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We consider an electronic three-level system with two dipole-allowed transitions that are resonantly excited with two single-mode quantum fields, respectively. The interaction is described with a Jaynes-Cummings type model. In such a fully-quantized system, quantum correlations between initially independent quantum fields are found to arise. Their theoretical analysis is an important but challenging task since each field appears in a mixed state and the known criteria of entanglement are not suitable for such a multi-partite case. Here, we present a detailed insight into the formation of such correlations by using the cluster-expansion approach. With this approach, the hierarchy problem that arises due to the light-matter interaction can be truncated and analyzed by classifying many-body quantities systematically into clusters and omitting clusters above a predefined size. This leads to explicit expressions for the correlated part of high-order N-particle operators, which do not allow for further factorization. In our case, we consider N-particle operators that are composed of at least one bosonic operator of the respective fields, where the number of bosonic operators is limited by the chosen maximum cluster size. The obtained correlated parts are processed into a single measure for the correlation between the fields. We perform simulations based on the obtained equations for the expectation value of the correlated parts, which allow a deeper insight into the formation of quantum correlations and to study the contribution and behavior of different cluster sizes. Numerical results for the correlation between the two quantum fields are presented and discussed.
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