151
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Tian H, Chen G. An efficient solution of Liouville-von Neumann equation that is applicable to zero and finite temperatures. J Chem Phys 2013. [PMID: 23205988 DOI: 10.1063/1.4767460] [Citation(s) in RCA: 29] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Application of quantum dissipation theory to electronic dynamics has been limited to model systems with few energy levels, and its numerical solutions are mostly restricted to high temperatures. A highly accurate and efficient numerical algorithm, which is based on the Chebyshev spectral method, is developed to integrate a single-particle Liouville-von Neumann equation, and the two long-standing limitations of quantum dissipation theory are resolved in the context of quantum transport. Its computational time scales to O(N(3)) with N being the number of orbitals involved, which leads to a reality for the quantum mechanical simulation of real open systems containing hundreds or thousands of atomic orbitals. More importantly, the algorithm spans both finite and zero temperatures. Numerical calculations are carried out to simulate the transient current through a metallic wire containing up to 1000 orbitals.
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Affiliation(s)
- Heng Tian
- Department of Chemistry, The University of Hong Kong, Hong Kong.
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152
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Zhang Y, Yam CY, Chen G. Dissipative time-dependent quantum transport theory. J Chem Phys 2013; 138:164121. [DOI: 10.1063/1.4802592] [Citation(s) in RCA: 24] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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153
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Pouthier V. The reduced dynamics of an exciton coupled to a phonon bath: A new approach combining the Lang-Firsov transformation and the perturbation theory. J Chem Phys 2013; 138:044108. [DOI: 10.1063/1.4789017] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/03/2023] Open
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154
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Zhong X, Zhao Y. Non-Markovian stochastic Schrödinger equation at finite temperatures for charge carrier dynamics in organic crystals. J Chem Phys 2013; 138:014111. [DOI: 10.1063/1.4773319] [Citation(s) in RCA: 49] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/11/2023] Open
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155
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Li Z, Tong N, Zheng X, Hou D, Wei J, Hu J, Yan Y. Hierarchical Liouville-space approach for accurate and universal characterization of quantum impurity systems. PHYSICAL REVIEW LETTERS 2012; 109:266403. [PMID: 23368590 DOI: 10.1103/physrevlett.109.266403] [Citation(s) in RCA: 69] [Impact Index Per Article: 5.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/17/2012] [Revised: 10/07/2012] [Indexed: 06/01/2023]
Abstract
A hierarchical equations of motion based numerical approach is developed for accurate and efficient evaluation of dynamical observables of strongly correlated quantum impurity systems. This approach is capable of describing quantitatively Kondo resonance and Fermi-liquid characteristics, achieving the accuracy of the latest high-level numerical renormalization group approach, as demonstrated on single-impurity Anderson model systems. Its application to a two-impurity Anderson model results in differential conductance versus external bias, which correctly reproduces the continuous transition from Kondo states of individual impurity to singlet spin states formed between two impurities. The outstanding performance on characterizing both equilibrium and nonequilibrium properties of quantum impurity systems makes the hierarchical equations of motion approach potentially useful for addressing strongly correlated lattice systems in the framework of dynamical mean-field theory.
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Affiliation(s)
- ZhenHua Li
- Department of Physics, Renmin University of China, Beijing 100872, China
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156
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Popescu B, Woiczikowski PB, Elstner M, Kleinekathöfer U. Time-dependent view of sequential transport through molecules with rapidly fluctuating bridges. PHYSICAL REVIEW LETTERS 2012; 109:176802. [PMID: 23215213 DOI: 10.1103/physrevlett.109.176802] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/28/2012] [Indexed: 06/01/2023]
Abstract
Molecules in junctions often fluctuate considerably, especially when subject to the influence of solvent molecules. These fluctuations in site energies and couplings can be sampled, for example, by using molecular dynamics simulations, and can lead to incoherent effects in charge transport. To this end, a popular snapshot-averaged Landauer approach is compared to a time-dependent Green's function scheme. Since sequential transport dominates in systems with rapidly varying bridges, schemes not taking the time order of conformations into account, such as the Landauer approach, are inappropriate.
