Determination of the full statistics of quantum observables using the maximum entropy method

TL;DR

This paper introduces a maximum entropy method to reconstruct full quantum observable statistics from limited moments, demonstrated on a quantum-dot-microcavity laser, revealing new insights into photon distribution characteristics.

Contribution

The paper develops a general numerical approach using maximum entropy to obtain unbiased quantum statistics from moments, applicable to various physical systems.

Findings

Photon statistics above threshold are Gaussian, not Poisson.

The method accurately reconstructs full statistics from moments.

Entropy and Lagrange multipliers offer physical insights.

Abstract

Numerical methods for the description of nonequilibrium many-particle quantum systems such as equation of motion techniques often cannot compute the full statistics of observables but only moments of it, such as mean, variance and higher-order moments. We employ here the maximum entropy method to numerically construct unbiased statistics based on the knowledge of moments. We verify the feasibility of the proposed method by numerical simulation of a simple birth-death model for quantum-dot-microcavity lasers, where the full photon and carrier statistics are available for comparison. We show that not only the constructed statistics but also the computed entropy and the Lagrange multipliers, which appear here as a byproduct, provide valuable insight into the physics of the considered system. For example, the entropy reveals that, in contrast to common wisdom, the photon statistics of the microcavity laser above threshold is better described by a Gaussian distribution than by a Poisson distribution. Our approach is general and can be applied to many other systems emerging in physics and related fields.