Engineering multipartite coupling in doubly pumped parametric down-conversion processes

We investigate the quantum state generated by optical parametric down-conversion in a χ(2) medium driven by two noncollinear light modes. The analysis shows the emergence of multimode, namely, three- or four-mode, entangled states in a subset of the spatiotemporal modes generated by the process. These appear as bright spots against the background fluorescence, providing an interesting analogy with the phenomenology recently observed in two-dimensional nonlinear photonic crystals. We study two realistic setups. (1) Noncritical phase matching in a periodically poled lithium tantalate slab, characterized by a three-mode entangled state. (2) A type I setup in a beta-barium borate crystal, where the spatial walk-off between the two pumps can be exploited to make a transition to a four-mode entangled state. In both cases, we show that the properties of the state can be controlled by modulating the relative intensity of two pump waves, making the device a versatile tool for quantum state engineering.