The spin-boson model with a structured environment: A comparison of approaches

TL;DR

This paper compares various analytical and numerical approaches to the spin-boson model with a structured environment characterized by a Breit-Wigner spectral density, revealing how the environment's peak influences the qubit dynamics and tunneling properties.

Contribution

The study systematically compares multiple methods for analyzing the structured spin-boson model, clarifying their applicability and agreement in the regime of a peaked spectral density.

Findings

For large peak frequency, the dynamics resemble an Ohmic model with reduced tunnel splitting.

The sign of the tunnel splitting scaling changes when the bare splitting crosses the peak frequency.

Good agreement between analytical approximations and numerical results.

Abstract

In the spin-boson model, the properties of the oscillator bath are fully characterized by the spectral density of oscillators $J(\omega)$. We study the case when this function is of Breit-Wigner shape and has a sharp peak at a frequency $\Omega$ with width $\Gamma\ll\Omega$. We use a number of approaches such as the weak-coupling Bloch-Redfield equation, the non-interacting blip approximation (NIBA) and the flow-equation renormalization scheme. We show, that if $\Omega$ is much larger than the qubit energy scales, the dynamics corresponds to an Ohmic spin-boson model with a strongly reduced tunnel splitting. We also show that the direction of the scaling of the tunnel splitting changes sign when the bare splitting crosses $\Omega$. We find good agreement between our analytical approximations and numerical results. We illuminate how and why different approaches to the model account for these features and discuss the interpretation of this model in the context of an application to quantum computation and read-out.