Understand Polaron Transformation in Quantum Optics: Wilson-Rae, Imamoğlu

[Cedric Chia]

TL;DR Summary

Help me understand the "polaron transformation" frequently encountered in quantum optics

I'm trying to understand the so-called polaron transformation as frequently encountered in quantum optics. Take the following paper as example: "Quantum dot cavity-QED in the presence of strong electron-phonon interactions" by I. Wilson-Rae and A. Imamoğlu. We have the spin-phonon model with cavity desribed by the following Hamiltonian
$$
H=\hbar\omega_{eg}\sigma_{ee}+\hbar\omega_ca^{\dagger}a+\hbar g(\sigma_{eg}a+a^{\dagger}\sigma_{ge})+\sum_k\hbar\omega_kb_k^{\dagger}b_k+\sigma_{ee}\sum_k\hbar\lambda_k(b_k+b_k^{\dagger})+\hbar\Omega_p(\sigma_{eg}e^{-i\omega t}+\sigma_{ge}e^{i\omega t})
$$

The polaron transformation is defined as
$$
H'=e^sHe^{-s}
$$
where $s=\sigma_{ee}\sum_k\frac{\lambda_k}{\omega_k}(b_k^{\dagger}-b_k)$. I have no difficulty transforming $H$ into $H'$ using the results I found
$$
e^s\sigma_{ge}e^{-s}=\sigma_{ge}\exp{\sum_k}\frac{\lambda_k}{\nu_k}(b_k^{\dagger}-b_k)
$$
and
$$
e^sb_k e^{-s}=b_k-\frac{\lambda_k}{\omega_k}\sigma_{ee}
$$
But I just don't know how to get the following transformed Hamiltonian in terms of $\left<B\right>$, $B_+$, $B_-$ and also $X_g$ and $X_u$:
$$
H^{\prime}=H_{s y s}^{\prime}+H_{i n t}^{\prime}+H_{b a t h}^{\prime}
$$
with
$$
\begin{align}
H_{b a t h}^{\prime}&=\sum_{k} \omega_{k} b_{k}^{\dagger} b_{k}\\
H_{s y s}^{\prime}&=\hbar \omega \sigma_{00}+\hbar \omega_{c} \sigma_{11}+\hbar\left(\omega_{e g}-\Delta\right) \sigma_{22}+\langle B\rangle X_{g}\\
H_{i n t}^{\prime}&=X_{g} \xi_{g}+X_{u} \xi_{u}
\end{align}
$$
with the definition of $\left<B\right>$, $B_+$, $B_-$, $X_g$ and $X_u$ defined as follows:
\begin{align}
X_{g}&=\hbar\left[g\left(\sigma_{21}+\sigma_{12}\right)+\Omega_{p}\left(\sigma_{20}+\sigma_{02}\right)\right]\\
X_{u}&=i \hbar\left[g\left(\sigma_{12}-\sigma_{21}\right)+\Omega_{p}\left(\sigma_{02}-\sigma_{20}\right)\right]\\
B_{\pm}&=\exp \left(\pm \sum_{k} \frac{\lambda_{k}}{\omega_{k}}\left(b_{k}-b_{k}^{\dagger}\right)\right)\\
\xi_{g}&=\frac{1}{2}\left(B_{+}+B_{-}-2\langle B\rangle\right)\\
\xi_{u}&=\frac{1}{2 i}\left(B_{+}-B_{-}\right)
\end{align}

What is the physical intuition of introducing these operators? In particular, why do we need to introduce $X_u$ and $\xi_u$ with an imaginary number $i$ at the front where in the original Hamiltonian $H$ there wasn't even any $i$?

This polaron transformation approach has been adopted by many recent studies so I really want to understand what's happening clearly but I couldn't find any lecture notes or textbooks on this. I would appreciate any help, explanation or book/paper recommendation greatly. Thank you.

 

[eljose79]

Hello,

The polaron transformation is a mathematical tool used to simplify the Hamiltonian in order to better understand the physical system. In this case, the Hamiltonian is describing the interaction between a quantum dot, a cavity, and a bath of phonons. The polaron transformation helps to decouple the different degrees of freedom and allows for a clearer understanding of the system.

The introduction of the operators $X_g$ and $X_u$ is necessary in order to simplify the transformed Hamiltonian and express it in terms of the desired variables $\langle B\rangle$, $B_+$, and $B_-$. These operators are defined in such a way that they represent the average value of the bath operators, as well as the fluctuations around that average. This is important because the interaction between the quantum dot and the bath is typically much weaker than the interaction between the quantum dot and the cavity. By separating out the bath variables, we can focus on the more interesting and relevant dynamics of the system.

As for the imaginary number $i$, it is introduced in order to simplify the transformed Hamiltonian and express it in terms of the desired variables. It is a mathematical tool used to simplify the equations, and it does not have a direct physical interpretation in this context.

I would recommend looking into the book "Quantum Optics: An Introduction" by Mark Fox, as it covers the polaron transformation and its applications in quantum optics. I hope this helps and clarifies the physical intuition behind the operators introduced in the transformed Hamiltonian.