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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 Lindblad equation ： ウィキペディア英語版 Lindblad equation In quantum mechanics, Kossakowski–Lindblad equation (after Andrzej Kossakowski and Göran Lindblad) or master equation in ''Lindblad form'' is the most general type of Markovian and time-homogeneous master equation describing non-unitary evolution of the density matrix that is trace-preserving and completely positive for any initial condition. Lindblad master equation for an -dimensional system's reduced density matrix can be written: :$\backslash dot\backslash rho=-()+\backslash sum\_^\; h\_\backslash left(L\_n\backslash rho\; L\_m^\backslash dagger-\backslash frac\backslash left(\backslash rho\; L\_m^\backslash dagger\; L\_n\; +\; L\_m^\backslash dagger\; L\_n\backslash rho\backslash right)\backslash right)$ where is a (Hermitian) Hamiltonian part, the are an arbitrary linear basis of the operators on the system's Hilbert space, and the are constants which determine the dynamics. The coefficient matrix must be positive to ensure that the equation is trace-preserving and completely positive. The summation only runs to because we have taken to be proportional to the identity operator, in which case the summand vanishes. Our convention implies that the are traceless for . The terms in the summation where can be described in terms of the Lindblad superoperator, :$L(C)\backslash rho=C\backslash rho\; C^\backslash dagger\; -\backslash frac\backslash left(C^\backslash dagger\; C\; \backslash rho\; +\backslash rho\; C^\backslash dagger\; C\backslash right).$ If the terms are all zero, then this is quantum Liouville equation (for a closed system), which is the quantum analog of the classical Liouville equation. A related equation describes the time evolution of the expectation values of observables, it is given by the Ehrenfest theorem. Note that is ''not'' necessarily equal to the self-Hamiltonian of the system. It may also incorporate effective unitary dynamics arising from the system-environment interaction. Lindblad equations is also called the following equations for quantum observables: : $\backslash frac\; d\; A\; =\; -\backslash frac\; 1\; (A)\; +\; \backslash frac\; 1\; \backslash sum^\backslash infty\_\; \backslash big(V^\backslash dagger\_k\; (V\_k)\; +\; (A)\; V\_k\; \backslash big),$ where $A$ is a quantum observable. ==Diagonalization== Since the matrix is positive, it can be diagonalized with a unitary transformation : :$u^\backslash dagger\; h\; u\; =\; \backslash begin$ \gamma_1 & 0 & \cdots & 0 \\ 0 & \gamma_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \gamma_ \end where the eigenvalues are non-negative. If we define another orthonormal operator basis :$A\_i\; =\; \backslash sum\_^\; u\_\; L\_j$ we can rewrite Lindblad equation in ''diagonal'' form :$\backslash dot\backslash rho=-()+\backslash sum\_^\; \backslash gamma\_i\backslash left(A\_i\backslash rho\; A\_i^\backslash dagger\; -\backslash frac\; \backslash rho\; A\_i^\backslash dagger\; A\_i\; -\backslash frac\; A\_i^\backslash dagger\; A\_i\; \backslash rho\; \backslash right).$ This equation is invariant under a unitary transformation of Lindblad operators and constants, :$\backslash sqrt\; A\_i\; \backslash to\; \backslash sqrt\; A\_i\text{'}\; =\; \backslash sum\_^\; v\_\; \backslash sqrt\; A\_j\; ,$ and also under the inhomogeneous transformation :$A\_i\; \backslash to\; A\_i\text{'}\; =\; A\_i\; +\; a\_i,$ :$H\; \backslash to\; H\text{'}\; =\; H\; +\; \backslash frac\; \backslash sum\_^\; \backslash gamma\_j\; \backslash left\; (a\_j^$ * A_j - a_j A_J^\dagger \right ). However, the first transformation destroys the orthonormality of the operators (unless all the are equal) and the second transformation destroys the tracelessness. Therefore, up to degeneracies among the , the of the diagonal form of the Lindblad equation are uniquely determined by the dynamics so long as we require them to be orthonormal and traceless. 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「Lindblad 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.