11 The time scale is renormalized because of the critical scal- ing in the vicinity of the transition. and gT → g . The idea of momentum shell RG is that we treat high momentum components of θ and p (or equiv- alently ˙ θ ) perturbatively, while not making any approx- imations about the low momentum components. Our goal is to find renormalization of the equations of motion governing the low momentum components. So we split where the Fourier expansion of θ $^{>}$( r , t ) only contains mo- menta in the shell Λ ′ ≡ Λ − δ Λ < | k | < Λ and θ $^{<}$( r , t ) contains all other Fourier components: We will treat θ > (and correspondingly p $^{>}$) pertur- bartively in g since the nonlinear term should only weakly couple to the high frequency field. We expand the high- momentum field as Here θ > 0 ( k , t ) is the solution of the equations of motion, with g set to zero: where ω$_{k}$ = v | k | , and the velocity v is v = √$_{λµ}$$_{. In the}$ next leading order we have where and we used Λ a = 1. Note that in the last equation in the argument of the sinus we changed θ$_{0}$ to θ < 0 because the contribution from θ > 0 is smaller by the factor δ Λ / Λ. So we see that in the leading order in g the high mo- mentum component of θ oscillates with time at very high frequency ω$_{Λ}$ . In the next order in g the high momentum component also acquires a low frequency component (as we will discuss below). Next we consider the equation of motion (A1) expand- ing it up to the second order in θ $^{>}$: We thank A. Castro Neto for useful discussions. Work of A.P. was supported by NSF DMR-0907039, AFOSR, and Sloan Foundation. L.M. acknowledges support from NRC/NIST, NSF Physics Frontier Grant PHY-0822671 and Boston University visitor’s program. In this Appendix we derive the RG Eqs. 40 and 41, which can also be written as a second order differential equation for θ Because of the nonlinearity high-momentum modes cou- ple to the low momentum modes leading to the renor- malization of the couplings governing the dynamics of To simplify the derivation, here and throughout the Ap- pendix, we formally change notations λT → λ , µT → µ , 1 µ d 2 dt $_{2}$θ ( r , t ) ≈ λ Δ θ ( r , t ) + g a 2 cos( θ $^{<}$( r , t )) θ $^{>}$( r , t ) + g a 2 sin θ $^{<}$( r , t ) ( 1 − ( θ $^{>}$( r , t )) 2 2 ) . (A9) 1 µ d 2 dt 2 θ = λ △ θ + g a 2 sin θ. (A1) Appendix A Acknowledgments In conclusion, we have studied the dynamics of the rel- ative phase of a bilayer of superfluids in 2D, after the hop- ping between them has been turned off rapidly. We find that on short time scales the dynamics of the correlation function shows a ’light-cone’-like behavior. Depending on the parameters of the system, the light cone dynam- ics can result in a phase that shows supercritical algebraic scaling, and can therefore be thought of as a superheated superfluid. On long time scales the system relaxes to a disordered state via vortex unbinding, which constitutes a reverse-Kibble-Zurek mechanism. The properties of the dynamical process can be understood with a renormal- ization group approach. We find that the dynamical evo- lution of the system resembles the RG flow of the equi- librium system. In particular, using the RG equations we found two possible scenarios of the system reaching the steady state: (i) if initial quantum and thermal fluc- tuations are weak the vortices are irrelevant and long time long distance behavior is governed by the algebraic fixed point. The only role of vortices is then renormal- ization of the superfluid stiffness and the sound velocity. (ii) If the initial fluctuations are strong then the vor- tices become relevant and proliferate resulting in a nor- mal (non-superfluid) steady state. In this case RG gives the time scale of vortex unbinding, which exponentially diverges as the system approaches the non-equilibrium phase transition. The behavior of the relative of phase of two superfluids can be accurately studied by interference experiments of ultra-cold atom systems, and therefore our predictions are of direct relevance to experiment. VI. CONCLUSIONS θ ( r , t ) = θ $^{<}$( r , t ) + θ $^{>}$( r , t ) , (A2) θ $^{<}$( r ) = 1 √$_{V}$ ∑ k< Λ ′ exp( i kr ) θ$_{k}$ (A3) θ $^{>}$( r ) = 1 √$_{V}$ ∑ Λ ′ <k< Λ exp( i kr ) θ$_{k}$. (A4) θ $^{>}$( k , t ) = θ > 0 ( k , t ) + θ > 1 ( k , t ) . (A5) θ > 0 ( k , t ) = µ ω$_{Λ}$ p > 0 , $_{k}$sin( ω$_{Λ}$t ) + θ > 0 , $_{k}$cos( ω$_{Λ}$t ) , (A6) θ > 1 ( k , t ) = gω$_{Λ}$ λ ∫ t 0 dτ F$_{1}$ ( k , τ ) sin( ω$_{Λ}$ ( t − τ )) , (A7) F$_{1}$ ( k , τ ) = ∫ d $^{2}$r exp[ − i kr ] sin( θ < 0 ( r , τ )) , (A8)