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  • Letter
  • Open Access

Bloch-Landau-Zener oscillations in a quasi-periodic potential

Henrique C. Prates and Vladimir V. Konotop

  • Departamento de Física and Centro de Física Teórica e Computacional, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, Edifício C8, Lisboa 1749-016, Portugal

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Phys. Rev. Research 6, L022011 – Published 15 April, 2024

DOI: https://doi.org/10.1103/PhysRevResearch.6.L022011

Abstract

Bloch oscillations and Landau-Zener tunneling are ubiquitous phenomena which are sustained by a band-gap spectrum of a periodic Hamiltonian and can be observed in dynamics of a quantum particle or a wavepacket in a periodic potential under action of a linear force. Such physical setting remains meaningful for aperiodic potentials too, although band-gap structure does not exist anymore. Here we consider the dynamics of noninteracting atoms and Bose-Einstein condensates in a quasiperiodic one-dimensional optical lattice subjected to a weak linear force. Excited states with energies below the mobility edge, and thus localized in space, are considered. We show that the observed oscillatory behavior is enabled by tunneling between the initial state and a state (or several states) located nearby in the coordinate-energy space. The states involved in such Bloch-Landau-Zener oscillations are determined by the selection rule consisting of the condition of their spatial proximity and condition of quasiresonances occurring at avoided crossings of the energy levels. The latter condition is formulated mathematically using the Gershgorin circle theorem. The effect of the inter-atomic interactions on the dynamics can also be predicted on the bases of the developed theory. The reported results can be observed in any physical system allowing for observation of the Bloch oscillations, upon introducing incommensurablity in the governing Hamiltonian.

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References (71)

