- 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
  • Letter
  • Open Access

Coexistence of extended and localized states in finite-sized mosaic Wannier-Stark lattices

Jun Gao1,*, Ivan M. Khaymovich2,3,†, Adrian Iovan1, Xiao-Wei Wang4, Govind Krishna1, Ze-Sheng Xu1, Emrah Tortumlu1, Alexander V. Balatsky2,5, Val Zwiller1 et al.

Ali W. Elshaari1,‡

  • 1Department of Applied Physics, KTH Royal Institute of Technology, Albanova University Centre, Roslagstullsbacken 21, 106 91 Stockholm, Sweden
  • 2Nordita, Stockholm University and KTH Royal Institute of Technology, Hannes Alfvéns väg 12, SE-106 91 Stockholm, Sweden
  • 3Institute for Physics of Microstructures, Russian Academy of Sciences, 603950 Nizhny Novgorod, GSP-105, Russia
  • 4Center for Integrated Quantum Information Technologies (IQIT), School of Physics and Astronomy and State Key Laboratory of Advanced Optical Communication Systems and Networks, Shanghai Jiao Tong University, Shanghai 200240, China
  • 5Department of Physics, University of Connecticut, Storrs, Connecticut 06269, USA

  • *junga@kth.se
  • †ivan.khaymovich@gmail.com
  • ‡elshaari@kth.se
PDF

Phys. Rev. B 108, L140202 – Published 10 October, 2023

DOI: https://doi.org/10.1103/PhysRevB.108.L140202

Abstract

Quantum transport and localization are fundamental concepts in condensed matter physics. It is commonly believed that in one-dimensional systems, the existence of mobility edges is highly dependent on disorder. Recently, there has been a debate over the existence of an exact mobility edge in a modulated mosaic model without quenched disorder, the so-called mosaic Wannier-Stark lattice. Here, we experimentally implement such disorder-free mosaic photonic lattices using a silicon photonics platform. By creating a synthetic electric field, we could observe energy-dependent coexistence of both extended and localized states in a finite number of waveguides. The Wannier-Stark ladder emerges when the resulting potential is strong enough, and can be directly probed by exciting different spatial modes of the lattice. Our studies provide the experimental proof of coexisting sets of strongly localized and conducting (though weakly localized) states in finite-sized mosaic Wannier-Stark lattices, which hold the potential to encode high-dimensional quantum resources with compact and robust structures.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (69)

