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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 superoperator ： ウィキペディア英語版 Lindblad superoperator The Lindblad superoperator is often used to express the quantum master equation for a dissipative system. In the canonical formulation of quantum mechanics, a system's time evolution is governed by unitary dynamics. This implies that phase coherence is maintained throughout the process, and is a consequence of the fact that all participating degrees of freedom are considered. However, any real physical system is not absolutely isolated, and will interact with its environment. This interaction with degrees of freedom external to the system results in dissipation of energy into the surroundings, and randomization of phase. This latter effect is the reason quantum mechanics is difficult to observe on a macroscopic scale. More so, understanding the interaction of a quantum system with its environment is necessary to understanding many commonly observed phenomena like the spontaneous emission of atoms, or the performance of many quantum technological devices, like the laser. Certain mathematical techniques have been introduced to treat the interaction of a quantum system with its environment. One of these is the use of the density matrix, and its associated master equation. While in principle this approach to solving quantum dynamics is equivalent to the Schrödinger picture or Heisenberg picture, it allows more easily for the inclusion of incoherent processes, which represent environmental interactions. The density operator has the property that it can represent a classical mixture of quantum states, and is thus vital to accurately describe the dynamics of so-called open quantum systems. For a collapse operator $C$, the Lindblad superoperator, acting on the density matrix $\backslash rho$, is :$L(C)\backslash rho\; =\; C\backslash rho\; C^\backslash dagger\; -\backslash frac\backslash left(\; C^\backslash dagger\; C\; \backslash rho\; +\; \backslash rho\; C^\backslash dagger\; C\backslash right)$ Such a term is found regularly in the Lindblad equation as used in quantum optics, where it can express absorption or emission of photons from a reservoir. For example, the master equation for a single mode optical resonator (e.g. a Fabry–Perot cavity) coupled to a thermal bath is :$\backslash dot=-i(a^\backslash dagger\; a,\backslash rho)+L(\backslash sqrta)\backslash rho\; +\; L(\backslash sqrt$ is the thermal occupation number of the photons in the bath, as given by the Bose–Einstein distribution. ==Derivation from Liouvillian dynamics== The derivation〔Carmichael, Howard. ''An Open Systems Approach to Quantum Optics''. Springer Verlag, 1991〕 assumes a quantum system with a finite number of degrees of freedom coupled to a bath containing an infinite number of degrees of freedom. The system and bath each possess a Hamiltonian written in terms of operators acting only on the respective subspace of the total Hilbert space. These Hamiltonians govern the internal dynamics of the uncoupled system and bath. There is a third Hamiltonian that contains products of system and bath operators, thus coupling the system and bath. The most general form of this Hamiltonian is :$H=\; H\_S\; +\; H\_B\; +\; H\_\; \backslash ,$ The dynamics of the entire system can be described by the Liouville equation of motion, $\backslash dot=-i()$. This equation, containing an infinite number of degrees of freedom, is impossible to solve analytically except in very particular cases. What's more, under certain approximations, the bath degrees of freedom need not be considered, and an effective master equation can be derived in terms of the system density matrix, $\backslash rho=\backslash operatorname\_B\; \backslash chi$. The problem can be analyzed more easily by moving into the interaction picture, defined by the unitary transformation $\backslash tilde=\; UMU^\backslash dagger$, where $M$ is an arbitrary operator, and $U=e^$. It is straightforward to confirm that the Liouville equation becomes :$\backslash dot\_,\backslash tilde"\; TITLE="\backslash tilde\_,\backslash tilde">)\; \backslash ,$ where the Hamiltonian $\backslash tilde\_=e^\; H\_\; e^$ is explicitly time dependent. This equation can be integrated directly to give :$\backslash tilde(t)=\backslash tilde(0)\; -i\backslash int^t\_0\; dt\text{'}\; ()$ This implicit equation for $\backslash tilde$ can be substituted back into the Liouville equation to obtain an exact differo-integral equation :$\backslash dot\_,\backslash tilde(0)"\; TITLE="\backslash tilde\_,\backslash tilde(0)">)\; -\; \backslash int^t\_0\; dt\text{'}\; ]\backslash \}$ This equation is exact for the time dynamics of the system density matrix but requires full knowledge of the dynamics of the bath degrees of freedom. A simplifying assumption called the Born approximation rests on the largeness of the bath and the relative weakness of the coupling, which is to say the coupling of the system to the bath should not significantly alter the bath eigenstates. In this case the full density matrix is factorable for all times as $\backslash tilde(t)=\backslash tilde(t)R\_0$. The master equation becomes :$\backslash dot\_R\backslash$ This is the final form of the master equation we need. 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 superoperator」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.