Dissipative dynamics of a two - level system resonantly coupled to a harmonic mode

TL;DR

This paper introduces an approximation method for modeling the dissipative dynamics of a two-level system coupled to a harmonic mode, especially near resonance, providing accurate analytical and numerical insights.

Contribution

It develops a new approximation scheme using displaced harmonic oscillator states to better understand the TSS-HO system dynamics near resonance.

Findings

Effective Hamiltonian accurately describes system dynamics

Displaced HO states improve Hilbert space truncation

Analytical solutions valid for weak coupling and low temperatures

Abstract

We propose an approximation scheme to describe the dynamics of the spin-boson model when the spectral density of the environment shows a peak at a characteristic frequency $\Omega$ which can be very close (or even equal) to the spin Zeeman frequency $\Delta$. Mapping the problem onto a two-state system (TSS) coupled to a harmonic oscillator (HO) with frequency $\omega_0$ we show that the representation of displaced HO states provides an appropriate basis to truncate the Hilbert space of the TSS-HO system and therefore a better picture of the system dynamics. We derive an effective Hamiltonian for the TSS-HO system, and show it furnishes a very good approximation for the system dynamics even when its two subsystems are moderately coupled. Finally, assuming the regime of weak HO-bath coupling and low temperatures, we are able to analytically evaluate the dissipative TSS dynamics.