  Since the realization of high-quality microwave cavities coupled to quantum
dots, one can envisage the possibility to investigate the coherent interaction
of light and matter in semiconductor quantum devices. Here we study a parallel
double quantum dot device operating as single-electron splitter interferometer,
with each dot coupled to a local photon cavity. We explore how quantum
correlation and entanglement between the two separated cavities are generated
by the coherent transport of a single electron passing simultaneously through
the two different dots. We calculate the covariance of the cavity occupations
by use of a diagrammatic perturbative expansion based on Keldysh Green's
functions to the fourth order in the dot-cavity interaction strength, taking
into account vertex diagrams. In this way, we demonstrate the creation of
entanglement by showing that the classical Cauchy-Schwarz inequality is
violated if the energy levels of the two dots are almost degenerate. For large
level detunings or a single dot coupled to two cavities, we show that the
inequality is not violated.
