Power law tails and non Markovian dynamics in open quantum systems: An exact solution from Keldysh field theory

TL;DR

This paper derives exact solutions for open quantum systems with non-analytic bath spectral functions, revealing power law tails in dynamics that deviate from traditional Markovian behavior, and demonstrates these effects in lattice models with interactions.

Contribution

The authors provide an exact analytical framework using Keldysh formalism to study non-Markovian dynamics with power law tails in open quantum systems with non-analytic spectral functions.

Findings

Green's functions exhibit power law decay $ o |t-t'|^{-3/2}$

Current-current correlators show long-time tails $ o |t-t'|^{-3}$

Power law tails persist with interactions, with shifted crossover times

Abstract

The Born-Markov approximation is widely used to study dynamics of open quantum systems coupled to external baths. Using Keldysh formalism, we show that the dynamics of a system of bosons (fermions) linearly coupled to non-interacting bosonic (fermionic) bath falls outside this paradigm if the bath spectral function has non-analyticities as a function of frequency. In this case, we show that the dissipative and noise kernels governing the dynamics have distinct power law tails. The Green's functions show a short time "quasi" Markovian exponential decay before crossing over to a power law tail governed by the non-analyticity of the spectral function. We study a system of bosons (fermions) hopping on a one dimensional lattice, where each site is coupled linearly to an independent bath of non-interacting bosons (fermions). We obtain exact expressions for the Green's functions of this system which show power law decay $\sim |t-t'|^{-3/2}$. We use these to calculate density and current profile, as well as unequal time current-current correlators. While the density and current profiles show interesting quantitative deviations from Markovian results, the current-current correlators show qualitatively distinct long time power law tails $|t-t'|^{-3}$ characteristic of non-Markovian dynamics. We show that the power law decays survive in presence of inter-particle interaction in the system, but the cross-over time scale is shifted to larger values with increasing interaction strength.