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An exactly solvable model of a trapped interacting Bose-Einstein condensate(BEC) coupled in the dipole approximation to a quantized light mode in a cavityis presented. The model can be seen as a generalization of theharmonic-interaction model for a trapped BEC coupled to a bosonic bath. Afterobtaining the ground-state energy and wavefunction in closed form, we focus oncomputing the correlations in the system. The reduced one-particle densitymatrices of the bosons and the cavity are constructed and diagonalizedanalytically, and the von Neumann entanglement entropy of the BEC and thecavity is also expressed explicitly as a function of the number and mass of thebosons, frequencies of the trap and cavity, and the cavity-boson couplingstrength. The results allow one to study the impact of the cavity on the bosonsand vice versa on an equal footing. As an application we investigate a specificcase of basic interest for itself, namely, non-interacting bosons in a cavity.We find that both the bosons and the cavity develop correlations in acomplementary manner while increasing the coupling between them. Whereas thecavity wavepacket broadens in Fock space, the BEC density saturates in realspace. On the other hand, while the cavity depletion saturates, and hence doesthe BEC-cavity entanglement entropy, the BEC becomes strongly correlated andeventually increasingly fragmented. The latter phenomenon implies single-trapfragmentation of otherwise ideal bosons, where their induced long-rangeinteraction is mediated by the cavity. Finally, as a complimentaryinvestigation, the mean-field equations for the BEC-cavity system are solvedanalytically as well, and the breakdown of mean-field theory for the cavity andthe bosons with increasing coupling is discussed. Further applications areenvisaged.

Entanglement and correlations in an exactly-solvable model of a Bose-Einstein condensate in a cavity