March  2018, 8(1): 155-176. doi: 10.3934/mcrf.2018007

Frequency-sparse optimal quantum control

1. 

Faculty of Mathematics, Technical University Munich, Boltzmannstr. 3, D-85747 Garching, Germany

2. 

Institute of Mathematics, FU Berlin, Arnimallee 6, D-14195 Berlin, Germany

3. 

Institute of Mathematics and Scientific Computing, Karl-Franzens-Universität Graz, Heinrichstr. 36, A-8010 Graz, Austria

4. 

Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenbergerstraße 69, A-4040 Linz, Austria

* Corresponding author: Karl Kunisch

Received  April 2017 Revised  October 2017 Published  January 2018

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.

Citation: Gero Friesecke, Felix Henneke, Karl Kunisch. Frequency-sparse optimal quantum control. Mathematical Control & Related Fields, 2018, 8 (1) : 155-176. doi: 10.3934/mcrf.2018007
References:
[1]

A. AugerA. BenH. YedderE. CancesC. L. BrisC. M. DionA. Keller and O. Atabek, Optimal laser control of molecular systems: Methodology and results, Math. Models Methods Appl. Sci, 12 (2012), 1281-1315.   Google Scholar

[2] G. G. Balint-KurtiS. Zou and A. Brown, Optimal control theory for manipulating molecular processes, John Wiley & Sons, Inc., 2008.   Google Scholar
[3]

J. M. BallJ. E. Marsden and M. Slemrod, Controllability for distributed bilinear systems, SIAM J. Control Optim., 20 (1982), 575-597.   Google Scholar

[4]

H. BergmannH. Theuer and B. W. Shore, Coherent population transfer among quantum states of atoms and molecules, Rev. Mod. Phys., 70 (1998), 1003-1025.   Google Scholar

[5]

E. CasasR. Herzog and G. Wachsmuth, Optimality conditions and error analysis of semilinear elliptic control problems with $L^1$ cost functional, SIAM J. Optim., 22 (2012), 795-820.   Google Scholar

[6]

G. Ciaramella and A. Borzì, A {LONE} code for the sparse control of quantum systems, Computer Physics Communications, 200 (2016), 312-323.   Google Scholar

[7]

G. Ciaramella and A. Borzì, Quantum optimal control problems with a sparsity cost functional, Numer. Funct. Anal. Optim., 37 (2016), 938-965.   Google Scholar

[8]

F. H. Clarke, Optimization and Nonsmooth Analysis, vol. 5 of Classics in Applied Mathematics, 2nd edition, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990.  Google Scholar

[9]

C. Clason and K. Kunisch, A duality-based approach to elliptic control problems in non-reflexive Banach spaces, ESAIM Control Optim. Calc. Var., 17 (2011), 243-266.   Google Scholar

[10]

D. D'Alessandro, Introduction to Quantum Control and Dynamics, Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series, Chapman & Hall/CRC, Boca Raton, FL, 2008.  Google Scholar

[11]

S. J. GlaserU. BoscainT. CalarcoC. P. KochW. KöckenbergerR. KosloffI. KuprovB. LuyS. SchirmerT. Schulte-HerbrüggenD. Sugny and F. K. Wilhelm, Training Schrödinger's cat: Quantum optimal control, The European Physical Journal D, 69 (2015), 1-24.   Google Scholar

[12]

M. HellgrenE. Räsänen and E. K. U. Gross, Optimal control of strong-field ionization with time-dependent density-functional theory, Phys. Rev. A, 88 (2013), 013414.   Google Scholar

[13]

F. Henneke and M. Liebmann, A generalized Suzuki-Trotter type method in optimal control of coupled Schrödinger equations, Computing and Visualization in Science, 17 (2015), 277-293.   Google Scholar

[14]

R. HerzogG. Stadler and G. Wachsmuth, Directional sparsity in optimal control of partial differential equations, SIAM J. Control Optim., 50 (2012), 943-963.   Google Scholar

[15]

M. HintermüllerD. MarahrensP. A. Markowich and C. Sparber, Optimal bilinear control of Gross-Pitaevskii equations, SIAM J. Control Optim., 51 (2013), 2509-2543.   Google Scholar

[16]

P. v. d. HoffS. ThallmairM. KowalewskiR. Siemering and R. D. Vivie-Riedle, Optimal control theory -closing the gap between theory and experiment, Phys. Chem. Chem. Phys., 14 (2012), 14460-14485.   Google Scholar

