Spatial compression of a particle state in a parabolic potential by spin measurements

We propose a scheme for engineering compressed spatial states in a two-dimensional parabolic potential with a spin-orbit coupling by selective spin measurements. This sequence of measurements results in a coordinate-dependent density matrix with probability maxima achieved at a set of lines or at a two dimensional lattice. The resultant probability density depends on the spin-orbit coupling and the potential parameters and allows one to obtain a broad class of localized pure states on demand. The proposed scheme can be realized in spin-orbit coupled Bose-Einstein condensates.