Theory for frequent measurements of spontaneous emissions in non-Markovian environment: beyond Lorentzian spectrum

TL;DR

This paper extends quantum trajectory theory to non-Markovian environments with arbitrary spectra, providing analytical and numerical tools for modeling spontaneous emissions under finite bandwidth conditions.

Contribution

It generalizes QT theory beyond Lorentzian spectra to arbitrary spectra in non-Markovian environments, including proofs of scaling behavior and simulation methods.

Findings

Proved existence of a perfect scaling behavior between environment bandwidth and detection interval.

Derived analytical results for specific examples to aid QT simulations.

Proposed numerical schemes for cases lacking analytical solutions.

Abstract

The measurement-result-conditioned evolution of a system (e.g. an atom) with spontaneous emissions of photons is well described by the quantum trajectory (QT) theory. In this work we generalize the associated QT theory from infinitely wide bandwidth Markovian environment to the case of finite bandwidth non-Markovian environment. In particular, we generalize the treatment for arbitrary spectrum, which is not restricted by the specific Lorentzian case. We rigorously prove a general existence of a perfect scaling behavior jointly defined by the bandwidth of environment and the time interval between successive photon detections. For a couple of examples, we obtain analytic results to facilitate QT simulations based on the Monte-Carlo algorithm. For the case where analytical result is not available, numerical scheme is proposed for practical simulations.