An efficient solution of Liouville-von Neumann equation that is applicable to zero and finite temperatures

Tian, Heng; Chen, Guanhua

doi:10.1063/1.4767460

Cited by 32 publications

(32 citation statements)

References 21 publications

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“…30 and similar to the method of Ref. 51, truncating at a certain K ch limits the approach to a simulation time dependent on the truncation order K ch . The uniform Chebyshev decomposition scheme shows an exponential convergence, generating extremely accurate results given the criteria for convergence are fulfilled 36 .…”

Section: Functionsmentioning

confidence: 99%

Chebyshev hierarchical equations of motion for systems with arbitrary spectral densities and temperatures

Rahman

Kleinekathöfer

2019

The Journal of Chemical Physics

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The time evolution in open quantum systems, such as a molecular aggregate in contact with a thermal bath, still poses a complex and challenging problem. The influence of the thermal noise can be treated using a plethora of schemes, several of which decompose the corresponding correlation functions in terms of weighted sums of exponential functions. One such scheme is based on the hierarchical equations of motion (HEOM), which is built using only certain forms of bath correlation functions. In the case where the environment is described by a complex spectral density or is at a very low temperature, approaches utilizing the exponential decomposition become very inefficient. Here we utilize an alternative decomposition scheme for the bath correlation function based on Chebyshev polynomials and Bessel functions to derive a hierarchical equations of motion approach up to an arbitrary order in the environmental coupling. These hierarchical equations are similar in structure to the popular exponential HEOM scheme, but are formulated using the derivatives of the Bessel functions. The proposed scheme is tested up to the fourth order in perturbation theory for a two-level system and compared to benchmark calculations for the case of zero-temperature quantum Ohmic and super-Ohmic noise. Furthermore, the benefits and shortcomings of the present Chebyshev-based hierarchical equations are discussed.

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Section: Functionsmentioning

confidence: 99%

Chebyshev hierarchical equations of motion for systems with arbitrary spectral densities and temperatures

Rahman

Kleinekathöfer

2019

The Journal of Chemical Physics

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“…It should be noted that the density operators for a closed system should satisfy the following requirements throughout the whole evolution [3]: (a) Hermitian, so that all the probabilities are real, (b) trace-preserving, because the sum of the probabilities over any complete set must be one, (c) positive, otherwise some probabilities might be negative. Different methods such as Liouville-von Neumann equation have been developed for the solution of the spin density operators [2][3][4][5][6][7]. In the special case of two-level systems, the time evolution of the density matrix elements can be described by the classical Bloch equation [8].…”

Section: Introductionmentioning

confidence: 99%

A Novel Approximate Numerical Method for Solving the Evolution of the Spin Density Operators

Hatefi

Sarreshtedari

Sabooni

2020

IJOP

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An approximate numerical method is proposed and discussed for solving the evolution of the spin density operator when the quantum system has an interaction with an external electromagnetic field. In this method by separating the relaxation and field interaction processes at small steps, instead of solving the conventional Liouville-von Neumann or Bloch differential equations, the time evolution of the density operator is efficiently obtained by a two-stage numerical algorithm. Here we have compared the results of this approach with Bloch equation results for a twolevel quantum system. The proposed approach has potential applications in calculation of the time evolution for different atomic system including nuclear or electron spin resonance systems.

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“…10,22 Beyond the WBL, the RSDM based HEOM have been applied to simulate time-dependent quantum transport in tightbinding model system. 23,24 However, in the derivation of RSDM based HEOM, a localized orthonormal basis set is assumed. In practical first principles calculation, atomic orbital basis, which are the eigenfunctions of the single-electron Hamiltonian of individual atoms, are commonly used due to their localized nature and clear chemical meaning.…”

Section: Introductionmentioning

confidence: 99%

Time-dependent density functional theory quantum transport simulation in non-orthogonal basis

Kwok

Xie

Yam

et al. 2013

The Journal of Chemical Physics

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Real-time, local basis-set implementation of time-dependent density functional theory for excited state dynamics simulations A model study of quantum dot polarizability calculations using time-dependent density functional methods Basing on the earlier works on the hierarchical equations of motion for quantum transport, we present in this paper a first principles scheme for time-dependent quantum transport by combining timedependent density functional theory (TDDFT) and Keldysh's non-equilibrium Green's function formalism. This scheme is beyond the wide band limit approximation and is directly applicable to the case of non-orthogonal basis without the need of basis transformation. The overlap between the basis in the lead and the device region is treated properly by including it in the self-energy and it can be shown that this approach is equivalent to a lead-device orthogonalization. This scheme has been implemented at both TDDFT and density functional tight-binding level. Simulation results are presented to demonstrate our method and comparison with wide band limit approximation is made. Finally, the sparsity of the matrices and computational complexity of this method are analyzed.

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An efficient solution of Liouville-von Neumann equation that is applicable to zero and finite temperatures

Cited by 32 publications

References 21 publications

Chebyshev hierarchical equations of motion for systems with arbitrary spectral densities and temperatures

Chebyshev hierarchical equations of motion for systems with arbitrary spectral densities and temperatures

A Novel Approximate Numerical Method for Solving the Evolution of the Spin Density Operators

Time-dependent density functional theory quantum transport simulation in non-orthogonal basis

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