## 1 Introduction  
Recent progress in time-domain techniques has made it possible to study the dynamics of elementary excitations under different non-equilibrium conditions and whence also the appearance of emergent order in real time [1; 2; 3; 4; 5] With low laser excitation intensities, the system is perturbed only slightly out of equilibrium, and the response is comparable to the information that one can obtain from frequency domain spectroscopies (the perturbative, or weak excitation regime). Different excitations can be directly distinguished by their respective lifetimes, so with different probes (including optical polarization analysis, X-ray or electron diffraction) very detailed information can be obtained on the system dynamics near equilibrium in many different materials, strongly correlated superconductors being perhaps the most researched. Beyond the low-excitation intensity regime, the possibility of driving the system through a phase transition (in the non-perturbative regime) brings the exciting prospect of monitoring the changing dynamics of the elementary excitations and thus the emergence of long-range order through the transition [6; 7; 8; 9]. The early studies using multiple-pulse optical techniques have already revealed that the response in the emergent state is complicated by the appearance of inhomogeneities--particularly topological defects [7]--that are quite difficult to study with conventional time resolved techniques that calculate an average over multiple transition outcomes.  
In this paper we will discuss the importance of such topological defects in rapid phase transitions, both in the context of the Kibble-Zurek phenomenon in second-order symmetry breaking transitions [10; 11; 12], and in charge-driven topological transitions. In short-laser-pulse induced phase transitions, the time-evolution of the system is so rapid that different areas of the sample emerge from fluctuations through the phase transition with uncorrelated phase \(\phi\) of the complex order parameter \(\psi=Ae^{i\phi}\). By necessity, therefore, domains and associated topological defects must form between these areas, as noted originally for 2 nd order transitions by [PERSON] [10; 11; 12]. However, the principle is quite general and extends to topological transitions as well, although the details are still not well understood.  
We recently found that the emerging domain structure in such transitions is not necessarily random. In certain systems, the domains emerging after a non-equilibrium topological transition become long-range ordered [13]. Rapid phase transition experiments thus reveal quite rich complexity that requires new experimental methods beyond stroboscopic averaging in order to reveal the intricate details of the new emergent states. In this paper, we will discuss a few examples illustrating the physics on the basis of recent experiments on charge density wave and charge-ordered systems through 2 nd order symmetry-breaking in rare-earth tritellurides and topological transitions in 1T-TaS\({}_{2}\), respectively.  
Finally, the field of topological states emerging through non-equilibrium transitions has gained wider importance because ultrafast switching between different electronically ordered states can be used to store information, leading to ultra-low energy non-volatile ultrafast memory devices for which there is currently a great need. Comparison between charge density wave switching devices with existing ones, including phase-change memories and various magnetic storage devices, shows superiority in multiple areas that leads to realistic prospects for commercialization. Obviously, the search using ultrafast techniques for better and more convenient materials and systems is imperative, stimulating further research in the field.