  1. F. Bloch, Über die Quantenmechanik der Electronen in Kristallgittern, Z. Phys. 52, 555 (1929).
  2. C. Zener, A theory of the electrical breakdown of solid dielectrics, Proc. R. Soc. London A 137, 696 (1932).
  3. L. D. Landau, On the theory of transfer of energy at collisions I, Phys. Z. Sowjet. 1, 88 (1932) [Collected Papers of L.D. Landau, edited by D. Ter Haar (Pergamon, 1965), pp. 52–59].
  4. J. Feldmann, K. Leo, J. Shah, B. A. B. Miller, J. E. Cunningham, T. Meier, G. von Plessen, A. Schulze, P. Thomas, and S. Schmitt-Rink, Optical investigation of Bloch oscillations in a semiconductor superlattice, Phys. Rev. B 46, 7252 (1992).
  5. K. Leo, H. Bolivar, F. Brüggemann, R. Schwedler, and K. Köhler, Observation of Bloch oscillations in a semiconductor superlattice, Solid State Commun. 84, 943 (1992)
  6. C. Waschke, H. G. Roskos, R. Schwedler, K. Leo, H. Kurz, and K. Köhler, Coherent submillimeter-wave emission from Bloch oscillations in a semiconductor superlattice, Phys. Rev. Lett. 70, 3319 (1993).
  7. C. F. Bharucha, K. W. Madison, P. R. Morrow, S. R. Wilkinson, B. Sundaram, and M. G. Raizen, Observation of atomic tunneling from an accelerating optical potential, Phys. Rev. A 55, R857(R) (1997).
  8. E. Peik, M. B. Dahan, I. Bouchoule, Y. Castin, and C. Salomon, Bloch oscillations of atoms, adiabatic rapid passage, and monokinetic atomic beams, Phys. Rev. A 55, 2989 (1997).
  9. E. Peik, M. B. Dahan, I. Bouchoule, Y. Castin, and C. Salomon, Bloch oscillations and an accelerator for cold atoms, Appl. Phys. B 65, 685 (1997).
  10. B. P. Anderson and M. A. Kasevich, Macroscopic quantum interference from atomic tunnel arrays, Science 282, 1686 (1998).
  11. O. Morsch, J. H. Müller, M. Cristiani, D. Ciampini, and E. Arimondo, Bloch oscillations and mean-field effects of bose-einstein condensates in 1D optical lattices, Phys. Rev. Lett. 87, 140402 (2001).
  12. M. Cristiani, O. Morsch, J. H. Müller, D. Ciampini, and E. Arimondo, Experimental properties of Bose-Einstein condensates in one-dimensional optical lattices: Bloch oscillations, Landau-Zener tunneling, and mean-field effects, Phys. Rev. A 65, 063612 (2002).
  13. G. Ferrari, N. Poli, F. Sorrentino, and G. M. Tino, Long-lived Bloch oscillations with bosonic Sr atoms and application to gravity measurement at the micrometer scale, Phys. Rev. Lett. 97, 060402 (2006).
  14. M. Gustavsson, E. Haller, M. J. Mark, J. G. Danzl, G. Rojas-Kopeinig, and H.-C. Nägerl, Control of interaction-induced dephasing of Bloch oscillations, Phys. Rev. Lett. 100, 080404 (2008).
  15. S. Kling, T. Salger, C. Grossert, and M. Weitz, Atomic Bloch-zener oscillations and stückelberg interferometry in optical lattices, Phys. Rev. Lett. 105, 215301 (2010).
  16. Z. A. Geiger, K. M. Fujiwara, K. Singh, R. Senaratne, S. V. Rajagopal, M. Lipatov, T. Shimasaki, R. Driben, V. V. Konotop, T. Meier, and D. M. Weld, Observation and uses of position-space Bloch oscillations in an ultracold gas, Phys. Rev. Lett. 120, 213201 (2018).
  17. B. M. Breid, D. Witthaut and H. J. Korsch, Bloch-Zener oscillations, New J. Phys. 8, 110 (2006).
  18. U. Peschel, T. Pertsch, and F. Lederer, Optical Bloch oscillations in waveguide arrays, Opt. Lett. 23, 1701 (1998).
  19. T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, Optical Bloch oscillations in temperature tuned waveguide arrays, Phys. Rev. Lett. 83, 4752 (1999).
  20. R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Experimental observation of linear and nonlinear optical Bloch oscillations, Phys. Rev. Lett. 83, 4756 (1999).
  21. F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, and A. Tünnermann, Bloch-Zener oscillations in binary superlattices, Phys. Rev. Lett. 102, 076802 (2009).
  22. R. Sapienza, P. Costantino, and D. Wiersma, Optical analogue of electronic Bloch oscillations, Phys. Rev. Lett. 91, 263902 (2003).
  23. L. Yuan and S. Fan, Bloch oscillation and unidirectional translation of frequency in a dynamically modulated ring resonator, Optica 3, 1014 (2016).
