Nonequilibrium dynamics of coupled Luttinger liquids

In this work we consider the dynamics of two tunnel-coupled chains after a quench in the tunneling strength is performed and the two systems are allowed to evolve independently. We describe the form of the initial state, comparing with previous results concerning the dynamics after the splitting of a one-dimensional gas of bosons into two phase-coherent systems. We compute different correlation functions, among which are those that are relevant for interference measurements, and discuss the emergence of effective temperatures also in connection with previous works.

Figure 1

Schematic figure representing the quench from high to low tunnel-coupled systems. The two states are characterized, respectively, by a given tunneling strength $t_{\perp }^0$ and $t_{\perp }$ and Bogoliubov angles ${\phi }_k^0$ and ${\phi }_k$.

Figure 2

Correlation functions $C_{\theta }(x,t)$ (23), for different time and length scales. Here $K=1$, $u=1$, and $\alpha m_0=1$. (Top) $C_{\theta }(x,t)$ as a function of the space distance for four different times $t{m}_0=0.1,1,2,\text{and}3$, shown, from top to bottom, respectively, with green, blue, red, and yellow lines. The black line is the asymptotic stationary correlation function. (Bottom) $C_{\theta }(x,t)$ as a function of the time, for three different space distances $x{m}_0=1,3,\text{and}5$, shown, respectively, with solid, dashed, and dot-dashed lines.

Figure 3

(Top) Asymptotic correlation functions $C_{\theta }(x,t\to \infty )=exp\{-\frac{1}{2}{lim}_{t\to \infty }\langle {[{\theta }_A(x,t)-{\theta }_A(0,t)]}^2\rangle \}$, with ${lim}_{t\to \infty }\langle {[{\theta }_A(x,t)-{\theta }_A(0,t)]}^2\rangle$ given in (25), and thermal correlation function $C_{\theta }\left(x\right)=exp\{-\frac{1}{2}{\langle {[{\theta }_A\left(x\right)-{\theta }_A\left(0\right)]}^2\rangle }_{T_{\mathrm{eff}}}\}$, with ${\langle {[{\theta }_A\left(x\right)-{\theta }_A\left(0\right)]}^2\rangle }_{T_{\mathrm{eff}}}$ given in (27) shown with black and blue lines, respectively. (Bottom) Asymptotic correlation function of the relative density (29) (black line) compared with the equilibrium one (30) (blue line). In both plots $\alpha m_0=1$, $K=1$, $u=1$, and $T_{\mathrm{eff}}=\frac{m_0}{4}$. In both cases the two curves look indistinguishable.

Figure 4

Correlation functions of the density $C_n(x,t)$ (top) and of the current $C_j(x,t)$ (bottom) as a function of the space distance for different times. Times $t{m}_0=1,2,3,\text{and}6$ are shown, respectively, with blue long-dash-dotted, red dash-dotted, green dashed, and yellow dotted lines. In both plots the black solid line is the asymptotic result in the stationary limit. In both figures we set $\alpha m_0=1$, $K=1$, and $u=1$. For the parameters used here the stationary curves in the two plots are the same.