[17]

K. Ito and K. Kunisch, Optimal bilinear control of an abstract Schrödinger equation, SIAM Journal on Control and Optimization, 46 (2007), 274-287.   Google Scholar

[18]

A. F. IzmailovA. L. Pogosyan and M. V. Solodov, Semismooth SQP method for equalityconstrained optimization problems with an application to the lifted reformulation of mathematical programs with complementarity constraints, Optimization Methods and Software, 26 (2011), 847-872.   Google Scholar

[19]

K. KormannS. Holmgren and H. Karlsson, A Fourier-coefficient based solution of an optimal control problem in quantum chemistry, Journal of Optimization Theory and Applications, 174 (2010), 491-506.   Google Scholar

[20]

K. KunischK. Pieper and B. Vexler, Measure valued directional sparsity for parabolic optimal control problems, SIAM J. Control Optim., 52 (2014), 3078-3108.   Google Scholar

[21]

S. Lang, Real and Functional Analysis, vol. 142 of Graduate Texts in Mathematics, 3rd edition, Springer-Verlag, New York, 1993.  Google Scholar

[22]

M. LapertD. SugnyR. Tehini and G. Turinici, Monotonically convergent optimal control theory of quantum systems with spectral constraints on the control field, Physical Review A: Atomic, Molecular and Optical Physics, 79 (2009), 063411.   Google Scholar

[23]

X. J. Li and J. M. Yong, Optimal Control Theory for Infinite-Dimensional Systems Systems & Control: Foundations & Applications, Birkhäuser Boston Inc., Boston, MA, 1995.  Google Scholar

[24]

L. Meziani, On the dual space $C^*_0(S, X)$, Acta Math. Univ. Comenian. (N.S.), 78 (2009), 153-160.   Google Scholar

[25]

J. Nocedal and S. J. Wright, Numerical Optimization, 2nd edition, Springer Series in Operations Research and Financial Engineering, Springer, New York, 2006.  Google Scholar

[26]

A. P. PeirceM. A. Dahleh and H. Rabitz, Optimal control of quantum-mechanical systems: Existence, numerical approximation, and applications, Phys. Rev. A (3), 37 (1988), 4950-4964.   Google Scholar

[27] M. Reed and B. Simon, Methods of Modern Mathematical Physics. Ⅱ. Fourier analysis, Self-adjointness, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975.   Google Scholar
[28]

Q. RenG. G. Balint-KurtiF. R. ManbyM. ArtamonovT.-S. Ho and H. Rabitz, Quantum control of molecular vibrational and rotational excitations in a homonuclear diatomic molecule: A full three-dimensional treatment with polarization forces, The Journal of Chemical Physics, 124 (2006), 014111.   Google Scholar

[29]

S. RuetzelC. StolzenbergerF. DimlerD. J. Tannor and T. Brixner, Adaptive coherent control using the von Neumann basis, Phys. Chem. Chem. Phys., 13 (2011), 8627-8636.   Google Scholar

[30]

J. ScheuerX. KongR. S. SaidJ. ChenA. KurzL. MarsegliaJ. DuP. R. HemmerS. MontangeroT. CalarcoB. Naydenov and F. Jelezko, Precise qubit control beyond the rotating wave approximation, New Journal of Physics, 16 (2014), 093022.   Google Scholar

[31]

S. SharmaH. Singh and G. G. Balint-Kurti, Genetic algorithm optimization of laser pulses for molecular quantum state excitation, The Journal of Chemical Physics, 132 (2010), 064108.   Google Scholar

[32] B. W. Shore, The Theory of Coherent Atomic Excitation, Wiley, New York, NY, 1990.   Google Scholar
[33]

G. Stadler, Elliptic optimal control problems with $L^1$-control cost and applications for the placement of control devices, Comput. Optim. Appl., 44 (2009), 159-181.   Google Scholar

[34]

S. Teufel, Adiabatic Perturbation Theory in Quantum Dynamics, vol. 1821 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2003.  Google Scholar

[35]

G. TuriniciC. Le Bris and H. Rabitz, Efficient algorithms for the laboratory discovery of optimal quantum controls, Phys. Rev. E, 70 (2004), 016704.   Google Scholar

[36]

G. von Winckel and A. Borzì, Computational techniques for a quantum control problem with $H^1$-cost, Inverse Problems, 24 (2008), 034007, 23pp.  Google Scholar

