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arXiv:1204.4872 · Physics Letters A

An explicit criterion for existence of the Magnus solution for a coupled spin system under a time-dependent radiofrequency pulse

Xijia Miao

Published Jul 1, 2000• 5 citations
arXivHTMLPDF
DOI
10.1016/s0375-9601(00)00373-x

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The Journal of Chemical Physics•Nov 1, 1986•31 citations
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.
In this article the dynamics of the qubits states based on solution of the time-dependent Schrödinger equation is investigated. Using the Magnus method we obtain an explicit interpolation representation for the propagator, which allows to find wave function at an arbitrary time. To illustrate the effectiveness of the approach, the population of the levels a single and two coupled qubits have been calculated by applying the Magnus propagator and the result have been compared with the numerical solution of the Schrödinger equation. As a measure of the approximation of the wave function, we calculate fidelity, which indicates proximity when the exact and approximate evolution operator acts on the initial state. We discuss the possibility of extending the developed methods to multi-qubits system, when high-speed calculation methods of the operators of evolution is particularly relevant.
Two difficulties arise in the Magnus expansion for the evolution operator of a periodic time dependent Hamiltonian when the eigenvalues of the Hamiltonian are separated by more than 2π/τ, where τ is the period: (1) The Magnus expansion generally does not converge, and (2) the equilibrium properties of the system cannot be determined by statistical mechanical techniques analogous to those for time independent systems. A transformation is introduced that folds the eigenvalue spectrum into the interval −π/τ<λi<π/τ, yet retains the periodicity of the Hamiltonian, thereby alleviating the above difficulties. The method is compared to the approach taken by Shirley [Phys. Rev. B 138, 979 (1965)] for the Floquet solution to the evolution operator. It is applied to a dipole coupled spin system in an oscillating transverse field and the implications of the folding transformation on the spin thermodynamics of this system and the ‘‘Magnus paradox’’ are discussed.