Fractality and macroscopic entanglement in two-component Bose-Einstein condensates

Spin coherent states are the matter equivalent of optical coherent states, where a large number of two component particles form a macroscopic state displaying quantum coherence. Here we give a detailed study of entanglement generated between two spin-$1/2$ Bose-Einstein condensates (BECs) due to an S₁ᶻS₂ᶻ interaction. The states that are generated show a remarkably rich structure showing fractal characteristics. In the limit of large particle number N, the entanglement shows a strong dependence upon whether the entangling gate times are a rational or irrational multiple of π/4, with a fractal dimension of d≈1.7. We discuss the robustness of various states under decoherence and show that despite the large number of particles in a typical BEC, entanglement on a macroscopic scale should be observable as long as the gate times are less than /J√N, where J is the effective BEC-BEC coupling energy. Such states are anticipated to be useful for various quantum information applications such as quantum teleportation and quantum algorithms.