<PERSON> approximation at low temperatures
I'm interested in a two-level system in contact with a reservoir of harmonic oscilators, described by the following Hamiltonian: $$ \hat{H}=\hat{H}S+\hat{H}_R+\hat{H}{RS}=\hbar\omega_0\hat{a}^\dagger \hat{a}+\sum_j\hbar\omega_j\hat{r}_j^\dagger\hat{r}_j+\sum_j\hbar\left(\kappa_j^*\hat{a}\hat{r}_j^\dagger+\kappa_j\hat{a}^\dagger\hat{r}_j\right), $$ so that if $\hat{R}_0$ is the reservoir density operator $$ \hat{R}_0=\prod_j e^{\hbar\omega_j\hat{r}_j^\dagger\hat{r}_j/k_BT}\left(1-e^{-\hbar\omega_j/k_BT}\right) $$ one can find a correlation function for the reservoir given by $$ C_R(\tau)=\sum_j|\kappa_j|^2e^{i\omega_j\tau}\bar{n}(\omega_j,T), $$ with $$ \bar{n}(\omega_j,T)=\frac{1}{e^{\hbar\omega_j/k_BT}-1}. $$
Now, if the reservoir is made of light, by using the density of states for a QED cavity, one can find that the correlation function is $$ C_R(\tau)\propto\int_0^\infty d\omega \omega^3 e^{i\omega\tau}\bar{n}(\omega,T)\propto\zeta_H\left(4,1-\frac{i\tau}{\beta}\right). $$
In the bove expression, $\zeta_H(a,x)$ is the Hurwitz-zeta function, $\beta=1/T$ and I used natural units, i.e., $\hbar=c=k_B=1$.
Now, by defining a time-scale for the reservoir at $T=300$K $t_{R}=\hbar/k_B(300K)$, the real parto of correlation function (normalized to $C_R(0)$) for $T=300K$, $T=10$K, and $T=1$K looks like the following figures:
As can be noticed, for $T=300$K the decay time of the reservor is $\tau_R^{300K}\sim t_{R}\approx0.25\times10^{-13}$s. 955
On the other hand, for $T=10$K, $\tau_R^{10K}\sim60t_{R}\approx1.5\times10^{-12}$, and $\tau_R^{1K}\approx0.25\times10^{-10}$. The first two decay times are much less than the decay of an atom $t_s\sim 10^{-8}$s, while $t_s\sim10^{2}\tau_R^{1K}$.
My question is: What is the reason to say that <PERSON> approximation is only valid at high temperatures? The above analysis (if is correct) ensures that the decay of the atom is slow compared to the decay for $T=300$K and $T=10$K (the case $T=1$K is hard to justify). Then, in both cases $\hat{\rho}(t-\tau)\approx\hat{\rho}(t)$, because $t_s\gg\tau_R^{10K}>\tau_R^{300K}$.
Is this correct? Or $T=10$K is understood as high temperature?
I will thankful for any comment/correction. 394