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Frequency-sparse optimal quantum control

  • * Corresponding author: Karl Kunisch

    * Corresponding author: Karl Kunisch
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  • A new class of cost functionals for optimal control of quantum systems which produces controls which are sparse in frequency and smooth in time is proposed. This is achieved by penalizing a suitable time-frequency representation of the control field, rather than the control field itself, and by employing norms which are of $L^1$ or measure form with respect to frequency but smooth with respect to time.

    We prove existence of optimal controls for the resulting nonsmooth optimization problem, derive necessary optimality conditions, and rigorously establish the frequency-sparsity of the optimizers. More precisely, we show that the time-frequency representation of the control field, which a priori admits a continuum of frequencies, is supported on only finitely many frequencies. These results cover important systems of physical interest, including (infinite-dimensional) Schrödinger dynamics on multiple potential energy surfaces as arising in laser control of chemical reactions. Numerical simulations confirm that the optimal controls, unlike those obtained with the usual $L^2$ costs, concentrate on just a few frequencies, even in the infinite-dimensional case of laser-controlled chemical reactions.

    Mathematics Subject Classification: Primary: 49J20, 35Q40.


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  • Figure 1.  Schematic representation of laser-controlled chemical reaction dynamics. The nuclei of a molecule move on different potential energy surfaces depending on the electronic state, and the laser induces transitions between these states. Blue: Potential energy surfaces. Magenta: Initial wave function of the nuclei. Cyan: Target region

    Figure 2.  Optimal control when the cost is chosen as a measure norm with respect to frequency and the $H^1_0$ norm with respect to time (system: Example 2.1, cost: Example 3.1). (A) Control field $Bu(t)$ as a function of time. (B) The contributions due to the two active frequencies of the optimal field. (C) Time-frequency representation $u(\omega, t)$ (color indicates absolute value).

    Figure 3.  Detailed numerical illustration of frequency sparsity of the optimal control from Figure 2. In Figure 3(A) it is depicted that the numerical optimal control (dots in (A)), drops by three orders of magnitude below the threshold given by the numerical regularization parameter $\theta$ (dashed line in (A)), away from the coincidence set $\lVert (B^* {\mathop{\rm Re}\nolimits}\langle \varphi, \tilde H \psi \rangle)(\omega) \rVert_{\mathcal U} = \alpha$, precisely as theoretically predicted by equation (34). In Figure 3(B) we illustrate the quantifiers of Theorem 5.1, which asserts that the optimal control should vanish off the frequencies where the norm $\lVert (B^* {\mathop{\rm Re}\nolimits}\langle \varphi, \tilde H \psi \rangle)(\omega) \rVert_{\mathcal U}$ (dots in (B)) reaches the cost parameter $\alpha$ (solid line in (B))

    Figure 4.  Optimal controls for Schrödinger dynamics on two potential energy surfaces (Example 2.2). Rows: Different choices of cost functionals and control operators. Columns: Time, frequency, and time-frequency representation of the optimal controls (i.e. $(Bu)(t)$, $\lvert u \rvert(\omega)$ and $u(\omega, t)$). The dashed red line in the middle column, rows 1 to 3, indicates the Huber threshold, and the nonzero contributions below it are an artifact of the Huber regularization (see Theorem 5.1). In the rightmost column, the absolute values of the optimal measures are plotted in the time-frequency plane. Note that in column 1, 2, and 4 far fewer frequencies are active compared to the standard $L^2$ control (column 5)

    Table 1.  Cost parameters $\alpha$ and expectation values $f(\bar u) = \frac{1}{2} \langle \bar \psi, \mathcal O \bar \psi \rangle$ for different control spaces. A value of $f(\bar u) = 2.5 \cdot 10^{-2}$ corresponds to a $95\%$ achievement of the control goal

    control space $\alpha$ $f(\bar u)$
    $\mathcal M(\Omega; H^1_0)$ $0.03$ $4.49 \cdot 10^{-3}$
    $\mathcal M(\Omega; L^2)$ $0.03$ $2.69 \cdot 10^{-2}$
    $\mathcal M(\Omega; \mathbb {C})$ $0.06$ $2.24 \cdot 10^{-2}$
    $\mathcal M(\Omega\times[0, T]; \mathbb {C})$ $0.02$ $1.89 \cdot 10^{-2}$
    $L^2(0, T)$ $0.0001$ $5.39 \cdot 10^{-6}$
     | Show Table
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