  24. H. Flayac, D. D. Solnyshkov, and G. Malpuech, Bloch oscillations of exciton-polaritons and photons for the generation of an alternating terahertz spin signal, Phys. Rev. B 84, 125314 (2011).
  25. A. Block, C. Etrich, T. Limboeck, F. Bleckmann, E. Soergel, C. Rockstuhl, and S. Linden, Bloch oscillations in plasmonic waveguide arrays, Nat. Commun. 5, 3843 (2014).
  26. H. Sanchis-Alepuz, Y. Kosevich, and J. Sánchez-Dehesa, Acoustic analogue of electronic Bloch oscillations, Phys. Rev. Lett. 98, 134301 (2007).
  27. N. D. Lanzillotti-Kimura, A. Fainstein, B. Perrin, B. Jusserand, O. Mauguin, and L. Largeau, and A. Lemaître, Bloch oscillations of THz acoustic phonons in coupled nanocavity structures, Phys. Rev. Lett. 104, 197402 (2010).
  28. S. Longhi, Bloch oscillations in complex crystals with PT symmetry, Phys. Rev. Lett. 103, 123601 (2009).
  29. Y.-L. Xu, W. S. Fegadolli, L. Gan, M.-H. Lu, X.-P. Liu, Z.-Y. Li, A. Scherer, and Y.-F. Chen, Experimental realization of Bloch oscillations in a parity-time synthetic silicon photonic lattice, Nat. Commun. 7, 11319 (2016).
  30. G. Nenciu, Dynamics of band electrons in electric and magnetic fields: Rigorous justification of the effective Hamiltonians, Rev. Mod. Phys. 63, 91 (1991).
  31. A. Kolovsky and H. Korsch, Bloch oscillations of cold atoms in optical lattices, Int. J. Mod. Phys. B 18, 1235 (2004).
  32. F. Domínguez-Adame, V. A. Malyshev, F. A. B. F. de Moura, and M. L. Lyra, Bloch-like oscillations in a one-dimensional lattice with long-range correlated disorder, Phys. Rev. Lett. 91, 197402 (2003).
  33. T. Schulte, S. Drenkelforth, G. Kleine Büning, W. Ertmer, J. Arlt, M. Lewenstein, and L. Santos, Dynamics of Bloch oscillations in disordered lattice potentials, Phys. Rev. A 77, 023610 (2008).
  34. S. Drenkelforth, G. Kleine Büning, J. Will1, T. Schulte, N. Murray, W. Ertmer, L. Santos, and J. J. Arlt, Damped Bloch oscillations of Bose–Einstein condensates in disordered potential gradients, New J. Phys. 10, 045027 (2008).
  35. S. Stützer, Y. V. Kartashov, V. A. Vysloukh, V. V. Konotop, S. Nolte, L. Torner, and A. Szameit, Hybrid Bloch–Anderson localization of light, Opt. Lett. 38, 1488 (2013).
  36. F. A. B. F. de Moura, M. L. Lyra, F. Domínguez-Adame, and V. A. Malyshev, Bloch oscillations in an aperiodic one-dimensional potential, Phys. Rev. B 71, 104303 (2005).
  37. F. A. B. F. de Moura, L. P. Viana, M. L. Lyra, V. A. Malyshev, and F. Domínguez-Adame, Bias driven coherent carrier dynamics in a two-dimensional aperiodic potential, Phys. Lett. A 372, 6694 (2008).
  38. S. Walter, D. Schneble, and A. C. Durst, Bloch oscillations in lattice potentials with controlled aperiodicity, Phys. Rev. A 81, 033623 (2010).
  39. G. Wang, Fragmentation of Bloch oscillations in quasiperiodic waveguide arrays, J. Opt. 16, 015502 (2014).
  40. J. B. Reeves, B. Gadway, T. Bergeman, I. Danshita, and D. Schneble, Superfluid Bloch dynamics in an incommensurate optical lattice, New J. Phys. 16, 065011 (2014).
  41. A. Yar, B. Sarwar, S. B. A. Shah, and K. Sabeeh, Bloch oscillations in moiré flat band systems, Phys. Lett. A 478, 128899 (2023).
  42. X. Kang and Z. Wang, Optical Bloch oscillation and resonant Zener tunneling in one-dimensional quasi-period structures containing single negative materials, Opt. Commun. 282, 355 (2009).
  43. N. Mott, The mobility edge since 1967, J. Phys. C. 20, 3075 (1987).
  44. P. Sarnak, Spectral behavior of quasi periodic potentials, Commun. Math. Phys. 84, 377 (1982).
  45. I. M. Suslov, Localization in one-dimensional incommensurate systems, Zh. Eksp. Teor. Fiz. 83, 1079 (1982) [Sov. Phys. JETP 56, 612 (1982)].
  46. S. Ostlund, R. Pandit, D. Rand H. J. Schellnhuber, and E. D. Siggia, One-Dimensional Schrödinger Equation with an Almost Periodic Potential, Phys. Rev. Lett. 50, 1873 (1983).
  47. J. Fröhlich, T. Spencer, and P. Wittwer, Localization for a class of one dimensional quasi-periodic Schrödinger operators, Commun. Math. Phys. 132, 5 (1990).
  48. S. Gershgorin, Über die Abgrenzung der Eigenwerte einer Matrix, Izv. Akad. Nauk. USSR Otd. Fiz.-Mat. Nauk 6, 749 (1931); [Bulletin de l'Académie des Sciences de l'URSS. Classe des sciences mathématiques et na, no. 6, 749–754 (1931)].
