  Entangled photon pairs are essential for many applications in quantum
technologies. Recent theoretical studies demonstrated that different types of
entangled Bell states can be created in a constantly driven four-level quantum
emitter-cavity system. Unlike other candidates for the realization of the
four-level emitter, semiconductor quantum dots unavoidably interact with their
environment, resulting in carrier-phonon interactions. Surprisingly, phonons
change the entanglement of emitted photon pairs in a qualitative way, already
at low temperatures on the order of 4 K. While one type of Bell state can still
be generated using small driving strengths, the other type is suppressed due to
phonon interactions in strongly-confined quantum dots. The degree of
entanglement decreases with rising temperature and driving strength until it
vanishes at a certain parameter value. Because it remains zero afterward, we
encounter a phonon-induced transition between entangled and nonentangled photon
emission that resembles a phase transition. The transition occurs at
temperatures below 30 K and, independent of the driving strength, the
concurrence as a function of the reduced temperature is found to obey a power
law with exponent one near the transition point.
