Abstract Predicting the dynamics of quantum systems far from equilibrium represents one of the most challenging problems in theoretical many-body physics1, 2. While the evolution of a many-body system is in general intractable in all its details, relevant observables can become insensitive to microscopic system parameters and initial conditions. This is the basis of the phenomenon of universality. Far from equilibrium, universality is identified through the scaling of the spatio-temporal evolution of the system, captured by universal exponents and functions. Theoretically, this has been studied in examples as different as the reheating process in inflationary Universe cosmology3, 4, the dynamics of nuclear collision experiments described by quantum chromodynamics5, 6, and the post-quench dynamics in dilute quantum gases in non-relativistic quantum field theory7–11. However, an experimental demonstration of such scaling evolution in space and time in a quantum many-body system has been lacking. Here we observe the emergence of universal dynamics by evaluating spatially resolved spin correlations in a quasi-one-dimensional spinor Bose–Einstein condensate12–16. For long evolution times we extract the scaling properties from the spatial correlations of the spin excitations. From this we find the dynamics to be governed by an emergent conserved quantity and the transport of spin excitations towards low momentum scales. Our results establish an important class of non-stationary systems whose dynamics is encoded in time-independent scaling exponents and functions, signalling the existence of non-thermal fixed points10, 17, 18. We confirm that the non-thermal scaling phenomenon involves no fine-tuning of parameters, by preparing different initial conditions and observing the same scaling behaviour. Our analogue quantum simulation approach provides the basis with which to reveal the underlying mechanisms and characteristics of non-thermal universality classes. One may use this universality to learn, from experiments with ultracold gases, about fundamental aspects of dynamics studied in cosmology and quantum chromodynamics.
We explore the far-from-equilibrium dynamics of Bose gases in a universal regime associated to nonthermal fixed points. While previous investigations concentrated on scaling functions and exponents describing equal-time correlations, we compute the additional scaling functions and dynamic exponent
z characterizing the frequency dependence or dispersion from unequal-time correlations. This allows us to compare the characteristic condensation and correlation times from a finite-size scaling analysis depending on the system's volume.
We compute the far-from-equilibrium dynamics of relativistic scalar quantum fields in
3+1 space-time dimensions starting from over-occupied initial conditions. We determine universal scaling exponents and functions for two-point correlators and the four-vertex in a self-similar regime in time and space or momenta. The scaling form of the momentum-dependent four-vertex exhibits a dramatic fall-off toward low momenta. Comparing spectral functions (commutators) and statistical correlations (anticommutators) of field operators allows us to detect strong violations of the fluctuation-dissipation relation in this non-perturbative infrared regime. Based on a self-consistent expansion in the number of field components to next-to-leading order, a wide range of interaction strengths is analyzed and compared to weak-coupling estimates in effective kinetic theory and classical-statistical field theory.
Turbulent scaling phenomena are studied in an ultracold Bose gas away from thermal equilibrium. Fixed points of the dynamical evolution are characterized in terms of universal scaling exponents of correlation functions. The scaling behavior is determined analytically in the framework of quantum field theory, using a nonperturbative approximation of the two-particle irreducible effective action. While perturbative Kolmogorov scaling is recovered at higher energies, scaling solutions with anomalously large exponents arise in the infrared regime of the turbulence spectrum. The extraordinary enhancement in the momentum dependence of long-range correlations could be experimentally accessible in dilute ultracold atomic gases. Such experiments have the potential to provide insight into dynamical phenomena directly relevant also in other present-day focus areas like heavy-ion collisions and early-universe cosmology.
Predicting universal transport coefficients in far-from-equilibrium quantum systems remains a fundamental challenge. A paradigmatic example is the non-thermal fixed point (NTFP) of isolated Bose gases, where coherence spreads as
ℓ2(t)=Cℏt/m with a universal constant
C. While the scaling exponent
z=2 is well established, the amplitude
C has remained elusive because the underlying particle cascade
n(k)∼k−4 leads to a divergent kinetic energy, threatening the very existence of a constant speed limit. Here we resolve this paradox and present the first analytical, parameter-free prediction of a universal amplitude
C. A deep interplay between symmetry and dissipation is uncovered. The emergent weak U(1) symmetry at the NTFP enforces a conserved total current, forcing the low-energy phase dynamics to obey a diffusive Langevin equation with noise entering as the divergence of a stochastic current. This structure, combined with dynamical decoherence of high-momentum modes, yields a universal power-law momentum distribution
f~(v)∼(1+v2)−3 (with
v=kℓ) that naturally regularizes the ultraviolet divergence. From this, a parameter-free geometric baseline
C=3 is obtained, independent of microscopic details. The experimental value
C=3.4(3) [Martirosyan et al., Nature 647, 608 (2025)] is then shown to be quantitatively consistent with universal logarithmic corrections arising from a marginally irrelevant coupling at the fixed point. A new paradigm is thus established for predicting transport coefficients in strongly correlated non-equilibrium systems: symmetry constraints determine the low-energy effective theory, dynamical decoherence provides a natural ultraviolet completion, and scaling analysis delivers testable predictions moving beyond scaling exponents to quantitative amplitude prediction.
Abstract Understanding the relaxation dynamics of isolated quantum systems remains a central challenge in nonequilibrium physics. In this short review, we discuss recent experimental and theoretical advances in universal dynamics of turbulent Bose gases, with particular emphasis on nonthermal fixed points and wave turbulence in our experimental system at the São Carlos Institute of Physics of the University of São Paulo. We highlight the emergence of self-similar scaling behavior and its connection to particle and energy transport across momentum scales. In addition, we discuss an approach in which a differential equation has a universal scaling solution, providing a practical framework for extracting universal exponents from limited regions of the momentum distribution. Furthermore, we recap experimental observations of direct and inverse cascades in a three–dimensional trapped Bose–Einstein condensate, revealing distinct relaxation stages governed by universal scaling laws. These results demonstrate that turbulence plays a key role in far–from–equilibrium dynamics, offering a unified perspective on transport processes and thermalization in quantum many–body systems.