Lindblad equation
= Lindblad equation
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The <Lindblad equation> describes the generator of a time-homogeneous, <completely positive>, trace-preserving quantum dynamical semigroup in a finite-dimensional system. In units $\hbar=1$, the <Hamiltonian> $H$ supplies coherent evolution and the <Lindblad operators> $L_\alpha$ specify dissipative contributions.
A Markovian continuous-time quantum channel obeys
$$
\dot\rho=-i[H,\rho]+\sum_\alpha\left(L_\alpha\rho L_\alpha^\dagger-\frac12\{L_\alpha^\dagger L_\alpha,\rho\}\right).
$$
The <Lindblad dissipator> combines its gain term and two <anticommutator> terms so that the <trace> remains one. For $L=\sqrt{\gamma/2}\,Z$ and $H=0$, diagonal entries in the $Z$ basis stay fixed while an off-diagonal element obeys $\dot\rho_{01}=-\gamma\rho_{01}$: <dephasing> without population decay. The semigroup assumption is a specific model of environmental memory loss. <On the generators of quantum dynamical semigroups by G. Lindblad>[1976, generators of completely positive dynamical semigroups]{ref}