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arXiv:1607.07736 · Physical Review A

Generalized Hartree-Fock-Bogoliubov description of the Fröhlich polaron

Ben Kain, Hong Yuan Ling

Published Jul 29, 2016• 30 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.94.013621

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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\mathbf{k}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.
Abstract Bose polarons are quasiparticles formed through the interaction between impurities and Bose–Einstein condensates. In this paper, we derive an effective Fröhlich Hamiltonian using the generalized Bogoliubov transformation. The effective Fröhlich Hamiltonian encompasses two types of effective interactions: impurity-density (ID) coupling and impurity-spin (IS) coupling. Furthermore, we employ the Lee–Low–Pines variational approach to investigate the relevant properties of Bose polarons induced by the ID and IS coupling. These properties include the ground state energy, effective mass, and average number of virtual phonons. Our findings reveal that the contribution resulting from IS couplings to the ground energy decreases to zero near the miscible–immiscible boundary. Additionally, the increase of the IS coupling induces a greater number of virtual phonons, impeding the movement of impurities and leading to a significant increase in the effective mass of Bose polarons.
F. Grusdt et al. [New J. Phys. 19, 103035 (2017)] recently made a renormalization-group study of a one-dimensional Bose polaron in cold atoms. Their study went beyond the usual Fr\"ohlich description, which includes only single-phonon processes, by including two-phonon processes in which two phonons are simultaneously absorbed or emitted during impurity scattering [Y. E. Shchadilova et al., Phys. Rev. Lett. 117, 113002 (2016)]. We study this same beyond-Fr\"ohlich model, but in the static impurity limit where the ground state is described by a multimode squeezed state instead of the multimode coherent state in the static Fr\"ohlich model. We solve the system exactly by applying the generalized Bogoliubov transformation, an approach that can be straightforwardly adapted to higher dimensions. Using our exact solution, we obtain a polaron energy free of infrared divergences and construct analytically the polaron phase diagram. We find that the repulsive polaron is stable on the positive side of the impurity-boson interaction but is always thermodynamically unstable on the negative side of the impurity-boson interaction, featuring a bound state, whose binding energy we obtain analytically. We find that the attractive polaron is always dynamically unstable, featuring a pair of imaginary energies which we obtain analytically. We expect the multimode squeezed state to help with studies that go not only beyond the Fr\"ohlich paradigm but also beyond Bogoliubov theory, just as the multimode coherent state has helped with the study of Fr\"ohlich polarons.
The system of a charged impurity in an interacting Bose gas has gained significant attention due to the long-range ion-atom interactions and the study of transport properties. Here, the ground state energy of a charged Bose polaron is calculated within the Bogoliubov approximation for both the Fröhlich and beyond-Fröhlich Hamiltonians using a generalized Feynman variational path-integral approach, which obtained accurate results for other polaron problems. The generalized approach, which was used to improve the energy result for the neutral polaron, has resulted in a minor improvement, indicating that Feynman's approach is sufficient when the impurity-boson interaction is long-range. Beyond-Fröhlich corrections results in the emergence of a divergence in the polaronic energy indicating a transition between the repulsive and attractive polaron regime. The path-integral approach with the beyond-Fröhlich Hamiltonian is also compared to a field-theory calculation from Christensen et al, 2021. The validity of the Bogoliubov approximation is investigated. The optical absorption has also been calculated within the Bogoliubov approximation for weak ion-atom interactions, and the effect of finite temperature has been studied. We show that the coupling of the ion to an oscillating external electric field offers a straigtforward experimental probe for the charged polaron in a Bose gas, different from but complementary to existing spectroscopic techniques.
In this paper we study an impurity in Bose-Einstein-Condensate system at T=0KT=0KT=0K and suppose the contact forms for boson-boson and boson-impurity interactions. Using Bogoliubov theory and a further approximation corresponding to only think over the forward scattering of impurity by bosons, we derive a reduced Hamiltonian whose form is the same as the Fröhlich Hamiltonian for large polaron. By using Lee-Low-Pines (LLP) theory for large polaron, we obtain the effective mass of impurity, the phonon number carried by impurity and the energy related to the existence of impurity. In addition, we also discuss the valid range for forward-scattering approximation.
The Fr\"ohlich model describes the interaction of a mobile impurity with a surrounding bath of phonons which leads to the formation of a quasiparticle, the polaron. In this paper an efficient renormalization group approach is presented which provides a description of Fr\"ohlich polarons in all regimes ranging from weak to strong coupling. The extended renormalization group approach introduced here is capable of predicting ground state properties for arbitrarily small impurity masses. This allows us to obtain the full phase diagram of the Fr\"ohlich Hamiltonian, which we present concretely for the Bogoliubov-Fr\"ohlich model originally introduced to describe ultracold impurities in a Bose-Einstein condensate. For this model, whose regime of validity in ultracold quantum gases is the subject of debate, we benchmark our method by comparison of the ground state energy to recent diagrammatic quantum Monte Carlo calculations.