  49. M. Y. Azbel, Quantum particle in one-dimensional potentials with incommensurate periods, Phys. Rev. Lett. 43, 1954 (1979).
  50. R. B. Diener, G. A. Georgakis, J. Zhong, M. Raizen, and Q. Niu, Transition between extended and localized states in a one-dimensional incommensurate optical lattice, Phys. Rev. A 64, 033416 (2001).
  51. H. C. Prates, D. A. Zezyulin, and V. V. Konotop, Bose-Einstein condensates in quasiperiodic lattices: Bosonic Josephson junction, self-trapping, and multimode dynamics, Phys. Rev. Res. 4, 033219 (2022).
  52. H. Yao, H. Khoudli, L. Bresque, and L. Sanchez-Palencia, Critical behavior and fractality in shallow one-dimensional quasiperiodic potentials, Phys. Rev. Lett. 123, 070405 (2019).
  53. D. J. Boers, B. Goedeke, D. Hinrichs, and M. Holthaus, Mobility edges in bichromatic optical lattices, Phys. Rev. A 75, 063404 (2007).
  54. S. Aubry and G. André, Analyticity breaking and Anderson localization in incommensurate lattices, Ann. Isr. Phys. Soc. 3, 133 (1980).
  55. M. Kohmoto, Metal-insulator transition and scaling for incommensurate systems, Phys. Rev. Lett. 51, 1198 (1983).
  56. M. Modugno, Exponential localization in one-dimensional quasi-periodic optical lattices, New J. Phys. 11, 033023 (2009).
  57. J. Biddle, B. Wang, D. J. Priour, and S. Das Sarma, Localization in one-dimensional incommensurate lattices beyond the Aubry-André model, Phys. Rev. A 80, 021603(R) (2009).
  58. L. Morales-Molina, E. Doerner, C. Danieli, and S. Flach, Resonant extended states in driven quasiperiodic lattices: Aubry-Andre localization by design, Phys. Rev. A 90, 043630 (2014).
  59. H. P. Lüschen, S. Scherg, T. Kohlert, M. Schreiber, P. Bordia, X. Li, S. Das Sarma, and I. Bloch, Single-particle mobility edge in a one-dimensional quasiperiodic optical lattice, Phys. Rev. Lett. 120, 160404 (2018).
  60. B. Simon, Almost periodic Schrödinger operators: A Review, Adv. App. Math. 3, 463 (1982).
  61. S. Surace Jr., The Schrödinger equation with a quasi-periodic potential, Trans. Amer. Math. Soc 320, 321 (1990).
  62. A. L. Gaunt, T. F. Schmidutz, I. Gotlibovych, R. P. Smith, and Z. Hadzibabic, Bose-Einstein condensation of atoms in a uniform potential, Phys. Rev. Lett. 110, 200406 (2013).
  63. N. Navon, R. P. Smith, and Z. Hadzibabic, Quantum gases in optical boxes, Nat. Phys. 17, 1334 (2021).
  64. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.6.L022011 for comparison of systems of different sizes and for more details of calculations.
  65. D. A. Zezyulin and V. V. Konotop, Localization of ultracold atoms in Zeeman lattices with incommensurate spin-orbit coupling, Phys. Rev. A 105, 063323 (2022).
  66. P. Voisin, J. Bleuse, C. Bouche, S. Gaillard, C. Alibert, and A. Regreny, Observation of the wannier-stark quantization in a semiconductor superlattice, Phys. Rev. Lett. 61, 1639 (1988).
  67. S. R. Wilkinson, C. F. Bharucha, K. W. Madison, Qian Niu, and M. G. Raizen, Observation of atomic wannier-stark ladders in an accelerating optical potential, Phys. Rev. Lett. 76, 4512 (1996).
  68. M. Helm, W. Hilber, G. Strasser, R. De Meester, F. M. Peeters, and A. Wacker, Continuum wannier-stark ladders strongly coupled by zener resonances in semiconductor superlattices, Phys. Rev. Lett. 82, 3120 (1999).
  69. M. Glück, A. R. Kolovsky, and H. J. Korsch, Wannier-Stark resonances in optical and semiconductor superlattices, Phys. Rep. 366, 103 (2002).
  70. R. Ghosh and M. Žnidarič, Resonance-induced growth of number entropy in strongly disordered systems, Phys. Rev. B 105, 144203 (2022).
  71. A. Morningstar, L. Colmenarez, V. Khemani, D. J. Luitz, and D. A. Huse, Avalanches and many-body resonances in many-body localized systems, Phys. Rev. B 105, 174205 (2022).

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Phys. Rev. Research 6, L022011– Published 15 April, 2024
  • Received 23 October 2023
  • Revised 12 February 2024
  • Accepted 29 March 2024
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