Optimized adiabatic-impulse protocol preserving Kibble-Zurek scaling with attenuated anti-Kibble-Zurek behavior

Abstract

We propose an optimized adiabatic-impulse (OAI) protocol that achieves much shorter evolution time while preserving the Kibble-Zurek scaling. Near the critical regime, the control field is linearly ramped across the quantum critical point at a rate characterized by a quench time $τ_Q$. Away from the critical regime, the evolution is designed to follow the threshold of adiabatic breakdown, which we characterize by an adiabatic coefficient $ζ\proptoτ_Q^α$. As a consequence, the total evolution time exhibits a sublinear power-law dependence on $τ_Q$, and the conventional linear quench protocol is recovered in the limit $α\rightarrow\infty$. We apply the OAI protocol to the transverse Ising chain and numerically determine the minimum value of $ζ$. We further investigate the nonequilibrium dynamics in the presence of a noisy field that can induce anti-Kibble-Zurek behavior, leading to more defects for slower ramps. Within the OAI protocol, the optimal quench time that minimizes defects obeys an altered universal power-law scaling with the noise strength. Finally, we generalize the OAI protocol to incorporate nonlinear Kibble-Zurek scaling.

Figures (7)

• Figure 1: OAI protocol in the transverse Ising chain with fixed parameters $g_i=2$, $g_f=0$, and $\tau_Q=1000$. (a) and (b) show the two timescales, $|\epsilon(t)/\dot{\epsilon}(t)|$ and $\tau(t)$, for several representative values of $\zeta$, evaluated far from and near the critical point, respectively. In (c), we compare the evolution of the dimensionless parameter $\epsilon(t)$ in the OAI protocol with that in the LQ protocol. $\epsilon(t)$ diverges at $t=\pm\theta$, where the definition of $\theta$ is provided below Eq. (\ref{['sol-epsi']}).
• Figure 2: Final defect density in the transverse Ising chain when the OAI protocol is applied. In (a), we set $g_i=2$ and $g_f=0$. The defect density (colored dingbats) gradually approaches the KZ scaling (black dashed line) as the adiabatic coefficient $\zeta$ increases. The inset of (a) compares the time evolution of the defect density for the OAI and LQ protocols at $\tau_Q=200$. For the OAI protocol we choose $\zeta=32$, while LQ protocol takes the form in Eq. (\ref{['linearq']}). The inset demonstrates the adiabaticity of the OAI protocol away from criticality. In (b), we fix $\tau_Q=200$, $\zeta=32$ and $g_f=0$, and the defect density data for different $g_i$ collapse onto the KZ scaling. The selected values of $g_i$ in (b) are indicated by the corresponding colored dingbats in the inset of (b).
• Figure 3: Scaled defect density as a function of $\zeta$ for several quench times, with $g_i=2$ and $g_f=0$. (a) demonstrates that $n/n_\text{KZ}\rightarrow 1$ for sufficiently large $\zeta$. (b) presents a log-log plot of $n/n_\text{KZ}-1$ versus $\zeta$, revealing a power-law relationship. We fix $n/n_\text{KZ}-1=2$ (horizontal dashed line) and extract the corresponding intersections, yielding $\zeta_1=1.072$ ($\tau_Q=1000$), $\zeta_2=1.274$ ($\tau_Q=2000$), $\zeta_3=1.467$ ($\tau_Q=3000$), $\zeta_4=1.513$ ($\tau_Q=4000$), $\zeta_5=1.601$ ($\tau_Q=5000$), and $\zeta_6=1.675$ ($\tau_Q=6000$). The relationship between $\zeta$ and $\tau_Q$ is summarized in the inset of (b). By fitting these data, we approximately obtain $\zeta\propto\tau_Q ^{1/4}$. In (c), the horizontal axis is rescaled as $\tau_Q^{-1/4}\zeta$. The data reveal the scaling behavior of $F$ defined in Eq. (\ref{['zeta-fit']}), $F\approx 0.113(\tau_Q^{-1/4}\zeta)^{-1.732}$. In the inset of (c) depicts the excitation probability for $\tau_Q=10$ and $\zeta=0.8$, indicating that $F$ originates from short wave oscillations.
• Figure 4: (a) Defect density $n$ versus $\tau_Q$ under a weak noisy control field, with $\zeta=\tau_Q^{1/4}$ (implying $\alpha=1/4$), $g_i=2$ and $g_f=0$. For sufficiently larger $\tau_Q$, the KZ scaling is suppressed and $n$ exhibits the AKZ behavior. In the inset of (a), the data collapse onto the black solid line given by Eq. (\ref{['AKZ']}), $n \propto (W^{16/5}\tau_Q)^{5/8}$. (b) Scaling behavior of the optimal quench time $\tilde{\tau}_Q$ versus $W$. Numerical results of $\tilde{\tau}_Q$ are shown as colored dingbats for different values of $\zeta$, while the corresponding fitting results are indicated by colored dashed lines. The data for $\zeta\rightarrow\infty$ are obtained by calculating Eq. (\ref{['app-tdBdG']}) in the LQ protocol.
• Figure 5: Bar chart comparing the scaling exponent of the optimal quench time $\tilde{\tau}_Q$ obtained from theory and numerical fitting. Burnt-orange bars show the theoretical prediction $r=\frac{4}{2+\alpha}$ defined in Eq. (\ref{['tauQScaling']}), while the corresponding exponent $s'$ is extracted from Fig. \ref{['plot-AKZ']}(b). The relative deviation $|s-s'|/s$ (in percent) is given above each pair of bars. For $\zeta\ll\tau_Q$, the deviation is relatively small, approximately $4\%$. In the limit $\zeta\rightarrow\infty$, the OAI protocol reduces to the LQ protocol, and the deviation becomes negligible.