Monotonically convergent optimal control theory of quantum systems with spectral constraints on the control field

We propose a monotonically convergent algorithm which can enforce spectral constraints on the control field (and extends to arbitrary filters). The procedure differs from standard algorithms in that at each iteration, the control field is taken as a linear combination of the control field (computed by the standard algorithm) and the filtered field. The parameter of the linear combination is chosen to respect the monotonic behavior of the algorithm and to be as close to the filtered field as possible. We test the efficiency of this method on molecular alignment. Using bandpass filters, we show how to select particular rotational transitions to reach high alignment efficiency. We also consider spectral constraints corresponding to experimental conditions using pulse-shaping techniques. We determine an optimal solution that could be implemented experimentally with this technique.

Figure 1

Plot as a function of the adimensional time $t/t_{per}$ of (a) the expectation value $⟨{\text{cos}}^2\theta ⟩$ obtained from the algorithm with filtering, (b) the corresponding optimal filtered field, and (c) the standard optimal field. (d) Plot as a function of the number of iterations $k$ of the adimensional cost $J_k$ (solid line) and of the parameter ${\mu }_k$ (dashed line) for the algorithm with filtering.

Figure 2

Normalized square modulus of the Fourier transform (a) of the trial field, (b) of the optimal field obtained by the standard algorithm, and (c) of the optimal field obtained by the algorithm with filtering. The vertical dashed lines correspond to the rotational frequencies between two successive rotational levels.

Figure 3

Plot as a function of the adimensional time $t/t_f$ of (a) the optimal electric field and (b) the projection onto the target state ${\rho }_{opt}$. (c) Plot as a function of the number of iteration $k$ of the parameter ${\mu }_k$. Solid, dashed, and dotted lines correspond to $T=10$, 7, and 5 K, respectively. The number of pixels is equal to 128.

Figure 4

Same as Fig. 3 but for $T=5\text{K}$ and different numbers of pixels. Solid, dashed, and dotted lines correspond to $N=128$, 64, and 256 pixels, respectively.