Quantum thermalisation equation had missing link
The discovery could provide a better understanding of how quantum systems thermalise
Like ink dissolving in water, systems thermalise towards equilibrium. CQT Fellow Dario Poletti and his collaborators discovered a new link that could give a more complete picture of quantum thermalisation. Image credit: Shutterstock.com/SunnyMeansSixteen.
Researchers have uncovered a surprising connection in an equation used for 30 years to predict how quantum systems ‘thermalise’. The finding by CQT Fellow Dario Poletti and his collaborators is published in Physical Review X on 14 March 2025.
Dario is also Associate Professor and Head of Cluster at Singapore University of Technology and Design (SUTD) and a researcher under the MajuLab International Research Laboratory.
Thermalisation is a familiar concept: a body warmer or colder than its surroundings tends towards the environmental average, like a drink chilling in the fridge, or a lizard warming on a hot rock. While the physics that describes such processes at the macroscopic scale is well established, what happens in systems at the quantum scale is less well understood.
Dario and his collaborators looked at a popular framework to describe thermalisation in isolated quantum systems known as the ‘eigenstate thermalisation hypothesis’, or ETH. ETH might apply, for example, to model the behaviour of a gas of cold atoms or an array of superconducting qubits.
ETH is formulated as an equation with two parts, and those two parts were thought to be independent in principle. Until now.
In linking ETH’s two parts, the new work delivers a more complete picture of quantum thermalisation. Future experiments testing the idea might also show routes to engineer systems to have desired thermalisation behaviours.
WTH is ETH?
For a macroscopic view of thermalisation, imagine a gas in a box. Without following the individual motions of particles, the average force generated as the particles collide against the wall can be described as pressure. This observable will change smoothly as temperature changes, the pressure rising as the gas gets hotter.
ETH postulates that isolated quantum systems can be described in a similar way. It assumes there is some observable property of the eigenstates of the quantum system, such as average magnetisation, which will change smoothly with changes in the system’s overall energy. It also accounts for the thermalisation of correlations in the quantum system.
Mathematically, ETH has one function that describes the local observable and correlations when the system is in equilibrium, and a second function that describes them when the system is out of equilibrium, capturing the transient dynamics. The former function depends on the energy of the eigenstates while the latter depends on the energy differences between eigenstates. In matrix formulation, these correspond to the diagonal and off-diagonal parts of the matrix. These ideas were first proposed separately by Mark Srednicki and Josh Deutsch in the 1990s.
The missing link
Dario and colleagues noticed something new. “We realised that we could predict the thermalisation, which should be predicted by the dynamic portion of the ETH, just by looking at the equilibrium properties. It looked more and more likely that there must be a link between the two,” he says.
The idea emerged from discussion among the paper’s coauthors. “It was really serendipitous,” says Dario. He and Xu Xiansong, a former SUTD Research Fellow and group member, were discussing with Wang Jiaozi at the University of Osnabrück a numerical way of studying thermalisation. During a visit to Leonardo Mazza’s group at LPTMS at Université Paris-Saclay, Dario met Luca Capizzi who quickly found the mathematical expression for the connection.
Dario says, “The five of us put all of our forces together and came up with this more complete result.” Specifically, they found that the dependence of the first term on the energy determined the exponent of the second term describing the dynamics.
For example, if the first term shows a quadratic dependence in energy, the researchers could predict that thermalisation would be faster than if the first term had linear dependence in energy.
The researchers evaluated their findings with numerical simulations of two one-dimensional spin models – a spin-1 Ising chain with a tilted field and a spin-1 Ising chain with transverse field. The numerical simulations were consistent with their theoretical predictions.
They now hope to collaborate with experimentalists to verify the finding.