  1. P. W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev. 109, 1492 (1958).
  2. F. Evers and A. D. Mirlin, Anderson transitions, Rev. Mod. Phys. 80, 1355 (2008).
  3. S. Aubry and G. André, Analyticity breaking and Anderson localization in incommensurate lattices, Ann. Israel Phys. Soc. 3, 133 (1980).
  4. P. G. Harper, Single band motion of conduction electrons in a uniform magnetic field, Proc. Phys. Soc., London Sect. A 68, 874 (1955).
  5. Y. Lahini, R. Pugatch, F. Pozzi, M. Sorel, R. Morandotti, N. Davidson, and Y. Silberberg, Observation of a localization transition in quasiperiodic photonic lattices, Phys. Rev. Lett. 103, 013901 (2009).
  6. R. E. Prange, D. R. Grempel, and S. Fishman, Wave functions at a mobility edge: An example of a singular continuous spectrum, Phys. Rev. B 28, 7370 (1983).
  7. J. Biddle, B. Wang, D. J. Priour Jr., and S. Das Sarma, Localization in one-dimensional incommensurate lattices beyond the Aubry-André model, Phys. Rev. A 80, 021603(R) (2009).
  8. J. Biddle and S. Das Sarma, Predicted mobility edges in one-dimensional incommensurate optical lattices: An exactly solvable model of Anderson localization, Phys. Rev. Lett. 104, 070601 (2010).
  9. S. Gopalakrishnan, Self-dual quasiperiodic systems with power-law hopping, Phys. Rev. B 96, 054202 (2017).
  10. X. Deng, S. Ray, S. Sinha, G. V. Shlyapnikov, and L. Santos, One-dimensional quasicrystals with power-law hopping, Phys. Rev. Lett. 123, 025301 (2019).
  11. M. Saha, S. K. Maiti, and A. Purkayastha, Anomalous transport through algebraically localized states in one dimension, Phys. Rev. B 100, 174201 (2019).
  12. X. Deng, V. E. Kravtsov, G. V. Shlyapnikov, and L. Santos, Duality in power-law localization in disordered one-dimensional systems, Phys. Rev. Lett. 120, 110602 (2018).
  13. P. A. Nosov, I. M. Khaymovich, and V. E. Kravtsov, Correlation-induced localization, Phys. Rev. B 99, 104203 (2019).
  14. A. G. Kutlin and I. M. Khaymovich, Renormalization to localization without a small parameter, SciPost Phys. 8, 049 (2020).
  15. X. Deng, A. L. Burin, and I. M. Khaymovich, Anisotropy-mediated reentrant localization, SciPost Phys. 13, 116 (2022).
  16. S. Das Sarma, S. He, and X. C. Xie, Mobility edge in a model one-dimensional potential, Phys. Rev. Lett. 61, 2144 (1988).
  17. S. Das Sarma, S. He, and X. C. Xie, Localization, mobility edges, and metal-insulator transition in a class of one-dimensional slowly varying deterministic potentials, Phys. Rev. B 41, 5544 (1990).
  18. S. Ganeshan, J. H. Pixley, and S. Das Sarma, Nearest neighbor tight binding models with an exact mobility ddge in one dimension, Phys. Rev. Lett. 114, 146601 (2015).
  19. T. Liu, X. Xia, S. Longhi, and L. Sanchez-Palencia, Anomalous mobility edges in one-dimensional quasiperiodic models, SciPost Phys. 12, 027 (2022).
  20. X. Li, X. Li, and S. Das Sarma, Mobility edges in one dimensional bichromatic incommensurate potentials, Phys. Rev. B 96, 085119 (2017).
  21. 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).
  22. H. Yin, J. Hu, A.-C. Ji, G. Juzeliunas, X.-J. Liu, and Q. Sun, Localization driven superradiant instability, Phys. Rev. Lett. 124, 113601 (2020).
  23. S. Roy, I. M. Khaymovich, A. Das, and R. Moessner, Multifractality without fine-tuning in a Floquet quasiperiodic chain, SciPost Phys. 4, 025 (2018).
  24. S. Ray, A. Ghosh, and S. Sinha, Drive-induced delocalization in the Aubry-André model, Phys. Rev. E 97, 010101(R) (2018).
  25. M. Sarkar, R. Ghosh, A. Sen, and K. Sengupta, Mobility edge and multifractality in a periodically driven Aubry-André model, Phys. Rev. B 103, 184309 (2021).
  26. M. Gonçalves, P. Ribeiro, and I. M. Khaymovich, Quasiperiodicity hinders ergodic Floquet eigenstates, Phys. Rev. B 108, 104201 (2023).
  27. C. Danieli, J. D. Bodyfelt, and S. Flach, Flat-band engineering of mobility edges, Phys. Rev. B 91, 235134 (2015).
