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Science Technology and Engineering

Strict Numerical Solution of Spin-boson Model of Single Model

Ying-Hua Huang

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

A new complete basis set is adopted to solve the single-mode spin-boson model through exact numerical diagonalization.The results show that the use of this basis can significantly simplify the calculation and avoid computing the complicated special functions in the original treatment.Moreover,The evolutions of the corresponding energy levels have been calculated,as well as the tunneling splitting with the environment temperature.

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The temperature influence on quantum tunneling in the spin-boson model

Zhide Chen, Ying-Hua Huang, Wong, Hang

The existing studies of the temperature effect of the spin-boson model show that the tunneling splitting will increase with temperature, no matter how weak the couplings are between the bath and the spin. However, the small polaron theory said that in the weak coupling and low temperature regime, the tunneling is, in fact, dominated by the diagonal transitions whereas this diagonal contribution to the tunneling should be faded away with increasing temperature. Taking advantage of the analysis originated from the Feynman-Vernon's influence functional theory, the influence on the tunneling by the phonon bath can be expressed as a product of the effects of every single phonon mode, which can be studied by numerical exact diagonalization. We find that, in the weak coupling and low temperature regime, all the spin-single-mode systems exhibit the same decreasing dependence of tunneling on increasing temperature. In turn, with the conclusion of influence functional theory, the tunneling of the spin-boson model should decrease and it is independent of the bath structure. In the strong coupling regime, however, the temperature effect reverses from suppressing to enhancing the tunneling with the increase of temperature. Discrepancies between the old theories and the small polaron theory are also explained.

arXiv:0812.3722•Dec 19, 2008•0 citations
A new complete basis set is adopted to solve the single-mode spin-boson model through exact numerical diagonalization.The results show that the use of this basis can significantly simplify the calculation and avoid computing the complicated special functions in the original treatment.Moreover,The evolutions of the corresponding energy levels have been calculated,as well as the tunneling splitting with the environment temperature.
An efficient and stable numerical scheme based on Hill’s determinant is proposed for computing the whole spectrum of the single-mode spin-boson model. Its shown that for the low-lying levels a simple version of Hill’s determinant method works well, while for high levels a normalization of the determinant is necessary and this normalization results in an entire function whose zeros correspond to the energy levels. It is very convenient to use this entire function to compute all the energy levels numerically. Some high levels of the model are computed and compared with Irish’s general rotating-wave approximation. A scaling law in the large photon number limit is numerically tested and several statistical properties of the spectrum are investigated.
The steady-state evolution of single-phonon spin-boson model is observed through the method of MATLAB numerical calculation,and analysed several evolutionary patterns while the coupling strength range from 0 to 2 as well as tunneling splitting equals 0.1.The result shows that the coupling strength has exerted a significant influence on states evolution,and there is an exclusive evolution discipline in a certain coupling strength interval,especially when the coupling strength is greater than 0.5 two-cycle phenomenon emerges.The average probability of each energy state in the time interval changes with the coupling strength is further calculated,and the result gives several evolution turning points.
The existing studies of the temperature effect of the spin-boson model show that the tunneling splitting will increase with temperature, no matter how weak the couplings are between the bath and the spin. However, the small polaron theory said that in the weak coupling and low temperature regime, the tunneling is, in fact, dominated by the diagonal transitions whereas this diagonal contribution to the tunneling should be faded away with increasing temperature. Taking advantage of the analysis originated from the Feynman-Vernon's influence functional theory, the influence on the tunneling by the phonon bath can be expressed as a product of the effects of every single phonon mode, which can be studied by numerical exact diagonalization. We find that, in the weak coupling and low temperature regime, all the spin-single-mode systems exhibit the same decreasing dependence of tunneling on increasing temperature. In turn, with the conclusion of influence functional theory, the tunneling of the spin-boson model should decrease and it is independent of the bath structure. In the strong coupling regime, however, the temperature effect reverses from suppressing to enhancing the tunneling with the increase of temperature. Discrepancies between the old theories and the small polaron theory are also explained.