Measurement-assisted Landau-Zener transitions

Nonselective quantum measurements, i.e., measurements without reading the results, are often considered as a resource for manipulating quantum systems. In this work, we investigate optimal acceleration of the Landau-Zener (LZ) transitions by nonselective quantum measurements. We use the measurements of a population of a diabatic state of the LZ system at certain time instants as control and find the optimal time instants which maximize the LZ transition. We find surprising nonmonotonic behavior of the maximal transition probability with increase of the coupling parameter when the number of measurements is large. This transition probability gives an optimal approximation to the fundamental quantum Zeno effect (which corresponds to continuous measurements) by a fixed number of discrete measurements. The difficulty for the analysis is that the transition probability as a function of time instants has a huge number of local maxima. We resolve this problem both analytically by asymptotic analysis and numerically by the development of efficient algorithms mainly based on the dynamic programming. The proposed numerical methods can be applied, besides this problem, to a wide class of measurement-based optimal control problems.

Figure 1

Energy levels of the LZ system.

Figure 2

Transition probability for one measurement at the instant $t$ (left) and for three measurements (right) at the instants $-t$, 0, and $t,t\ge 0$, for $\gamma =0.1$ (top curves), $\gamma =1$ (bottom thick curves), and $\gamma =5$ (bottom thin curves).

Figure 3

Transition probability for two measurements at the instants $t_1,t_2$ (left column) and for five measurements (right column) at the instants $-t_2,-t_1$, 0, $t_1,t_2,0\le t_1\le t_2$, for $\gamma =0.1$ (top plots), $\gamma =1$ (middle plots), and $\gamma =5$ (bottom plots). The ranges of $t_1$ and $t_2$ for the middle and bottom plots are the same as for the top plots.

Figure 4

Maximal transition probability vs number of measurements $N$ for different $\gamma$. Left (from top to bottom): $\gamma =0.01,0.05,0.1,0.2,0.5,0.75$. Right (from bottom to top): $\gamma =0.75,1,1.2,1.5,2,5$. Inset: a part of the same graph. Results for $\gamma \le 0.5$ were obtained using the algorithm for small $\gamma$ (see Sec. 5) and verified by a more precise and general, but more computationally costly method of Sec. 4. Differences in the maximal transition probabilities found by these two algorithms are less than $0.01$. Results for $\gamma >0.5$ were obtained using the general method of Sec. 4, but the differential evolution algorithm combined with the analysis of Sec. 6 also gives good results. The difference in the maximal transition probabilities is less than $0.01$ for $\gamma \ge 1$ and less than or of the order of $0.01$ for $0.5\le \gamma <1$. The curve for $\gamma =5$, as well as for larger $\gamma$ not shown here, almost coincides with the theoretical upper bound curve given by (20).

Figure 5

Maximal transition probability vs $\gamma$ for a fixed number of measurements; from bottom to top, $N=0,1,3,5,10,15$. Curves for $N=0$ and $N=1$ are drawn using the exact LZ formula (7) and formula (12). Other calculations are the same as described in the caption of Fig. 4.

Figure 6

$f_N$ vs number of measurements $N$. The target function is $\langle 0\left|\rho \right(+\infty \left)\right|0\rangle =1-2\pi \gamma f_N+O\left({\gamma }^2\right)$ [see (17)]. The case of no measurements corresponds to $f_0=1$ and the first-order approximation $1-2\pi \gamma$ of the LZ formula (for small $\gamma$ or antiadiabatic regime).

Figure 7

Points on the Kornu spiral that correspond to the optimal instants of measurements for $N=12$ (sticks) and $N=13$ (balls), in the first-order approximation for small $\gamma$ or antiadiabatic regime. Points corresponding to $\pm \infty$ are also shown in the centers of the spires. Note that the optimal instants of measurements are symmetric with respect to zero for $N=12$, but they are nonsymmetric for $N=13$. This symmetry is the case for the majority of $N$. “Symmetry breaking” occurs when a new spire of the Kornu spiral becomes populated, as is indicated by a ball without stick on a new spire in the top right corner of the graph.