Dynamics of a qubit coupled to a dissipative nonlinear quantum oscillator: an effective bath approach

TL;DR

This paper develops an effective bath approach to analyze a qubit coupled to a nonlinear quantum oscillator and its dissipative environment, providing analytical insights into the qubit's dynamics.

Contribution

It introduces an approximate mapping method to derive the spectral density of an effective bath for a qubit coupled to a nonlinear oscillator, using linear response theory.

Findings

Analytical formula for the qubit's population difference.

Good agreement with Bloch-Redfield master equation predictions.

Effective spectral density derived from the oscillator's susceptibility.

Abstract

We consider a qubit coupled to a nonlinear quantum oscillator, the latter coupled to an Ohmic bath, and investigate the qubit dynamics. This composed system can be mapped onto that of a qubit coupled to an effective bath. An approximate mapping procedure to determine the spectral density of the effective bath is given. Specifically, within a linear response approximation the effective spectral density is given by the knowledge of the linear susceptibility of the nonlinear quantum oscillator. To determine the actual form of the susceptibility, we consider its periodically driven counterpart, the problem of the quantum Duffing oscillator within linear response theory in the driving amplitude. Knowing the effective spectral density, the qubit dynamics is investigated. In particular, an analytic formula for the qubit's population difference is derived. Within the regime of validity of our theory, a very good agreement is found with predictions obtained from a Bloch-Redfield master equation approach applied to the composite qubit-nonlinear oscillator system.