Optimal suppression of defect generation during a passage across a quantum critical point

The dynamics of quantum phase transitions are inevitably accompanied by the formation of defects when crossing a quantum critical point. For a generic class of quantum critical systems, we solve the problem of minimizing the production of defects through the use of a gradient-based deterministic optimal control algorithm. By considering a finite-size quantum Ising model with a tunable global transverse field, we show that an optimal power-law quench of the transverse field across the Ising critical point works well at minimizing the number of defects, in spite of being drawn from a subset of quench profiles. These power-law quenches are shown to be inherently robust against noise. The optimized defect density exhibits a transition at a critical ratio of the quench duration to the system size, which we argue coincides with the intrinsic speed limit for quantum evolution.

Figure 1

The defect density $\rho \left(T\right)$ (blue circles) obtained by applying a power-law quench with a power $r^{*}$ obtained through the gradient algorithm, compared with the results for a linear quench [34] (red squares), the truncated assisted driving with $M=10$ [12] (green triangles), and the local adiabatic evolution [9] (black triangles). All the solid lines are plotted using $N=100$. The inset displays the optimal power $r^{*}$ found by the gradient algorithm for different $\tau$ and system sizes. For $\tau >{\tau }_c,r^{*}$ takes on a uniquely defined large value that increases with $N$ and decreases with $\tau$. When $\tau <{\tau }_c$, the value for $r^{*}$ found by the gradient algorithm is not unique due to traps in the control landscape, and this is reflected in the erratic behavior of $r^{*}\left(\tau \right)$.

Figure 2

The excitation probability $P_k$ for different quench durations $T$ with $N=100$. Shown are the results for both a linear quench with $r=1.0$ (dashed lines) and the optimal power-law quench with $r=r^{*}$ (solid lines). For $T>T_c$, optimization of the control pulse causes $P_{k_N}$ to drop sharply; this occurs concurrently with the drastic improvement in $\rho \left(T\right)$.

Figure 3

The quantum speed limit $T_{\mathrm{QSL}}^{\prime }\left(k\right)/N$ is plotted for three system sizes $N=24$, 50, and 100. The limiting value of $T_{\mathrm{QSL}}^{\prime }\left(k_N\right)/N=1/8$ is plotted as the dashed line.