be written as

[rCl]{} \[PhiE\_damp\]\_E &=& .

In general, even though both $\Phi$ and $\Phi_E$ only depend on the diagonal entries of $\rho$, they are not directly proportional to each other. However, taking the limit where $T\gg\omega$, we may approximate Eq. (\[eq:phi\_damp\_Q3\]) to $$\label{Phi_Phi_E_high_T}
\Phi \simeq\frac{1}{(1+1/J)}\frac{\Phi_E}{T}.$$ If we then take both $T\gg\omega$ and $J\rightarrow\infty$, we recover the von Neumann result (\[Phi\_vN\]). Thus, in the classical limit, we recover the usual thermodynamic results of the von Neumann framework, which is a key consistency requirement of such a theory.

\[sec:applications\] Applications
=================================

Spontaneous emission
--------------------

We now present several applications of the amplitude damping results, focusing on the case of spin-1/2 particle. We start with the case of spontaneous emission. In Figs. \[fig:spont\_emiss\](a) and \[fig:spont\_emiss\](b), we show the entropy production as a function of time for a system starting in the excited state, for two different temperatures. As can be seen, as the temperature goes down, the von Neumann entropy production gradually starts to diverge, whereas the Wehrl entropy production rate remains well behaved. We also study the total entropy produced in the process, defined as $$\label{Sigma}
\Sigma = \int\limits_0^\infty  \Pi(t){\,\mathrm{d}}t.$$ Results as a function of temperature are shown in Fig. \[fig:spont\_emiss\](c).

