1
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Yalouz S, Pouthier V. Extended star graph as a light-harvesting-complex prototype: Excitonic absorption speedup by peripheral energy defect tuning. Phys Rev E 2022; 106:064313. [PMID: 36671156 DOI: 10.1103/physreve.106.064313] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/14/2022] [Accepted: 12/06/2022] [Indexed: 12/23/2022]
Abstract
We study the quantum dynamics of a photoexcitation uniformly distributed at the periphery of an extended star network (with N_{B} branches of length L_{B}). More specifically, we address here the question of the energy absorption at the core of the network and how this process can be improved (or not) by the inclusion of peripheral defects with a tunable energy amplitude Δ. Our numerical simulations reveal the existence of optimal value of energy defect Δ^{*} which depends on the network architecture. Around this value, the absorption process presents a strong speedup (i.e., reduction of the absorption time) provided that L_{B}≤L_{B}^{*} with L_{B}^{*}≈12.5/ln(N_{B}). Analytical and numerical developments are then conducted to interpret this feature. We show that the origin of this speedup takes place in the hybridization of two upper-band excitonic eigenstates. This hybridization is important when L_{B}≤L_{B}^{*} and vanishes almost totally when L_{B}>L_{B}^{*}. These structural rules we draw here could represent a potential guide for the practical design of molecular nanonetwork dedicated to the realization of efficient photoexcitation absorption.
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Affiliation(s)
- Saad Yalouz
- Laboratoire de Chimie Quantique, Institut de Chimie, CNRS/Université de Strasbourg, 4 rue Blaise Pascal, 67000 Strasbourg, France
| | - Vincent Pouthier
- Institut UTINAM, Université de Franche-Comté, CNRS UMR 6213, 25030 Besançon, France
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2
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Cavazzoni S, Razzoli L, Bordone P, Paris MGA. Perturbed graphs achieve unit transport efficiency without environmental noise. Phys Rev E 2022; 106:024118. [PMID: 36110011 DOI: 10.1103/physreve.106.024118] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/20/2022] [Accepted: 08/01/2022] [Indexed: 06/15/2023]
Abstract
Coherent transport of an excitation through a network corresponds to continuous-time quantum walk on a graph, and the transport properties of the system may be radically different depending on the graph and on the initial state. The transport efficiency, i.e., the integrated probability of trapping at a certain vertex, is a measure of the success rate of the transfer process. Purely coherent quantum transport is known to be less efficient than the observed excitation transport, e.g., in biological systems, and there is evidence that environmental noise is indeed crucial for excitation transport. At variance with this picture, we here address purely coherent transport on highly symmetric graphs, and show analytically that it is possible to enhance the transport efficiency without environmental noise, i.e., using only a minimal perturbation of the graph. In particular, we show that adding an extra weight to one or two edges, depending on whether the initial state is localized or in a superposition of two vertex states, breaks the inherent symmetries of the graph and may be sufficient to achieve unit transport efficiency. We also briefly discuss the conditions to obtain a null transport efficiency, i.e., to avoid trapping.
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Affiliation(s)
- Simone Cavazzoni
- Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Università di Modena e Reggio Emilia, I-41125 Modena, Italy
| | - Luca Razzoli
- Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Università di Modena e Reggio Emilia, I-41125 Modena, Italy
- Dipartimento di Scienza e Alta Tecnologia, Università degli Studi dell'Insubria, I-22100 Como, Italy
| | - Paolo Bordone
- Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Università di Modena e Reggio Emilia, I-41125 Modena, Italy
- Centro S3, CNR-Istituto di Nanoscienze, I-41125 Modena, Italy
| | - Matteo G A Paris
- Quantum Technology Lab, Dipartimento di Fisica Aldo Pontremoli, Università degli Studi di Milano, I-20133 Milano, Italy
- INFN, Sezione di Milano, I-20133 Milano, Italy
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3
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Pouthier V. Two-exciton bound state quantum self-trapping in an extended star graph. J Chem Phys 2022; 156:155101. [DOI: 10.1063/5.0087200] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The attractive Bose-Hubbard model is applied for describing the quantum self-trapping in an extended star graph. In the strong coupling limit and when two excitons are created on the core of the star, the dynamics is dominated by pair states whose properties is governed by the branch number $N$. When $N=2$, the star reduces to a linear chain so that the energy does not self-localize. Conversely, when $N\geq3$, a restructuring of the eigenstates arises and a low-energy state occurs describing a pair localized on the core of the star. Preferentially excited, this localized state gives rise to a quantum self-trapping of the energy, a process that intensifies as $N$ increases.
