Maximum-entropy method applied to the many-particle hierarchy problem in quantum-dot-microcavity systems

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The study of light-matter interaction in semiconductor quantum dots embedded into optical microcavities is a topical research field in condensed matter physics with many potential applications, such as ultra-low threshold micro- and nanolasers, single-photon sources, and sources of entangled photon pairs. The theoretical description of such driven-dissipative quantum many-particle systems in terms of the reduced density operator is, however, only feasible for small or highly symmetric systems. Approaches based on equations of motion of relevant expectation values are numerically much more efficient, but require to truncate the many-particle hierarchy at a suitable level and therefore only provide a subset of moments instead of the full statistics. In this project, we propose to apply the maximum entropy method, which was originally introduced in equilibrium statistical mechanics, to the many-particle hierarchy problem of non-equilibrium systems in two different ways. The first method still uses the results of conventional equations-of-motion approaches and allows to approximately determine the full statistics und substatistics such as the photon statistics of a microcavity laser. The second method goes much further by replacing the equations-of-motion approaches for stationary non-equilibrium problems by a novel scheme which has three important advantages: (i) it does not require any factorization scheme to truncate the many-particle hierarchy, (ii) avoids the numerical integration of equations of motion, and (iii) gives access to the full statistics. The purpose of the project is to study in detail both methods with focus on semiconductor quantum-dot microcavity systems. Once completed, we expect not only to have developed an highly efficient scheme to solve driven-dissipative quantum many-particle problems, but also to have gained a deeper understanding of the many-particle hierarchy and its truncation.