  28. A. Ahmed, A. Ramachandran, I. M. Khaymovich, and A. Sharma, Flat band based multifractality in the all-band-flat diamond chain, Phys. Rev. B 106, 205119 (2022).
  29. S. Lee, A. Andreanov, and S. Flach, Critical-to-insulator transitions and fractality edges in perturbed flat bands, Phys. Rev. B 107, 014204 (2023).
  30. Y. Kim, T. Čadež, A. Andreanov, and S. Flach, Flat band induced metal-insulator transitions for weak magnetic flux and spin-orbit disorder, Phys. Rev. B 107, 174202 (2023).
  31. Y. Wang, X. Xia, L. Zhang, H. Yao, S. Chen, J. You, Q. Zhou, and X.-J. Liu, One-dimensional quasiperiodic mosaic lattice with exact mobility edges, Phys. Rev. Lett. 125, 196604 (2020).
  32. Y. Liu, Y. Wang, X.-J. Liu, Q. Zhou, and S. Chen, Exact mobility edges, PT -symmetry breaking, and skin effect in onedimensional non-Hermitian quasicrystals, Phys. Rev. B 103, 014203 (2021).
  33. 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).
  34. F. A. An, K. Padavić, E. J. Meier, S. Hegde, S. Ganeshan, J. H. Pixley, S. Vishveshwara, and B. Gadway, Interactions and mobility edges: observing the generalized Aubry-André model, Phys. Rev. Lett. 126, 040603 (2021).
  35. Y. Wang, J.-H. Zhang, Y. Li, J. Wu, W. Liu, F. Mei, Y. Hu, L. Xiao, J. Ma, C. Chin, and S. Jia, Observation of interaction-induced mobility edge in an atomic Aubry-André wire, Phys. Rev. Lett. 129, 103401 (2022).
  36. D. Dwiputra and F. P. Zen, Single-particle mobility edge without disorder, Phys. Rev. B 105, L081110 (2022).
  37. G. H. Wannier, Dynamics of band electrons in electric and magnetic fields, Rev. Mod. Phys. 34, 645 (1962).
  38. H. Fukuyama, R. A. Bari, and H. C. Fogedby, Tightly bound electrons in a uniform electric field, Phys. Rev. B 8, 5579 (1973).
  39. D. Emin and C. F. Hart, Existence of Wannier-Stark localization, Phys. Rev. B 36, 7353 (1987).
  40. S. Longhi, Absence of mobility edges in mosaic Wannier-Stark lattices, Phys. Rev. B 108, 064206 (2023).
  41. X. Wei, L. Wu, K. Feng, T. Liu, and Y. Zhang, Exact mobility edges in finite-height Wannier-Stark ladders, arXiv:2308.15516 (2023).
  42. L. Lu, J. D. Joannopoulos, and M. Soljacic, Topological photonics, Nat. Photon. 8, 821 (2014).
  43. T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, Topological photonics, Rev. Mod. Phys. 91, 015006 (2019).
  44. D. Smirnova, D. Leykam, Y. Chong, and Y. Kivshar, Nonlinear topological photonics, Appl. Phys. Rev. 7, 021306 (2020).
  45. D. T. H. Tan, Topological silicon photonics, Adv. Photonics Res. 2, 2100010 (2021).
  46. J. Wang, F. Sciarrino, A. Laing, and M. G. Thompson, Integrated photonic quantum technologies, Nat. Photon. 14, 273 (2020).
  47. A. W. Elshaari, W. Pernice, K. Srinivasan, O. Benson, and V. Zwiller, Hybrid integrated quantum photonic circuits, Nat. Photon. 14, 285 (2020).
  48. J. Chang, J. Gao, I. Esmaeil Zadeh, A. W. Elshaari, and V. Zwiller, Nanowire-based integrated photonics for quantum information and quantum sensing, Nanophotonics 12, 339 (2023).
  49. M. Zhang, L. Feng, M. Li, Y. Chen, L. Zhang, D. He, G. Guo, G. Guo, X. Ren, and D. Dai, Supercompact photonic quantum logic gate on a silicon chip, Phys. Rev. Lett. 126, 130501 (2021).
  50. Y. Chi, J. Huang, Z. Zhang, J. Mao, Z. Zhou, X. Chen, C. Zhai, J. Bao, T. Dai, H. Yuan, M. Zhang, D. Dai, B. Tang, Y. Yang, Z. Li, Y. Ding, L. K. Oxenløwe, M. G. Thompson, J. L. O'Brien, Y. Li, Q. Gong et al., A programmable qudit-based quantum processor, Nat. Commun. 13, 1166 (2022).
  51. Y. Lahini, Y. Bromberg, D. N. Christodoulides, and Y. Silberberg, Quantum correlations in two-particle Anderson localization, Phys. Rev. Lett. 105, 163905 (2010).
  52. M. Segev, Y. Silberberg, and D. N. Christodoulides, Anderson localization of light, Nat. Photon. 7, 197 (2013).