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Cuadra L, Nieto-Borge JC. Approaching Disordered Quantum Dot Systems by Complex Networks with Spatial and Physical-Based Constraints. NANOMATERIALS 2021; 11:nano11082056. [PMID: 34443887 PMCID: PMC8400585 DOI: 10.3390/nano11082056] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 07/01/2021] [Revised: 07/22/2021] [Accepted: 08/03/2021] [Indexed: 01/01/2023]
Abstract
This paper focuses on modeling a disordered system of quantum dots (QDs) by using complex networks with spatial and physical-based constraints. The first constraint is that, although QDs (=nodes) are randomly distributed in a metric space, they have to fulfill the condition that there is a minimum inter-dot distance that cannot be violated (to minimize electron localization). The second constraint arises from our process of weighted link formation, which is consistent with the laws of quantum physics and statistics: it not only takes into account the overlap integrals but also Boltzmann factors to include the fact that an electron can hop from one QD to another with a different energy level. Boltzmann factors and coherence naturally arise from the Lindblad master equation. The weighted adjacency matrix leads to a Laplacian matrix and a time evolution operator that allows the computation of the electron probability distribution and quantum transport efficiency. The results suggest that there is an optimal inter-dot distance that helps reduce electron localization in QD clusters and make the wave function better extended. As a potential application, we provide recommendations for improving QD intermediate-band solar cells.
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Affiliation(s)
- Lucas Cuadra
- Department of Signal Processing and Communications, University of Alcalá, 28801 Alcalá de Henares, Spain
- Department of Physics and Mathematics, University of Alcalá, 28801 Alcalá de Henares, Spain;
- Correspondence:
| | - José Carlos Nieto-Borge
- Department of Physics and Mathematics, University of Alcalá, 28801 Alcalá de Henares, Spain;
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5
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Cuadra L, Nieto-Borge JC. Modeling Quantum Dot Systems as Random Geometric Graphs with Probability Amplitude-Based Weighted Links. NANOMATERIALS (BASEL, SWITZERLAND) 2021; 11:375. [PMID: 33540687 PMCID: PMC7912992 DOI: 10.3390/nano11020375] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 12/21/2020] [Revised: 01/18/2021] [Accepted: 01/27/2021] [Indexed: 12/28/2022]
Abstract
This paper focuses on modeling a disorder ensemble of quantum dots (QDs) as a special kind of Random Geometric Graphs (RGG) with weighted links. We compute any link weight as the overlap integral (or electron probability amplitude) between the QDs (=nodes) involved. This naturally leads to a weighted adjacency matrix, a Laplacian matrix, and a time evolution operator that have meaning in Quantum Mechanics. The model prohibits the existence of long-range links (shortcuts) between distant nodes because the electron cannot tunnel between two QDs that are too far away in the array. The spatial network generated by the proposed model captures inner properties of the QD system, which cannot be deduced from the simple interactions of their isolated components. It predicts the system quantum state, its time evolution, and the emergence of quantum transport when the network becomes connected.
