Open Quantum Systems

\[\mathcal{K}(t) \rho=-i[H, \rho]+\sum_i \gamma_i\left[A_i \rho A_i^{\dagger}-\frac{1}{2}\left\{A_i^{\dagger} A_i, \rho\right\}\right] \]

with \(H(t)\) the Hermitian Hamiltonian for the open quantum system without the couplings to the bath. \(\{\cdot, \cdot\}\) denotes the anti-commutator. If all \(\gamma_i\) and \(A_i\) are time independent, and all \(\gamma_i\) are positive, equation (2) is the conventional master equation of the Lindblad form, which describes the conventional Markovian process.