Quantum Propagator Dynamics of a Harmonic Oscillator in a Multimode Harmonic Oscillators Environment using White Noise Functional Analysis

TL;DR

This paper employs white noise analysis to derive the propagator of a harmonic oscillator coupled to a multimode harmonic oscillator environment, simplifying the mathematical description of open quantum system dynamics.

Contribution

It introduces a novel method using white noise analysis to derive system propagators by decoupling normal modes, streamlining open quantum system analysis.

Findings

Propagator expressed as a product of N simple harmonic oscillator propagators.

Normal modes include degenerate and non-degenerate frequencies.

Method simplifies the mathematical treatment of open quantum systems.

Abstract

White noise analysis is used to derive the propagator of an open quantum system consisting of a harmonic oscillator which is coupled to an environment consisting of N multimode harmonic oscillators. The quantum propagators are obtained after solving for the normal modes of the system-environment interaction in order to decouple the coordinates in the Lagrangian describing the dynamics of the system, the environment and their interaction with each other. The decoupled Lagrangian is then used in the path integral corresponding to the propagator of the system, with the path integral evaluated using white noise analysis. The resulting propagator is then found to consist of a product of N simple harmonic oscillator propagators. N-2 of these propagators correspond to the degenerate normal mode frequencies of the system-environment interaction, while the other 2 correspond to the non-degenerate normal mode frequencies. This method of deriving the propagators greatly simplifies the task of mathematically describing the dynamics of the open quantum system. The ease of use of this method suggests that, together with its inherent mathematical rigor, white noise analysis can be a powerful tool in analyzing the dynamics of an open quantum system.