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Affiliation(s)
- Lucas Cuadra
- Department of Signal Processing and Communications, University of Alcalá, 28801 Alcalá de Henares, Spain
- Department of Physics and Mathematics, University of Alcalá, 28801 Alcalá de Henares, Spain;
| | - José Carlos Nieto-Borge
- Department of Physics and Mathematics, University of Alcalá, 28801 Alcalá de Henares, Spain;
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6
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Razzoli L, Paris MGA, Bordone P. Transport Efficiency of Continuous-Time Quantum Walks on Graphs. ENTROPY 2021; 23:e23010085. [PMID: 33435338 PMCID: PMC7826824 DOI: 10.3390/e23010085] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 11/27/2020] [Revised: 01/05/2021] [Accepted: 01/07/2021] [Indexed: 11/16/2022]
Abstract
Continuous-time quantum walk describes the propagation of a quantum particle (or an excitation) evolving continuously in time on a graph. As such, it provides a natural framework for modeling transport processes, e.g., in light-harvesting systems. In particular, the transport properties strongly depend on the initial state and specific features of the graph under investigation. In this paper, we address the role of graph topology, and investigate the transport properties of graphs with different regularity, symmetry, and connectivity. We neglect disorder and decoherence, and assume a single trap vertex that is accountable for the loss processes. In particular, for each graph, we analytically determine the subspace of states having maximum transport efficiency. Our results provide a set of benchmarks for environment-assisted quantum transport, and suggest that connectivity is a poor indicator for transport efficiency. Indeed, we observe some specific correlations between transport efficiency and connectivity for certain graphs, but, in general, they are uncorrelated.
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Affiliation(s)
- Luca Razzoli
- Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Università di Modena e Reggio Emilia, I-41125 Modena, Italy
- Correspondence: (L.R.); (P.B.)
| | - Matteo G. A. Paris
- Quantum Technology Lab, Dipartimento di Fisica Aldo Pontremoli, Università Degli Studi di Milano, I-20133 Milano, Italy;
- INFN, Sezione di Milano, I-20133 Milano, Italy
| | - Paolo Bordone
- Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Università di Modena e Reggio Emilia, I-41125 Modena, Italy
- Centro S3, CNR-Istituto di Nanoscienze, I-41125 Modena, Italy
- Correspondence: (L.R.); (P.B.)
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7
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Zurn P, Bassett DS. Network architectures supporting learnability. Philos Trans R Soc Lond B Biol Sci 2020; 375:20190323. [PMID: 32089113 PMCID: PMC7061954 DOI: 10.1098/rstb.2019.0323] [Citation(s) in RCA: 12] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Journal Information] [Subscribe] [Scholar Register] [Accepted: 12/31/2019] [Indexed: 12/25/2022] Open
Abstract
Human learners acquire complex interconnected networks of relational knowledge. The capacity for such learning naturally depends on two factors: the architecture (or informational structure) of the knowledge network itself and the architecture of the computational unit-the brain-that encodes and processes the information. That is, learning is reliant on integrated network architectures at two levels: the epistemic and the computational, or the conceptual and the neural. Motivated by a wish to understand conventional human knowledge, here, we discuss emerging work assessing network constraints on the learnability of relational knowledge, and theories from statistical physics that instantiate the principles of thermodynamics and information theory to offer an explanatory model for such constraints. We then highlight similarities between those constraints on the learnability of relational networks, at one level, and the physical constraints on the development of interconnected patterns in neural systems, at another level, both leading to hierarchically modular networks. To support our discussion of these similarities, we employ an operational distinction between the modeller (e.g. the human brain), the model (e.g. a single human's knowledge) and the modelled (e.g. the information present in our experiences). We then turn to a philosophical discussion of whether and how we can extend our observations to a claim regarding explanation and mechanism for knowledge acquisition. What relation between hierarchical networks, at the conceptual and neural levels, best facilitate learning? Are the architectures of optimally learnable networks a topological reflection of the architectures of comparably developed neural networks? Finally, we contribute to a unified approach to hierarchies and levels in biological networks by proposing several epistemological norms for analysing the computational brain and social epistemes, and for developing pedagogical principles conducive to curious thought. This article is part of the theme issue 'Unifying the essential concepts of biological networks: biological insights and philosophical foundations'.
