In quantum mechanics, the Gorini–Kossakowski–Sudarshan–Lindblad equation (GKSL equation, named after Vittorio Gorini, Andrzej Kossakowski, George Sudarshan and Göran Lindblad), master equation in Lindblad form, quantum Liouvillian, or Lindbladian is one of the general forms of Markovian master equations describing open quantum systems. It generalizes the Schrödinger equation to open quantum systems; that is, systems in contacts with their surroundings. The resulting dynamics is no longer unitary, but still satisfies the property of being trace-preserving and completely positive for any initial condition.^[1]

The Schrödinger equation or, actually, the von Neumann equation, is a special case of the GKSL equation, which has led to some speculation that quantum mechanics may be productively extended and expanded through further application and analysis of the Lindblad equation.^[2] The Schrödinger equation deals with state vectors, which can only describe pure quantum states and are thus less general than density matrices, which can describe mixed states as well.

Motivation

In the canonical formulation of quantum mechanics, a system's time evolution is governed by unitary dynamics. This implies that there is no decay and 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, causing decay and randomization of phase. More so, understanding the interaction of a quantum system with its environment is necessary for understanding many commonly observed phenomena like the spontaneous emission of light from excited 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.

Definition

The Lindblad master equation for system's density matrix ρ can be written as^[1] (for a pedagogical introduction you may refer to^[3])

${\displaystyle {\dot {\rho }}=-{i \over \hbar }+\sum _{i}^{}\gamma _{i}\left(L_{i}\rho L_{i}^{\dagger }-{\frac {1}{2}}\left\{L_{i}^{\dagger }L_{i},\rho \right\}\right)}$

where ${\displaystyle \{a,b\}=ab+ba}$ is the anticommutator, ${\displaystyle H}$ is the system Hamiltonian, describing the unitary aspects of the dynamics, and ${\displaystyle L_{i}}$ are a set of jump operators describing the dissipative part of the dynamics. The shape of the jump operators describes how the environment acts on the system, and must ultimately be determined from microscopic models of the system-environment dynamics. Finally, ${\displaystyle \gamma _{i}\geq 0}$ are a set of non-negative coefficients called damping rates. If all ${\displaystyle \gamma _{i}=0}$ one recovers the von Neumann equation ${\displaystyle {\dot {\rho }}=-(i/\hbar )}$ describing unitary dynamics, which is the quantum analog of the classical Liouville equation.

More generally, the GKSL equation has the form

${\displaystyle {\dot {\rho }}=-{i \over \hbar }+\sum _{n,m}h_{nm}\left(A_{n}\rho A_{m}^{\dagger }-{\frac {1}{2}}\left\{A_{m}^{\dagger }A_{n},\rho \right\}\right)}$

where ${\displaystyle \{A_{m}\}}$ are arbitrary operators and h is a positive semidefinite matrix. The latter is a strict requirement to ensure the dynamics is trace-preserving and completely positive. The number of ${\displaystyle A_{m}}$ operators is arbitrary, and they do not have to satisfy any special properties. But if the system is ${\displaystyle N}$-dimensional, it can be shown^[1] that the master equation can be fully described by a set of ${\displaystyle N^{2}-1}$ operators, provided they form a basis for the space of operators.

Since the matrix h is positive semidefinite, it can be diagonalized with a unitary transformation u:

${\displaystyle u^{\dagger }hu={\begin{bmatrix}\gamma _{1}&0&\cdots &0\\0&\gamma _{2}&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &\gamma _{N^{2}-1}\end{bmatrix}}}$

where the eigenvalues γ_i are non-negative. If we define another orthonormal operator basis

${\displaystyle L_{i}=\sum _{j}u_{ji}A_{j}}$

This reduces the master equation to the same form as before:

${\displaystyle {\dot {\rho }}=-{i \over \hbar }+\sum _{i}^{}\gamma _{i}\left(L_{i}\rho L_{i}^{\dagger }-{\frac {1}{2}}\left\{L_{i}^{\dagger }L_{i},\rho \right\}\right)}$

Quantum dynamical semigroup

The maps generated by a Lindbladian for various times are collectively referred to as a quantum dynamical semigroup—a family of quantum dynamical maps ${\displaystyle \phi _{t}}$ on the space of density matrices indexed by a single time parameter ${\displaystyle t\geq 0}$ that obey the semigroup property

${\displaystyle \phi _{s}(\phi _{t}(\rho ))=\phi _{t+s}(\rho ),\qquad t,s\geq 0.}$

The Lindblad equation can be obtained by

${\displaystyle {\mathcal {L}}(\rho )=\mathrm {lim} _{\Delta t\to 0}{\frac {\phi _{\Delta t}(\rho )-\phi _{0}(\rho )}{\Delta t}}}$

Zdroj:https://en.wikipedia.org?pojem=Lindbladian
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