Physics-Uspekhi

Magnus expansion paradoxes in the study of equilibrium magnetization and entanglement in multi-pulse spin locking

E. I. Kuznetsova, Edward B. Fel’dman, Dima E. Feldman

Published May 26, 2016• 17 citations

Abstract

Divergence of the Magnus expansion leads to paradoxes in the spin dynamics of solid-state NMR and in quantum informatics. This review presents results on quasi-equilibrium magnetization in a system of dipole-dipole (DD) coupled spins at times in multiple-pulse spin locking ( is the transverse spin relaxation time and is the rotating-frame spin-lattice relaxation time). It is shown how contradictions between the results obtained with the Magnus expansion and experimental data can be removed. Systems of two and three DD coupled spins in multi-pulse spin locking are considered, and the entanglement evolution is investigated using both the Magnus expansion and the exact solution. The critical temperature for an entangled state is also found.

DOI: 10.3367/ufne.2016.02.037753

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Physics-Uspekhi

Magnus expansion paradoxes in the study of equilibrium magnetization and entanglement in multi-pulse spin locking

E. I. Kuznetsova, Edward B. Fel’dman, Dima E. Feldman

Published May 26, 2016• 17 citations

Publisher

Abstract

DOI: 10.3367/ufne.2016.02.037753

Related papers

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\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.

We investigate analytically and numerically the multiple-quantum (MQ) NMR dynamics in systems of nuclear spins

1∕2

coupled by the dipole-dipole interactions in the case of the dipolar ordered initial state. We suggest two different methods of MQ NMR. One of them is based on the measurement of the dipolar temperature in the quasiequilibrium state which is established after the time of order ${\ensuremath{\omega}}_{\mathit{loc}}^{\ensuremath{-}1}$ (${\ensuremath{\omega}}_{\mathit{loc}}$ is the dipolar local field) after the MQ NMR experiment. The other method uses an additional resonance ${(\ensuremath{\pi}∕4)}_{y}$ pulse after the preparation period of the standard MQ NMR experiment in solids. Many-spin clusters and correlations are created faster in such experiments than in the usual MQ NMR experiments and can be used for the investigation of many-spin dynamics of nuclear spins in solids.