# Photon Counting Statistics of a Microwave Cavity Coupled with Double Quantum Dots
## 1 Introduction  
Counting statistics is now at the center of attention in the quantum statistical description of a given process, counting the random number of events up to a given point. For example, one can count up to a given time the number of photons emitted by a quantum optical system and the number of electrons transported through a nanoscopic structure. The distribution of waiting times between elementary physical events is an important tool to investigate temporal correlations and the counting statistics of stochastic processes [1]. Short-time physics and correlations can be described by waiting time distributions (WTDs), which give the distribution of the time interval between two successive events. Additionally, most counting statistical properties can be obtained from the WTDs alone in Markovian and renewal systems [2]. WTDs play a significant role in various branches of science and technology--for instance, in quantum optics [3; 4] and nanososcopic electron transport [5; 6]. Moreover, the theory of large deviations (LD) is concerned with the exponential decay of probabilities of large fluctuations in random systems. For a long time limit, the measurement of random systems will focusaround the average value, and large deviations away from this value are exponentially suppressed [7]. The LD method provides a deep description of a stochastic counting process, and it can be applied to characterize photon counting processes in different quantum optical systems [8; 9] and electron transport properties through mesoscopic conductors [10; 11]. WTDs and the LD method are particularly useful and represent a fundamental theory and tools in investigating nanososcopic devices. Experimentally, obtaining all cumulants of photon or electron counting distribution requires time-resolved single-particle detection techniques. By capacitively coupling a single-electron transistor to a quantum dot (QD), real-time counting statistics has been carried out experimentally by detecting single electron tunneling through a quantum dot [12; 13]. The single-electron detection has also been used to measure electron-electron interference in the QD system [14], as well as the full counting statistics of superconducting junctions [15]. Furthermore, full counting statistics and spin dynamics in a quantum dot have been investigated using resonance fluorescence [16]. Usually, most experiments have been carried out at ultralow temperatures in the milliKelvin range; however, recently, single-electron detection has begun to be operated at room temperature by the optical blinking of a nearby semiconductor nanocrystal [17]. As for the single photon detector, a quantum nondemolition detector for propagating microwave photons was realized [18]. Such single-photon detectors have a wide range of applications within quantum thermodynamics [19] and quantum information processing [20]. Further progress in the measurements of photon counting statistics in quantum coherent optics is anticipated in the coming years, and with the rapid progress in single-photon detection, the measurement of single photon counting statistics could soon be within reach. Theoretically, [PERSON] was the first one to study the WTDs of mesoscopic quantum transport by defining jump operators from a quantum master equation [21]. To date, WTDs have been investigated for double-quantum dots [22; 23], quantum dot spin-valves [24], non-Markovian transport [25], superconducting devices [26; 27], and coherent conductors [6; 28; 29]. Recently, the photon counting statistical characteristics of photon emission from a microwave cavity, such as WTDs and LD statistics, have been investigated [30]. Furthermore, photon counting statistics for a microwave cavity with multiple heat baths have also been studied.