Super and Sub-Poissonian photon statistics for single molecule spectroscopy

TL;DR

This paper provides an exact analytical study of photon emission statistics from a single molecule under spectral diffusion, revealing a transition from quantum sub-Poissonian to classical super-Poissonian behavior depending on modulation speed.

Contribution

It introduces a stochastic spectral diffusion model combined with a generating function approach to derive exact formulas for line shape and photon statistics, including Mandel's Q parameter.

Findings

Line shape exhibits motional narrowing and power broadening.

Photon statistics transition from sub-Poissonian to super-Poissonian with modulation speed.

Optimal Rabi frequency enhances quantum sub-Poissonian emission.

Abstract

We investigate the distribution of the number of photons emitted by a single molecule undergoing a spectral diffusion process and interacting with a continuous wave laser field. The spectral diffusion is modeled based on a stochastic approach, in the spirit of the Anderson-Kubo line shape theory. Using a generating function formalism we solve the generalized optical Bloch equations, and obtain an exact analytical formula for the line shape and Mandel's Q parameter. The line shape exhibits well known behaviors, including motional narrowing when the stochastic modulation is fast, and power broadening. The Mandel parameter, describing the line shape fluctuations, exhibits a transition from a Quantum sub-Poissonian behavior in the fast modulation limit, to a classical super-Poissonian behavior found in the slow modulation limit. Our result is applicable for weak and strong laser field, namely for arbitrary Rabi frequency. We show how to choose the Rabi frequency in such a way that the Quantum sub-Poissonian nature of the emission process becomes strongest. A lower bound on $Q$ is found, and simple limiting behaviors are investigated. A non-trivial behavior is obtained in the intermediate modulation limit, when the time scales for spectral diffusion and the life time of the excited state, become similar. A comparison is made between our results, and previous ones derived based on the semi-classical generalized Wiener--Khintchine theorem.