Exploring dynamical phase transitions and prethermalization with quantum noise of excitations

Dynamical phase transitions can occur in isolated quantum systems that are brought out of equilibrium by sudden parameter changes. We discuss the characterization of such dynamical phase transitions based on the statistics of produced excitations. We consider both the $O\left(N\right)$ model in the large-$N$ limit and a spin model with long-range interactions and show that the dynamical criticality of their prethermal steady states manifests most dramatically not in the average number of excitations but in their higher moments. We argue that the growth of defect fluctuations carries unique signatures of the dynamical criticality, irrespective of the precise details of the model. Our theoretical results should be relevant to quantum quench experiments with ultracold bosonic atoms in optical lattices.

Figure 1

Schematic of the double quantum quench protocol. The system is prepared in the disordered (Mott) phase in a deep optical lattice (a). By reducing the lattice depth, the system is quenched to the ordered (superfluid) phase and evolves in time (b). Finally, by rapidly ramping up the optical lattice, the dynamics is frozen and the defect density is measured (c). In (d) the lattice depth over time is shown.

Figure 2

(a) The quantum noise, i.e., fluctuations, in the number of defects, shows qualitatively different behavior for quantum quenches above, to, and below the dynamical phase transition (DPT). While it saturates for quenches to the dynamically disordered phase (above the transition), blue dashed line, it grows logarithmically for quenches to transition, red solid line, and as a power law for quenches to the dynamically ordered phase (below the transition), black dotted line. In contrast, the number of excitations in the steady state shows a nonanalytic behavior as a function of the quench parameter at the dynamical critical point (b).

Figure 3

Saturation value of the defect density $N(t\to \infty )/V, V=L^d$, red solid line, for quenches from $r_{0,i}=5$ to $r_{0,f}$ and $\lambda =10$. $N(t\to \infty )/V$ is nonanalytic at the dynamical phase transition, indicated by the thick gray line. Inset: The first derivative of the asymptotic defect density $-\frac{1}{V}\frac{dN(t\to \infty )}{d{r}_{0,f}}$ exhibits a pronounced kink at the transition.

Figure 4

(a) Variance per unit volume ${\sigma }^2/V, V=L^d$, in a log-linear scale for quenches above or at the dynamical transition, i.e., $r_{0,f}\ge r_{0,f}^c$ in $d=3$, for different values of the predicted asymptotic effective parameter $r^{☆}$; see Eq. (6). The inset shows ratios of different cumulants $C_n(t\to \infty )$ as a function of $r^{☆}$, solid lines, and compares them to the corresponding equilibrium cumulants at finite temperature, dashed lines. (b) Variance per unit volume ${\sigma }^2/V$ in a log-log scale for quenches below the dynamical transition, i.e., $r_{0,f}<r_{0,f}^c$ in $d=3$. $\delta r_{0,f}=r_{0,f}^c-r_{0,f}$ measures the distance from the dynamical critical point. In all plots $\lambda =10$ and $r_i=5$.

Figure 5

Long-time saturation value of the first two cumulants for a finite volume as a function of the quench parameter $r_{0,f}$. They show signatures of the dynamical phase transition, indicated by the thick gray solid line.

Figure 6

First two cumulants of the time-averaged statistics of the magnetization as a function of the final quench parameter ${\lambda }_f$ starting from the ground state at ${\lambda }_i=0.1$.

Figure 7

Statistics of the global defect density in the nonintegrable, one-dimensional Bose-Hubbard model. First and second cumulant as a function of the quench parameter $J_f/U_f$, left, and their ratios, right, at filling (a) $n=1$ and (b) $n=2$. Vertical lines indicate the equilibrium phase transition, solid gray line, and the ratio $J_f/U_f$ at which the equilibrium gap in the Mott phase corresponds to the energy density pumped into the system, dashed line. The thin solid lines show the thermal value of the cumulants at the self-consistently determined effective temperature.

Figure 8

Comparison between $r\left(t\right)$ obtained by numerical integration of Eq. (A1) for quenches to different $r_{0,f}>r_{0,f}^c$ (curves of different colors) and the asymptotic value predicted by Eq. (A4) (black dashed lines) for $d=3$ (a) and $d=4$ (b).

Figure 9

Fluctuations of excitations for quenches (a) above, (b) to, and (c) below the dynamic phase transition for various parameters as indicated in the figure caption and legend.

Figure 10

(a) Variance per unit volume ${\sigma }^2/V$ in log-linear scale for quenches above or at the dynamical transition, i.e., $r_{0,f}\ge r_{0,f}^c$ in $d=4$, for different values of the predicted asymptotic effective parameter $r^{☆}$; see Eq. (A4). (b) Variance per unit volume ${\sigma }^2/V, V=L^d$, in log-log scale for quenches below the dynamical transition, i.e., $r_{0,f}<r_{0,f}^c$ in $d=4$. $\delta r_{0,f}=r_{0,f}^c-r_{0,f}$ measures the distance from the dynamical critical point. In all plots $\lambda =10$ and $r_i=5$.