Algorithmic cooling of nuclear spins using long-lived singlet order

Abstract

Algorithmic cooling methods manipulate an open quantum system in order to lower its temperature below that of the environment. We achieve significant cooling of an ensemble of nuclear spin-pair systems by exploiting the long-lived nuclear singlet state, which is an antisymmetric quantum superposition of the "up" and "down" Zeeman states. The effect is demonstrated by nuclear magnetic resonance (NMR) experiments on a molecular system containing a coupled pair of near-equivalent 13 C nuclei. The populations of the system are subjected to a repeating sequence of cyclic permutations separated by relaxation intervals. The long-lived nuclear singlet order is pumped well beyond the unitary limit. The pumped singlet order is converted into nuclear magnetization which is enhanced by 21% relative to its thermal equilibrium value.

Representation of population exchange at level anti-crossings

“…will mostly consider spin order transfer pathways with a single permutation. Nevertheless, we also discuss cases where more complex permutations come into play (Rodin et al, 2020).…”

“…Basis "rotation" becomes an important concern in some NMR experiments: an example is given 265 by our recent work (Rodin et al, 2020) on "algorithmic cooling" of a spin system exploiting long-lived singlet order. The protocol for algorithmic cooling requires specific permutations of state populations in a four-level system, which are carried out by using NMR pulses with adiabatically increased or decreased field strength (which make use of adiabatic passage through LACs).…”

“…Such pulses not only swap state populations but also rotate the basis of spin eigenstates. Consequently, additional pulses are required to 270 compensate for this effect (Rodin et al, 2020). Examples, in which basis rotation is taking place, are discussed below.…”

Representation of population exchange at level anti-crossings

“…A related field is singlet-state NMR ; Carravetta and , dealing with slowly relaxing symmetry-protected spin states, which can be used to probe various slow processes and to store nonequilibrium spin polarization. In many molecules Stevanato et al, 2015;Sheberstov et al, 2019;Zhou et al, 2017;Wang et al, 2017;Buratto et al, 2014;Vasos et al, 2009;Zhang et al, 2015;Franzoni et al, 2012;DeVience et al, 2013) singlet-order relaxes much longer than spin magnetization for the reason that it is immune to some relaxation mechanisms, for instance, in a two-spin system dipolar relaxation cannot drive singlettriplet transitions because the dipole-dipole interaction is invariant to the exchange of the two spins (Pileio, 2010). In singlet-state NMR experiments, spin magnetization is converted into singlet order by a suitable pulse sequence; singletstate readout is also done by singlet-to-magnetization conversion using special pulse sequences.…”

“…In the cases of PHIP and singlet-state NMR a consideration of LACs often becomes important, in particular, in molecules with pairs of nearly equivalent spins (Ivanov et al, 2014;Pravdivtsev et al, 2013b;Franzoni et al, 2013Franzoni et al, , 2012Sheberstov et al, 2019;Stevanato et al, 2015;De-Vience et al, 2013;Theis et al, 2014a), such that the symmetry breaking is due to a very small chemical shift difference of the nuclei or due to their magnetic non-equivalence, i.e., due to slightly different couplings to other spins. Such symmetry breaking is usually a minor effect, giving rise to spin mixing only under special conditions, which correspond to LACs.…”