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Affiliation(s)
- Bogdan Popescu
- School of Engineering and Science, Jacobs University Bremen, Campus Ring 1, 28759 Bremen, Germany
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157
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Zhang WM, Lo PY, Xiong HN, Tu MWY, Nori F. General non-Markovian dynamics of open quantum systems. PHYSICAL REVIEW LETTERS 2012; 109:170402. [PMID: 23215166 DOI: 10.1103/physrevlett.109.170402] [Citation(s) in RCA: 22] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/25/2012] [Indexed: 06/01/2023]
Abstract
We present a general theory of non-Markovian dynamics for open systems of noninteracting fermions (bosons) linearly coupled to thermal environments of noninteracting fermions (bosons). We explore the non-Markovian dynamics by connecting the exact master equations with the nonequilibirum Green's functions. Environmental backactions are fully taken into account. The non-Markovian dynamics consists of nonexponential decays and dissipationless oscillations. Nonexponential decays are induced by the discontinuity in the imaginary part of the self-energy corrections. Dissipationless oscillations arise from band gaps or the finite band structure of spectral densities. The exact analytic solutions for various non-Markovian thermal environments show that non-Markovian dynamics can be largely understood from the environmental-modified spectra of open systems.
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Affiliation(s)
- Wei-Min Zhang
- Department of Physics, National Cheng Kung University, Tainan 70101, Taiwan.
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158
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Ding JJ, Xu RX, Yan Y. Optimizing hierarchical equations of motion for quantum dissipation and quantifying quantum bath effects on quantum transfer mechanisms. J Chem Phys 2012; 136:224103. [PMID: 22713032 DOI: 10.1063/1.4724193] [Citation(s) in RCA: 45] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We present an optimized hierarchical equations of motion theory for quantum dissipation in multiple Brownian oscillators bath environment, followed by a mechanistic study on a model donor-bridge-acceptor system. We show that the optimal hierarchy construction, via the memory-frequency decomposition for any specified Brownian oscillators bath, is generally achievable through a universal pre-screening search. The algorithm goes by identifying the candidates for the best be just some selected Padé spectrum decomposition based schemes, together with a priori accuracy control criterions on the sole approximation, the white-noise residue ansatz, involved in the hierarchical construction. Beside the universal screening search, we also analytically identify the best for the case of Drude dissipation and that for the Brownian oscillators environment without strongly underdamped bath vibrations. For the mechanistic study, we quantify the quantum nature of bath influence and further address the issue of localization versus delocalization. Proposed are a reduced system entropy measure and a state-resolved constructive versus destructive interference measure. Their performances on quantifying the correlated system-environment coherence are exemplified in conjunction with the optimized hierarchical equations of motion evaluation of the model system dynamics, at some representing bath parameters and temperatures. Analysis also reveals the localization to delocalization transition as temperature decreases.
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Affiliation(s)
- Jin-Jin Ding
- Hefei National Laboratory for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei, Anhui 230026, China
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159
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Xie H, Jiang F, Tian H, Zheng X, Kwok Y, Chen S, Yam C, Yan Y, Chen G. Time-dependent quantum transport: An efficient method based on Liouville-von-Neumann equation for single-electron density matrix. J Chem Phys 2012; 137:044113. [DOI: 10.1063/1.4737864] [Citation(s) in RCA: 45] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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160
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White AJ, Galperin M. Inelastic transport: a pseudoparticle approach. Phys Chem Chem Phys 2012; 14:13809-19. [DOI: 10.1039/c2cp41017f] [Citation(s) in RCA: 35] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/03/2023]
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161
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Ding JJ, Xu J, Hu J, Xu RX, Yan Y. Optimized hierarchical equations of motion theory for Drude dissipation and efficient implementation to nonlinear spectroscopies. J Chem Phys 2011; 135:164107. [DOI: 10.1063/1.3653479] [Citation(s) in RCA: 51] [Impact Index Per Article: 3.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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162
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Hu J, Luo M, Jiang F, Xu RX, Yan Y. Padé spectrum decompositions of quantum distribution functions and optimal hierarchical equations of motion construction for quantum open systems. J Chem Phys 2011; 134:244106. [PMID: 21721611 DOI: 10.1063/1.3602466] [Citation(s) in RCA: 150] [Impact Index Per Article: 11.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Padé spectrum decomposition is an optimal sum-over-poles expansion scheme of Fermi function and Bose function [J. Hu, R. X. Xu, and Y. J. Yan, J. Chem. Phys. 133, 101106 (2010)]. In this work, we report two additional members to this family, from which the best among all sum-over-poles methods could be chosen for different cases of application. Methods are developed for determining these three Padé spectrum decomposition expansions at machine precision via simple algorithms. We exemplify the applications of present development with optimal construction of hierarchical equations-of-motion formulations for nonperturbative quantum dissipation and quantum transport dynamics. Numerical demonstrations are given for two systems. One is the transient transport current to an interacting quantum-dots system, together with the involved high-order co-tunneling dynamics. Another is the non-Markovian dynamics of a spin-boson system.
