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Lawrie T, Tanner G, Chronopoulos D. A quantum graph approach to metamaterial design. Sci Rep 2022; 12:18006. [PMID: 36289310 PMCID: PMC9605957 DOI: 10.1038/s41598-022-22265-2] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/07/2022] [Accepted: 10/12/2022] [Indexed: 11/09/2022] Open
Abstract
Since the turn of the century, metamaterials have gained a large amount of attention due to their potential for possessing highly nontrivial and exotic properties—such as cloaking or perfect lensing. There has been a great push to create reliable mathematical models that accurately describe the required material composition. Here, we consider a quantum graph approach to metamaterial design. An infinite square periodic quantum graph, constructed from vertices and edges, acts as a paradigm for a 2D metamaterial. Wave transport occurs along the edges with vertices acting as scatterers modelling sub-wavelength resonant elements. These resonant elements are constructed with the help of finite quantum graphs attached to each vertex of the lattice with customisable properties controlled by a unitary scattering matrix. The metamaterial properties are understood and engineered by manipulating the band diagram of the periodic structure. The engineered properties are then demonstrated in terms of the reflection and transmission behaviour of Gaussian beam solutions at an interface between two different metamaterials. We extend this treatment to N layered metamaterials using the Transfer Matrix Method. We demonstrate both positive and negative refraction and beam steering. Our proposed quantum graph modelling technique is very flexible and can be easily adjusted making it an ideal design tool for creating metamaterials with exotic band diagram properties or testing promising multi-layer set ups and wave steering effects.
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Affiliation(s)
- Tristan Lawrie
- grid.4563.40000 0004 1936 8868School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD UK
| | - Gregor Tanner
- grid.4563.40000 0004 1936 8868School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD UK
| | - Dimitrios Chronopoulos
- grid.5596.f0000 0001 0668 7884Department of Mechanical Engineering and Mecha(tro)nic System Dynamics (LMSD), KU Leuven, 9000 Leuven, Belgium
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Chakraborty S, Novo L, Di Giorgio S, Omar Y. Optimal Quantum Spatial Search on Random Temporal Networks. PHYSICAL REVIEW LETTERS 2017; 119:220503. [PMID: 29286791 DOI: 10.1103/physrevlett.119.220503] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/27/2017] [Indexed: 06/07/2023]
Abstract
To investigate the performance of quantum information tasks on networks whose topology changes in time, we study the spatial search algorithm by continuous time quantum walk to find a marked node on a random temporal network. We consider a network of n nodes constituted by a time-ordered sequence of Erdös-Rényi random graphs G(n,p), where p is the probability that any two given nodes are connected: After every time interval τ, a new graph G(n,p) replaces the previous one. We prove analytically that, for any given p, there is always a range of values of τ for which the running time of the algorithm is optimal, i.e., O(sqrt[n]), even when search on the individual static graphs constituting the temporal network is suboptimal. On the other hand, there are regimes of τ where the algorithm is suboptimal even when each of the underlying static graphs are sufficiently connected to perform optimal search on them. From this first study of quantum spatial search on a time-dependent network, it emerges that the nontrivial interplay between temporality and connectivity is key to the algorithmic performance. Moreover, our work can be extended to establish high-fidelity qubit transfer between any two nodes of the network. Overall, our findings show that one can exploit temporality to achieve optimal quantum information tasks on dynamical random networks.
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Affiliation(s)
- Shantanav Chakraborty
- Instituto de Telecomunicações, Physics of Information and Quantum Technologies Group, Lisbon, Portugal and Instituto Superior Técnico, Universidade de Lisboa, Portugal
| | - Leonardo Novo
- Instituto de Telecomunicações, Physics of Information and Quantum Technologies Group, Lisbon, Portugal and Instituto Superior Técnico, Universidade de Lisboa, Portugal
| | - Serena Di Giorgio
- Instituto de Telecomunicações, Physics of Information and Quantum Technologies Group, Lisbon, Portugal and Instituto Superior Técnico, Universidade de Lisboa, Portugal
| | - Yasser Omar
- Instituto de Telecomunicações, Physics of Information and Quantum Technologies Group, Lisbon, Portugal and Instituto Superior Técnico, Universidade de Lisboa, Portugal
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Walschaers M, Mulet R, Wellens T, Buchleitner A. Statistical theory of designed quantum transport across disordered networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:042137. [PMID: 25974468 DOI: 10.1103/physreve.91.042137] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/09/2014] [Indexed: 06/04/2023]
Abstract
We explain how centrosymmetry, together with a dominant doublet of energy eigenstates in the local density of states, can guarantee interference-assisted, strongly enhanced, strictly coherent quantum excitation transport between two predefined sites of a random network of two-level systems. Starting from a generalization of the chaos-assisted tunnelling mechanism, we formulate a random matrix theoretical framework for the analytical prediction of the transfer time distribution, of lower bounds of the transfer efficiency, and of the scaling behavior of characteristic statistical properties with the size of the network. We show that these analytical predictions compare well to numerical simulations, using Hamiltonians sampled from the Gaussian orthogonal ensemble.
