6 0 5 1 0 0 10 20 30 0 20 40 60 0 5 1 0 0 10 20 30 0 20 40 60 0 5 1 0 0 10 20 30 0 5 10 15 20 T$_{0}$ T$_{0}$ T$_{0}$ t t t n$_{v}$ n$_{v}$ n$_{v}$ 0 2 4 6 8 1 2 3 4 5 6 T * T$_{c}$ T$_{0}$ a) b) d) c) 10 10 10 for each mode the Wigner function is equivalent to that of a harmonic oscillator at finite ’temperature’ T ∗ $_{k}$, which is in general mode-dependent: where r ∗ k = 1 / tanh( ω$_{k}$/ 2 T ∗ $_{k}$). Solving for T ∗ k gives For large V β $^{2}$, which corresponds to initially strong cou- pling between two superfluids, this simplifies to a single value, independent of k : FIG. 6: a) T $^{∗}$, as given in Eq. 19, for κ = 1 , 3 , 10, from top to bottom, and for V = 20. The line T c = π/ 2 was added to indicate the critical temperature. b) – d) Simulations for these values of κ and V . We plot the number of vortices n v as a function of time t and initial temperature T 0 . r ∗ k ω$_{k}$ = r$_{k,}$$_{0}$ 2 ω$_{k,}$$_{0}$ + r$_{k,}$$_{0}$ω$_{k,}$$_{0}$ 2 ω 2 k (17) T ∗ k = ω$_{k}$ 2 tanh − 1 ( 2 ω$_{k}$ω$_{k,}$ 0 ω 2 $_{k}$+ ω 2 k, 0 tanh( ω$_{k,}$$_{0}$/ 2 T ) . (18) T ∗ = √ V β 2 4 tanh( V β $^{2}$/ 2 T ) . (19) For small initial temperatures T we have T ∗ ≈ √ V β $^{2}$/ 4 ( T ⋆ = 2 J$_{⊥}$/J in terms of the original Hubbard param- eters), that is, the temperature is fully determined by the initial coupling energy. The coupling energy be- tween the two layers is transferred into the in-plane ki- netic energy. We remind again that this result is valid as long as J$_{⊥}$ ≲ Un , otherwise the dependence of T ⋆ on J$_{⊥}$ saturates and for the infinite coupling limit we have T ⋆ ∼ Un/J . For large T we have T ∗ ≈ T/ 2. This result is a reflection of the doubling of the degrees of freedom when two layers are uncoupled. In Fig. 6 a) we show dependence T $^{⋆}$( T ) evaluated according to Eq. (19), for V = 20, and for κ = 1 , 3 , 10 corresponding to lowering J$_{⊥}$ . For κ = 1, T ∗ is always above the critical tempera- ture T$_{c}$ = π/ 2, for κ = 10, it crosses it. We therefore ex- pect to see very little vortex formation for small temper- atures for κ = 10, and many vortices for all temperatures for κ = 1. The intermediate value κ = 3 approximately describes the transition between these limits. To make this point more clear in Fig. 6 b)–d) we plot full non- linear TWA simulation for each of the three cases. We run the quench for different temperatures T , and plot the number of vortices n$_{v}$ in the system as a function of time. This number is obtained by counting the vor- tices (indicated by red plaquettes in Fig. 2), and then by averaging over many runs. We find that for κ = 1 the number of vortices is virtually unchanged implying that the dynamics is completely dominated by quantum fluctuations, whereas for κ = 10 this number drops to zero when the temperature is lowered. These results are consistent with the emergent temperature T ∗ obtained within the linearized approach. wave excitations. The same mechanism of reaching a steady state is much more general and is likely re- lated to the existence of the maximum group velocity in Schr¨ odinger systems as was proven by Lieb and Robin- son $^{32}$. In this section we demonstrate the presence of the light-cone dynamics in another solvable model, the transverse Ising chain 26,33 described by the Hamiltonian where J$_{I}$ is an overall energy scale, g describes the strength of the transverse field, and σ x,z are the Pauli matrices. We follow the calculational procedure in Ref. [33]. First, we use a Jordan-Wigner transformation where c$_{i}$ are Fermi operators, and n$_{i}$ = c † $_{i}$c$_{i}$ . This transformation leads to a fermionic representation of the Hamiltonian, that can be further diagonalized using the Bogoliubov transformation where c$_{k}$ is the Fourier transform of c$_{i}$ , u$_{k,g}$ and v$_{k,g}$ are given by cos( θ$_{k,g}$/ 2) and sin( θ$_{k,g}$/ 2), where θ$_{k,g}$ = arctan(sin k/ ( g − cos k )). The resulting dispersion is In this section we demonstrate that light cone dynam- ics is not just characteristic for the system we are inter- ested in, which is characterized by low energy bosonic IV. LIGHT-CONE DYNAMICS IN THE TRANSVERSE ISING MODEL H$_{I}$ = − J$_{I}$ ∑ i ( σ x i σ x i +1 + gσ z $_{i}$) , (20) σ z i = 1 − 2 n$_{i}$ (21) σ x i = ∏ j<i (1 − 2 n$_{j}$ )( c$_{i}$ + c † $_{i}$) , (22) γ$_{k,g}$ = u$_{k,g}$c$_{k}$ − iv$_{k,g}$c † − k (23) ϵ$_{k,g}$ = 2 J$_{I}$ √ g 2 − 2 g cos k + 1 . (24)