Protecting coherence in optimal control theory: State-dependent constraint approach

Optimal control theory is developed for the task of obtaining an objective in a subspace of the Hilbert space while avoiding population transfer to other subspaces. The objective, a state-to-state transition or a unitary transformation, is carried out without loss of coherence, provided the system in the allowed subspace is decoupled from its environment. An optimization functional is introduced that leads to monotonic convergence of the algorithm. This approach becomes necessary for molecular systems which are subject to processes implying loss of coherence such as ionization or predissociation. In the subspaces corresponding to lossy channels, controllability is hampered or even completely lost. A functional constraint that depends on the state of the system at each instant in time keeps the system out of the lossy channels. We outline the resulting algorithm and discuss its convergence properties. The functionality of the algorithm is demonstrated for the examples of a state-to-state transition and of a unitary transformation for a model of cold ${\text{Rb}}_2$.

Figure 1

Schematic representation of the ${\text{Rb}}_2$ model used in the calculations. The dotted lines indicate the position of the vibrational manifolds considered in each electronic state.

Figure 2

(Color online) $J_{norm}$ (violet solid line) and $I_P$ (green dashed line) as a function of the number of iterations for ${\lambda }_b=0$ corresponding to optimization without state-dependent constraint (upper panel) and ${\lambda }_{b}T=-32$ (lower panel). Note the different scale of the $x$-axes in the two panels.

Figure 3

(Color online) Population in the forbidden subspace as a function of time for the system driven by the optimal field obtained with ${\lambda }_b=0$ after 17 iterations (blue dashed line) and with ${\lambda }_{b}T=-32$ after 500 iterations (red solid line). The inset shows the population in the central time interval in a semi-logarithmic plot.

Figure 4

(Color online) Population in levels $v=0$, $v=1$, $v^{\prime }=10$ (in the allowed subspace) and $v^{″}=6$ (in the forbidden subspace) as a function of time for the system driven by the optimal field obtained with ${\lambda }_b=0$ after 17 iterations (blue dashed line) and with ${\lambda }_{b}T=-32$ after 500 iterations (red solid line). Also shown is the population for the evolution with the guess field (black dotted line). Note the different scale of the $y$-axis for $v^{″}=6$.

Figure 5

(Color online) Spectral amplitude as a function of the frequency $\omega$ for the optimized field obtained with ${\lambda }_b=0$ after 17 iterations (blue dashed line) and with ${\lambda }_{b}T=-32$ after 500 iterations (red solid line). The dashed vertical lines indicate the transition frequencies ${\omega }_{v=0\to v^{\prime }=10}$ and ${\omega }_{v=1\to v^{\prime }=10}$. The transition frequencies between the levels of the $X{{}^1\Sigma }_g^{+}$ electronic ground state and the ${{}^1\Sigma }_u^{+}$ intermediate state are represented by green circles, and the transition frequencies between the ${{}^1\Sigma }_u^{+}$ intermediate state and the ${{}^1\Pi }_g$ upper state by violet squares.

Figure 6

(Color online) Population in the forbidden subspace as a function of time for the system driven by the optimal field obtained with ${\lambda }_b=0$ after 50 iterations (blue dashed line) and with ${\lambda }_{b}T=-8$ after 500 iterations (red solid line) when the system is initially in level $v=0$.

Figure 7

(Color online) Comparison of final results at time $t=T$ as a function of the lifetime of the upper excited state. (a) State-to-state transition: Population in level $v=1$, $P_{v=1}\left(T\right)$ for the optimized field obtained with (red solid line) and without the state-dependent constraint (blue dashed line), and total population remaining in the system, $P_s\left(T\right)$, for the optimized field obtained with (red dashed-dotted line) and without the state-dependent constraint (blue dotted line). (b) Unitary transformation: Normalized final time objective, $\left|\tau \right|/N_r$ for a field obtained by optimization of the unitary transformation with (red solid line) and without state-dependent constraint (blue dashed line). The vertical dotted line indicates the overall pulse duration.