Rigorous Born Approximation and beyond for the Spin-Boson Model

TL;DR

This paper provides an exact calculation of the spin-boson model dynamics within the Born approximation, revealing non-Markovian effects and coherence loss, and extends the analysis to higher-order corrections with computational tools.

Contribution

It introduces an exact lowest-order Born approximation calculation for the spin-boson model and explores fourth-order corrections using computer algebra, advancing understanding of system-bath interactions.

Findings

Non-Markovian effects cause qualitative coherence changes.

Coherence decays at a rate proportional to √α in the biased case.

Higher-order corrections match equilibrium state predictions.

Abstract

Within the lowest-order Born approximation, we present an exact calculation of the time dynamics of the spin-boson model in the ohmic regime. We observe non-Markovian effects at zero temperature that scale with the system-bath coupling strength and cause qualitative changes in the evolution of coherence at intermediate times of order of the oscillation period. These changes could significantly affect the performance of these systems as qubits. In the biased case, we find a prompt loss of coherence at these intermediate times, whose decay rate is set by $\sqrt{\alpha}$, where $\alpha$ is the coupling strength to the environment. We also explore the calculation of the next order Born approximation: we show that, at the expense of very large computational complexity, interesting physical quantities can be rigorously computed at fourth order using computer algebra, presented completely in an accompanying Mathematica file. We compute the $O(\alpha)$ corrections to the long time behavior of the system density matrix; the result is identical to the reduced density matrix of the equilibrium state to the same order in $\alpha$. All these calculations indicate precision experimental tests that could confirm or refute the validity of the spin-boson model in a variety of systems.