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Affiliation(s)
- Mattia Walschaers
- Physikalisches Institut, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Str. 3, D-79104 Freiburg, Germany
- Instituut voor Theoretische Fysica, University of Leuven, Celestijnenlaan 200D, B-3001 Heverlee, Belgium
| | - Roberto Mulet
- Physikalisches Institut, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Str. 3, D-79104 Freiburg, Germany
- Complex Systems Group, Department of Theoretical Physics, University of Havana, Cuba
| | - Thomas Wellens
- Physikalisches Institut, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Str. 3, D-79104 Freiburg, Germany
| | - Andreas Buchleitner
- Physikalisches Institut, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Str. 3, D-79104 Freiburg, Germany
- Freiburg Institute for Advanced Studies, Albert-Ludwigs-Universität Freiburg, Albertstr. 19, D-79104 Freiburg, Germany
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Böhm J, Bellec M, Mortessagne F, Kuhl U, Barkhofen S, Gehler S, Stöckmann HJ, Foulger I, Gnutzmann S, Tanner G. Microwave experiments simulating quantum search and directed transport in artificial graphene. PHYSICAL REVIEW LETTERS 2015; 114:110501. [PMID: 25839247 DOI: 10.1103/physrevlett.114.110501] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/08/2014] [Indexed: 06/04/2023]
Abstract
A series of quantum search algorithms have been proposed recently providing an algebraic speedup compared to classical search algorithms from N to √N, where N is the number of items in the search space. In particular, devising searches on regular lattices has become popular in extending Grover's original algorithm to spatial searching. Working in a tight-binding setup, it could be demonstrated, theoretically, that a search is possible in the physically relevant dimensions 2 and 3 if the lattice spectrum possesses Dirac points. We present here a proof of principle experiment implementing wave search algorithms and directed wave transport in a graphene lattice arrangement. The idea is based on bringing localized search states into resonance with an extended lattice state in an energy region of low spectral density-namely, at or near the Dirac point. The experiment is implemented using classical waves in a microwave setup containing weakly coupled dielectric resonators placed in a honeycomb arrangement, i.e., artificial graphene. Furthermore, we investigate the scaling behavior experimentally using linear chains.
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Affiliation(s)
- Julian Böhm
- Laboratoire de Physique de la Matière Condensée, UMR 7336, Université Nice Sophia Antipolis, CNRS, Parc Valrose, 06100 Nice, France
| | - Matthieu Bellec
- Laboratoire de Physique de la Matière Condensée, UMR 7336, Université Nice Sophia Antipolis, CNRS, Parc Valrose, 06100 Nice, France
| | - Fabrice Mortessagne
- Laboratoire de Physique de la Matière Condensée, UMR 7336, Université Nice Sophia Antipolis, CNRS, Parc Valrose, 06100 Nice, France
| | - Ulrich Kuhl
- Laboratoire de Physique de la Matière Condensée, UMR 7336, Université Nice Sophia Antipolis, CNRS, Parc Valrose, 06100 Nice, France
| | - Sonja Barkhofen
- AG Quantenchaos, Fachbereich Physik der Philipps-Universität Marburg, D-35032 Marburg, Germany
- Applied Physics, University of Paderborn, Warburger Strasse 100, 33098 Paderborn, Germany
| | - Stefan Gehler
- AG Quantenchaos, Fachbereich Physik der Philipps-Universität Marburg, D-35032 Marburg, Germany
- Department of Energy Management and Power System Operation, University of Kassel, D-34121 Kassel, Germany
| | - Hans-Jürgen Stöckmann
- AG Quantenchaos, Fachbereich Physik der Philipps-Universität Marburg, D-35032 Marburg, Germany
| | - Iain Foulger
- School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom
| | - Sven Gnutzmann
- School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom
| | - Gregor Tanner
- School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom
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Foulger I, Gnutzmann S, Tanner G. Quantum search on graphene lattices. PHYSICAL REVIEW LETTERS 2014; 112:070504. [PMID: 24579580 DOI: 10.1103/physrevlett.112.070504] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/18/2013] [Indexed: 06/03/2023]
Abstract
We present a continuous-time quantum search algorithm on a graphene lattice. This provides the sought-after implementation of an efficient continuous-time quantum search on a two-dimensional lattice. The search uses the linearity of the dispersion relation near the Dirac point and can find a marked site on a graphene lattice faster than the corresponding classical search. The algorithm can also be used for state transfer and communication.
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Affiliation(s)
- Iain Foulger
- School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom
| | - Sven Gnutzmann
- School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom
| | - Gregor Tanner
- School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom
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