put on a lattice with a lattice constant r$_{l}$ we obtain 5 H$_{1}$ = Ω$_{0}$ ∑ i ( − r$_{l}$ r$_{0}$ ∑ j$_{i}$ ( φ$_{i}$ − φ$_{j}$$_{i}$ ) 2 2 + g$_{⊥}$r 2 l 2Ω$_{0}$ φ 2 i + r$_{0}$ 2 r$_{l}$ n 2 i ) . (7) This expression can be also obtained directly linearizing the original Hamiltonian (1). The index j$_{i}$ here describes the four neighboring sites of the site i , and n$_{i}$ is the filling fraction, related to the density via n$_{i}$ = nr 2 l . Ω$_{0}$ is related to the phonon velocity as Ω$_{0}$ = v/r$_{l}$ . We therefore find that the squeezing parameter κ/π is given by κ/π = r$_{l}$/r$_{0}$ , i.e. it is the ratio of the discretization length scale r$_{l}$ and the short-range cut-off r$_{0}$ of the system. g$_{⊥}$ is related to V ( t ) by g$_{⊥}$r 2 l / 2 = Ω$_{0}$ V ( t ). We now consider the time evolution of φ and n under (6). It is convenient to go to momentum representation where different modes decouple from each other. Assum- ing also that we are interested in momenta smaller than 1 /r$_{l}$ , where the lattice effects are not important we obtain the following equations of motion: FIG. 4: 〈 δφ $^{2}$〉 of the linearized system, for T = 1 and V β 2 = 20, as function of the lattice site, and vt . d dt n$_{k}$ = − Ω$_{0}$ ( r$_{l}$ϵ 2 k r$_{0}$ + 2 V ) φ$_{−}$$_{k}$ (8) d dt φ$_{k}$ = Ω$_{0}$ r$_{0}$ r$_{l}$ n$_{−}$$_{k}$. (9) where ϵ 2 k = 4 sin 2 k$_{x}$/ 2 + 4 sin 2 k$_{y}$/ 2, and k is dimen- sionless, k = − π...π . We rescale the time variable as ˜ t = Ω$_{0}$ t/ ℏ . The initial dispersion is then given by The dispersion ω$_{k}$ after the quench is simply ω 2 k = ϵ 2 $_{k}$, in these units. We solve these equations and calculate the equal-time correlation function at time t after the quench. We use where δφ = φ ( 0 , t ) − φ ( x , t ). The averaging is now triv- ially done using the Wigner distribution (4). If we put the system back to the lattice we then find for x ≪ 2 vt , where T ∗ is an effective temperature that is estimated below, and C$_{1}$ is a numerical prefactor. Out- side of the light cone ( x ≫ 2 vt ) the function G ( x , t ) only depends on time t but not on the distance x : FIG. 5: The correlation function of the linearized system, for T = 1 and V β 2 = 20, as function of the lattice site, and vt . ω 2 k, 0 = ϵ 2 k + 2 V r$_{0}$/r$_{l}$. (10) G ( x , t ) = 〈 exp( iφ ( 0 , t )) exp( − iφ ( x , t )) 〉 (11) = exp( −〈 δφ $^{2}$〉 / 2) , (12) G ( x , t ) = C$_{2}$ | t | − T ∗ / 4 T$_{KT}$ , (15) 〈 δφ $^{2}$〉 = ∑ k (2 − 2 cos kx ) × ( r$_{k,}$$_{0}$ 2 ω$_{k,}$$_{0}$ cos $^{2}$( ω$_{k}$t ) + r$_{k,}$$_{0}$ω$_{k,}$$_{0}$ 2 ω 2 k sin $^{2}$( ω$_{k}$t ) . (13) 〈 φ 2 $_{k}$( t → ∞ ) 〉 = r$_{k,}$$_{0}$ 4 ω$_{k,}$$_{0}$ + r$_{k,}$$_{0}$ω$_{k,}$$_{0}$ 4 ω 2 k . (16) G ( x , t ) ≈ C$_{1}$ | x | − T ∗ / 4 T$_{KT}$ (14) where T ∗ is the same effective temperature. At the light cone boundary x ≈ 2 vt the two asymptotics for the cor- relation function (14) and (15) approximately coincide. However, we note that the prefactor C$_{2}$ is in general dif- ferent from C$_{1}$v − T ∗ / 4 T$_{KT}$ as it is evident from the exis- tence of a wavefront that is visible in Figs. 4 and 5. The temperature that emerges inside the light cone can be estimated by considering the quadratures of φ at long times: We find that the whole Wigner function in the non- interacting evolution remains Gaussian. It means that The quantities r$_{k,}$$_{0}$ and ω$_{k,}$$_{0}$ are defined as before. We now calculate the Green’s function in the lin- earized regime numerically using Eqs. (12) and (13). We choose the discretization r$_{l}$ = r$_{0}$ , the initial tempera- ture T/ Ω$_{0}$ = 1, and the initial coupling V β 2 = 20. In Fig. 4 we plot 〈 δφ $^{2}$〉 and in Fig. 5 we plot the correla- tion function. In both plots the light-cone dynamics is clearly visible. Because of the translational invariance the correlation function that emerges in the light-cone only depends on the relative distance is given by