Open quantum dynamics

Dynamics of open quantum systems requries to solve the quantum master equaiton

\[i \dot{\rho} = (\mathcal{L}_0 + \mathcal{L}_1) \rho\]

Lindblad quantum master equaiton

\[H = \frac{\Delta}{2} \sigma_x + \sum_k \sigma_z {g_k a_k^\dagger + g_k^*a_k} + \sum_k \omega_k a^\dagger_k a_k\]

The environmental influence to the system dynamics is encoded in the so-called spectral density,

\[J(\omega) = \sum_k |{g_k}|^2 \delta(\omega - \omega_k).\]

We here implement the Lorentz-Drude form

\[J(\omega) = \frac{2\lambda \omega \gamma}{\omega^2 + \gamma^2}\]

with a single exponential for the time-correlation function.

\[D(t) = \pi^{-1} \int_0^\infty d \omega J(\omega)(\coth(\beta\omega/2) - i \sin(\omega t)) \approx \lambda (2T - i \gamma) e^{-\gamma t}\]