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.
Author indications on fulfilling journal expectations
Provide a novel and synergetic link between different research areas.
Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
Detail a groundbreaking theoretical/experimental/computational discovery
Present a breakthrough on a previously-identified and long-standing research stumbling block
List of changes
- We have revised the article in many places and updated the references based on the two reports. Please, see our point-to-point responses, which also describe the corresponding changes. - In addition, we have corrected a few minor errors as listed below. (The equation numbers below refer to the revised manuscript.) - The vector of first moments was mistakenly defined with an extra minus sign in Eq. (6). We have corrected this throughout the text. - We have corrected minor errors in Eqs. (33), (34), (54), and (56). - In Fig. 5, the covariance curve for the BS system contained an extraneous prefactor of 0.0025, which has been corrected. - We have clarified that the expressions in Appendix A.4 refer to the normal-ordered Riccati equation in the first paragraph of the corresponding section.
Current status:
In refereeing
Reports on this Submission
Report #1 by
Anonymous
(Referee 2) on 2025-3-5
(Invited Report)
Report
The authors have addressed my comments properly. I recommend that the paper is accepted.
Recommendation
Publish (meets expectations and criteria for this Journal)
