Large-scale thermalization, prethermalization, and impact of temperature in the quench dynamics of two unequal Luttinger liquids

We study the effect of a quantum quench between two tunnel coupled Tomonaga-Luttinger liquids (TLLs) with different speed of sound and interaction parameter. The quench dynamics is induced by switching off the tunneling and letting the two systems evolve independently. We fully diagonalize the problem within a quadratic approximation for the initial tunneling. Both the case of zero and finite temperature in the initial state are considered. We focus on correlation functions associated with the antisymmetric and symmetric combinations of the two TLLs (relevant for interference measurements), which turn out to be coupled due to the asymmetry in the two systems' Hamiltonians. The presence of different speeds of sound leads to multiple light cones separating different decaying regimes. In particular, in the large time limit, we are able to identify a prethermal regime where the two-point correlation functions of vertex operators of symmetric and antisymmetric sector can be characterized by two emerging effective temperatures, eventually drifting towards a final stationary regime that we dubbed quasithermal, well approximated at large scale by a thermal-like state, where these correlators become time independent and are characterized by a unique correlation length. If the initial state is at equilibrium at nonzero temperature $T_0$, all the effective temperatures acquire a linear correction in $T_0$, leading to faster decay of the correlation functions. Such effects can play a crucial role for the correct description of currently running cold atoms experiments.

Figure 1

Schematic picture of the system studied in this paper. It consists of two unequal Luttinger liquids, with sound velocities $u_i$ and Luttinger parameters $K_i$ ($i=1,2$). We quench the system by switching off the tunneling $g$, starting from a nonzero value (both ground state and finite temperature thermal states are considered as initial states). This corresponds to suddenly raising the barrier of the double well potential separating the two sides.

Figure 2

Logarithm of the correlation functions $C_{+}(x,t)$ [top row: (a), (b), (e)] and $C_{-}(x,t)$ [bottom row: (c), (d), (f)] after a quench from the ground state of the Hamiltonian (6) ($T_0=0$), as a function of distance $x$ and time $t$. The exact expressions from Eq. (25) [left panels: (a), (c)] are compared with the approximation in Eq. (39) [central panels: (b), (d)]. The parameters used are $u_1=6, u_2=5, K_1=20, K_2=10, g=40$, and $\alpha =1$. The position of the light cones derived in Eq. (39) are also shown. The dashed lines correspond to the light cones at $x=2u_{1}t,(u_1+u_2)t,2u_{2}t$ (in the chosen regime $u_1\approx u_2$ they are close to each other), separating the transient from the prethermal regime. The dot-dashed line corresponds to the last light cone at $x=|u_1-u_2|t$, separating prethermal and final quasithermal regime. In the right panels (e),(f) we show a slice of the plots at time $t=10$ comparing the exact numerical results with Eq. (39) (with an arbitrary amplitude) and for $C_{+}(x,t)$ the expression corrected by a power law decay in $x$ as in (40).

Figure 3

Space decay of (a) $C_{+}(x,t)$ and (b) $C_{-}(x,t)$ from (25) after a quench from the ground state of the Hamiltonian (6) ($T_0=0$), at different times $t=10,20,100$, compared with the equilibrium correlations at temperature $T^{\mathrm{eff}}$ and $T_{+}^{\mathrm{eff}}$ for $C_{+}(x,t)$ and $T^{\mathrm{eff}}$ and $T_{-}^{\mathrm{eff}}$ for $C_{-}(x,t)$. The parameters used are $u_1=6, u_2=5, K_1=20, K_2=10, g=40$, and $\alpha =1$.

Figure 4

Space decay of $C_{-}(x,t)$ and $C_{+}(x,t)$ from (25) after a quench from the ground state of the Hamiltonian (6) ($T_0=0$), at time $t=100$, compared with the equilibrium correlations at temperature $T^{\mathrm{eff}}, T_{-}^{\mathrm{eff}}$, and $T_{+}^{\mathrm{eff}}$. The parameters used are $u_1=6, u_2=5, K_1=20, K_2=10, g=40$, and $\alpha =1$.

Figure 5

Space decay of $C_{-}(x,t,T_0)$ from (25) with $T_0=0.5$, at different times $t=10,20,100$, compared with the equilibrium correlations at temperature $T^{\mathrm{eff}}$ and $T_{-}^{\mathrm{eff}}$. The parameters used are $u_1=6, u_2=5, K_1=20, K_2=10, g=40$, and $\alpha =1$.

Figure 6

(a) Time dependence of $C_{-}(x,t)$ from (25) with $T_0=0.5$, at different points in space $x=50,100,150$. (b) Inverse correlation length [obtained as the spatial derivative of the exponent of Eq. (25)] at distance $x=100$ as a function of the time, compared with the equilibrium correlation length at temperature $T^{\mathrm{eff}}$ and $T_{-}^{\mathrm{eff}}$. The parameters used are $u_1=6, u_2=5, K_1=20, K_2=10, g=40$, and $\alpha =1$.