Bi-directional universal dynamics in a spinor Bose gas close to a non-thermal fixed point

We numerically study the universal scaling dynamics of an isolated one-dimensional ferromagnetic spin-1 Bose gas. Preparing the system in a far-from-equilibrium initial state, simultaneous coarsening and refining is found to enable and characterize the approach to a non-thermal fixed point. A macroscopic length scale which scales in time according to $L_{\Lambda}(t)\sim t^{\, \beta}$, with $\beta\simeq 1/4$, quantifies the coarsening of the size of spin textures. At the same time kink-like defects populating these textures undergo a refining process measured by a shrinking microscopic length scale $L_{\lambda}\sim t^{\, \beta'}$, with $\beta'\simeq-0.17$. The combination of these scaling evolutions enables particle and energy conservation in the isolated system and constitutes a bi-directional transport in momentum space. The value of the coarsening exponent $\beta$ suggests the dynamics to belong to the universality class of diffusive coarsening of the one-dimensional XY-model. However, the universal momentum distribution function exhibiting non-linear transport marks the distinction between diffusive coarsening and the approach of a non-thermal fixed point in the isolated system considered here. This underlines the importance of the universal scaling function in classifying non-thermal fixed points. Present-day experiments with quantum gases are expected to have access to the predicted bi-directional scaling.