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Affiliation(s)
- Perry Zurn
- Department of Philosophy, American University, Washington, DC 20016, USA
| | - Danielle S. Bassett
- Department of Bioengineering, School of Engineering and Applied Science, University of Pennsylvania, Philadelphia, PA 19104, USA
- Department of Physics and Astronomy, College of Arts and Sciences, University of Pennsylvania, Philadelphia, PA 19104, USA
- Department of Electrical and Systems Engineering, School of Engineering and Applied Science, University of Pennsylvania, Philadelphia, PA 19104, USA
- Department of Neurology, Perelman School of Medicine, University of Pennsylvania, Philadelphia, PA 19104, USA
- Department of Psychiatry, Perelman School of Medicine, University of Pennsylvania, Philadelphia, PA 19104, USA
- Santa Fe Institute, Santa Fe, NM 87501, USA
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8
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Yalouz S, Pouthier V. Continuous-time quantum walk on an extended star graph: Disorder-enhanced trapping process. Phys Rev E 2020; 101:012310. [PMID: 32069559 DOI: 10.1103/physreve.101.012310] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/16/2019] [Indexed: 06/10/2023]
Abstract
Using a tight binding model, we investigate the dynamics of an exciton on a disordered extended star graph whose central site acts as an energy trap. When compared with what happens in an ordered network, our results reveal that the disorder drastically improves the excitonic absorption that becomes complete. Moreover, we show the occurrence of an optimal disorder for which the absorption time is strongly minimized, a surprising effect that originates in a disorder-induced restructuring process of the exciton eigenstates. Finally, we also show the existence of an optimal value of the absorption rate that reduces even more the absorption time. The resulting superoptimized trapping process is interpreted as a positive interplay between both the disorder and the so-called superradiance transition.
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Affiliation(s)
- Saad Yalouz
- Institut UTINAM, Université de Franche-Comté, CNRS UMR 6213, 25030 Besançon Cedex, France
| | - Vincent Pouthier
- Institut UTINAM, Université de Franche-Comté, CNRS UMR 6213, 25030 Besançon Cedex, France
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9
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Fast and slow dynamics for classical and quantum walks on mean-field small world networks. Sci Rep 2019; 9:19143. [PMID: 31844101 PMCID: PMC6914773 DOI: 10.1038/s41598-019-55580-2] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/18/2019] [Accepted: 11/12/2019] [Indexed: 11/08/2022] Open
Abstract
This work investigates the dynamical properties of classical and quantum random walks on mean-field small-world (MFSW) networks in the continuous time version. The adopted formalism profits from the large number of exact mathematical properties of their adjacency and Laplacian matrices. Exact expressions for both transition probabilities in terms of Bessel functions are derived. Results are compared to numerical results obtained by working directly the Hamiltonian of the model. For the classical evolution, any infinitesimal amount of disorder causes an exponential decay to the asymptotic equilibrium state, in contrast to the polynomial behavior for the homogeneous case. The typical quantum oscillatory evolution has been characterized by local maxima. It indicates polynomial decay to equilibrium for any degree of disorder. The main finding of the work is the identification of a faster classical spreading as compared to the quantum counterpart. It stays in opposition to the well known diffusive and ballistic for, respectively, the classical and quantum spreading in the linear chain.
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10
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Lechman JB, Bond SD, Bolintineanu DS, Grest GS, Yarrington CD, Silbert LE. Random walks on jammed networks: Spectral properties. Phys Rev E 2019; 100:012905. [PMID: 31499918 DOI: 10.1103/physreve.100.012905] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/08/2018] [Indexed: 06/10/2023]
Abstract
Using random walk analyses we explore diffusive transport on networks obtained from contacts between isotropically compressed, monodisperse, frictionless sphere packings generated over a range of pressures in the vicinity of the jamming transition p→0. For conductive particles in an insulating medium, conduction is determined by the particle contact network with nodes representing particle centers and edges contacts between particles. The transition rate is not homogeneous, but is distributed inhomogeneously due to the randomness of packing and concomitant disorder of the contact network, e.g., the distribution of the coordination number. A narrow escape time scale is used to write a Markov process for random walks on the particle contact network. This stochastic process is analyzed in terms of spectral density of the random, sparse, Euclidean and real, symmetric, positive, semidefinite transition rate matrix. Results show network structures derived from jammed particles have properties similar to ordered, euclidean lattices but also some unique properties that distinguish them from other structures that are in some sense more homogeneous. In particular, the distribution of eigenvalues of the transition rate matrix follow a power law with spectral dimension 3. However, quantitative details of the statistics of the eigenvectors show subtle differences with homogeneous lattices and allow us to distinguish between topological and geometric sources of disorder in the network.