[37]

G. von WinckelA. Borzì and S. Volkwein, A globalized Newton method for the accurate solution of a dipole quantum control problem, SIAM J. Sci. Comput., 31 (2009/10), 4176-4203.   Google Scholar

[38]

G. Vossen and H. Maurer, On $L^1$-minimization in optimal control and applications to robotics, Optimal Control Appl. Methods, 27 (2006), 301-321.   Google Scholar

show all references

References:
[1]

A. AugerA. BenH. YedderE. CancesC. L. BrisC. M. DionA. Keller and O. Atabek, Optimal laser control of molecular systems: Methodology and results, Math. Models Methods Appl. Sci, 12 (2012), 1281-1315.   Google Scholar

[2] G. G. Balint-KurtiS. Zou and A. Brown, Optimal control theory for manipulating molecular processes, John Wiley & Sons, Inc., 2008.   Google Scholar
[3]

J. M. BallJ. E. Marsden and M. Slemrod, Controllability for distributed bilinear systems, SIAM J. Control Optim., 20 (1982), 575-597.   Google Scholar

[4]

H. BergmannH. Theuer and B. W. Shore, Coherent population transfer among quantum states of atoms and molecules, Rev. Mod. Phys., 70 (1998), 1003-1025.   Google Scholar

[5]

E. CasasR. Herzog and G. Wachsmuth, Optimality conditions and error analysis of semilinear elliptic control problems with $L^1$ cost functional, SIAM J. Optim., 22 (2012), 795-820.   Google Scholar

[6]

G. Ciaramella and A. Borzì, A {LONE} code for the sparse control of quantum systems, Computer Physics Communications, 200 (2016), 312-323.   Google Scholar

[7]

G. Ciaramella and A. Borzì, Quantum optimal control problems with a sparsity cost functional, Numer. Funct. Anal. Optim., 37 (2016), 938-965.   Google Scholar

[8]

F. H. Clarke, Optimization and Nonsmooth Analysis, vol. 5 of Classics in Applied Mathematics, 2nd edition, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990.  Google Scholar

[9]

C. Clason and K. Kunisch, A duality-based approach to elliptic control problems in non-reflexive Banach spaces, ESAIM Control Optim. Calc. Var., 17 (2011), 243-266.   Google Scholar

[10]

D. D'Alessandro, Introduction to Quantum Control and Dynamics, Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series, Chapman & Hall/CRC, Boca Raton, FL, 2008.  Google Scholar

[11]

S. J. GlaserU. BoscainT. CalarcoC. P. KochW. KöckenbergerR. KosloffI. KuprovB. LuyS. SchirmerT. Schulte-HerbrüggenD. Sugny and F. K. Wilhelm, Training Schrödinger's cat: Quantum optimal control, The European Physical Journal D, 69 (2015), 1-24.   Google Scholar

[12]

M. HellgrenE. Räsänen and E. K. U. Gross, Optimal control of strong-field ionization with time-dependent density-functional theory, Phys. Rev. A, 88 (2013), 013414.   Google Scholar

[13]

F. Henneke and M. Liebmann, A generalized Suzuki-Trotter type method in optimal control of coupled Schrödinger equations, Computing and Visualization in Science, 17 (2015), 277-293.   Google Scholar

[14]

R. HerzogG. Stadler and G. Wachsmuth, Directional sparsity in optimal control of partial differential equations, SIAM J. Control Optim., 50 (2012), 943-963.   Google Scholar

[15]

M. HintermüllerD. MarahrensP. A. Markowich and C. Sparber, Optimal bilinear control of Gross-Pitaevskii equations, SIAM J. Control Optim., 51 (2013), 2509-2543.   Google Scholar

[16]

P. v. d. HoffS. ThallmairM. KowalewskiR. Siemering and R. D. Vivie-Riedle, Optimal control theory -closing the gap between theory and experiment, Phys. Chem. Chem. Phys., 14 (2012), 14460-14485.   Google Scholar

[17]

K. Ito and K. Kunisch, Optimal bilinear control of an abstract Schrödinger equation, SIAM Journal on Control and Optimization, 46 (2007), 274-287.   Google Scholar

[18]

A. F. IzmailovA. L. Pogosyan and M. V. Solodov, Semismooth SQP method for equalityconstrained optimization problems with an application to the lifted reformulation of mathematical programs with complementarity constraints, Optimization Methods and Software, 26 (2011), 847-872.   Google Scholar

