Photon counting statistics in Gaussian bosonic networks

TL;DR

This paper develops a comprehensive quantum mechanical framework to analyze photon counting statistics in Gaussian bosonic networks, including driven cavities with beamsplitter and two-mode-squeezing interactions, with applications to entanglement and photon flow control.

Contribution

It introduces a general theory using phase-space methods and Riccati equations to compute photon counting statistics in complex bosonic networks, extending previous approaches.

Findings

Photon cross-correlations reveal entanglement between cavities.

Synthetic flux influences photon flow direction in circulator networks.

The framework enables systematic analysis of photon statistics in Gaussian networks.

Abstract

The statistics of transmitted photons in microwave cavities play a foundational role in microwave quantum optics and its technological applications. By utilizing quantum mechanical phase-space methods, we here develop a general theory of the photon counting statistics in Gaussian bosonic networks consisting of driven cavities with beamsplitter interactions and two-mode-squeezing. The dynamics of the network can be captured by a Lyapunov equation for the covariance matrix of the cavity fields, which generalizes to a Riccati equation, when counting fields are included. By solving the Riccati equation, we obtain the statistics of emitted and absorbed photons as well as the time-dependent correlations encoded in waiting time distributions and second-order coherence functions. To illustrate our theoretical framework, we first apply it to a simple linear network consisting of two coupled cavities, for which we evaluate the photon cross-correlations and discuss connections between the photon emission statistics and the entanglement between the cavities. We then consider a bosonic circulator consisting of three coupled cavities, for which we investigate how a synthetic flux may affect the direction of the photon flow, similarly to recent experiments. Our general framework paves the way for systematic investigations of the photon counting statistics in Gaussian bosonic networks.