Light cone dynamics and reverse Kibble-Zurek mechanism in two-dimensional superfluids following a quantum quench

We study the dynamics of the relative phase of a bilayer of two-dimensional superfluids after the two superfluids have been decoupled. We find that on short time scales the relative phase shows “light cone”-like dynamics and creates a metastable superfluid state, which can be supercritical. We also demonstrate similar light cone dynamics for the transverse field Ising model. On longer time scales the supercritical state relaxes to a disordered state due to dynamical vortex unbinding. This scenario of dynamically suppressed vortex proliferation constitutes a reverse-Kibble-Zurek effect. We study this effect both numerically using truncated Wigner approximation and analytically within a newly suggested time dependent renormalization group approach (RG). In particular, within RG we show that there are two possible fixed points for the real-time evolution corresponding to the superfluid and normal steady states. So depending on the initial conditions and the microscopic parameters of the Hamiltonian the system undergoes a nonequilibrium phase transition of the Kosterlitz-Thouless type. The time scales for the vortex unbinding near the critical point are exponentially divergent, similar to the equilibrium case.

Figure 1

Illustration of the Kibble-Zurek (KZ) mechanism, which describes ramping across a phase transition from the disordered phase and its counterpart, the reverse-Kibble-Zurek (rKZ) effect. The latter describes ramping across a transition from the ordered side. Its defining feature is the dynamical suppression of vortex unbinding, which happens on a much longer time scale than the appearance of phononic excitations. We propose to study the rKZ in a bilayer of 2D superfluids of ultracold atoms by decoupling the superfluids and measuring the dynamics of the relative phase.

Figure 2

We simulate the dynamics of the relative phase of two 2D superfluids by solving the equations of motion and by averaging over the Wigner distribution of the initial state. A single run is shown here for $V=100$, $\kappa =10$, and $T=2$, at the times $t=0,5,10,20,40,100$. Vortices are marked red, antivortices blue.

Figure 3

Plot of short-time behavior of the correlation function as a function of time and space, at temperature $T=3$, for $\kappa =10$ and $V=20$. The dynamics separate into instantaneous, damped oscillatory behavior, and a “light cone” -like pulse.

Figure 4

$\langle \delta {\varphi }^2\rangle$ of the linearized system, for $T=1$ and $V{\beta }^2=20$, as a function of the lattice site, and $\mathit{vt}$.

Figure 5

The correlation function of the linearized system, for $T=1$ and $V{\beta }^2=20$, as a function of the lattice site, and $\mathit{vt}$.

Figure 6

(a) $T^{*}$, as given in Eq. (19), for $\kappa =1,3,10$, from top to bottom and for $V=20$. The line $T_c=\pi /2$ was added to indicate the critical temperature. (b)–(d) Simulations for these values of $\kappa$ and $V$. We plot the number of vortices $n_v$ as a function of time $t$ and initial temperature $T_0$.

Figure 7

The correlation function $\langle {\sigma }_0^z(t){\sigma }_r^z(t)\rangle$ for a quench from $g=3$ to $g^{\text{'}}=1$, in (a) and from $g=3$ to $g^{\text{'}}=0.5$ in (b), as a function of time $t$, specifcally of $2Jt$, and the spatial distance $r$.

Figure 8

Long-time behavior of the correlation function for $T=1$, $\kappa =8$, and $V=80$. The correlation function first develops algebraic scaling, so the system forms a metastable quasisuperfluid state. On longer time scales the correlation function shows exponential decay. The coherence is lost due to dynamical vortex unbinding.

Figure 9

Time dependence of the exponent $\tau$ extracted from fitting the long-time correlation function $G(x,t)$, for different initial couplings. In all four examples we use $T=1$ and $\kappa =8$. The initial couplings $V$ are chosen as $V=80,70,50$, and $20$, corresponding to curves I to IV. The curve I corresponds to the example shown in Fig. 8. In this case the correlation function can be well fitted with an algebraic function for up to $t\approx 60$, after that $G(x,t)$ is better fitted by an exponential function with a decay length of the order of the lattice constant. For the other cases, $G(x,t)$ is well fitted with an algebraic function throughout the whole time interval.

Figure 10

Schematic representation of a renormalization step in the real-time RG approach. In each step we renormalize simultaneously the space and time variables. This “moves” the equations of motion from $(\mathbf{r},t)$ to $({\mathbf{r}}^{\text{'}},t^{\text{'}})$. We correct for the integrated-out degrees of freedom to second order in $g$, which renormalizes the parameters $g$ and $\lambda$ according to Eqs. (40) and (41).