Invariance of quantum optimal control fields to experimental parameters

We consider an ensemble of closed finite-level systems described by a density matrix where the goal is to find an optimal control field to maximize the expectation value of an observable. The eigenvalues of the initial density matrix are assumed to depend on an experimental parameter (e.g., the temperature), whereas the eigenvalues of the observable may depend on an additional application-specific experimental parameter. We show that an optimal control will remain optimal for all such experimental parameters, if the relative ordering, by magnitude, of the eigenvalues of the initial density matrix as well as of the observable is unaltered regardless of the parameter values. More generally, we show like invariance of a control associated with any particular critical point on the corresponding control landscape. The invariance of control laser fields with respect to temperature is illustrated for vibrational excitation of diatomic molecules and photoassociation of atoms.

Figure 1

(a) Ratio $J[T,\varepsilon ]/J_{\max }\left[T\right]$ as a function of the temperature. Each curve was constructed for a different control field. The curve with filled circles corresponds to the nearly optimal control field ${\varepsilon }^{*}\left(t\right)$ determined at $T=500$K, and then utilized again at all other temperatures $T$. The other curves in (a) with nonoptimal fields do not exhibit temperature invariance. The near invariance to $T$ for the curve created with ${\varepsilon }^{*}\left(t\right)$ occurs, despite the fact that the value of $J_{\max }\left[T\right]$ in (b) varies significantly as a function of $T$.

Figure 2

(a) Ratio $J[T,\varepsilon ]/J_{\mathrm{saddle}}\left(T\right)$ as a function of the temperature. Each curve was constructed for a different control field identified at 500 K and then utilized at other temperatures. The curve with filled circles corresponds to the control field ${\varepsilon }^s\left(t\right)$ closest to the saddle. The nearly constant value for $J[T,{\varepsilon }^{\mathrm{s}}]/J_{\mathrm{saddle}}\left(T\right)$ with respect to temperature occurs, despite the saddle point value $J_{\mathrm{saddle}}\left[T\right]$ in (b) having a significant nonmonotonic variation with temperature $T$. In contrast, the other curves in (a) obtained with nonsaddle creating fields do not show temperature invariance.

Figure 3

(a) Ratio $J[T,\varepsilon ]/J_{\max }\left[T\right]$ as a function of the temperature for photoassociation. Each curve was obtained for a different control field determined during the optimization process performed at $T=30$ K. The curve with filled circles corresponds to the control field ${\varepsilon }^{*}\left(t\right)$ producing a yield of $\approx 0.94$ at $T=30$ K. All of the fields asymptote to a constant value at high temperature, but only the case of ${\varepsilon }^{*}\left(t\right)$ approaches the optimal limit of $1.0$. (b) The value of $J_{\max }\left[T\right]$ changes significantly as a function of $T$.