Effects of colored noise on Landau-Zener Transitions: Two and Three-Level Systems

TL;DR

This paper analyzes how classical Gaussian noise affects Landau-Zener transitions in two- and three-level quantum systems, providing analytical solutions for different noise regimes and exploring their physical implications.

Contribution

It offers analytical solutions for Landau-Zener transition probabilities under fast and slow noise, connecting Bloch and Schrödinger descriptions and extending results to higher spin systems.

Findings

Fast noise induces transition probabilities via Bloch's equation averaging.

Slow noise leads to transition probabilities following the Arrhenius activation law.

The paper establishes relations between S=1/2 and S=1 transition probabilities influenced by noise.

Abstract

We investigate the Landau-Zener transition in two- and three- level systems subject to a classical Gaussian noise. Two complementary limits of the noise being fast and slow compared to characteristic Landau-Zener tunnel times are discussed. The analytical solution of a density matrix (Bloch) equation is given for a long time asymptotic of transition probability. It is demonstrated that the transition probability induced/assisted by the fast noise can be obtained through a procedure of {\it Bloch's equation averaging} with further reduction it to a master equation. In contrast to the case of fast noise, the transition probability for LZ transition induced/assisted by the slow classical noise can be obtained by averaging a {\it solution} of Bloch's equation over the noise realization. As a result, the transition probability is described by the activation Arrhenius law. The approximate solution of the Bloch's equation at finite times is written in terms of Fresnel's integrals and interpreted in terms of interference pattern. We discuss consequences of a local isomorphism between SU(2) and SO(3) groups and connections between Schr\"odinger and Bloch descriptions of spin dynamics. Based on this isomorphism we establish the relations between S=1/2 and S=1 transition probabilities influenced by the noise. A possibility to use the slow noise as a probe for tunnel time is discussed.