How fast can quantum coherence spread?
Christoph Eigen and Zoran Hadzibabic
Cavendish physicists have observed a fundamental rate limit for the spreading of quantum coherence, the process by which a system syncs into a single quantum state. The existence of such a limit has profound implications for disparate fields, from quantum computing to the evolution of the early universe.
The discovery of fundamental limits on the rates of physical processes has repeatedly reshaped our understanding of nature. The speed of light constrains how quickly information can propagate through space-time, while other limits bound the rates of quantum evolution or the spreading of correlations in many-body systems. Such limits are not just curiosities: they reveal which quantities truly set the pace of a physical process, and they can have profound technological implications.
In our recent work, we uncovered a new such limit: a universal limit for the rate at which coherence spreads during the formation of a Bose–Einstein condensate.
A Bose-Einstein condensate (BEC) is a state of matter that appears when an ensemble of bosons – particles that can share the same quantum state – is cooled to ultralow temperatures, typically below a millionth of a degree above absolute zero. In this regime, the particles no longer behave as independent objects. Instead, they form a unified macroscopic quantum object that behaves coherently and exhibits wave-like properties.
Since their first experimental realisation in 1995, BECs have become a routine part of ultracold-atom laboratories around the world. Yet, a foundational question has remained largely unanswered: once a condensate begins to form, how long does it take for coherence to spread across truly macroscopic length scales?
Photograph of our experimental apparatus used to create homogeneous Bose gases consisting of ultra cold potassium atoms, with lead authors Gevorg Martirosyan (left) and Martin Gazo (right) tweaking the setup.
We began by preparing a gas with very low energy per particle and then drove it far from thermodynamic equilibrium using external magnetic forces. In this way, we created initial states that were highly disordered and incoherent, but still ‘cold’ enough that, in equilibrium, they should feature a sizable BEC. We then let the cloud relax in isolation, with its subsequent evolution driven only by interactions between the atoms.
Images: Cartoon illustration of coherence growth during Bose-Einstein condensation. Starting from an initially disordered phase field (at t = 0), locally coherent regions emerge and grow over time, gradually establishing phase coherence across the box (which is cylindrical in our experiments). The colour scale indicates the local quantum phase from 0 to 2 π.
To monitor the relaxation dynamics, we measured the momentum distribution of the gas at different evolution times, repeating the experiment many times under otherwise identical conditions. As the system relaxed, we observed a gradual narrowing of the momentum distribution, signalling the growth of coherence in real space.
More quantitatively, we extracted the coherence length, denoted ℓ, which characterises the distance over which different parts of the gas ‘know’ about one another’s quantum phase. During condensate formation, coherence does not appear everywhere at once. Instead, locally coherent regions form, grow, and merge, in a process known as coarsening. We found that, after an initial transient stage, the square of the coherence length grows linearly in time. This behaviour is a hallmark of universal scaling dynamics far from equilibrium.
The most striking result emerged when we varied the microscopic details of the system, for example the strength of the interparticle interactions that drive thermalisation. As expected from decades of cold-atom intuition, stronger interactions make the condensate appear sooner: they accelerate the initial approach to local equilibrium.
However, once the system enters the universal coarsening regime, the rate at which the square of the coherence length, ℓ2, grows is always the same. Initial conditions, density, interaction strength, and even the system size, affect how the system reaches the universal regime, but not the rate at which coherence ultimately spreads. Our measurements reveal a universal limit D = dℓ²/dt = (3.4+0.3) ħ/m, set solely by the ratio of the reduced Planck constant (ħ) and the particle mass (m). For our potassium-39 atoms, this corresponds to 5.5 μm² per millisecond.
This result has striking consequences for the formation of coherence over truly macroscopic distances. Since the coherence length grows only as the square root of time, extending coherence over large length scales rapidly becomes slow. At the rate we measure, it would take centuries for coherence to spread by this mechanism through a potassium cloud the size of a swimming pool. Even for particles as light as neutrons, for coherence to spread across planetary distances would take far longer than the age of the Universe.
"This result has striking consequences for the formation of coherence over truly macroscopic distances."
Uncovering a universal limit on the rate at which coherence can spread. (a) Momentum-space snapshots of a relaxing gas for two interaction strengths differing by a factor of four. As coherence develops, the broad incoherent distribution narrows into a sharp central peak. In line with decades of ultracold-atom intuition, stronger interactions do accelerate the early relaxation, bringing the gas into the coherent regime sooner. (b) However, the evolution of the extracted coherence area ℓ² reveals two distinct stages of the dynamics: the initial, non-universal regime that depends on the interaction strength (open symbols), and the coarsening regime in which ℓ² grows linearly, and at essentially the same rate in both cases.
Our work connects two complementary theoretical pictures of how order develops in a quantum fluid. In one, coherence spreads through wave-like excitations of the phase field. In the other, the growth of order is linked to the decay of a turbulent spaghetti-like tangle of vortex filaments. Our measurements do not yet distinguish which microscopic mechanism dominates, or whether the two are naturally intertwined. What they do show, is that the macroscopic coarsening rate is universal.
The implications reach well beyond ultracold atoms. Far-from-equilibrium ordering appears in many areas of physics, from superfluids and magnets to quark-gluon plasmas and early-universe cosmology. Ultracold gases provide a uniquely clean and tunable platform for studying these questions, because interactions, density, geometry, and dimensionality can all be controlled precisely. Our experiment offers a quantitative benchmark for theories of universality far from equilibrium: the search for simple laws governing the evolution of complex many-body systems.
Looking forwards, a major challenge is to determine how general this limit is.
Similar measurements in lower-dimensional gases, strongly interacting systems, mixtures, driven systems, or lattice geometries could reveal whether analogous bounds govern the growth of coherence in other forms of quantum matter. Direct probes of vortices and other phase defects are underway, and could help clarify the microscopic mechanism behind the observed coarsening.
There are also practical implications. Many quantum technologies rely on producing and maintaining coherence over large distances or across many degrees of freedom. Our work shows that, at least when coherence is generated by relaxation in an isolated system, its growth is subject to an intrinsic many-body constraint. This is relevant not only for engineered quantum systems, but also for large natural quantum systems, including neutron stars and possible dark-matter candidates.
Reference: Martirosyan et al., ‘A universal speed limit for spreading of coherence.’ Nature (2025). DOI: 10.1038/s41586-025 09735-z
Christoph Eigen is Assistant Research Professor in the Quantum Gases and Collective Phenomena Group.
Zoran Hadzibabic is Professor of Physics and head of the Quantum Gases and Collective Phenomena Group.