spectra and universality of level fluctuation laws. Phys. Rev. Lett. 52, 1 (1984).
Google Scholar
Sridhar, S. Experimental observation of scarred eigenfunctions of chaotic microwave cavities. Phys. Rev. Lett. 67, 785 (1991).
Google Scholar
Stein, J. & Stöckmann, H.-J. Experimental determination of billiard wave functions. Phys. Rev. Lett. 68, 2867 (1992).
Google Scholar
Chinnery, P. A. & Humphrey, V. F. Experimental visualization of acoustic resonances within a stadium-shaped cavity. Phys. Rev. E 53, 272 (1996).
Google Scholar
Kudrolli, A., Abraham, M. C. & Gollub, J. P. Scarred patterns in surface waves. Phys. Rev. E 63, 026208 (2001).
Google Scholar
Manoharan, H., Lutz, C. & Eigler, D. Quantum mirages formed by coherent projection of electronic structure. Nature 403, 512–515 (2000).
Google Scholar
Crommie, M. F., Lutz, C. P. & Eigler, D. M. Confinement of electrons to quantum corrals on a metal surface. Science 262, 218–220 (1993).
Google Scholar
Ghahari, F. et al. An on/off Berry phase switch in circular graphene resonators. Science 356, 845–849 (2017).
Google Scholar
Behn, W. A. et al. Measuring and tuning the potential landscape of electrostatically defined quantum dots in graphene. Nano Lett. 21, 5013–5020 (2021).
Google Scholar
Ge, Z. et al. Giant orbital magnetic moments and paramagnetic shift in artificial relativistic atoms and molecules. Nat. Nanotechnol. 18, 250–256 (2023).
Google Scholar
Zhao, Y. et al. Creating and probing electron whispering-gallery modes in graphene. Science 348, 672–675 (2015).
Google Scholar
Gutiérrez, C., Brown, L., Kim, C.-J., Park, J. & Pasupathy, A. N. Klein tunnelling and electron trapping in nanometre-scale graphene quantum dots. Nat. Phys. 12, 1069–1075 (2016).
Google Scholar
Zheng, Q., Zhuang, Y.-C., Sun, Q.-F. & He, L. Coexistence of electron whispering-gallery modes and atomic collapse states in graphene/WSe2 heterostructure quantum dots. Nat. Commun. 13, 1597 (2022).
Google Scholar
Akis, R., Ferry, D. & Bird, J. Wave function scarring effects in open stadium shaped quantum dots. Phys. Rev. Lett. 79, 123 (1997).
Google Scholar
Novoselov, K. S. et al. Two-dimensional gas of massless Dirac fermions in graphene. Nature 438, 197–200 (2005).
Google Scholar
Katsnelson, M. I., Novoselov, K. S. & Geim, A. K. Chiral tunnelling and the Klein paradox in graphene. Nat. Phys. 2, 620–625 (2006).
Google Scholar
Berry, M. V. & Mondragon, R. Neutrino billiards: time-reversal symmetry-breaking without magnetic fields. Proc. R. Soc. Lond. A 412, 53–74 (1987).
Google Scholar
Chen, S. et al. Electron optics with pn junctions in ballistic graphene. Science 353, 1522–1525 (2016).
Google Scholar
Cao, H. & Wiersig, J. Dielectric microcavities: model systems for wave chaos and non-Hermitian physics. Rev. Mod. Phys. 87, 61–111 (2015).
Google Scholar
Young, A. F. & Kim, P. Quantum interference and Klein tunnelling in graphene heterojunctions. Nat. Phys. 5, 222–226 (2009).
Google Scholar
Ge, Z. Wavefunction Mapping and Magnetic Field Response of Electrostatically Defined Graphene Quantum Dots. PhD thesis, Univ. California, Santa Cruz (2023).
Zomer, P., Dash, S., Tombros, N. & Van Wees, B. A transfer technique for high mobility graphene devices on commercially available hexagonal boron nitride. Appl. Phys. Lett. 99, 232104 (2011).
Google Scholar
Goossens, A. et al. Mechanical cleaning of graphene. Appl. Phys. Lett. 100, 073110 (2012).
Google Scholar
Ge, Z. et al. Source data for “Direct visualization of relativistic quantum scars”. Zenodo. https://doi.org/10.5281/zenodo.13751637 (2024).