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The time scale is renormalized because of the critical scal- ing in the vicinity of the transition. 

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. 

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 

To simplify the derivation, here and throughout the Ap- pendix, we formally change notations λT → λ , µT → µ , 

## VI. CONCLUSIONS 

## Acknowledgments 

## Appendix A 

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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: 

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where ω$_{k}$ = v | k | , and the velocity v is v = √$_{λµ}$$_{. In the}$ next leading order we have 

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where 

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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 $^{>}$: 

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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 

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