Entanglement in massive coupled oscillators

This article investigates entanglement of the motional states of massive coupled oscillators.The specific realization of an idealized diatomic molecule in one-dimension isconsidered, but the techniques developed apply to any massive particles with two degreesof freedom and a quadratic Hamiltonian. We present two methods, one analyticand one approximate, to calculate the interatomic entanglement for Gaussian and non-Gaussian pure states as measured by the purity of the reduced density matrix. Thecases of free and trapped molecules and hetero- and homonuclear molecules are treated.In general, when the trap frequency and the molecular frequency are very different, andwhen the atomic masses are equal, the atoms are highly-entangled for molecular coherentstates and number states. Surprisingly, while the interatomic entanglement can be quitelarge even for molecular coherent states, the covariance of atomic position and momentumobservables can be entirely explained by a classical model with appropriately chosenstatistical uncertainty.