Describing how matter behaves at the quantum-mechanical level is notoriously hard, because the equations get so difficult to solve once there is more than a handful of particles involved. But a new experiment shows that the fine details might not matter too much – and that, if we “squint” at a many-particle quantum system to blur them, how the system changes over time can look surprisingly like the familiar classical process of diffusion.

Figuring out the exact trajectories of particles of ink in a glass of water as they are battered this way and that by water molecules is difficult. But you do not need to keep track of every molecule because. Fick’s law of diffusion says that the flow of material is simply proportional to its concentration gradient.

Fick’s law is an example of coarse-graining that is commonly used in hydrodynamics. For example, a fluid can be considered a collection of little “parcels”, each containing countless molecules, which move frictionally past one another.

### Local blobs

Researchers have recently sought to describe quantum-mechanical many-particle systems using such an approach. Say you have a material containing a bunch of quantum spins that interact with one another, you create a local “blob” of oriented spins and calculate how it will then spread through the system.

“Hydrodynamics is generally the study of how a system goes from local equilibrium to global equilibrium,” says Joel Moore of the University of California at Berkeley. The equations of fluid mechanics, he says, assume that any detailed information about the initial state – where the particles are and how they are moving – is quickly lost once they have experienced just a few interactions (collisions) with others. “Then the fluid equations describe everything on longer time scales, from microseconds to years, very accurately.”

To study an analogous system of quantum spins, Moore has collaborated with Berkeley physicist Norman Yao and others. They examined a tiny single crystal of diamond containing two types of spins, both arising from the unpaired electrons of defects in the carbon-crystal lattice. One kind of defect, called a P1 centre and consisting of a nitrogen atom in place of a carbon, was dispersed randomly through the lattice at a concentration of 100 ppm. The other defect, called an NV centre, consists of a nitrogen substitution next to an empty space in the lattice, was dispersed at a concentration about two hundred times lower.

### Spins with feelings

These spins can “feel” one another over long distances of many times the atomic separation. To see how the dynamics of the spins evolved, the researchers used the NV defects both to set up a perturbation and to probe the response. They could polarize these spins in one region using a laser pulse, and then transfer that localized disturbance to the more populous P1 spins by coupling the two using a magnetic field to bring them into resonance. Then they monitored how this perturbation spread through the P1 spins as the system moves towards its global equilibrium configuration.

“The lore is that if you take a uniform initial state and make a packet of excess energy or spin somewhere in the system, this packet will spread out according to some differential equation,” says Yao. For a quantum system, we might expect that to be the Schrödinger equation. But it would be very challenging to describe the process that way for all the interacting spins.

The measurements showed, however, that the overall dynamics were well described by a simpler equation that looked a lot like Fick’s diffusion. In other words, this strictly quantum process turns out to share much the same dynamics as a classical one. If you just measure the spin density at a slightly coarse-grained resolution, explains Yao, then “the differential equation that describes these dynamics can be much simpler than the Schrödinger equation, and instead like the diffusion equation”.

### Not a perfect match

However, the behaviour of the spins was not a perfect match for diffusion. This, says Moore, is partly because the spins, unlike colliding particles, can feel one another over long distances. But it also seems to stem from the fact that the P1 defects are not quite all equivalent: the atoms around each defect might have slightly different local arrangements, producing some random disorder.

Yao says that other theoretical studies suggest that other processes of “quantum hydrodynamics” can take different forms, equivalent to other types of coarse-grained classical dynamics. For example, some systems in which spins interact in 1D chains are predicted to have similar dynamics to the so-called Kardar–Parisi–Zhang equation, which governs some kinds of surface growth processes and the way shock waves propagate.

What this suggests is that, at a coarse-grained level, dynamical processes on many-body systems might be rather insensitive to whether they are governed by quantum or classical physics. There is a kind of universality that depends more on the general interactions between the coarse-grained components than on their microscopic details – an echo, perhaps, of the universality seen in magnetic spin systems and classical fluids close to a critical phase transition.

### Active field

“Understanding this emergence in detail is challenging for microscopic entities that are quantum, and equally so, in different ways, for microscopics that are classical,” says David Huse of Princeton University in New Jersey. This, he says, “is a long-standing enterprise in the fundamentals of statistical physics, that has become rather active recently for quantum microscopics because of the improved ability to explore it in the lab”.READ MOREClassical approach extends the range of noisy quantum computers

The experimental work “is an impressive feat”, says Arijeet Pal of University College London. “The demonstration that there is a different hydrodynamic regime from conventional diffusion is an essential first step towards understanding the different dynamical universality classes allowed by quantum physics.”

The research is described in *Nature*.