Interaction picture of a phonon-electron coupling

[jorgerp24601]

Homework Statement

I'm working on my undergraduate thesis about nitrogen vacancies and a part of it consists in emulating the results obtained in Appendix A of the paper titled "Coupling a Surface Acoustic Wave to an Electron Spin in Diamond via a Dark State" (in case further detail is needed), and I'm currently stuck in the part where the transformation to the interaction picture occurs.

Relevant Equations

Basically what I need to calculate is the interaction picture of a hamiltonian of the form:
$$H=\hbar \omega_m \hat{b}^+\hat{b}-\hbar\nu|1><1|-\hbar\frac{g^2}{\omega_m}|e><e|+\hbar\frac{\Omega}{2}\left(e^{-i\omega t+\frac{g}{\omega_m}(\hat{b}^+-\hat{b})}|e><1|+H.c. \right)$$

Where b is the bosonic anihilation operator.

What I have tried to do is to separate the exponential of the unitary transformation operator to the interaction picture into three different Hilbert "subspaces" like:
$$e^{i\frac{H_0}{\hbar}t}=e^{i\omega_m \hat{b}^+\hat{b}}\otimes e^{-i\hbar\nu|1><1|} \otimes e^{-i\frac{g}{\omega_m}|e><e|t}$$

Therefore, each one of the operators in the tensor product act only in their corresponden subspace. In that case, the result is:
$$H_I=\hbar\frac{\Omega}{2}e^{i\omega_m \hat{b}^+\hat{b}}\left(e^{i\left(\nu-\frac{g^2}{\omega_m}-\omega\right)}e^{\frac{g}{\omega_m}(\hat{b}^+-\hat{b})}|e><2|+H.c\right) e^{-i\omega_m \hat{b}\hat{b}^+}$$

So, I think the next and final step I have to do is to evaluate:
$$e^{i\omega_m \hat{b}^+\hat{b}t}e^{\frac{g}{\omega_m}(\hat{b}^+-\hat{b})}e^{-i\omega_m \hat{b}\hat{b}^+t}$$

And there is where I'm currently stuck. According to the paper and if my reasoning is right, that must be:
$$e^{i\omega_m \hat{b}^+\hat{b}t}e^{\frac{g}{\omega_m}(\hat{b}^+-\hat{b})}e^{-i\omega_m \hat{b}\hat{b}^+t}=e^{-\frac{g}{\omega_m}\left(\hat{b}^+e^{i\omega_m t}-\hat{b}e^{-i\omega_m t}\right)}$$

However, I have tried expanding the exponentials in a Taylor series and that didn't seem to work or at least it got messy to the point I couldn't follow what I was doing. I don't know if I'm doing wrong or if the negative sign before g/wm in the final result is a mistake.

Thank you so much to whoever is reading this and thank you even more to whoever helps me!

 

[mighty2000]

Thank you for sharing your approach and progress on this problem. It seems like you are on the right track in separating the exponential operator into three different Hilbert subspaces. However, I would suggest double-checking your final result for the interaction Hamiltonian, as the negative sign in front of the term involving g/wm may be a mistake.

As for evaluating the exponential operator, instead of expanding it into a Taylor series, you can try using the Baker-Campbell-Hausdorff formula. This formula can help simplify the exponential of a sum of operators, which may be useful in your case. You can also try using the commutation relations between the operators in the exponential to simplify the expression.

If you are still having trouble, I would suggest consulting with your colleagues or a mentor for further assistance. They may have some helpful insights or suggestions on how to proceed. Additionally, you can also try looking for similar problems or examples in textbooks or online resources to get some inspiration.

I hope this helps and good luck with your research!