Valley entanglement of carriers in monolayers of transition-metal dichalcogenides

The entanglement of two quantum systems or ensembles is usually generated as a result of coupling between them. This coupling can be mediated by classical electromagnetic fields. At the same time, one can also entangle \textit{non-interacting} quantum systems by a quantum field. Here we consider the optical excitation of electron-hole or exciton states near the band gap of a transition-metal dichalcogenide monolayer in two valleys K' and K with opposite valley indices. We show that a linearly polarized single-photon field in a cavity or a stationary stream of linearly polarized single photons gives rise to an efficient entanglement of non-interacting carriers in different valleys, i.e. the generation of electron states entangled with respect to the valley degree of freedom. An intuitive explanation of this effect is that the carriers ``view'' linearly polarized photons as entangled left- and right-circularly polarized photon states. Valley entanglement of carriers gives rise to peculiar properties of the reemitted optical field and photocurrent fluctuations.