Abstract
We investigate a system of two identical and distinguishable spins 1/2, with a direct magnetic dipole–dipole interaction, in an external magnetic field. Constraining the hyperfine tensor to exhibit axial symmetry generates the notable symmetry properties of the corresponding Hamiltonian model. In fact, we show that the reduction of the anisotropy induces the invariance of the Hamiltonian in the subspace of the Hilbert space of the two spins in which invariably assumes its highest eigenvalue of 2. By means of appropriate mapping, it is then possible to choose initial density matrices of the two-spin system that evolve in such a way as to exactly simulate the time evolution of a pseudo-qutrit, in the sense that the the actual two-spin system nests the subdynamics of a qutrit regardless of the strength of the magnetic field. The occurrence of this dynamic similitude is investigated using two types of representation for the initial density matrix of the two spins. We show that the qutrit state emerges when the initial polarizations and probability vectors of the two spins are equal to each other. Further restrictions on the components of the probability vectors are reported and discussed.
Highlights
The new routes to using spin systems in such fields of investigations [4,5,6,7,8] have raised a vast gamut of physical problems which have not been considered in standard NMR and ESR applications of spin systems and their hyperfine interactions
There is, in particular, a renewed interest in the dynamic problem of two interacting identical spins as considered from the perspective of quantum information technology, which differs from the statement of the problem in traditional NMR research
The knowledge of the exact eigenvalues of Ĥas enables the evaluation of the corresponding eigenvectors, and the investigation of the dynamics of the two-spin system, as well as that of the nested qutrit; in the rest of this study, we will turn our attention to general connections between the properties exhibited by the density matrix of the two spins and that of the emerged qutrit
Summary
It is interesting to tackle this problem first for a system of two interacting and neighboring identical nuclear spins in a crystal lattice We assume that their spatial configuration is such that their physical properties are experimentally accessible individually. Diatomic molecules with identical nuclei (e.g., the hydrogen molecule H2 at sufficiently low, ideally zero, temperature), or approximately, by a pair of adjacent protons or other identical isotopes in more complicated molecules Such molecules should be frozen in order to exclude the averaging of the direct nuclear magnetic dipole–dipole interaction over the different rotational states of the molecule. It is interesting to solve the physical problem of two interacting identical nuclear spins that are nearest neighbors in a crystal lattice (at zero or a sufficiently low temperature) or in some other “frozen” configuration.
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