Title: Waiting time statistics for a double quantum dot coupled with an optical cavity

Abstract: A double quantum dot coupled to an optical cavity is a prototypical example of a non-trivial open quantum system. Recent experimental and theoretical studies show that this system is a candidate for single-photon detection in the microwave domain. This motivates studies that go beyond just the average current, and also take into account the full counting statistics of photon and electron detections. With this in mind, here we provide a detailed analysis of the waiting time statistics of this system within the quantum jump unravelling, which allows us to extract analytical expressions for the success and failure probabilities, as well as for the inter detection times. Furthermore, by comparing single and multi-photon scenarios, we infer a hierarchy of occurrence probabilities for the different events, highlighting the role of photon interference events in the detection probabilities. Our results therefore provide a direct illustration of how waiting time statistics can be used to optimize a timely and relevant metrological task.