Quantum Entanglement in Quasi-Equilibrium States in the Multipulse Spin Locking of the Nuclear Magnetic Resonance

Abstract

Quantum entanglement is studied NMR multipulse spin locking, when a system of spins coupled by the dipole–dipole interaction in a strong magnetic field is irradiated by a sequence of resonant high-frequency $$\varphi $$ -pulses with the same time delay $$2\tau $$ between successive pulses. For times $$t \sim {{T}_{2}}$$ ( $${{T}_{2}} \approx \omega _{{{\text{loc}}}}^{{ - 1}}$$ , $${{\omega }_{{{\text{loc}}}}}$$ is determined by the dipole–dipole interaction), quasi-equilibrium with a structure depending on the relation between the pulsed field $${{\omega }_{1}} = \varphi {\text{/}}2\tau $$ and $${{\omega }_{{{\text{loc}}}}}$$ is established in the system. For $${{\omega }_{1}} \lessapprox {{\omega }_{{{\text{loc}}}}}$$ , a one-temperature quasi-equilibrium state arises, while a quasi-equilibrium with the dipole and Zeeman temperatures appears for $${{\omega }_{1}} \gg {{\omega }_{{{\text{loc}}}}}$$ . The temperature dependence of entanglement is investigated for $${{\omega }_{1}} \lessapprox {{\omega }_{{{\text{loc}}}}}$$ . For $${{\omega }_{1}} \gg {{\omega }_{{{\text{loc}}}}}$$ , the further evolution of the system is determined by the Provotorov equations and leads to the equalization of the dipole and Zeeman temperatures. It is shown that the entanglement is absent in this case.