[19]

K. KormannS. Holmgren and H. Karlsson, A Fourier-coefficient based solution of an optimal control problem in quantum chemistry, Journal of Optimization Theory and Applications, 174 (2010), 491-506.   Google Scholar

[20]

K. KunischK. Pieper and B. Vexler, Measure valued directional sparsity for parabolic optimal control problems, SIAM J. Control Optim., 52 (2014), 3078-3108.   Google Scholar

[21]

S. Lang, Real and Functional Analysis, vol. 142 of Graduate Texts in Mathematics, 3rd edition, Springer-Verlag, New York, 1993.  Google Scholar

[22]

M. LapertD. SugnyR. Tehini and G. Turinici, Monotonically convergent optimal control theory of quantum systems with spectral constraints on the control field, Physical Review A: Atomic, Molecular and Optical Physics, 79 (2009), 063411.   Google Scholar

[23]

X. J. Li and J. M. Yong, Optimal Control Theory for Infinite-Dimensional Systems Systems & Control: Foundations & Applications, Birkhäuser Boston Inc., Boston, MA, 1995.  Google Scholar

[24]

L. Meziani, On the dual space $C^*_0(S, X)$, Acta Math. Univ. Comenian. (N.S.), 78 (2009), 153-160.   Google Scholar

[25]

J. Nocedal and S. J. Wright, Numerical Optimization, 2nd edition, Springer Series in Operations Research and Financial Engineering, Springer, New York, 2006.  Google Scholar

[26]

A. P. PeirceM. A. Dahleh and H. Rabitz, Optimal control of quantum-mechanical systems: Existence, numerical approximation, and applications, Phys. Rev. A (3), 37 (1988), 4950-4964.   Google Scholar

[27] M. Reed and B. Simon, Methods of Modern Mathematical Physics. Ⅱ. Fourier analysis, Self-adjointness, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975.   Google Scholar
[28]

Q. RenG. G. Balint-KurtiF. R. ManbyM. ArtamonovT.-S. Ho and H. Rabitz, Quantum control of molecular vibrational and rotational excitations in a homonuclear diatomic molecule: A full three-dimensional treatment with polarization forces, The Journal of Chemical Physics, 124 (2006), 014111.   Google Scholar

[29]

S. RuetzelC. StolzenbergerF. DimlerD. J. Tannor and T. Brixner, Adaptive coherent control using the von Neumann basis, Phys. Chem. Chem. Phys., 13 (2011), 8627-8636.   Google Scholar

[30]

J. ScheuerX. KongR. S. SaidJ. ChenA. KurzL. MarsegliaJ. DuP. R. HemmerS. MontangeroT. CalarcoB. Naydenov and F. Jelezko, Precise qubit control beyond the rotating wave approximation, New Journal of Physics, 16 (2014), 093022.   Google Scholar

[31]

S. SharmaH. Singh and G. G. Balint-Kurti, Genetic algorithm optimization of laser pulses for molecular quantum state excitation, The Journal of Chemical Physics, 132 (2010), 064108.   Google Scholar

[32] B. W. Shore, The Theory of Coherent Atomic Excitation, Wiley, New York, NY, 1990.   Google Scholar
[33]

G. Stadler, Elliptic optimal control problems with $L^1$-control cost and applications for the placement of control devices, Comput. Optim. Appl., 44 (2009), 159-181.   Google Scholar

[34]

S. Teufel, Adiabatic Perturbation Theory in Quantum Dynamics, vol. 1821 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2003.  Google Scholar

[35]

G. TuriniciC. Le Bris and H. Rabitz, Efficient algorithms for the laboratory discovery of optimal quantum controls, Phys. Rev. E, 70 (2004), 016704.   Google Scholar

[36]

G. von Winckel and A. Borzì, Computational techniques for a quantum control problem with $H^1$-cost, Inverse Problems, 24 (2008), 034007, 23pp.  Google Scholar

[37]

G. von WinckelA. Borzì and S. Volkwein, A globalized Newton method for the accurate solution of a dipole quantum control problem, SIAM J. Sci. Comput., 31 (2009/10), 4176-4203.   Google Scholar

[38]

G. Vossen and H. Maurer, On $L^1$-minimization in optimal control and applications to robotics, Optimal Control Appl. Methods, 27 (2006), 301-321.   Google Scholar

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