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Affiliation(s)
- Jie Hu
- Department of Chemistry, Hong Kong University of Science and Technology, Kowloon, Hong Kong
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163
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Xu J, Xu RX, Abramavicius D, Zhang HD, Yan YJ. Advancing Hierarchical Equations of Motion for Efficient Evaluation of Coherent Two-dimensional Spectroscopy. CHINESE J CHEM PHYS 2011. [DOI: 10.1088/1674-0068/24/05/497-506] [Citation(s) in RCA: 25] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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164
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Zhu KB, Xu RX, Zhang HY, Hu J, Yan YJ. Hierarchical Dynamics of Correlated System−Environment Coherence and Optical Spectroscopy. J Phys Chem B 2011; 115:5678-84. [DOI: 10.1021/jp2002244] [Citation(s) in RCA: 32] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- Kun-Bo Zhu
- State Key Laboratory of Supramolecular Structure and Materials, Jilin University, Changchun 130012, China
- Department of Chemistry, Hong Kong University of Science and Technology, Kowloon, Hong Kong SAR, China
| | - Rui-Xue Xu
- Department of Chemistry, Hong Kong University of Science and Technology, Kowloon, Hong Kong SAR, China
- Hefei National Laboratory for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Hou Yu Zhang
- State Key Laboratory of Supramolecular Structure and Materials, Jilin University, Changchun 130012, China
- Department of Chemistry, Hong Kong University of Science and Technology, Kowloon, Hong Kong SAR, China
| | - Jie Hu
- Department of Chemistry, Hong Kong University of Science and Technology, Kowloon, Hong Kong SAR, China
| | - Yi Jing Yan
- State Key Laboratory of Supramolecular Structure and Materials, Jilin University, Changchun 130012, China
- Department of Chemistry, Hong Kong University of Science and Technology, Kowloon, Hong Kong SAR, China
- Hefei National Laboratory for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei, Anhui 230026, China
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165
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Hu J, Xu RX, Yan Y. Communication: Padé spectrum decomposition of Fermi function and Bose function. J Chem Phys 2011; 133:101106. [PMID: 20849157 DOI: 10.1063/1.3484491] [Citation(s) in RCA: 157] [Impact Index Per Article: 12.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Padé approximant is exploited for an efficient sum-over-poles decomposition of Fermi and Bose functions. The resulting poles are all pure imaginary and can therefore be used to define Padé frequencies, in analogy with the celebrated Matsubara frequencies. The proposed Padé spectrum decomposition is shown to be equivalent to a truncated continued fraction. It converges significantly faster than other schemes such as the Matsubara expansion at all temperatures. By introducing the characteristic validity length as the measure of approximant, we analyze the convergence properties of different schemes thoroughly. Our results qualify the present scheme the best among all sum-over-poles approaches. Thus, it is of great value in efficient numerical evaluations of integrals involving Fermi/Bose function in various condensed-phase matter problems.
