Dynamical symmetry breaking with optimal control: Reducing the number of pieces

We analyze the production of defects during the dynamical crossing of a mean-field phase transition with a real order parameter. When the parameter that brings the system across the critical point changes in time according to a power-law schedule, we recover the predictions dictated by the well-known Kibble-Zurek theory. For a fixed duration of the evolution, we show that the average number of defects can be drastically reduced for a very large but finite system, by optimizing the time dependence of the driving using optimal control techniques. Furthermore, the optimized protocol is robust against small fluctuations.

Figure 1

Time dependence of the function $\varepsilon \left(t\right)$ from Eq. (4) for $\alpha =1$ (top) and $\alpha =2$ (bottom) for ${\tau }_Q=20$ (solid line), 50 (dashed line), and 100 (dotted line).

Figure 2

Scaling of the average number of defects $N_D$ (points) as a function of $1/{\tau }_Q$ in log-log scale for $\alpha =1$ (top) and $\alpha =4$ (bottom). Also shown are the best-fit lines according to prediction (6).

Figure 3

Average number of defects for fixed transition time $T$ as a function of the crossing exponent $\alpha$: (a) $T=20$, (b) $T=40$, (c) $T=60$, (d) $T=80$. The error bars are taken as the standard deviations of each set of data at fixed $T$ and $\alpha$. The solid lines connect the points and are only a guide to the eye.

Figure 4

Top panel: Optimal traversing exponents ${\alpha }_{\text{opt}}$ (symbols) found from the minimization of the data in Fig. 3 versus the total time $T$ in semilogarithmic scale. The solid line represents the best-fitting function ${\tilde{\alpha }}_{\text{opt}}$. Bottom panel: Average number of defects $N_{D,\text{opt}}$ (symbols) as a function of the total time $T$ in log-log scale. The solid line is the best-fitting function ${\tilde{N}}_{D,\text{opt}}$ [see Eq. (12)].

Figure 5

Optimal control results. Comparison of the optimized pulses $\varepsilon \left(t\right)$ for $n_{\text{max}}=4$ (dashed line) with the original power-law dependence (solid line). We set $T=20$ and $\alpha =0.6$.

Figure 6

Optimized density of defects for $T=10$ (top panel) and $T=20$ (bottom panel) versus the number of frequencies $n_{\text{max}}$ in the optimization algorithm. The different symbols are results obtained for different systems sizes $N$ ranging from $2^8$ to $2^{14}$. Straight lines connecting symbols are only a guide to the eye.