Quantum Dynamics in Noisy Backgrounds: from sampling to dissipation and fluctuations

TL;DR

This paper explores the effects of Gaussian white noise on quantum systems, deriving an effective non-Hermitian Hamiltonian that captures dissipation and fluctuations, and demonstrating localization phenomena.

Contribution

It introduces a method to analyze noisy quantum dynamics using functional integration, resulting in an explicit non-Hermitian Hamiltonian for the system.

Findings

Derivation of an effective complex frequency harmonic oscillator Hamiltonian

Identification of space and time localization due to noise

Analytical computation of the evolution operator for the noisy system

Abstract

We investigate the dynamics of a quantum system coupled linearly to Gaussian white noise using functional methods. By performing the integration over the noisy field in the evolution operator, we get an equivalent non-Hermitian Hamiltonian, which evolves the quantum state with a dissipative dynamics. We also show that if the integration over the noisy field is done for the time evolution of the density matrix, a gain contribution from the fluctuations, can be accessed in addition to the loss one from the non-hermitian Hamiltonian dynamics. We illustrate our study by computing analytically the effective non-Hermitian Hamiltonian, which we found to be the complex frequency harmonic oscillator, with a known evolution operator. It leads to space and time localisation, a common feature of noisy quantum systems in general applications.