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Affiliation(s)
- Jie Hu
- Department of Chemistry, Hong Kong University of Science and Technology, Kowloon, Hong Kong SAR, China
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166
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167
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Zheng X, Chen G, Mo Y, Koo S, Tian H, Yam C, Yan Y. Time-dependent density functional theory for quantum transport. J Chem Phys 2010; 133:114101. [DOI: 10.1063/1.3475566] [Citation(s) in RCA: 87] [Impact Index Per Article: 6.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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168
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Tian BL, Ding JJ, Xu RX, Yan Y. Biexponential theory of Drude dissipation via hierarchical quantum master equation. J Chem Phys 2010; 133:114112. [DOI: 10.1063/1.3491270] [Citation(s) in RCA: 28] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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169
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Xu J, Xu RX, Luo M, Yan Y. Hierarchical theory of quantum dissipation: Partial fraction decomposition scheme. Chem Phys 2010. [DOI: 10.1016/j.chemphys.2009.12.002] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
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170
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Hughes KH, Christ CD, Burghardt I. Effective-mode representation of non-Markovian dynamics: a hierarchical approximation of the spectral density. II. Application to environment-induced nonadiabatic dynamics. J Chem Phys 2009; 131:124108. [PMID: 19791853 DOI: 10.1063/1.3226343] [Citation(s) in RCA: 78] [Impact Index Per Article: 5.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/18/2022] Open
Abstract
The non-Markovian approach developed in the companion paper [Hughes et al., J. Chem. Phys. 131, 024109 (2009)], which employs a hierarchical series of approximate spectral densities, is extended to the treatment of nonadiabatic dynamics of coupled electronic states. We focus on a spin-boson-type Hamiltonian including a subset of system vibrational modes which are treated without any approximation, while a set of bath modes is transformed to a chain of effective modes and treated in a reduced-dimensional space. Only the first member of the chain is coupled to the electronic subsystem. The chain construction can be truncated at successive orders and is terminated by a Markovian closure acting on the end of the chain. From this Mori-type construction, a hierarchy of approximate spectral densities is obtained which approach the true bath spectral density with increasing accuracy. Applications are presented for the dynamics of a vibronic subsystem comprising a high-frequency mode and interacting with a low-frequency bath. The bath is shown to have a striking effect on the nonadiabatic dynamics, which can be rationalized in the effective-mode picture. A reduced two-dimensional subspace is constructed which accounts for the essential features of the nonadiabatic process induced by the effective environmental mode. Electronic coherence is found to be preserved on the shortest time scale determined by the effective mode, while decoherence sets in on a longer time scale. Numerical simulations are carried out using either an explicit wave function representation of the system and overall bath or else an explicit representation of the system and effective-mode part in conjunction with a Caldeira-Leggett master equation.
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Affiliation(s)
- Keith H Hughes
- School of Chemistry, University of Wales Bangor, Bangor, Gwynedd LL57 2UW, United Kingdom.
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171
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Xu RX, Tian BL, Xu J, Shi Q, Yan Y. Hierarchical quantum master equation with semiclassical Drude dissipation. J Chem Phys 2009; 131:214111. [DOI: 10.1063/1.3268922] [Citation(s) in RCA: 32] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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172
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Hughes KH, Christ CD, Burghardt I. Effective-mode representation of non-Markovian dynamics: a hierarchical approximation of the spectral density. I. Application to single surface dynamics. J Chem Phys 2009; 131:024109. [PMID: 19603972 DOI: 10.1063/1.3159671] [Citation(s) in RCA: 65] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/12/2022] Open
Abstract
An approach to non-Markovian system-environment dynamics is described which is based on the construction of a hierarchy of coupled effective environmental modes that is terminated by coupling the final member of the hierarchy to a Markovian bath. For an arbitrary environment, which is linearly coupled to the subsystem, the discretized spectral density is replaced by a series of approximate spectral densities involving an increasing number of effective modes. This series of approximants, which are constructed analytically in this paper, guarantees the accurate representation of the overall system-plus-bath dynamics up to increasing times. The hierarchical structure is manifested in the approximate spectral densities in the form of the imaginary part of a continued fraction similar to Mori theory. The results are described for cases where the hierarchy is truncated at the first-, second-, and third-order level. It is demonstrated that the results generated from a reduced density matrix equation of motion and large dimensional system-plus-bath wavepacket calculations are in excellent agreement. For the reduced density matrix calculations, the system and hierarchy of effective modes are treated explicitly and the effects of the bath on the final member of the hierarchy are described by the Caldeira-Leggett equation and its generalization to zero temperature.
