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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Redfield equation ： ウィキペディア英語版 Redfield equation In quantum mechanics, the Redfield equation is a Markovian master equation that describes the time evolution of the density matrix of a quantum system that is weakly coupled to an environment. There is a close connection to the Lindblad master equation. If a so-called secular approximation is done, where only certain resonant interactions with the environment are retained, every Redfield equation transforms into a master equation of Lindblad type. Redfield equations are trace-preserving and correctly produce a thermalized state for asymptotic propagation. However, in contrast to Lindblad equations, Redfield equations do not guarantee a positive time evolution of the density matrix. That is, it is possible to get negative populations during the time evolution. The Redfield equation approaches the correct dynamics for sufficiently weak coupling to the environment. The general form of the Redfield equation is :$\backslash frac\; \backslash rho(t)\; =\; -\backslash frac\; (\backslash rho(t))\; -\backslash frac\; \backslash sum\_m\; (S\_m,\; \backslash Lambda\_m\; \backslash rho(t)\; -\; \backslash rho(t)\; \backslash Lambda\_m^\backslash dagger)$ where $H$ is the Hermitian Hamiltonian, and the $S\_m,\; \backslash Lambda\_m$ are operators that describe the coupling to the environment. Their explicit form is given in the derivation below. ==Derivation== Let us consider a quantum system coupled to an environment with a total Hamiltonian of $H\_\backslash text\; =\; H\; +\; H\_\backslash text\; +\; H\_\backslash text$. Furthermore, we assume that the interaction Hamiltonian can be written as $H\_\backslash text\; =\; \backslash sum\_n\; S\_n\; E\_n$, where the $S\_n$ act only on the system degrees of freedom, the $E\_n$ only on the environment degrees of freedom. The starting point of Redfield theory is the Nakajima–Zwanzig equation with $\backslash mathcal$ projecting on the equilibrium density operator of the environment and $\backslash mathcal$ treated up to second order.〔Volkhard May, Oliver Kuehn: ''Charge and Energy Transfer Dynamics in Molecular Systems.'' Wiley-VCH, 2000 ISBN〕 An equivalent derivation starts with second-order perturbation theory in the interaction $H\_\backslash text$.〔Heinz-Peter Breuer, Francesco Petruccione: ''Theory of Open Quantum Systems.'' Oxford, 2002 ISBN 〕 In both cases, the resulting equation of motion for the density operator in the interaction picture (with $H\_\; =\; H\; +\; H\_\backslash text$) is :$\backslash frac\; \backslash rho\_I(t)\; =\; -\backslash frac\; \backslash sum\_\; \backslash int\_^t\; dt\text{'}\; \backslash biggl(C\_(t-t\text{'})\; \backslash Bigl(S\_(t\text{'})\; \backslash rho\_I(t\text{'})\backslash Bigr)\; -\; C\_^\backslash ast(t-t\text{'})\; \backslash Bigl(\backslash rho\_I(t\text{'})\; S\_(t\text{'})\backslash Bigr)\backslash biggr)$ Here, $t\_0$ is some initial time, where the total state of the system and bath is assumed to be factorized, and we have introduced the bath correlation function $C\_(t)\; =\; \backslash text(E\_(t)\; E\_\; \backslash rho\_\backslash text)$ in terms of the density operator of the environment in thermal equilibrium, $\backslash rho\_\backslash text$. This equation is non-local in time: To get the derivative of the reduced density operator at time t, we need its values at all past times. As such, it cannot be easily solved. To construct an approximate solution, note that there are two time scales: a typical relaxation time $\backslash tau\_r$ that gives the time scale on which the environment affects the system time evolution, and the coherence time of the environment, $\backslash tau\_c$ that gives the typical time scale on which the correlation functions decay. If the relation :$\backslash tau\_c\; \backslash ll\; \backslash tau\_r$ holds, then the integrand becomes approximately zero before the interaction-picture density operator changes significantly. In this case, the so-called Markov approximation $\backslash rho\_I(t\text{'})\; \backslash approx\; \backslash rho\_I(t)$ holds. If we also move $t\_0\; \backslash to\; -\backslash infty$ and change the integration variable $t\text{'}\; \backslash to\; \backslash tau\; =\; t\; -\; t\text{'}$, we end up with the Redfield master equation :$\backslash frac\; \backslash rho\_I(t)\; =\; -\backslash frac\; \backslash sum\_\; \backslash int\_0^\backslash infty\; d\backslash tau\; \backslash biggl(C\_(\backslash tau)\; \backslash Bigl(S\_(t-\backslash tau)\; \backslash rho\_I(t)\backslash Bigr)\; -\; C\_^\backslash ast(\backslash tau)\; \backslash Bigl(\backslash rho\_I(t)\; S\_(t-\backslash tau)\backslash Bigr)\backslash biggr)$ We can simplify this equation considerably if we use the shortcut $\backslash Lambda\_m\; =\; \backslash sum\_n\; \backslash int\_0^\backslash infty\; d\backslash tau\; C\_(\backslash tau)\; S\_(-\backslash tau)$. In the Schrödinger picture, the equation then reads :$\backslash frac\; \backslash rho(t)\; =\; -\backslash frac\; (\backslash rho(t))\; -\backslash frac\; \backslash sum\_m\; (S\_m,\; \backslash Lambda\_m\; \backslash rho(t)\; -\; \backslash rho(t)\; \backslash Lambda\_m^\backslash dagger)$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Redfield equation」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.