Polaron transformation in quantum optics

Question

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.

Answer 1

Where do canonical transformations come from?
Such transformations are often good guesses, based on experience with mathematics. There are however lines of thought that may lead to guessing a transformation. I will mention two that I am more experienced with:

• Schriffer-Wolff transformation is a perturbative transformation intended to remove the first-order tunneling terms from Anderson Hamiltonian (which transforms it into Kondo Hamiltonian). Although the context seems very different, it is the same thing that is done in Polaron transformation, where the electron-phonon coupling term disappears (the polaron and the phonon field become decoupled). SW derivation is also instructive in terms of how one arrives at transformation perturbatively. Applying this method to the polaron Hamiltonian one obtains the exact result.
• Canonical bosonization is a method used to map a system of interacting electrons onto a system of non-interacting bosons. It is very similar in spirit to the polaron transformation, except that we deal with and infinite number of polarons. The way it is represented in Schotte & Schotte paper is probably the closest to thee polaron spirit, but for more solid description of canonical bosonization see Haldane or Voit (there are also more recent works, and one should not confuse canonical bosonization with more modern versions, e.g., done via path integrals).

Physical intuition
Polaron transformation in quantum optics is not necessarily applied to a Hamiltonian containin a two-level system - e.g., in semiconductor Bragg lattices photon modes may be coupled to creation and annihilation of excitons.

Originally the transformation was developed in connection to dielectric materials, some properties of which can be described by introducing a concept of polaron an electron dressed by a cloud of optical phonons. This is literally what is achieved mathematically by performing the polaron transformation in the Hamiltonian. More generally polarons appear as quasiparticle poles in the Green's function decsribing such a material. Strictly speaking the transformation is usually used for description of what is known as small polaron, and contributions in its theory were made by such well-known physicists as Lev Landau and Richard Feynmann.

Answer 2

• Operator averages need to be taken in order to get to those expressions. Here, OA is carried out over the phonon modes (of the bath).
• One particular term, thereof, is taken to estimate the truncation error.