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Affiliation(s)
- Keith H Hughes
- School of Chemistry, Bangor University, Bangor, Gwynedd LL57 2UW, United Kingdom.
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173
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Mo Y, Zheng X, Chen G, Yan Y. Transient electronic dynamics of noninteracting open systems beyond linear response. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2009; 21:355301. [PMID: 21828633 DOI: 10.1088/0953-8984/21/35/355301] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/31/2023]
Abstract
We present analytical results on the transient current response of noninteracting open electronic systems under time-dependent external voltages in both linear- and nonlinear-response regimes. The derivations are based on an equation of motion formalism for the system reduced single-electron density matrix (Zheng et al 2007 Phys. Rev. B 75 195127). Dissipative interactions between the system and leads are treated by the nonequilibrium Green's function approach. The linear-response dynamics is characterized by the analytical admittance spectrum of the open system, through which the quantum coherent transport properties are mapped to equivalent classical circuits. The nonlinear-response current spectrum not only resolves the intrinsic energetic configuration of the system, but also reflects the unique dynamical features due to the transient characteristics of the applied voltages.
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Affiliation(s)
- Yan Mo
- Department of Chemistry, Hong Kong University of Science and Technology, Kowloon, Hong Kong. Department of Chemistry, The University of Hong Kong, Hong Kong
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174
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Shi Q, Chen L, Nan G, Xu R, Yan Y. Electron transfer dynamics: Zusman equation versus exact theory. J Chem Phys 2009; 130:164518. [DOI: 10.1063/1.3125003] [Citation(s) in RCA: 68] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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175
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Zheng X, Jin J, Welack S, Luo M, Yan Y. Numerical approach to time-dependent quantum transport and dynamical Kondo transition. J Chem Phys 2009; 130:164708. [DOI: 10.1063/1.3123526] [Citation(s) in RCA: 94] [Impact Index Per Article: 6.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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176
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Nan G, Shi Q, Shuai Z. Nonperturbative time-convolutionless quantum master equation from the path integral approach. J Chem Phys 2009; 130:134106. [DOI: 10.1063/1.3108521] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/14/2022] Open
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177
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Zheng X, Luo J, Jin J, Yan Y. Complex non-Markovian effect on time-dependent quantum transport. J Chem Phys 2009; 130:124508. [DOI: 10.1063/1.3095424] [Citation(s) in RCA: 42] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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178
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Shi Q, Chen L, Nan G, Xu RX, Yan Y. Efficient hierarchical Liouville space propagator to quantum dissipative dynamics. J Chem Phys 2009; 130:084105. [DOI: 10.1063/1.3077918] [Citation(s) in RCA: 190] [Impact Index Per Article: 12.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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179
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Xu RX, Tian BL, Xu J, Yan Y. Exact dynamics of driven Brownian oscillators. J Chem Phys 2009; 130:074107. [DOI: 10.1063/1.3078024] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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180
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Chen L, Zheng R, Shi Q, Yan Y. Optical line shapes of molecular aggregates: Hierarchical equations of motion method. J Chem Phys 2009; 131:094502. [DOI: 10.1063/1.3213013] [Citation(s) in RCA: 111] [Impact Index Per Article: 7.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/20/2022] Open
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181
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Lee MT, Zhang WM. Non-Markovian suppression of charge qubit decoherence in the quantum point contact measurement. J Chem Phys 2008; 129:224106. [DOI: 10.1063/1.3036114] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/27/2022] Open
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182
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Zheng X, Jin J, Yan Y. Dynamic electronic response of a quantum dot driven by time-dependent voltage. J Chem Phys 2008; 129:184112. [DOI: 10.1063/1.3010886] [Citation(s) in RCA: 59] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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