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

Duc Viet Hoang, Peter Pickl

TL;DR

This work analyzes the quantum dynamics of a single impurity immersed in a dense three-dimensional Fermi gas in a box with periodic boundaries. By employing a particle-hole transformation, patch decomposition, and almost-bosonic pair operators, the authors derive an effective Fröhlich-type Hamiltonian that linearly couples the impurity to a bosonized excitation field, capturing the polaron-like quasi-particle formation. They prove a rigorous effective time-evolution theorem showing that the microscopic dynamics are accurately approximated by the effective Hamiltonian on timescales $t = \mathcal{O}(k_{\text{F}}^{-1}\lambda^{-1})$ for couplings $\lambda \in [k_{\text{F}}^{-1/6},1]$, with explicit error bounds. In addition, they establish a coherent-state representation of the excitations, demonstrating that the time-evolved state can be approximated by a coupled coherent state $\psi_t = e^{iP(t)}W(\eta_t)\phi\otimes\Omega$ with quantitative control on the deviation, thereby providing a detailed description of polaron formation and collective excitations. The results hold for interaction strengths including order-one couplings and illuminate the role of the linear coupling term $\Phi(h_y)$ in the effective dynamics, advancing rigorous understanding of impurity dynamics in dense Fermi gases.

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 $k_\text{F}$, 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 $k_\text{F}^{-1}$.

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

TL;DR

This work analyzes the quantum dynamics of a single impurity immersed in a dense three-dimensional Fermi gas in a box with periodic boundaries. By employing a particle-hole transformation, patch decomposition, and almost-bosonic pair operators, the authors derive an effective Fröhlich-type Hamiltonian that linearly couples the impurity to a bosonized excitation field, capturing the polaron-like quasi-particle formation. They prove a rigorous effective time-evolution theorem showing that the microscopic dynamics are accurately approximated by the effective Hamiltonian on timescales for couplings , with explicit error bounds. In addition, they establish a coherent-state representation of the excitations, demonstrating that the time-evolved state can be approximated by a coupled coherent state with quantitative control on the deviation, thereby providing a detailed description of polaron formation and collective excitations. The results hold for interaction strengths including order-one couplings and illuminate the role of the linear coupling term in the effective dynamics, advancing rigorous understanding of impurity dynamics in dense Fermi gases.

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 , 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 .
Paper Structure (8 sections, 11 theorems, 52 equations, 1 figure)

This paper contains 8 sections, 11 theorems, 52 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The microscopic model
  3. Relevant parameters and time scales
  4. Preliminaries
  5. Main results
  6. Effective time evolution
  7. Effective coherent state
  8. Estimates within the bosonization framework

Key Result

Theorem 3.1

Assume that $\hat{V} \geq 0$ is compactly supported and satisfies $\hat{V} (-k)= \hat{V} (k)$ for all $k \in \mathbb{Z}^{3}$. Let $\lambda \in [k_{\textnormal{F}}^{-1/6}, 1]$ be the interaction parameter as introduced in eq:microscopic-Hamiltonian and take the number of patches to be $M=N^{\frac{16}

Figures (1)

  • Figure 1: Plot of $\| \eta_t \|^2$ using lem:eta-bounds with constant $\hat{V}$ and $\lambda=1$. As a qualitative feature one observes a parabolic growth followed by a logarithmic increase.

Theorems & Definitions (22)

  • Theorem 3.1: Effective dynamics of the system
  • Remark 3.2
  • Definition 3.3
  • Remark 3.4
  • Remark 3.5
  • Theorem 3.6: Effective coherent dynamics
  • Remark 3.7
  • Remark 3.8
  • Remark 3.9
  • Remark 3.10
  • ...and 12 more