A quantum Pascal pyramid and an extended de Moivre-Laplace theorem

Abstract

Pascal's triangle is widely used as a pedagogical tool to explain the "first-order" multiplet patterns that arise in the spectra of $I_N S$ coupled spin-1/2 systems in magnetic resonance. Various other combinatorial structures, which may be well-known in the broader field of quantum dynamics, appear to have largely escaped the attention of the magnetic resonance community with a few exceptions, despite potential usefulness. In this brief set of lecture notes, we describe a "quantum Pascal pyramid" (OEIS https://oeis.org/A268533) as a generalization of Pascal's triangle, which is shown to directly map the relationship between multispin operators of arbitrary spin product rank $q$ ($\hat{Z}_N^q$) and population operators for states with magnetic quantum number $m$ ($\hat{S}_N^m$), and - as a consequence - obtain the general form of the intensity ratios of multiplets associated with antiphase single-quantum coherences, with an expression given in terms of the Jacobi polynomials. An extension of the de Moivre-Laplace theorem, beyond the trivial case $q=0$, is applied to the $q$-th columns of the quantum Pascal pyramid, and is given in terms of a product of the $q$-th order Hermite polynomials and a Gaussian distribution, reproducing the well-known functional forms of the solutions of the quantum harmonic oscillator and the classical limit of Hermite-Gaussian modes in laser physics (Allen et al., $\textit{Phys. Rev. A.}$, $\textbf{45}$, 1992). This is used to approximate the Fourier-transformed spectra of $\hat{Z}_N^q$-associated multiplets of arbitrary complexity. Finally, an exercise is shown in which the first two columns of the quantum Pascal pyramid are used to calculate the previously known symmetry-constrained upper bound on $I_z \rightarrow S_z$ polarization transfer in $I_N S$ spin systems.