Quantum heat waves in a one-dimensional condensate

We study the dynamics of phase relaxation between a pair of one-dimensional condensates created by a bi-directional, supersonic ‘unzipping’ of a finite single condensate. We find that the system fractures into different extensive chunks of space-time, within which correlations appear thermal but correspond to different effective temperatures. Coherences between different eigen-modes are crucial for understanding the development of such thermal correlations; at no point in time can our system be described by a generalized Gibbs' ensemble despite nearly always appearing locally thermal. We rationalize a picture of propagating fronts of hot and cold sound waves, populated at effective, relativistically red- and blue-shifted temperatures to intuitively explain our findings. The disparity between these hot and cold temperatures vanishes for the case of instantaneous splitting but diverges in the limit where the splitting velocity approaches the speed of sound; in this limit, a sonic boom occurs wherein the system is excited only along an infinitely narrow, and infinitely hot beam. We expect our findings to apply generally to the study of superluminal perturbations in systems with emergent Lorentz symmetry.

Figure 1

Realization of the theoretical model using ultracold atoms. (a) A finite one-dimensional condensate, of length $L$, is prepared (at low temperatures) and (b) split into two halves along a supersonic ‘knife-edge’ that travels at velocity $v_s$; (c) the phase difference between the halves $\varphi (x,t)$ evolves as a Luttinger liquid and can be measured by interfering the two halves, as shown in (d).

Figure 2

(a) The energy density (large magnitude is shown in darker color) is plotted as a function of time on the $y$ axis (in units of ${\xi }_c/c)$ and position on the $x$ axis (in units of ${\xi }_c)$ for system-size $L=100{\xi }_c$, splitting velocity $v_s=2c$, Luttinger parameter $K=\rho {\xi }_c/2=10$, and healing length ${\xi }_c$. Regions of different energy densities or “heat waves” created by waves emitted from the splitter at relativistically red- and blue-Doppler-shifted temperatures (curly lines emanating from splitter) are seen; these are described by a temperature $x{T}_0$, where $x$ is noted on the plot. The regions at temperature $T_0{\gamma }_s$ immediately above the splitting front correspond to the regions described by previous work [40]. We also indicate the regions considered in Figs. 3–3 to use the decay of phase correlations. (b) The energy density is plotted as a function of position at time $t=60{\xi }_c/c$ [green line, as also indicated in (a)]. Thick dashed lines correspond to energy density as expected from a system at temperature $x{T}_0$. The dash-dot line corresponds to the average energy density which is seen to coincide well with the energy density of the state at temperature $T_0/{\gamma }_s$.

Figure 3

Decay of equal-time correlations $C(x_1,x_2)=\text{exp}[-i(\varphi \left(x_1\right)-\varphi \left(x_2\right))]$ in various space-time intervals; (a) $t=0,x_1=5{\xi }_c$; (b) $t=0,x_1=35{\xi }_c$; (c) $t=37.5{\xi }_c/c,x_1=-42.5{\xi }_c$; and (d) $t\in [0,L/c],x_1=-5{\xi }_c$. $C(x_1,x_2,t)$ is log-averaged (see main text) over a complete evolution cycle, $t\in [0,L/c]$ and plotted in (d); it is seen to agree with the decay corresponding to a system at temperature $T_f=T_0/{\gamma }_s$, describing the occupation of LL bosons. In (a), (b), and (c), equal-time correlations calculated from a dynamical solution of the problem (blue) are seen to decay in space exponentially and in agreement with the decay predicted by the local effective temperature of the region (red or green), as shown in Fig. 2; these local temperatures do not agree with $T_f$. The space-time intervals considered in (a), (b), and (c) are also shown in Fig. 2.

Figure 4

Comparison of dynamics and boundary conditions in the laboratory frame and Lorentz boosted frame.

Figure 5

Boundary conditions used: ${\partial }_t\varphi (L/2)=0$. The correlations $〈e^{-i(\varphi \left(x\right)-\varphi \left(x^{\prime }\right))}〉$ are plotted in a matrix format for $x,x^{\prime }\in [-30{\xi }_c,30{\xi }_c]$, system size $L=100{\xi }_c$, Luttinger parameter $K=\rho {\xi }_c/2=10$ at times $t_c\left(u_s\right)=L(1-u_s)/\left(2c\right)$ for which cross correlations are strongest in (a), (b), and (c) for different splitting velocities $v_s=u_s^{-1}$ (in units of $c)$. In (d), the development of the cross correlations is shown in time for $u_s=0.1c$.

Figure 6

Boundary conditions used: ${\partial }_x\varphi (L/2)=0$. The correlations $〈e^{-i(\varphi \left(x\right)-\varphi \left(x^{\prime }\right))}〉$ are plotted in a matrix format for $x,x^{\prime }\in [-30{\xi }_c,30{\xi }_c]$, system size $L=100{\xi }_c$, Luttinger parameter $K=\rho {\xi }_c/2=40$ at times $t_c\left(u_s\right)=L(1-u_s)/\left(2c\right)$ for which cross anticorrelations are strongest in (a) and (b) for two different splitting velocities $v_s=u_s^{-1}$ (in units of $c)$. In (c), correlations (from top to bottom) along the cuts (dashed lines; away from diagonal) seen in (a) are plotted (in the same color), showing a marked suppression of correlations at $x^{\prime }=-x$.

Figure 7

(a) shows the correlation function $C\left(x_{,}{x}_2\right)$, with $x_1=-10{\xi }_c$ (red dots) and $x_2\in [-10,10]{\xi }_c$ over times $t\in [0,50]{\xi }_c/c$ and $u_s=0.5c$. Nonthermal correlations are formed over supersonic ‘fronts’ (see as dark orange lines in the region) that move at speeds $c_{\pm }^{\prime }=2c/(1\pm u_s/c)$. (b) and (c) examine the emergence and fading, respectively, of these nonthermal correlations in the region $x\in [-10{\xi }_c,10{\xi }_c]$ at times that are highlighted in (a), and verify the speeds $c_{+}^{\prime }=4/3c$ and $c_{-}^{\prime }=4c$, as expected for $v_s=2c$, or $u_s=0.5c$. (d) provides another confirmation of the supersonic spread of correlations by examining the change of normalized correlations $\widehat{C}(x_1,x_2)$ (explained in the main text) over times $t\in [-6{\xi }_c,2{\xi }_c]$, and $x_1=12{\xi }_c$ (red dot), as shown in (a). Note $\widehat{C}(x_1,x_2)=1$ when points $x_1$ and $x_2$ are uncorrelated and $<1$ otherwise. System parameters are as in Fig. 2. Boundary conditions used: ${\partial }_t\varphi (L/2)=0$.

Figure 8

The calculation of $c_{-}$ and $c_{+}$, the speeds that govern the spread of correlations in the system, is illustrated. Pairs of oppositely traveling quasiparticles (dashed lines) are emitted by the splitter and entangle the system over the distance covered by them. The velocity of this spread is clearly $2c$ if the splitting is instantaneous. The spread of correlations can happen faster or slower than $2c$ in our system since the splitting front is itself moving.