content_image/422734.jpg content_image/422738.jpg content_image/422739.jpg content_image/422746.jpg content_image/422747.jpg content_image/422748.jpg content_image/422750.jpg content_image/422752.jpg content_image/422733.jpg content_image/422740.jpg content_image/422751.jpg content_image/422749.jpg
put on a lattice with a lattice constant r$_{l}$ we obtain 

<img src='content_image/422734.jpg'>

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: 

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 

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 

<img src='content_image/422738.jpg'>

<img src='content_image/422739.jpg'>

<img src='content_image/422746.jpg'>

<img src='content_image/422747.jpg'>

<img src='content_image/422748.jpg'>

<img src='content_image/422750.jpg'>

<img src='content_image/422752.jpg'>

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<img src='content_image/422740.jpg'>

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 : 

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: 

<img src='content_image/422751.jpg'>

We find that the whole Wigner function in the non- interacting evolution remains Gaussian. It means that 

<img src='content_image/422749.jpg'>

