Hyperpolarization and the Physical Boundary of Liouville Space

Abstract

Abstract. The quantum state of a spin ensemble is described by a density operator, which corresponds to a point in the Liouville space of orthogonal spin operators. Valid density operators are confined to a particular region of Liouville space, which we call the physical region, and which is bounded by multidimensional figures called simplexes. Each vertex of a simplex corresponds to a pure-state density operator. We provide examples for spins I = 1 / 2, I = 1, I = 3 / 2, and for coupled pairs of spins-1/2. We use the von Neumann entropy as a criterion for hyperpolarization. It is shown that the inhomogeneous master equation for spin dynamics leads to non-physical results in some cases, a problem that may be avoided by using the Lindbladian master equation.

Comment on mr-2021-26

“…The vertices of the triangle have coordinates {ρ 10 , ρ 20 } given by {±2 −1/2 , 6 −1/2 } and {0, −(2/3) 1/2 }; Each vertex corresponds to a purestate density operator, in which only one state is populated. Similar triangular bounds have been identified in the mathematics literature (Kimura and Kossakowski (2005); Goyal et al (2016)).…”

“…", etc. Surprisingly, these natural questions are rarely posed in the NMR world, although they have not escaped the attention of mathematical physicists and applied mathematicians (Byrd and Khaneja (2003); Kimura and Kossakowski (2005); Bengtsson and Zyczkowski (2006); Goyal et al (2016); Szymański et al (2018)).…”

Synergies between Hyperpolarized NMR and Microfluidics: A Review

“…We take "hyperpolarized" to mean any nuclear spin system strongly perturbed from its thermal equilibrium state that can yield enhanced NMR signals (a more quantitative analysis of this topic is given in Ref. [4]).…”