The Renormalised Bogoliubov-Fröhlich Hamiltonian

Marcin Napiórkowski

Jonas Lampart

The Bogoliubov–Fröhlich Hamiltonian models the interaction of an impurity with the excitations of a Bose–Einstein condensate. It has been observed that the dependence of the ground state energy on the ultraviolet (UV) cutoff differs significantly from what would be expected from similar well-known models. We give a detailed explanation of this UV behavior and provide an explicit representation of the renormalized Hamiltonian.

Marcin Napiórkowski

Laurent H. A. Simons, Michiel Wouters, Jacques Tempere

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.

T. Ichmoukhamedov, Jacques Tempere

The celebrated variational path integral approach to the polaron problem shows remarkable discrepancies with diagrammatic Monte Carlo (DiagMC) for the Bogoliubov-Fr\"ohlich Hamiltonian which describes an impurity weakly coupled to a Bose condensed atomic gas. It has been shown both by a renormalization group approach and by the method of correlated Gaussian wave functions that the model has a subtle UV divergence caused by quantum fluctuations, which are not captured within Feynman's approach. In this work we address the issues with Feynman's approach and show that by extending the model action to a more general form, and by considering higher-order corrections beyond the Jensen-Feynman inequality, a good agreement with DiagMC can be obtained.

M. Marinaro