Entanglement of polar molecules in pendular states

In proposals for quantum computers using arrays of trapped ultracold polar molecules as qubits, a strong external field with appreciable gradient is imposed in order to prevent quenching of the dipole moments by rotation and to distinguish among the qubit sites. That field induces the molecular dipoles to undergo pendular oscillations, which markedly affect the qubit states and the dipole-dipole interaction. We evaluate entanglement of the pendular qubit states for two linear dipoles, characterized by pairwise concurrence, as a function of the molecular dipole moment and rotational constant, strengths of the external field and the dipole-dipole coupling, and ambient temperature. We also evaluate a key frequency shift, $\triangle\omega$, produced by the dipole-dipole interaction. Under conditions envisioned for the proposed quantum computers, both the concurrence and $\triangle\omega$ become very small for the ground eigenstate. In principle, such weak entanglement can be sufficient for operation of logic gates, provided the resolution is high enough to detect the $\triangle\omega$ shift unambiguously. In practice, however, for many candidate polar molecules it appears a challenging task to attain adequate resolution. Simple approximate formulas fitted to our numerical results are provided from which the concurrence and $\triangle\omega$ shift can be obtained in terms of unitless reduced variables.