# Efficient variational approach to dynamics of a spatially extended bosonic Kondo model

## TL;DR

This paper introduces an efficient variational method for analyzing the dynamics of a quantum spin impurity coupled to a bosonic bath, enabling analytical treatment of complex out-of-equilibrium phenomena in impurity models.

## Contribution

The authors develop a variational approach using bosonic Gaussian states and symmetry considerations to simplify and analytically describe the dynamics of a bosonic Kondo impurity model.

## Key findings

- Analytical equations for variational time evolution derived
- Method applicable to a wide class of quantum impurity problems
- Comparison with Monte Carlo shows new features from orbital motion

## Abstract

We develop an efficient variational approach to studying dynamics of a localized quantum spin coupled to a bath of mobile spinful bosons. We use parity symmetry to decouple the impurity spin from the environment via a canonical transformation and reduce the problem to a model of the interacting bosonic bath. We describe coherent time evolution of the latter using bosonic Gaussian states as a variational ansatz. We provide full analytical expressions for equations describing variational time evolution that can be applied to study in- and out-of-equilibrium phenomena in a wide class of quantum impurity problems. In the accompanying paper [Y. Ashida {\it et al.}, Phys. Rev. Lett. 123, 183001 (2019)], we present a concrete application of this general formalism to the analysis of the Rydberg Central Spin Model, in which the spin-1/2 Rydberg impurity undergoes spin-changing collisions in a dense cloud of two-component ultracold bosons. To illustrate new features arising from orbital motion of the bath atoms, we compare our results to the Monte Carlo study of the model with spatially localized bosons in the bath, in which random positions of the atoms give rise to random couplings of the standard central spin model.