  53. A. Crespi, R. Osellame, R. Ramponi, V. Giovannetti, R. Fazio, L. Sansoni, F. De Nicola, F. Sciarrino, and P. Mataloni, Anderson localization of entangled photons in an integrated quantum walk, Nat. Photon. 7, 322 (2013).
  54. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevB.108.L140202 for details of calculations of the probabilities of resonances, which includes Refs. [13, 36, 40, 58, 59, 61, 66, 67, 68, 69].
  55. L. Chrostowski and M. Hochberg, Silicon Photonics Design: From Devices to Systems (Cambridge University Press, Cambridge, 2015).
  56. FDTD: 3D Electromagnetic Simulator, Lumerical Inc.
  57. Z.-Q. Jiao, J. Gao, W.-H. Zhou, X.-W. Wang, R.-J. Ren, X.-Y. Xu, L.-F. Qiao, Y. Wang, and X.-M. Jin, Two-dimensional quantum walks of correlated photons, Optica 8, 1129 (2021).
  58. J. Gao, Z.-S. Xu, D. A Smirnova, D. Leykam, S. Gyger, W.-H. Zhou, S. Steinhauer, V. Zwiller, and A. W. Elshaari, Observation of Anderson phase in a topological photonic circuit, Phys. Rev. Res. 4, 033222 (2022).
  59. Z.-S. Xu, J. Gao, G. Krishna, S. Steinhauer, V. Zwiller, and A. W. Elshaari, Direct measurement of topological invariants in photonic superlattices, Photon. Res. 10, 2901 (2022).
  60. D. J. Thouless, Electrons in disordered systems and the theory of localization, Phys. Rep. 13, 93 (1974).
  61. F. Borgonovi, F. Izrailev, L. Santos, and V. Zelevinsky, Quantum chaos and thermalization in isolated systems of interacting particles, Phys. Rep. 626, 1 (2016).
  62. J. Gao, L. Santos, G. Krishna, Z.-S. Xu, A. Iovan, S. Steinhauer, O. Gühne, P. Poole, D. Dalacu, V. Zwiller, and A. W. Elshaari, Scalable generation and detection of on-demand W states in nanophotonic circuits, Nano Lett. 23, 5350 (2023).
  63. G. Moody, V. J. Sorger, D. J. Blumenthal, P. W. Juodawlkis, W. Loh, C. Sorace-Agaskar, A. E. Jones, K. C. Balram, J. C. Matthews, A. Laing, M. Davanco, L. Chang, J. E. Bowers, N. Quack, C. Galland, I. Aharonovich, M. A. Wolff, C. Schuck, N. Sin-clair, M. Lončar, T. Komljenovic, D. Weld, S. Mookher-jea, S. Buckley, M. Radulaski, S. Reitzenstein, B. Pingault, B. Machielse, D. Mukhopadhyay, A. Akimov, A. Zheltikov, G. S. Agarwal, K. Srinivasan, J. Lu, H. X. Tang, W. Jiang, T. P. McKenna, A. H. Safavi-Naeini, S. Steinhauer, A. W. Elshaari, V. Zwiller, P. S. Davids, N. Martinez, M. Gehl, J. Chiaverini, K. K. Mehta, J. Romero, N. B. Lingaraju, A. M. Weiner, D. Peace, R. Cernansky, M. Lobino, E. Diamanti, L. T. Vidarte, and R. M. Camacho, Roadmap on integrated quantum photonics, Journal of Physics: Photonics 4, 012501 (2022).
  64. A. W. Elshaari, A. Skalli, S. Gyger, M. Nurizzo, L. Schweckert, I. E. Zadeh, M. Svedendahl, S. Steinhauer, and V. Zwiller, Deterministic integration of hBN emitter in silicon nitride photonic waveguide, Adv. Quantum Tech. 4, 2100032 (2021).
  65. A. W. Elshaari, I. E. Zadeh, A. Fognini, M. E. Reimer, D. Dalacu, P. Poole, V. Zwiller, and K. D. Jons, On-chip single photon filtering and multiplexing in hybrid quantum photonic circuits, Nat. Commun. 8, 379 (2017).
  66. S. Nevlacsil, M. Eggeling, P. Muellner, G. Koppitsch, M. Sagmeister, J. Kraft, and R. Hainberger, Broadband sin asymmetric directional coupler for 840 nm operation, OSA Continuum 1, 1324 (2018).
  67. P. A. Nosov and I. M. Khaymovich, Robustness of delocalization to the inclusion of soft constraints in long-range random models, Phys. Rev. B 99, 224208 (2019).
  68. A. G. Kutlin and I. M. Khaymovich, Emergent fractal phase in energy stratified random models, SciPost Phys. 11, 101 (2021).
  69. V. R. Motamarri, A. S. Gorsky, and I. M. Khaymovich, Localization and fractality in disordered Russian Doll model, SciPost Phys. 13, 117 (2022).

Outline

Information

Phys. Rev. B 108, L140202– Published 10 October, 2023
  • Received 17 March 2023
  • Revised 23 June 2023
  • Accepted 26 September 2023
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. Funded by Bibsam.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation