What is the spin-boson model for decoherence in a Penning trap?

[Malamala]

Hello! I read a bit about decoherence lately (I made a post few weeks ago about it and got some reading suggestions) and I would like to try to apply it to a practical situation I need, which is a Penning trap with a single ion at the center. For now I would like to account just for the decoherence due to the black body radiation and as far as I understand the spin-boson model would be the right one for this (at least to start with). However that involves certain coupling constant which I am not sure how to approximate for my given setup. Can someone point me towards some reading about this or previous calculations made for a (cylindrical) Penning trap such that I can start from there? Thank you!

 

[Twigg]

Sorry for the slow and short reply.

I'm not 100% sure how to calculate the decoherence rate, but I can tell you that the blackbody radiation shifts the energy difference $E_e - E_g$ by $\delta E = (\alpha_e - \alpha_g) \| \vec{E}_{BBR} \|^2$ where $\alpha_{e,g}$ is the polarizability of the ground (excited) state (at a particular frequency). This is for Stark shifted qubit states. There is an analogous expression for Zeeman states. You can get $\| \vec{E}_{BBR} \|^2$ from Planck's law for blackbody radiation (spectral irradiance) and your trap's geometry.

My gut feeling is that you can get the decoherence rate by taking the variance of this blackbody frequency shift ($\gamma = \sigma_\nu$, same idea as when you calculate the coherence time of a laser from bandwidth). Thus, the decoherence rate would bee $$\gamma = |\alpha_e - \alpha_g| \sqrt{\langle \| \vec{E}_{BBR} \|^4 \rangle - \langle \| \vec{E}_{BBR} \|^2 \rangle^2}$$ The quadratic term can be calculated from the blackbody partition function by looking at the expectation value of energy squared (just as you do when you calculate energy fluctuations in an ideal gas from the heat capacity).

Does that make sense?

 

[Twigg]

Also, I'm not sure which "spin-boson model" you're referring to. Is there a paper you're reading that you can link us to?