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Affiliation(s)
- Jeremy B Lechman
- Sandia National Laboratories, Albuquerque, New Mexico 87185, USA
| | - Stephen D Bond
- Sandia National Laboratories, Albuquerque, New Mexico 87185, USA
| | | | - Gary S Grest
- Sandia National Laboratories, Albuquerque, New Mexico 87185, USA
| | | | - Leonardo E Silbert
- School of Math, Science and Engineering, Central New Mexico Community College, Albuquerque, New Mexico 87106, USA
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11
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Tang H, Lin XF, Feng Z, Chen JY, Gao J, Sun K, Wang CY, Lai PC, Xu XY, Wang Y, Qiao LF, Yang AL, Jin XM. Experimental two-dimensional quantum walk on a photonic chip. SCIENCE ADVANCES 2018; 4:eaat3174. [PMID: 29756040 PMCID: PMC5947980 DOI: 10.1126/sciadv.aat3174] [Citation(s) in RCA: 48] [Impact Index Per Article: 8.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/13/2018] [Accepted: 03/21/2018] [Indexed: 05/12/2023]
Abstract
Quantum walks, in virtue of the coherent superposition and quantum interference, have exponential superiority over their classical counterpart in applications of quantum searching and quantum simulation. The quantum-enhanced power is highly related to the state space of quantum walks, which can be expanded by enlarging the photon number and/or the dimensions of the evolution network, but the former is considerably challenging due to probabilistic generation of single photons and multiplicative loss. We demonstrate a two-dimensional continuous-time quantum walk by using the external geometry of photonic waveguide arrays, rather than the inner degree of freedoms of photons. Using femtosecond laser direct writing, we construct a large-scale three-dimensional structure that forms a two-dimensional lattice with up to 49 × 49 nodes on a photonic chip. We demonstrate spatial two-dimensional quantum walks using heralded single photons and single photon-level imaging. We analyze the quantum transport properties via observing the ballistic evolution pattern and the variance profile, which agree well with simulation results. We further reveal the transient nature that is the unique feature for quantum walks of beyond one dimension. An architecture that allows a quantum walk to freely evolve in all directions and at a large scale, combining with defect and disorder control, may bring up powerful and versatile quantum walk machines for classically intractable problems.
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Affiliation(s)
- Hao Tang
- State Key Laboratory of Advanced Optical Communication Systems and Networks, Institute of Natural Sciences and Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
- Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Xiao-Feng Lin
- State Key Laboratory of Advanced Optical Communication Systems and Networks, Institute of Natural Sciences and Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
- Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Zhen Feng
- State Key Laboratory of Advanced Optical Communication Systems and Networks, Institute of Natural Sciences and Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
- Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Jing-Yuan Chen
- State Key Laboratory of Advanced Optical Communication Systems and Networks, Institute of Natural Sciences and Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
| | - Jun Gao
- State Key Laboratory of Advanced Optical Communication Systems and Networks, Institute of Natural Sciences and Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
- Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Ke Sun
- State Key Laboratory of Advanced Optical Communication Systems and Networks, Institute of Natural Sciences and Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
| | - Chao-Yue Wang
- State Key Laboratory of Advanced Optical Communication Systems and Networks, Institute of Natural Sciences and Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
| | - Peng-Cheng Lai
- State Key Laboratory of Advanced Optical Communication Systems and Networks, Institute of Natural Sciences and Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
| | - Xiao-Yun Xu
- State Key Laboratory of Advanced Optical Communication Systems and Networks, Institute of Natural Sciences and Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
- Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Yao Wang
- State Key Laboratory of Advanced Optical Communication Systems and Networks, Institute of Natural Sciences and Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
- Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Lu-Feng Qiao
- State Key Laboratory of Advanced Optical Communication Systems and Networks, Institute of Natural Sciences and Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
- Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Ai-Lin Yang
- State Key Laboratory of Advanced Optical Communication Systems and Networks, Institute of Natural Sciences and Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
- Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Xian-Min Jin
- State Key Laboratory of Advanced Optical Communication Systems and Networks, Institute of Natural Sciences and Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
- Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
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12
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Yalouz S, Pouthier V. Continuous-time quantum walk on an extended star graph: Trapping and superradiance transition. Phys Rev E 2018; 97:022304. [PMID: 29548191 DOI: 10.1103/physreve.97.022304] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/16/2017] [Indexed: 06/08/2023]
Abstract
A tight-binding model is introduced for describing the dynamics of an exciton on an extended star graph whose central node is occupied by a trap. On this graph, the exciton dynamics is governed by two kinds of eigenstates: many eigenstates are associated with degenerate real eigenvalues insensitive to the trap, whereas three decaying eigenstates characterized by complex energies contribute to the trapping process. It is shown that the excitonic population absorbed by the trap depends on the size of the graph, only. By contrast, both the size parameters and the absorption rate control the dynamics of the trapping. When these parameters are judiciously chosen, the efficiency of the transfer is optimized resulting in the minimization of the absorption time. Analysis of the eigenstates reveals that such a feature arises around the superradiance transition. Moreover, depending on the size of the network, two situations are highlighted where the transport efficiency is either superoptimized or suboptimized.
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Affiliation(s)
- Saad Yalouz
- Institut UTINAM, Univ. Bourgogne Franche-Comté, CNRS UMR 6213, 25030 Besançon Cedex, France
| | - Vincent Pouthier
- Institut UTINAM, Univ. Bourgogne Franche-Comté, CNRS UMR 6213, 25030 Besançon Cedex, France
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13
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Riascos AP, Mateos JL. Fractional dynamics on networks: emergence of anomalous diffusion and Lévy flights. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:032809. [PMID: 25314484 DOI: 10.1103/physreve.90.032809] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/01/2014] [Indexed: 05/14/2023]
Abstract
We introduce a formalism of fractional diffusion on networks based on a fractional Laplacian matrix that can be constructed directly from the eigenvalues and eigenvectors of the Laplacian matrix. This fractional approach allows random walks with long-range dynamics providing a general framework for anomalous diffusion and navigation, and inducing dynamically the small-world property on any network. We obtained exact results for the stationary probability distribution, the average fractional return probability, and a global time, showing that the efficiency to navigate the network is greater if we use a fractional random walk in comparison to a normal random walk. For the case of a ring, we obtain exact analytical results showing that the fractional transition and return probabilities follow a long-range power-law decay, leading to the emergence of Lévy flights on networks. Our general fractional diffusion formalism applies to regular, random, and complex networks and can be implemented from the spectral properties of the Laplacian matrix, providing an important tool to analyze anomalous diffusion on networks.
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Affiliation(s)
- A P Riascos
- Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364, 01000 México, D.F., México
| | - José L Mateos
- Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364, 01000 México, D.F., México
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14
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Darázs Z, Anishchenko A, Kiss T, Blumen A, Mülken O. Transport properties of continuous-time quantum walks on Sierpinski fractals. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:032113. [PMID: 25314401 DOI: 10.1103/physreve.90.032113] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/20/2014] [Indexed: 06/04/2023]
Abstract
We model quantum transport, described by continuous-time quantum walks (CTQWs), on deterministic Sierpinski fractals, differentiating between Sierpinski gaskets and Sierpinski carpets, along with their dual structures. The transport efficiencies are defined in terms of the exact and the average return probabilities, as well as by the mean survival probability when absorbing traps are present. In the case of gaskets, localization can be identified already for small networks (generations). For carpets, our numerical results indicate a trend towards localization, but only for relatively large structures. The comparison of gaskets and carpets further implies that, distinct from the corresponding classical continuous-time random walk, the spectral dimension does not fully determine the evolution of the CTQW.
