Maximum-entropy method applied to micro- and nanolasers

Abstract

During the last decades, the principle of maximum entropy established itself as a fundamental principle of consistent reasoning and hence has been successfully applied in many fields inside and outside of physics. Whenever only partial information is available it is an ideal tool to derive full statistical distributions or, in the case of quantum-mechanical descriptions, the full density matrix in a least-biased manner by exactly reproducing the given a-priori information while being maximally non-committal otherwise. In this thesis the maximum-entropy method is applied to the context of micro- and nanolasers. While tremendous improvements concerning the miniaturization and efficiency of these lasers have been achieved in the recent years, their light is still most often characterized in terms of the first statistical moments of the photon-number distribution, e.g., the light intensity and auto-correlation function. Lately, however, direct measurement of the full photon distribution became available experimentally, and theoretically its availability is highly desirable since it contains the full information about the system and therefore enables a deeper insight into the underlying physical processes. The application of the maximum-entropy method in this work is threefold. Firstly, a birthdeath model that describes the interaction of a cavity light-field with a quantum-dot system is considered as a benchmark. The full photon-number distribution which is available via conventional methods here is compared to the maximum-entropy distributions that are obtained when known photon moment values are used as input information. Secondly, a combination of the maximum-entropy method and equation-of-motion approaches is proposed where output quantities of the latter are used as input for the former. Apart from the access to the full photon-number distribution, the method also provides the possibility to define an unambiguous threshold even for cases where usual definitions fail. Lastly, the maximum-entropy method is used as a stand-alone approach without the need for moment values from external sources. Instead, knowledge of the stationarity of several distinct observables is used as an input to derive the least biased steady state. By doing so, the many-particle hierarchy problem that arises in conventional equation-of-motion techniques is circumvented. Moreover, the numerical determination of the full density matrix is enabled and access to all relevant expectation values and the full statistics is obtained. The results are compared to the numerically exact solution of the von Neumann-Lindblad equation for a four-level system resonantly coupled to a single cavity mode. Moreover, usage of the maximum-entropy method as a trial-and-error approach to identify the relevant processes of quantum systems is exemplified.