(9754) Fermions that are coupled to a bath of bosons

This is a variation of the so-called Polaron problem. We consider electrons in a lattice. Due to the interaction of the electrons with high-frequency phonons, they are "dressed" and acquire a larger effective mass. Optionally one can consider an electron in a "quantum dot" site \( j=0 \) that can decay into a lead that is formed of the \( j \neq 0 \) sites. There, the suppressed hopping frequency is known as the Franck-Condon matrix-element. Another version, with only two sites, is essentially a special case of the well studied spin-boson model of quantum-dissipation, where coherent oscillations of a two-level-system are affected by an external bath.

Consider the the following Hamiltonian that describes fermions (\( a_j \)) that are coupled to bath modes (\( b_k \)).

\( \mathcal{H} = \sum_j J_{i,j} a_i^{\dagger} a_j + \sum_k \omega_k b_k^{\dagger} b_k + a_0^{\dagger} a_0 \sum_{k} \gamma_k (b_k^{\dagger}+b_k) \).

Note that \( \epsilon_j = J_{j,j} \) are the on-site energies, while the off-diagonal elements generate hopping \( K \) between connected sites.

(1) Explain why the \(K=0\) model is trivial. Specify the constants of motion ("good quantum numbers"), and write what are the associated "blocks" of the Hamiltonian.

(2) Use the results of Ex9753 in order to define new \( c_k \) coordinates, and write what are the eigenvalues \( E_{\nu} \) of \( \mathcal{H} \) per-block.

(3) The transformation from \( (a_j,b_k) \) to \( (a_j,c_k) \) is not canonical. Explain why. Using the transformation matrix \( S \) of Ex9753 define a proper canonical transformation from \( (a_j,b_k) \) to \( (d_j,c_k) \).

(4) Write expression for the Hamiltonian in the \( (d_j,c_k) \) coordinates. Specifically, show that there is an extra Hubbard-like interaction term \(U\); that there is detuning of the dot binding energy; and that the dot-lead hopping term is modified.

(5) Assume that the bath oscillators have high-frequencies, hence they stay in the ground state during the dynamics. Under such assumption, write what is the dressed value of \( K \). Explain the implication for the effective mass in the Polaron problem.