Open Quantum System Stochastic Dynamics and the Rotating Wave Approximation

TL;DR

This paper investigates how the rotating wave approximation affects the stochastic dynamics of a two-level quantum system interacting with electromagnetic fields and noise, revealing that non-commutation impacts the Markovian nature and requires modifications in the master equation.

Contribution

It demonstrates that the RWA modifies stochastic terms and decay dynamics in quantum systems with non-commuting noise, providing a detailed analysis for both white and Ornstein-Uhlenbeck noise.

Findings

RWA preserves Markovian dynamics with white noise but alters dynamics with Ornstein-Uhlenbeck noise.

Non-commutation of RWA and noise Hamiltonian necessitates modifications in the master equation.

Time-dependent Lindblad terms emerge when non-commutation is properly considered.

Abstract

We study the stochastic dynamics of a two-level quantum system interacting with a stochastic magnetic field, and a single frequency electromagnetic field, with and without making the rotating wave approximation (RWA). The transformation to the rotating frame does not commute with the stochastic Hamiltonian if the stochastic field has nonvanishing components in the transverse direction. Hence, making the RWA modifies the stochastic terms in the Hamiltonian. Modification of the decay terms is also required in a master equation approach (i.e., the Liouville--von Neumann density matrix equation) for describing the dynamics. For isotropic Gaussian white noise, the RWA dynamics remains Markovian, although the Lindblad terms in the master equation for the density matrix become time-dependent when the non-commutation of the RWA transformation and the noise Hamiltonian is properly accounted for. We also treat Ornstein--Uhlenbeck noise, and find, in contra-distinction to the white noise case, a significant difference in the dynamics calculated with the RWA when the non-commutation of the RWA transformation and the noise Hamiltonian is taken into account. These findings are applicable to the modeling of any open quantum system coupled to an electromagentic field.