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Chinese Physics Letters

Variational Study on a Dissipative Two-Level System

Wei-Ting Zhu, Qing-Bao Ren, Qing‐Hu Chen

Published Sep 1, 2008• 0 citations
Publisher

Abstract

A new variational approach is proposed to study the ground-state of a two-level system coupled to a dispersionless phonon bath. By the extended coherent state, where the more phonon correlations are easily incorporated, we can obtain very accurate ground state energy and the tunnelling reduction factor in all regime of tunnelling matrix element d0 and coupling parameter s. The relative difference between the present ones and those by exact numerical diagonalization is less then 0.001%. In addition, some simple analytical results are given in the limits of δ 0 /s ↠ 0 and ∞.

DOI
10.1088/0256-307x/25/9/010

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A new variational approach is proposed to study the ground-state of a two-level system coupled to a dispersionless phonon bath. By the extended coherent state, where the more phonon correlations are easily incorporated, we can obtain very accurate ground state energy and the tunnelling reduction factor in all regime of tunnelling matrix element d0 and coupling parameter s. The relative difference between the present ones and those by exact numerical diagonalization is less then 0.001%. In addition, some simple analytical results are given in the limits of δ 0 /s ↠ 0 and ∞.
The ground state of a two-level system coupled to a dispersionless phonon bath is studied using the coherent approximation method. The ground-state energy and the tunneling reduction factor we obtain are in good agreement with the exact values in the whole parameter space. Since the real ground-state wave functions are well simulated, some other physical information about the system, such as the interaction energy and the interaction phonon numbers in each level and the whole system, are given. The interaction phonon numbers in the two-level system are found to increase with the coupling parameter s. In the intermediate coupling region, phenomenon of the population inversion and a maximum for the interaction phonon numbers versus ${\ensuremath{\delta}}_{0}/s$ have been found. Consisting with the exact results, our results show there is no evidence of the discontinuous localization-delocalization transition.
The ground state of a two-level system coupled to a dispersionless phonon bath is studied using both a connected moments expansion and a truncated Lanczos tridiagonal scheme. We consider the spin-boson Hamiltonian, $\ifmmode \hat{H}\else \^{H}\fi{}=\ensuremath{-}{\ensuremath{\delta}}_{o}{\ensuremath{\sigma}}_{x}+{\ensuremath{\sum}}_{\mathbf{k}}\ensuremath{\Elzxh}{\ensuremath{\omega}}_{\mathbf{k}}{a}_{\mathbf{k}}^{\ifmmode\dagger\else\textdagger\fi{}}{a}_{\mathbf{k}}+{\ensuremath{\sum}}_{\mathbf{k}}{g}_{\mathbf{k}}{(a}_{\mathbf{k}}^{\ifmmode\dagger\else\textdagger\fi{}}{+a}_{\mathbf{k}}){\ensuremath{\sigma}}_{z},$ where ${\ensuremath{\delta}}_{0}$ is the bare tunneling matrix element and gk{g}_{\mathbf{k}}gk​ represents the coupling to the k\mathbf{k}k phonon modes. Such systems have found relevance in applications to molecular polaron formation, exciton motion, and attenuation of sound in glasses. Our results are then compared to those of variational methods as well as an exact numerical diagonalization.
A variational approach is developed to study the groundstate (GS) of the two-site Holstein model. By the extended coherent state, where the more phonon correlations are easily incorporated, we can get the very accurate ground state energy for all electron-phonon coupling range in typical values of hopping integral t = 0.5,1.1, and 2.1 (in units of phonon frequency ω 0 ), which covers the crossover region from antiadiabatic limit to the adiabatic limit. Within a very wide t range [0,2.7], the exact results for the GS energy are obtained with the twelfth (fourteenth) order corrections to the zeroth order wave function. Moreover, the present approach is more concise than any other analytical ones in this field, and hopefully can be easily generalized to many other Holstein models.
The variational approach is applied to the study of the ground state of quantum tunneling system interacting with two phonons by use of a correlated-squeezed transformation. We fine: 1) For the two phonons bath, the ground state energy of the phonons with correlation effect is lower than that without correlation effect; 2) The effect of correlation varies with the mode parameters: In the resonant regime of the phonons and two-state system, the correlation effect is more significant and will be strengthened together with the increase of coupling coefficient.
A variational approach is developed to study the groundstate (GS) of the two-site Holstein model. By the extended coherent state, where the more phonon correlations are easily incorporated, we can get the very accurate ground state energy for all electron-phonon coupling range in typical values of hopping integral t = 0.5, 1.1, and 2.1 (in units of phonon frequency ωo ), which covers the crossover region from antiadiabatic limit to the adiabatic limit. Within a very wide t range [0, 2.7], the exact results for the GS energy are obtained with the twelfth (fourteenth) order corrections to the zeroth order wave function. Moreover, the present approach is more concise than any other analytical ones in this field, and hopefully can be easily generalized to many other Holstein models.