Qubit dynamics with classical noise

TL;DR

This paper analyzes how a qubit's state evolves under classical noise, showing convergence to a final state with rates depending on noise properties, and characterizing the resulting decoherence channels.

Contribution

It provides an explicit evaluation of the asymptotic qubit state under classical noise and links noise regimes to specific decoherence channels.

Findings

The noise-averaged qubit state converges polynomially in time.

Weak off-diagonal noise induces dephasing in the energy basis.

Strong off-diagonal noise causes dephasing in the delocalized basis.

Abstract

We study the evolution of a qubit evolving according to the Schr\"odinger equation with a Hamiltonian containing noise terms, modeled by random diagonal and off-diagonal matrix elements. We show that the noise-averaged qubit density matrix converges to a final state, in the limit of large times $t.$ The convergence speed is polynomial in $1/t$, with a power depending on the regularity of the noise probability density and its low frequency behaviour. We evaluate the final state explicitly. We show that in the regimes of weak and strong off-diagonal noise, the process implements the dephasing channel in the energy- (localized) and the delocalized basis, respectively.