Emergent entropy of exotic oscillators and squeezing in three-wave mixing process

Abstract We demonstrate the existence of entanglement in a system of harmonic oscillators placed in the noncommutative plane (“exotic oscillators”) by computing the entanglement entropy as measured by the von Neumann entropy of the reduced density matrix. It is explicitly verified that the entanglement arises from the noncommutativity, which controls the coupling strength between the spatial modes. This can easily be generalized to the case where the momentum components also satisfy noncommutative relations, so that the entire phase space becomes noncommutative. It is noted that the underlying mathematical structure is reminiscent of the Unruh effect and using the Landau problem in the presence of a harmonic interaction, a concrete physical realization has been illustrated. Finally, we show that phase-space noncommutativity can give rise to the non-classical effect of squeezing, which generally results from the non-linearity of a medium in a three-wave mixing process.