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Affiliation(s)
- Zoltán Darázs
- Wigner RCP, SZFKI, Konkoly-Thege Miklós út 29-33, H-1121 Budapest, Hungary and Eötvös University, Pázmány Péter sétány 1/A, H-1117 Budapest, Hungary
| | - Anastasiia Anishchenko
- Physikalisches Institut, Universität Freiburg, Hermann-Herder-Straße 3, 79104 Freiburg, Germany
| | - Tamás Kiss
- Wigner RCP, SZFKI, Konkoly-Thege Miklós út 29-33, H-1121 Budapest, Hungary
| | - Alexander Blumen
- Physikalisches Institut, Universität Freiburg, Hermann-Herder-Straße 3, 79104 Freiburg, Germany
| | - Oliver Mülken
- Physikalisches Institut, Universität Freiburg, Hermann-Herder-Straße 3, 79104 Freiburg, Germany
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15
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Pouthier V. Disorder-enhanced exciton delocalization in an extended dendrimer. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:022818. [PMID: 25215792 DOI: 10.1103/physreve.90.022818] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/17/2014] [Indexed: 06/03/2023]
Abstract
The exciton dynamics in a disordered extended dendrimer is investigated numerically. Because a homogeneous dendrimer exhibits few highly degenerate energy levels, a dynamical localization arises when the exciton is initially located on the periphery. However, it is shown that the disorder lifts the degeneracy and favors a delocalization-relocalization transition. Weak disorder enhances the delocalized nature of the exciton and improves any quantum communication, whereas strong disorder prevents the exciton from propagating in accordance with the well-known Anderson theory.
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Affiliation(s)
- Vincent Pouthier
- Institut UTINAM, Université de Franche-Comté, CNRS UMR 6213, 25030 Besançon Cedex, France
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16
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Xu XP. Coherent exciton transport and trapping on long-range interacting cycles. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:011117. [PMID: 19257011 DOI: 10.1103/physreve.79.011117] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/05/2008] [Indexed: 05/27/2023]
Abstract
We consider coherent exciton transport modeled by continuous-time quantum walks on long-range interacting cycles (LRICs), which are constructed by connecting all the two nodes of distance m in the cycle graph. LRIC has a symmetric structure and can be regarded as the extensions of the cycle graph (nearest-neighboring lattice). For small values of m , the classical and quantum return probabilities show power law behavior p(t) approximately t;{-0.5} and pi(t) approximately t;{-1} , respectively. However, for large values of m , the classical and quantum efficiency scales as p(t) approximately t;{-1} and pi(t) approximately t;{-2} . We give a theoretical explanation of this transition using the method of stationary phase approximation. In the long time limit, depending on the network size N and parameter m , the limiting probability distributions of quantum transport show various patterns. When the network size N is an even number, we find an asymmetric transition probability of quantum transport between the initial node and its opposite node. This asymmetry depends on the precise values of N and m . Finally, we study the transport processes in the presence of traps and find that the survival probability decays faster on networks of large m .
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Affiliation(s)
- Xin-Ping Xu
- Institute of Particle Physics, HuaZhong Normal University, Wuhan 430079, China.
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17
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Xu XP. Continuous-time quantum walks on one-dimensional regular networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 77:061127. [PMID: 18643237 DOI: 10.1103/physreve.77.061127] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/22/2007] [Revised: 03/24/2008] [Indexed: 05/26/2023]
Abstract
In this paper, we consider continuous-time quantum walks (CTQWs) on a one-dimensional ring lattice of N nodes in which every node is connected to its 2m nearest neighbors ( m on either side). In the framework of the Bloch function ansatz, we calculate the space-time transition probabilities between two nodes of the lattice. We find that the transport of CTQWs between two different nodes is faster than that of the classical continuous-time random walks (CTRWs). The transport speed, which is defined by the ratio of the shortest path length and propagating time, increases with the connectivity parameter m for both CTQWs and CTRWs. For fixed parameter m , the transport of CTRWs gets slower with the increase of the shortest distance while the transport (speed) of CTQWs turns out to be a constant value. In the long-time limit, depending on the network size N and connectivity parameter m , the limiting probability distributions of CTQWs show various patterns. When the network size N is an even number, the probability of being at the original node differs from that of being at the opposite node, which also depends on the precise value of parameter m .
