Spectrum analysis with quantum dynamical systems. II. Finite-time analysis

TL;DR

This paper numerically validates finite-time spectral photon counting for noise spectroscopy, showing it maintains significant advantages over homodyne detection in realistic measurement durations.

Contribution

It provides a finite-time analysis confirming the effectiveness of spectral photon counting, previously studied only in asymptotic limits, for noise spectroscopy.

Findings

Spectral photon counting outperforms homodyne detection at finite times.

Fisher information and estimation errors approach asymptotic limits smoothly.

Spectral photon counting retains substantial advantages over homodyne detection in finite measurement durations.

Abstract

The prequel to this work [Ng et al., Phys. Rev. A 93, 042121 (2016)] proposes the method of spectral photon counting to enhance noise spectroscopy with an optical interferometer. While the predicted enhancement over homodyne detection is promising, the results there are derived by taking an asymptotic limit of infinite observation time; their validity for a finite time remains unclear. To validate the theory, here we perform a numerical study of a finite-time model. Assuming that the signal is an Ornstein--Uhlenbeck process with an unknown variance parameter, we evaluate the Fisher information for homodyne detection, a lower bound on the Fisher information for spectral photon counting, and a quantum upper bound, all without taking the infinite-time limit. To confirm that the Fisher-information quantities are satisfactory precision measures, we also compute the errors of the maximum-likelihood estimator by Monte-Carlo simulations. The results demonstrate that the Fisher-information quantities and the estimation errors all smoothly approach their asymptotic limits, and the advantage of spectral photon counting over homodyne detection can remain substantial for finite times.