Optimal control of time-dependent targets

In this work, we investigate how and to which extent a quantum system can be driven along a prescribed path in Hilbert space by a suitably shaped laser pulse. To calculate the optimal, i.e., the variationally best pulse, a properly defined functional is maximized. This leads to a monotonically convergent algorithm which is computationally not more expensive than the standard optimal-control techniques to push a system, without specifying the path, from a given initial to a given final state. The method is successfully applied to drive the time-dependent density along a given trajectory in real space and to control the time-dependent occupation numbers of a two-level system and of a one-dimensional model for the hydrogen atom.

Figure 1

Target function and calculated occupation numbers for $J_1=0.90$, 0.95, 0.99, 0.9995 (a), optimal field $J_1=0.9995$ and extracted envelopes for $J_1=0.90$, 0.95 (b), and the value of the functional $J_1+J_2$ (c).

Figure 2

Target curves $a_0\left(t\right),a_1\left(t\right)$ and calculated occupation numbers (a), optimal field (b), and the value of $J_1$ and $J_1+J_2$ (c).

Figure 3

Target curves $b_0\left(t\right),b_1\left(t\right)$ and calculated occupation numbers (a), optimized field (b), and functional $J_1+J_2$ (c) for $\alpha =0.05$.

Figure 4

The influence of the penalty factor $\alpha$: occupation numbers (a) and optimal fields (b).

Figure 5

$⟨\widehat{x}⟩$ calculated with target and optimized wave function (a) and optimal field (b) (after 1000 iterations) for parameters $\alpha =0.5,\Delta t=0.005,{ϵ}_0={10}^{-3}$.

Figure 6

Comparison for the expectation value $⟨\widehat{x}⟩$ with the target trajectory (squares) for different exponents $n$.