Extensions of One Axis Twisting to more than One Degree of Freedom*

The creation and manipulation of quantum entanglement is central to improving precision measurements. A principal method of generating entanglement for use in atomic interferometry is the process of spin squeezing, whereupon the states become more sensitive to SU(2) rotations. This entanglement is typically generated via one axis twisting (OAT), where a many particle entangled state of one degree of freedom is generated by a non-linear Hamiltonian. We introduce a straightforward extension of OAT that creates squeezing and entanglement across two distinct degrees of freedom. We present this extension in the physical context of entanglement between collective atomic dipoles, and discrete collective momentum states. Our method may be considered as an SU(4) extension of the typical SU(2) one axis twisting. The states therefore prepared by this system are sensitive to SU(4) rotations, and therefore are a potential platform for realizing more intricate measurement schemes.