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arXiv:2406.08028 · Annales Henri Poincaré

Effective Polaron Dynamics of an Impurity Particle Interacting with a Fermi Gas

Duc Viet Hoang, Peter Pickl

Published Jan 13, 2025• 0 citations
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Abstract

Abstract We study the quantum dynamics of a homogeneous ideal Fermi gas coupled to an impurity particle on a three-dimensional box with periodic boundary condition. For large Fermi momentum $$k_{\text {F}}$$ k F , we prove that the effective dynamics is generated by a Fröhlich-type polaron Hamiltonian, which linearly couples the impurity particle to an almost-bosonic excitation field. Moreover, we prove that the effective dynamics can be approximated by an explicit coupled coherent state. Our method is applicable to a range of interaction couplings, in particular including interaction couplings of order 1 and time scales of the order $$k_{\text {F}}^{-1}$$ k F - 1 .

DOI
10.1007/s00023-024-01532-0

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Abstract We study the quantum dynamics of a homogeneous ideal Fermi gas coupled to an impurity particle on a three-dimensional box with periodic boundary condition. For large Fermi momentum kFk_{\text {F}}kF​ k F , we prove that the effective dynamics is generated by a Fröhlich-type polaron Hamiltonian, which linearly couples the impurity particle to an almost-bosonic excitation field. Moreover, we prove that the effective dynamics can be approximated by an explicit coupled coherent state. Our method is applicable to a range of interaction couplings, in particular including interaction couplings of order 1 and time scales of the order kF−1k_{\text {F}}^{-1}kF−1​ k F - 1 .
We consider the dynamics of the Bose polaron system, a dense quantum gas consisting of NNN bosons evolving in R3\mathbb{R}^3R3 in the presence of an impurity particle. The system is studied in the mean-field scaling with initially high density ρρρ and large volume ΛΛΛ of the gas. In the initial state, almost all bosons are in the Bose-Einstein condensate, with a few excitations. We derive from the microscopic dynamics, in the joint limit of large densities and volumes, with the constraint Λ3≪ρΛ^3 \ll ρΛ3≪ρ, the effective description by the translation-invariant Bogoliubov-Fröhlich Hamiltonian, which couples the quantum field of excitations linearly to the impurity particle.
We consider the dynamics of the Bose polaron system, a dense quantum gas consisting of NNN bosons evolving in R3\mathbb{R}^3R3 in the presence of an impurity particle. The system is studied in the mean-field scaling with initially high density ρ\rhoρ and large volume Λ\LambdaΛ of the gas. In the initial state, almost all bosons are in the Bose-Einstein condensate, with a few excitations. We derive from the microscopic dynamics, in the joint limit of large densities and volumes, with the constraint Λ3≪ρ\Lambda^3\ll\rhoΛ3≪ρ, the effective description by the translation-invariant Bogoliubov-Fröhlich Hamiltonian, which couples the quantum field of excitations linearly to the impurity particle.
We study the dynamics of a small number of impurity particles coupled to the ideal Fermi gas in a ddd-dimensional box. The impurities interact with the fermions via a two-body potential λv(x)λv(x)λv(x) where λλλ is a coupling constant and v(x)v(x)v(x) for instance a screened Coulomb potential. After taking the large-volume limit at positive Fermi momentum kFk_FkF​ we consider the regime of high density of the fermions, that is, kFk_FkF​ large compared to one. For coupling constants that scale like λ2∼kF(2−d)λ^2 \sim k_F^{(2-d)}λ2∼kF(2−d)​ we show that the impurity particles effectively decouple from the fermions but evolve with an attractive pair interaction among each other which is induced by fluctuations in the Fermi gas.
When a mobile impurity interacts with a surrounding bath of bosons, it forms a polaron. Numerous methods have been developed to calculate how the energy and the effective mass of the polaron are renormalized by the medium for equilibrium situations. Here, we address the much less studied nonequilibrium regime and investigate how polarons form dynamically in time. To this end, we develop a time-dependent renormalization-group approach which allows calculations of all dynamical properties of the system and takes into account the effects of quantum fluctuations in the polaron cloud. We apply this method to calculate trajectories of polarons following a sudden quench of the impurity-boson interaction strength, revealing how the polaronic cloud around the impurity forms in time. Such trajectories provide additional information about the polaron's properties which are challenging to extract directly from the spectral function measured experimentally using ultracold atoms. At strong couplings, our calculations predict the appearance of trajectories where the impurity wavers back at intermediate times as a result of quantum fluctuations. Our method is applicable to a broader class of nonequilibrium problems. As a check, we also apply it to calculate the spectral function and find good agreement with experimental results. At very strong couplings, we predict that quantum fluctuations lead to the appearance of a dark continuum with strongly suppressed spectral weight at low energies. While our calculations start from an effective Fr\"ohlich Hamiltonian describing impurities in a three-dimensional Bose-Einstein condensate, we also calculate the effects of additional terms in the Hamiltonian beyond the Fr\"ohlich paradigm. We demonstrate that the main effect of these additional terms on the attractive side of a Feshbach resonance is to renormalize the coupling strength of the effective Fr\"ohlich model.
An impurity atom immersed in an ultracold atomic Fermi gas can form a quasiparticle, so-called Fermi polaron, due to impurity-fermion interaction. We consider a three-dimensional homogeneous dipolar Fermi gas as a medium, where the interatomic dipole-dipole interaction (DDI) makes the Fermi surface deformed into a spheroidal shape, and, using a Chevy-type variational method, investigate the ground-state properties of the Fermi polaron: the effective mass, the momentum distribution of a particle-hole excitation, the drag parameter, and the medium-density modification around the impurity. These quantities are shown to exhibit spatial anisotropies in such a way as to reflect the momentum anisotropy of the background dipolar Fermi gas. We also give numerical results for the polaron properties at the unitarity limit of the impurity-fermion interaction in the case in which the impurity and fermion masses are equal. It has been found that the transverse effective mass and the transverse momentum drag parameter of the polaron both tend to decrease by $\ensuremath{\sim}10%$ when the DDI strength is raised from 0 up to around its critical value, while the longitudinal ones exhibit a very weak dependence on the DDI.