Entanglement of dipolar coupling spins

TL;DR

This paper demonstrates that dipolar-coupled spins can become entangled under thermodynamic equilibrium in low magnetic fields, with maximum entanglement when the field is perpendicular to the inter-nuclear vector, providing analytical expressions for concurrence.

Contribution

It reveals that entanglement can occur at equilibrium in low magnetic fields and derives exact formulas for concurrence considering realistic dipole interactions.

Findings

Entanglement appears at equilibrium in low magnetic fields.

Maximum concurrence occurs when the field is perpendicular to the inter-nuclear vector.

Analytical expression for concurrence was obtained for realistic dipole interactions.

Abstract

Entanglement of dipole-dipole interacting spins 1/2 is usually investigated when the energy of interaction with an external magnetic field (the Zeeman energy) is greater than the energy of dipole interactions by three orders. Under this condition only a non-equilibrium state of the spin system, realized by pulse radiofrequence irradiations, results in entanglement. The present paper deals with the opposite case: the dipolar interaction energy is the order of magnitude or even larger than the Zeeman one. It was shown that entanglement appears under the thermodynamic equilibrium conditions and the concurrence reaches the maximum when the external field is directed perpendicular to the vector connecting the nuclei. For this direction of the field and a system of two spins with the Hamiltonian accounting the realistic dipole-dipole interactions in low external magnetic field, the exact analytical expression for concurrence was also obtained. The condition of the entanglement appearance and the dependence of concurrence on the external magnetic field, temperature, and dipolar coupling constant were studied.