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Molecular Physics

Symmetry and isotropic coherence transfer

John Listerud, Steffen J. Glaser, Gary P. Drobny

Published Feb 20, 1993• 27 citations
Publisher

Abstract

A general treatment of spin dynamics during coherence transfer is presented as an example of a many-body problem under strong coupling. In a previous paper, coherence transfer by isotropic mixing (IM) between two coupled spins was examined. However, the greatest experimental interest in the technique of IM centers on the transfer of coherence between spins that are not directly coupled. The dynamics of coherence transfer in multiple spin systems are cast in a superoperator formulation, in which diagonalization of the mixing Liouvilian matrix allows derivation of the coherence transfer frequencies from the eigenvalues. Development of a particular basis of the spin density operator space facilitates diagonalization of the strong coupling Liouvillian. This basis is completely reduced with respect to rotation, permutation, and particle number. Treatment of the many-body aspect is based on an approach using the Young tableaux. The exact calculation of the dynamics of coherence transfer in a three spin system is accomplished using the tableaux method, and closed form solutions are compared with results obtained using numerical methods. The degree to which practical coherence transfer pulse sequences deviate from ideal isotropic mixing, and the effect that these deviations have on symmetry-induced selection rules is calculated numerically.

DOI
10.1080/00268979300100431

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A general treatment of spin dynamics during coherence transfer is presented as an example of a many-body problem under strong coupling. In a previous paper, coherence transfer by isotropic mixing (IM) between two coupled spins was examined. However, the greatest experimental interest in the technique of IM centers on the transfer of coherence between spins that are not directly coupled. The dynamics of coherence transfer in multiple spin systems are cast in a superoperator formulation, in which diagonalization of the mixing Liouvilian matrix allows derivation of the coherence transfer frequencies from the eigenvalues. Development of a particular basis of the spin density operator space facilitates diagonalization of the strong coupling Liouvillian. This basis is completely reduced with respect to rotation, permutation, and particle number. Treatment of the many-body aspect is based on an approach using the Young tableaux. The exact calculation of the dynamics of coherence transfer in a three spin system is accomplished using the tableaux method, and closed form solutions are compared with results obtained using numerical methods. The degree to which practical coherence transfer pulse sequences deviate from ideal isotropic mixing, and the effect that these deviations have on symmetry-induced selection rules is calculated numerically.
We study multiple-spin coherence transfers in linear Ising spin chains with nearest-neighbor couplings. These constitute a model for efficient information transfers in future quantum computing devices and for many multidimensional experiments for the assignment of complex spectra in nuclear magnetic resonance spectroscopy. We complement prior analytic techniques for multiple-spin coherence transfers with a systematic numerical study where we obtain strong evidence that a certain analytically motivated family of restricted controls is sufficient for time optimality. In the case of a linear three-spin system, additional evidence suggests that prior analytic pulse sequences using this family of restricted controls are time optimal even for arbitrary local controls. In addition, we compare the pulse sequences for linear Ising spin chains to pulse sequences for more realistic spin systems with additional long-range couplings between nonadjacent spins. We experimentally implement the derived pulse sequences in three- and four-spin systems and demonstrate that they are applicable in realistic settings under relaxation and experimental imperfections---in particular---by deriving broadband pulse sequences which are robust with respect to frequency offsets.
Exact diagonalization and other numerical studies of quantum spin systems are notoriously limited by the exponential growth of the Hilbert space dimension with system size. A common and well-known practice to reduce this increasing computational effort is to take advantage of the translational symmetry CN{C}_{N}CN​ in periodic systems. This represents a rather simple yet elegant application of the group theoretical symmetry projection operator technique. For isotropic exchange interactions, the spin-rotational symmetry SU(2)SU(2)SU(2) can be used, where the Hamiltonian matrix is block structured according to the total spin and magnetization quantum numbers. Rewriting the Heisenberg Hamiltonian in terms of irreducible tensor operators allows for an efficient and highly parallelizable implementation to calculate its matrix elements recursively in the spin-coupling basis. When combining both CN{C}_{N}CN​ and SU(2)SU(2)SU(2), mathematically, the symmetry projection technique leads to ready-to-use formulas. However, the evaluation of these formulas is very demanding in both computation time and memory consumption, problems which are said to outweigh the benefits of the symmetry-reduced matrix shape. We show a way to minimize the computational effort for selected systems and present the largest numerically accessible cases.
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The density matrix is a very convenient way to do calculations for a large number of spin systems and is developed from the fundamental ideas of quantum mechanics. The idea of pure and mixed states is introduced and used to develop an expression for the single-spin density matrix as it applies to NMR spectroscopy. Multiple-spin density matrices are produced by using the outer product of single-spin matrices and their commutation properties are examined. A very useful form of the scalar coupling density operator is developed and used, along with the chemical shift operator, to calculate the spectrum of a weakly coupled two-spin system. The density operator of spin-1 is explored briefly. Density matrix calculations in perturbed systems are investigated in preparation for relaxation calculations. Strong coupling is discussed and compared to the weak coupling approximation, using the full scalar coupling Hamiltonian and matrix diagonalization techniques.
Numerous examples in condensed matter physics possess an effective Hamiltonian structure involving two-body anharmonic interactions between elementary excitations. The effective second-quantised Hamiltonian used to model these systems is essentially of the same form except for the particulars of the interaction parameters and the statistics of the particles which are interacting. The author investigates such systems in the limit of strongly anharmonic interactions which may possibly lead to symmetry breaking. He introduces quantum field operators and calculates their equations of motion. The interaction coefficients are expanded in a Taylor series about the critical mode to second order to illustrate the method, but he gives the expansion to infinite order for completeness. It is demonstrated that both in first and second order the equations of motion are reduced to a system of highly nonlinear coupled partial differential equations (PDE) for the field operator. He shows that the form of the second-order equation is exact due to renormalisation theory, in the case where the Hamiltonian coefficients represent operators in real space which are isotropic, and incorporates symmetries and topologies which would appear in an expansion to infinite order. This special situation arises in a very wide range of physical examples. These equations can be solved exactly using a particular ansatz in conjunction with recent mathematical developments in the field of non-linear analysis. In particular, the method of symmetry reduction for PDE provides a complete set of symmetry variables for the type of equations used in the study.