Quantum mechanics notoriously doesn’t play by the rules of the classical world. One of the most famous and fundamental examples of quantum weirdness is the idea that two observable quantities, such as a particle’s position and momentum, can be incompatible, or not simultaneously measurable. If you want to measure them both, the order in which you make the measurements matters. In mathematical terms, their commutator—momentum times position minus position times momentum—is nonzero.

When considering a time-evolving system, it can be helpful to talk about the so-called out-of-time-order (OTO) commutator between two quantities measured at different times—say, position at time 0 and momentum at time *t* > 0. From a theoretical perspective, there’s nothing paradoxical about the idea of making a later measurement first and an earlier measurement second. One need only write down a factor of *e ^{i}*

^{H}*in between, where*

^{t}**H**is the system’s Hamiltonian. And perhaps surprisingly, experimenters too have tricks for turning back the clock on a quantum system—although they can be challenging to implement—so OTO commutators make both mathematical and physical sense.

OTO commutators are especially useful in describing how a quantum system progresses toward thermal equilibrium—for example, how quantum information propagates as an initially localized spin excitation diffuses along a chain of interacting spins. At first, the states of spins at opposite ends of the chain are compatible observables: Measuring one has no effect on the other. As time goes by and each spin’s influence spreads across the system, their OTO commutator steadily increases.

But quantum mechanics has more surprises in store. For various reasons, a quantum system might not ever fully thermalize, even at an effectively infinite temperature. In those cases, the OTO commutator increases more slowly—or not at all.

So far, studies of OTO commutators have been mostly confined to relatively small or simple systems. Theorists are limited by their computational resources, and experimenters by the number of trapped ions or solid-state spins they can coherently manipulate. Neither group has yet been able to explore all the peculiarities of quantum mechanics in a thermalizing system.

Now MIT’s Paola Cappellaro and her colleagues are finding that the answer might lie in a surprisingly simple experimental setup. Rather than a meticulously crafted chain of spins and intricate network of lasers, they use an inexpensive natural crystal—bought for $10 on eBay—and an ordinary NMR machine. The structure of the crystal, a material called fluorapatite, is shown in the figure; its fluorine atoms, shown in red, are arranged into columns that are well enough isolated from one another that they behave like long one-dimensional spin chains.

The researchers use established NMR methods to manipulate the effective Hamiltonian that the spin chains experience. (The principle of the technique, called average Hamiltonian theory, is described in the article by Clare Grey and Robert Tycko, *Physics Today*, September 2009, page 44.) They can thus tune the coupling between adjacent fluorine spins, turn on or off the disorder induced by other atoms in the crystal, and even run time backwards.

Notably, NMR is limited to making collective manipulations and methods; it’s not possible to control or observe the states of individual spins. But single-site addressability, it turns out, isn’t necessary. Collective control is enough to observe OTO commutators behaving in characteristically quantum ways. By tuning the effective Hamiltonian of their fluorapatite spin chains, Cappellaro and colleagues observed the distinct hallmarks of quantum systems that don’t thermalize and those that do. (K. X. Wei et al., *Phys. Rev. Lett.* **123**, 090605, 2019; see also K. X. Wei, C. Ramanathan, P. Cappellaro, *Phys. Rev. Lett.* **120**, 070501, 2018.)