Localization and fractality in inhomogeneous quantum walks with self-duality
We introduce and study a class of discrete-time quantum walks on a one-dimensional lattice. In contrast to the standard homogeneous quantum walks, coin operators are inhomogeneous and depend on their positions in this class of models. The models are shown to be self-dual with respect to the Fourier...
| Main Authors: | , |
|---|---|
| Other Authors: | |
| Format: | Article |
| Language: | en_US |
| Published: |
American Physical Society
2011
|
| Online Access: | http://hdl.handle.net/1721.1/61329 |
| _version_ | 1826201520834609152 |
|---|---|
| author | Shikano, Yutaka Katsura, Hosho |
| author2 | Massachusetts Institute of Technology. Department of Mechanical Engineering |
| author_facet | Massachusetts Institute of Technology. Department of Mechanical Engineering Shikano, Yutaka Katsura, Hosho |
| author_sort | Shikano, Yutaka |
| collection | MIT |
| description | We introduce and study a class of discrete-time quantum walks on a one-dimensional lattice. In contrast to the standard homogeneous quantum walks, coin operators are inhomogeneous and depend on their positions in this class of models. The models are shown to be self-dual with respect to the Fourier transform, which is analogous to the Aubry-André model describing the one-dimensional tight-binding model with a quasiperiodic potential. When the period of coin operators is incommensurate to the lattice spacing, we rigorously show that the limit distribution of the quantum walk is localized at the origin. We also numerically study the eigenvalues of the one-step time evolution operator and find the Hofstadter butterfly spectrum which indicates the fractal nature of this class of quantum walks. |
| first_indexed | 2024-09-23T11:53:03Z |
| format | Article |
| id | mit-1721.1/61329 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T11:53:03Z |
| publishDate | 2011 |
| publisher | American Physical Society |
| record_format | dspace |
| spelling | mit-1721.1/613292022-09-27T22:33:47Z Localization and fractality in inhomogeneous quantum walks with self-duality Shikano, Yutaka Katsura, Hosho Massachusetts Institute of Technology. Department of Mechanical Engineering Shikano, Yutaka Shikano, Yutaka We introduce and study a class of discrete-time quantum walks on a one-dimensional lattice. In contrast to the standard homogeneous quantum walks, coin operators are inhomogeneous and depend on their positions in this class of models. The models are shown to be self-dual with respect to the Fourier transform, which is analogous to the Aubry-André model describing the one-dimensional tight-binding model with a quasiperiodic potential. When the period of coin operators is incommensurate to the lattice spacing, we rigorously show that the limit distribution of the quantum walk is localized at the origin. We also numerically study the eigenvalues of the one-step time evolution operator and find the Hofstadter butterfly spectrum which indicates the fractal nature of this class of quantum walks. Japan Society for the Promotion of Science (Grant No. 21008624) National Science Foundation (U.S.) (Grant No. PHY05–51164) 2011-02-24T20:02:54Z 2011-02-24T20:02:54Z 2010-09 2010-05 Article http://purl.org/eprint/type/JournalArticle 1539-3755 1550-2376 http://hdl.handle.net/1721.1/61329 Shikano, Yutaka, and Hosho Katsura. “Localization and fractality in inhomogeneous quantum walks with self-duality.” Physical Review E 82.3 (2010): 031122. © 2010 The American Physical Society. en_US http://dx.doi.org/10.1103/PhysRevE.82.031122 Physical Review E Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf American Physical Society APS |
| spellingShingle | Shikano, Yutaka Katsura, Hosho Localization and fractality in inhomogeneous quantum walks with self-duality |
| title | Localization and fractality in inhomogeneous quantum walks with self-duality |
| title_full | Localization and fractality in inhomogeneous quantum walks with self-duality |
| title_fullStr | Localization and fractality in inhomogeneous quantum walks with self-duality |
| title_full_unstemmed | Localization and fractality in inhomogeneous quantum walks with self-duality |
| title_short | Localization and fractality in inhomogeneous quantum walks with self-duality |
| title_sort | localization and fractality in inhomogeneous quantum walks with self duality |
| url | http://hdl.handle.net/1721.1/61329 |
| work_keys_str_mv | AT shikanoyutaka localizationandfractalityininhomogeneousquantumwalkswithselfduality AT katsurahosho localizationandfractalityininhomogeneousquantumwalkswithselfduality |