2 sition to the exponential regime is due to vortex excita- tions. Above the transition, vortex-antivortex pairs are deconfined so that vortices and anti-vortices become un- bound. These excitations generate a much more disor- dered phase field, which leads to exponential scaling of the correlation function. to one of the phases in real time. Divergent time (and length) scales in equilibrium systems translate into diver- gent relaxation times required to reach thermalization in the non-equilibrium case. Physically this decay of the metastable superfluid state to the new equilibrium is very reminiscent of the Kibble- Zurek (KZ) effect. The latter describes a ramp across a phase transition, starting on the disordered side. If the ordered state supports topological excitations, like vortices, then one expects very slow relaxation of the re- sulting state to the equilibrium due to vortex-antivortex recombination. This scenario is illustrated in Fig. 1 a): In the disordered phase we have excitations such as phonons, as well as topological defects. When we ap- ply a fast ramp across the phase transition, the phonon excitations thermalize on very short time scales, while topological defects can exist on much longer time scales. The mechanism of relaxation in our case is exactly com- plimentary and can be termed as reverse Kibble-Zurek effect. Here, the ramp across a phase transition starts from the ordered side, as illustrated in Fig. 1 b). In the ordered phase both phonon excitations and vortices are suppressed. When the system is ramped across the tran- sition, phonons are generated on a fast time scale. How- ever, vortices are generated at much longer time scales leading to the long-lived supercritical superfluid state. We point out that in thermally isolated systems (like cold atom systems) it is much easier to observe reverse KZ ef- fect because the disordered phase usually corresponds to a higher temperature. In isolated systems it is relatively easy to increase temperature by quenching some param- eter, while decreasing temperature requires much more effort and can be done only in open systems. If we couple the two superfluids with a hopping term in the temperature regime of the critical temperature, the system forms a phase-locked state, see Refs. [20,21]. Here the correlation function of the relative phase scales as G ( x ) ∼ exp( −| x | /ξ$_{i}$ ) + C , where C = 0, and ξ$_{i}$ is the correlation length of the phase-locked state. In this state the relative phase is well-aligned over long distances; its fluctuations are strongly suppressed. We then turn off the hopping and study the evolution of the system. In this paper we show that the relaxational dynam- ics occurs in two stages. The first fast stage, which we will term light cone relaxation, establishes a metastable quasi-equilibrium state of phonons (or Bogoliubov exci- tations) characterized by effective non-equilibrium tem- perature (in principle this metastable state can be com- pletely non-thermal). During this stage the correla- tions between two arbitrary points in space x 1 and x $_{2}$, G ( x $_{1}$, x $_{2}$, t ), where t is the time after the quench, ini- tially decay in time, independent of their spatial separa- tion x = | x 1 − x 2 | because these points are not causally connected: G ( x $_{1}$, x $_{2}$, t ) ∼ 1 /t $^{α}$, where α is a power-law exponent related to the parameters of the system. At a later time t $^{⋆}$, when the condition 2 vt ⋆ = x is fulfilled, these correlations (approximately) freeze in time so that G ( x $_{1}$, x $_{2}$, t ) ∼ 1 /x $^{α}$. The exponent α thus defines the non-equilibrium phonon temperature in the system. Be- cause this first stage of dynamics involves only phonons, the exponent α can exceed the maximally allowed equi- librium value of one fourth, leading to a non-equilibrium super-critical metastable state, which can be thought of as a supercritical superfluid. It is analogous to an over- heated classical liquid, for which a liquid state can be sustained above the critical temperature if the creation of defects is avoided. We find that the power-law can be substantially above the critical scaling, and furthermore, that this metastable can be very long-lived. This paper is organized as follows: In Sect. II we in- troduce the numerical method that we use and find that at short time scales the system shows light cone dynam- ics. In Sect. III we consider the linearized dynamics of the bilayer system. Within this approximation both light cone dynamics and the emerging superfluid state can be understood. In Sect. IV we study the light cone dynam- ics of a solvable model, the transverse Ising model. In Sect. V we study dynamical vortex unbinding both with truncated Wigner approximation, and with a renormal- ization group approach. We note that a short version of this paper with some of the results was published ear- lier $^{18}$. Here we expand the earlier treatment, present ad- ditional results and derivations, and formulate the real time renormalization group approach which explains the numerical results. At longer time scales, vortex-antivortex pairs emerge and proliferate leading to the true equilibrium state. This process occurs at much longer time scales. We describe this thermalization process both numerically, using trun- cated Wigner approximation (TWA) and analytically. In particular, we show that thermalization (here cor- responding to the process of vortex-antivortex prolifer- ation) can be understood by extending renormalization group ideas to real time dynamics. By doing partial aver- aging over fast oscillating high energy degrees of freedom, we can rewrite the equations of motion of slower degrees of freedom through renormalized coupling constants. As in the case of equilibrium systems we observe two pos- sible scenarios corresponding to vortex-antivortex pairs being irrelevant (superfluid phase) or relevant (normal phase). Thus we are able to see how the system relaxes In this section we present the model that we use in our numerical approach. We consider two two-dimensional (quasi-)condensates that are aligned in parallel to each other, that are coupled by a hopping term which is then II. MICROSCOPIC MODEL AND THE TRUNCATED WIGNER APPROXIMATION