We adapt the generalized Hartree-Fock-Bogoliubov (HFB) method to an interacting many-phonon system free of impurities. The many-phonon system is obtained from applying the Lee-Low-Pine (LLP) transformation to the Fr\"ohlich model which describes a mobile impurity coupled to noninteracting phonons. We specialize our general HFB description of the Fr\"ohlich polaron to Bose polarons in quasi-one-dimensional cold-atom mixtures. The LLP-transformed many-phonon system distinguishes itself with an artificial phonon-phonon interaction which is very different from the usual two-body interaction. We use the quasi-one-dimensional model, which is free of an ultraviolet divergence that exists in higher dimensions, to better understand how this unique interaction affects polaron states and how the density and pair correlations inherent to the HFB method conspire to create a polaron ground state with an energy in good agreement with and far closer to the prediction from Feynman's variational path integral approach than mean-field theory where HFB correlations are absent.
We consider the Fermi polaron problem at zero temperature, where a single impurity interacts with noninteracting host fermions. We approach the problem starting with a Fr\"ohlich-like Hamiltonian where the impurity is described with canonical position and momentum operators. We apply the Lee-Low-Pine (LLP) transformation to change the fermionic Fr\"ohlich Hamiltonian into the fermionic LLP Hamiltonian, which describes a many-body system containing host fermions only. We adapt the self-consistent Hartree-Fock (HF) approach, first proposed by Edwards, to the fermionic LLP Hamiltonian in which a pair of host fermions with momenta
k and ${\mathbf{k}}^{\ensuremath{'}}$ interact with a potential proportional to $\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{k}}^{\ensuremath{'}}$. We apply the HF theory, which has the advantage of not restricting the number of particle-hole pairs, to repulsive Fermi polarons in one dimension. When the impurity and host fermion masses are equal our variational ansatz, where HF orbitals are expanded in terms of free-particle states, produces results in excellent agreement with McGuire's exact analytical results based on the Bethe ansatz. This work raises the prospect of using the HF ansatz and its time-dependent generalization as building blocks for developing all-coupling theories for both equilibrium and nonequilibrium Fermi polarons in higher dimensions.