The dynamics of quantum systems far from equilibrium represents one of the most challenging problems in theoretical many-body physics [1,2]. While the evolution 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 cosmology [3,4], the dynamics of nuclear collision experiments described by quantum chromodynamics [5,6], or the postquench dynamics in dilute quantum gases in nonrelativistic quantum field theory [7][8][9][10][11]. However, an experimental demonstration of such scaling evolution in space and time in a quantum many-body system is lacking so far. Here we observe the emergence of universal dynamics by evaluating spatially resolved spin correlations in a quasi one-dimensional spinor Bose-Einstein condensate [12][13][14][15][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 transport of an emergent conserved quantity 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 signaling the existence of non-thermal fixed points [10,17,18]. We confirm that the non-thermal scaling phenomenon involves no fine-tuning, by preparing different initial conditions and observing the same scaling behaviour. Our analog quantum simulation approach provides the basis to reveal the underlying mechanisms and characteristics of non-thermal universality classes. One may use this universality to learn, from experiments with ultra-cold gases, about fundamental aspects of dynamics studied in cosmology and quantum chromodynamics.Isolated quantum many-body systems offer particularly clean settings for studying fundamental properties of the underlying unitary time evolution [19]. For sys-tems initialised far from equilibrium different scenarios have been identified, including the occurence of manybody oscillations [20] and revivals [21], the manifestation of many-body localisation [22], and quasi-stationary behaviour in a prethermalised stage of the evolution [23].Here we observe a new scenario associated to the notion of non-thermal fixed points. This is illustrated schematically in Fig. 1a: Starting from a class of farfrom-equilibrium initial conditions, the system develops a universal scaling behaviour in time and space. This is a consequence of the effective loss of details about initial conditions and system parameters long before a quasistationary or equilibrium situation may be reached. The transient scaling behaviour is found to be governed by the transport of an emergent collectiv...

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Non-thermal fixed points in the evolution of a quantum many-body system quenched far out of equilibrium manifest themselves in a scaling evolution of correlations in space and time. We develop a low-energy effective theory of non-thermal fixed points in a bosonic quantum many-body system by integrating out long-wavelength density fluctuations. The system consists of N distinguishable spatially uniform Bose gases with U(N)symmetric interactions. The effective theory describes interacting Goldstone modes of the total and relativephase excitations. It is similar in character to the non-linear Luttinger-liquid description of low-energy phonons in a single dilute Bose gas, with the markable difference of a universal non-local coupling function depending, in the large-N limit, only on momentum, single-particle mass, and density of the gas. Our theory provides a perturbative description of the non-thermal fixed point, technically easy to apply to experimentally relevant cases with a small number of fields N. Numerical results for N = 3 allow us to characterize the analytical form of the scaling function and confirm the analytically predicted scaling exponents. The fixed point which is dominated by the relative phases is found to be Gaussian, while a non-Gaussian fixed point is anticipated to require scaling evolution with a distinctly lower power of time.