Ohmic spectral density I am witting a paper about the non-Markovian effects of open quantum systems (a qubit interacting with a bosonic environment). I am using a spectral density of the form below:
$$
J(\omega) = \frac{\omega^S}{\omega_C^{S-1}}e^{-\omega/\omega_C}
$$
I want to know what is the physical interpretation of the $S$ quantity? I just know that $S=1$ means ohmic environment, $S>1$ super ohmic and $S<1$ subohmic, but I would like more interpretation.
and also what is the best amount for this quantity?
for example in my work, I reached $S=20$? is this a logical result?
I would really appreciate if you could help me with your answer or send me useful links.
 A: I like to think of the spectral density as a filter for the bosonic field frequencies, it tells you "how much" of each frequency there is. In this case, if $S=1$ you have a linear increase for low frequencies ($\omega<\omega_C$) and an exponential decay for $\omega>\omega_C$. If you put $S=2$ the increase for low $\omega$ is parabolic instead of linear. As you can see in the plot ($\omega_C=1$), subohmic spectral densities (blueish) are kind of "low-pass" filters, meaning high frequencies are heavily damped. Superohmic are more like "band-pass" filters: they damp low and high frequencies but let through those in between. Both peak at $\omega=\omega_C S$.

There is no "best amount" it depends on the envirorment you are modelling, maybe there is an envirorment which is well modelled for $S=20$, it would be one which damps heavily all frequencies below $\omega_C$ whith a sharp high-pass cut-off at $\omega=\omega_C$ then peaking at $\omega=20\omega_C$ and finally decaying exponentialy (in the plot you cant see it because its off-range). I don't know if there are envirorments whith this kind of behaviour.
I think the main reference here is the book Quantum dissipative systems by Ulrich Weiss, it covers spectral densities extensively.
Related:

*

*https://www.physik.uni-augsburg.de/theo1/hanggi/Chapter_4.pdf (Chapter 4.2.3)


*https://www.researchgate.net/publication/225726556_Quantum_dissipative_systems (equation (147))


*http://www.rmki.kfki.hu/~diosi/tutorial/ohmmarkovtutor.pdf


*spectral functions


*What information is contained in the quantum spectral density?


*What is the relationship between the Drude form and the exponential form of Ohmic spectral density?


*Pure dephasing $\gamma_\phi$ in a master equation and noise power spectral densities


*http://www.scholarpedia.org/article/Caldeira-Leggett_model
