![\[fig:bloch\_dephasing\] Evolution of a spin-$1/2$ particle under a time-dependent magnetic field \[Eq. (\[H\_example\])\] and a dephasing bath \[Eq. (\[D\_dephasing\])\]. Left: trajectory in the Bloch sphere. Right: Wehrl and von Neumann entropy production rates. We assume the system initially starts in the state $|x_-\rangle = (|z_+\rangle - |z_-\rangle)/\sqrt{2}$ (with $\sigma_z|z_{\pm}\rangle=\pm|z_{\pm}\rangle$). The chosen parameters were $b_0/\lambda=5$, $b_1/\lambda=1$, and $\omega/\lambda=1$. ](plot2.pdf){width="100.00000%"}

\[sec:amplitude\_damping\] Amplitude damping
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Dissipator and relevant currents
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Next we consider the amplitude damping dissipator, which we define as $$\begin{aligned}
\label{damp_diss}
D(\rho) &= \gamma(\bar{n}+1) \left[ J_- \rho J_+ - \{ J_+ J_-, \rho\}/2 \right]\\
&+ \gamma \bar{n} \left[ J_+ \rho J_- -  \{ J_- J_+, \rho\}/2\right],
\end{aligned}$$ where $\bar{n}$ is the mean number of excitations in the environment. This dissipator targets the thermal Gibbs state $e^{-\beta H}/\operatorname{tr}[e^{-\beta H}]$ of the Hamiltonian $H = \omega J_z$, provided $\bar{n} = (e^{\beta \omega}-1)^{-1}$. When $T  \to 0 $, this state becomes the “south-pole” state $|J,-J\rangle$ when $\omega >0$, and the “north-pole” state $|J,J\rangle$ when $\omega<0$.

It is convenient to define the superoperator $$f(\rho) = (\bar{n}+1)  \rho J_+ - \bar{n} J_+ \rho$$ with which Eq. (\[damp\_diss\]) can be written as $$\label{amp_D}
D(\rho) = \frac{\gamma}{2} \left\{ [J_-, f(\rho)] - [J_+, f^\dagger(\rho)]\right\}.$$ The super-operator $f(\rho)$ represents a 