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Affiliation(s)
- Xin-Ping Xu
- Institute of Particle Physics, HuaZhong Normal University, Wuhan 430079, China.
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18
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Mülken O, Pernice V, Blumen A. Universal behavior of quantum walks with long-range steps. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 77:021117. [PMID: 18351997 DOI: 10.1103/physreve.77.021117] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/23/2007] [Indexed: 05/26/2023]
Abstract
Continuous-time quantum walks with long-range steps R(-gamma) (R being the distance between sites) on a discrete line behave in similar ways for all gamma > or =2 . This is in contrast to classical random walks, which for gamma>3 belong to a different universality class than for gamma < or =3 . We show that the average probabilities to be at the initial site after time t as well as the mean square displacements are of the same functional form for quantum walks with gamma =2, 4, and with nearest neighbor steps. We interpolate this result to arbitrary gamma > or =2 .
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Affiliation(s)
- Oliver Mülken
- Theoretische Polymerphysik, Universität Freiburg, Hermann-Herder-Strasse 3, Freiburg, Germany.
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19
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Baek SK, Yi SD, Kim BJ. Diffusion on a heptagonal lattice. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 77:022104. [PMID: 18352068 DOI: 10.1103/physreve.77.022104] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/21/2007] [Indexed: 05/26/2023]
Abstract
We study the diffusion phenomena on the negatively curved surface made up of congruent heptagons. Unlike the usual two-dimensional plane, this structure makes the boundary increase exponentially with the distance from the center, and hence the displacement of a classical random walker increases linearly in time. The diffusion of a quantum particle put on the heptagonal lattice is also studied in the framework of the tight-binding model Hamiltonian, and we again find the linear diffusion like the classical random walk. A comparison with diffusion on complex networks is also made.
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Affiliation(s)
- Seung Ki Baek
- Department of Physics, BK21 Physics Research Division, and Institute of Basic Science, Sungkyunkwan University, Suwon, Republic of Korea
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20
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Mülken O, Pernice V, Blumen A. Quantum transport on small-world networks: a continuous-time quantum walk approach. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:051125. [PMID: 18233641 DOI: 10.1103/physreve.76.051125] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/11/2007] [Revised: 07/31/2007] [Indexed: 05/25/2023]
Abstract
We consider the quantum mechanical transport of (coherent) excitons on small-world networks (SWNs). The SWNs are built from a one-dimensional ring of N nodes by randomly introducing B additional bonds between them. The exciton dynamics is modeled by continuous-time quantum walks, and we evaluate numerically the ensemble-averaged transition probability to reach any node of the network from the initially excited one. For sufficiently large B we find that the quantum mechanical transport through the SWNs is, first, very fast, given that the limiting value of the transition probability is reached very quickly, and second, that the transport does not lead to equipartition, given that on average the exciton is most likely to be found at the initial node.
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Affiliation(s)
- Oliver Mülken
- Theoretische Polymerphysik, Universität Freiburg, Hermann-Herder-Strasse 3, 79104 Freiburg, Germany.
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21
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Mülken O, Blumen A, Amthor T, Giese C, Reetz-Lamour M, Weidemüller M. Survival probabilities in coherent exciton transfer with trapping. PHYSICAL REVIEW LETTERS 2007; 99:090601. [PMID: 17930996 DOI: 10.1103/physrevlett.99.090601] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/25/2007] [Indexed: 05/25/2023]
Abstract
In the quest for signatures of coherent transport we consider exciton trapping in the continuous-time quantum walk framework. The survival probability displays different decay domains, related to distinct regions of the spectrum of the Hamiltonian. For linear systems and at intermediate times the decay obeys a power law, in contrast with the corresponding exponential decay found in incoherent continuous-time random walk situations. To differentiate between the coherent and incoherent mechanisms, we present an experimental protocol based on a frozen Rydberg gas structured by optical dipole traps.
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Affiliation(s)
- Oliver Mülken
- Physikalisches Institut, Universität Freiburg, Hermann-Herder-Strasse 3, 79104 Freiburg i.Br., Germany
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