Dissipative quantum mechanics beyond Bloch-Redfield: A consistent weak-coupling expansion of the ohmic spin boson model at arbitrary bias

TL;DR

This paper develops a comprehensive weak-coupling expansion for the ohmic spin boson model at arbitrary bias, extending beyond Bloch-Redfield, and provides analytical solutions for the time evolution of the system's density matrix.

Contribution

It introduces a consistent zero-temperature perturbation theory and real-time RG approach that captures the full time dynamics, including logarithmic corrections and bias effects, beyond previous methods.

Findings

Exponential decay of all time evolution terms.

Bias-dependent power-law behavior of oscillating modes.

Renormalized tunneling and Rabi frequency analysis.

Abstract

We study the time dynamics of the ohmic spin boson model at arbitrary bias $\epsilon$ and small coupling $\alpha$ to the bosonic bath. Using perturbation theory and the real-time renormalization group (RG) method we present a consistent zero-temperature weak-coupling expansion for the time evolution of the reduced density matrix one order beyond the Bloch-Redfield solution. We develop a renormalized perturbation theory and present an analytical solution covering the whole range from small to large times, including further results for exponentially small or large times. Resumming all secular terms in all orders of perturbation theory we find exponential decay for all terms of the time evolution. We determine the preexponential functions and find slowly varying logarithmic terms with the renormalized Rabi frequency $\Omega$ as energy scale together with strongly varying parts falling off asymptocially as $1/t$ in leading order, in contrast to the unbiased case. Resumming all logarithmic terms in all orders of perturbation theory via real-time RG we find the correct renormalized tunneling and a power-law behaviour for the oscillating modes with exponent crossing over from $2\alpha$ for exponentially small times to a bias-dependent value $2\alpha \epsilon^2/\Omega^2$ for exponentially large times. Furthermore, we present a degenerate perturbation theory to calculate consistently the purely decaying mode one order beyond Bloch-Redfield.