Variational approach for impurity dynamics at finite temperature

TL;DR

This paper introduces a variational method for modeling impurity dynamics at finite temperature, applicable to quantum media, and demonstrates its effectiveness through comparison with experiments and exact solutions.

Contribution

It develops a general variational principle for impurity dynamics at finite temperature, compatible with conservation laws and related to Green's function approaches.

Findings

Successfully models Ramsey interference experiments with Fermi gases.

Agrees well with exact solutions for fixed impurity responses.

Provides a framework for impurities with dynamical degrees of freedom.

Abstract

We present a general variational principle for the dynamics of impurity particles immersed in a quantum-mechanical medium. By working within the Heisenberg picture and constructing approximate time-dependent impurity operators, we can take the medium to be in any mixed state, such as a thermal state. Our variational method is consistent with all conservation laws and, in certain cases, it is equivalent to a finite-temperature Green's function approach. As a demonstration of our method, we consider the dynamics of heavy impurities that have suddenly been introduced into a Fermi gas at finite temperature. Using approximate time-dependent impurity operators involving only one particle-hole excitation of the Fermi sea, we find that we can successfully model the results of recent Ramsey interference experiments on $^{40}$K atoms in a $^6$Li Fermi gas [M.~Cetina et al., Science \textbf{354}, 96 (2016)]. We also show that our approximation agrees well with the exact solution for the Ramsey response of a fixed impurity at finite temperature. Our approach paves the way for the investigation of impurities with dynamical degrees of freedom in arbitrary quantum-mechanical mediums.