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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.

Qmechanic
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Zeinab
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1 Answers1

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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 $\omega=\omega_C S$.

enter image description here

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:

kl0z
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  • The second link (https://www.icmm.csic.es/mjcalderon/dissipativeQS.pdf) is broken. The correct link is- https://aip.scitation.org/doi/abs/10.1063/1.56342. details- Quantum Dissipative Systems (F. Guinea, E. Bascones, M.J. Calderon). – Prem Nov 02 '22 at 05:42
  • Thank you! Here is an open version. I've edited the answer accordingly. https://www.researchgate.net/publication/225726556_Quantum_dissipative_systems – kl0z Nov 03 '22 at 14:47