The effect of classical noise on a quantum two-level system

TL;DR

This paper investigates how classical noise affects a quantum two-level system, analyzing its long-term behavior, convergence properties, and deriving an approximate diffusion equation for transition probabilities.

Contribution

It introduces a detailed analysis of a quantum two-level system under classical noise, including invariant measures, convergence rates, and an approximate diffusion model.

Findings

Invariant measure of the system is unique.

Convergence speed to the invariant measure is determined for Ornstein-Uhlenbeck noise.

An approximate diffusion equation for transition probabilities is derived.

Abstract

We consider a quantum two-level system perturbed by classical noise. The noise is implemented as a stationary diffusion process in the off-diagonal matrix elements of the Hamiltonian, representing a transverse magnetic field. We determine the invariant measure of the system and prove its uniqueness. In the case of Ornstein-Uhlenbeck noise, we determine the speed of convergence to the invariant measure. Finally, we determine an approximate one-dimensional diffusion equation for the transition probabilities. The proofs use both spectral-theoretic and probabilistic methods.