Spatial multipartite entanglement and localization of entanglement

We present a simple model together with its physical implementation which allows one to generate multipartite entanglement between several spatial modes of the electromagnetic field. It is based on parametric down-conversion with $N$ pairs of symmetrically tilted plane waves serving as a pump. The characteristics of this spatial entanglement are investigated in the cases of zero as well as nonzero phase mismatch. Furthermore, the phenomenon of entanglement localization in just two spatial modes is studied in detail and shown to result in an enhancement of the entanglement by a factor $\sqrt{N}$.

Figure 1

Scheme for the generation of spatial multipartite entanglement with $2N$ symmetrically tilted pump waves (illustrated for $N=2$ and 4). The little circles represent the projections of the pump wave vectors in the $\mathit{xy}$ plane of the crystal entering face.

Figure 2

The plane depicted is the one passing through one of the $N$ pairs of opposed pumps on Fig. 1 (e.g., the $\mathit{yz}$ plane). The tilted pumps have wave vectors ${\mathbf{k}}_p(\pm {\mathbf{q}}_n)$ with $n$ ranging from $1$ to $N$. The transverse (vertical) components of these wave vectors are denoted $\pm {\mathbf{q}}_n$ [see below (1) for a definition].