Figure 1.
Conical intersection as a function of the controls $ (u,v)\in {\mathbb{R}}^2 $
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Stochastic impulse control Problem with state and time dependent cost functions
December
2020, 10(4): 877-911.
doi: 10.3934/mcrf.2020023
Semi-conical eigenvalue intersections and the ensemble controllability problem for quantum systems
| 1. | Inria & Université Côte d'Azur, INRA, CNRS, Sorbonne Université, Sophia Antipolis, France |
| 2. | CNRS & Sorbonne Université, Inria, Université de Paris, Laboratoire Jacques-Louis Lions, Paris, France |
| 3. | Inria & Sorbonne Université, Université de Paris, CNRS, Laboratoire Jacques-Louis Lions, Paris, France |
We study one-parametric perturbations of finite dimensional real Hamiltonians depending on two controls, and we show that generically in the space of Hamiltonians, conical intersections of eigenvalues can degenerate into semi-conical intersections of eigenvalues. Then, through the use of normal forms, we study the problem of ensemble controllability between the eigenstates of a generic Hamiltonian.
Keywords:
Ensemble control,
adiabatic approximation,
quantum control,
conical intersections of eigenvalues,
normal forms.
Mathematics Subject Classification:
81Q93, 93B05, 37G05, 37C20.
Citation:
Nicolas Augier, Ugo Boscain, Mario Sigalotti. Semi-conical eigenvalue intersections and the ensemble controllability problem for quantum systems. Mathematical Control and Related Fields,
2020, 10
(4)
: 877-911.
doi: 10.3934/mcrf.2020023
![]()
References:
| [1] |
A. Agrachev, D. Barilari and U. Boscain, A Comprehensive Introduction to Sub-Riemannian Geometry, vol. 181 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2020. |
| [2] |
A. Agrachev, Y. Baryshnikov and A. Sarychev,
Ensemble controllability by Lie algebraic methods, ESAIM Control Optim. Calc. Var., 22 (2016), 921-938.
doi: 10.1051/cocv/2016029. |
| [3] |
A. A. Agrachev and Y. L. Sachkov, Control Theory from the Geometric Viewpoint, vol. 87 of Encyclopaedia of Mathematical Sciences, Springer-Verlag, Berlin, 2004, Control Theory and Optimization, Ⅱ.
doi: 10.1007/978-3-662-06404-7. |
| [4] |
N. Augier, U. Boscain and M. Sigalotti,
Adiabatic ensemble control of a continuum of quantum systems, SIAM J. Control Optim., 56 (2018), 4045-4068.
doi: 10.1137/17M1140327. |
| [5] |
K. Beauchard, J.-M. Coron and P. Rouchon,
Controllability issues for continuous-spectrum systems and ensemble controllability of Bloch equations, Comm. Math. Phys., 296 (2010), 525-557.
doi: 10.1007/s00220-010-1008-9. |
| [6] |
U. Boscain, J.-P. Gauthier, F. Rossi and M. Sigalotti,
Approximate controllability, exact controllability, and conical eigenvalue intersections for quantum mechanical systems, Comm. Math. Phys., 333 (2015), 1225-1239.
doi: 10.1007/s00220-014-2195-6. |
| [7] |
U. V. Boscain, F. Chittaro, P. Mason and M. Sigalotti,
Adiabatic control of the Schrödinger equation via conical intersections of the eigenvalues, IEEE Trans. Automat. Control, 57 (2012), 1970-1983.
doi: 10.1109/TAC.2012.2195862. |
| [8] |
S. Chelkowski, A. D. Bandrauk and P. B. Corkum,
Efficient molecular dissociation by a chirped ultrashort infrared laser pulse, Phys. Rev. Lett., 65 (1990), 2355-2358.
doi: 10.1103/PhysRevLett.65.2355. |
| [9] |
C. Chen, D. Dong, R. Long, I. R. Petersen and H. A. Rabitz, Sampling-based learning control of inhomogeneous quantum ensembles, Phys. Rev. A, 89 (2014), 023402.
doi: 10.1103/PhysRevA.89.023402. |
| [10] |
F. C. Chittaro and J.-P. Gauthier,
Asymptotic ensemble stabilizability of the Bloch equation, Systems Control Lett., 113 (2018), 36-44.
doi: 10.1016/j.sysconle.2018.01.008. |
| [11] |
F. C. Chittaro and P. Mason,
Approximate controllability via adiabatic techniques for the three-inputs controlled Schrödinger equation, SIAM J. Control Optim., 55 (2017), 4202-4226.
doi: 10.1137/15M1041419. |
| [12] |
Y. Colin de Verdière, The level crossing problem in semi-classical analysis. Ⅰ. The symmetric case, in Proceedings of the International Conference in Honor of Frédéric Pham (Nice, 2002), 53 (2003), 1023–1054.
doi: 10.5802/aif.1973. |
| [13] |
Y. Colin de Verdière, The level crossing problem in semi-classical analysis. Ⅱ. The Hermitian case, Ann. Inst. Fourier (Grenoble), 54 (2004), 1423–1441, xv, xx–xxi.
doi: 10.5802/aif.2054. |
| [14] |
G. Dirr,
Ensemble controllability of bilinear systems, Oberwolfach Rep., 9 (2012), 661-732.
doi: 10.4171/OWR/2012/12. |
| [15] |
U. Gaubatz, P. Rudecki, S. Schiemann and K. Bergmann,
Population transfer between molecular vibrational levels by stimulated raman scattering with partially overlapping laser fields. a new concept and experimental results, The Journal of Chemical Physics, 92 (1990), 5363-5376.
doi: 10.1063/1.458514. |
| [16] |
S. J. Glaser, T. Schulte-Herbrüggen, M. Sieveking, O. Schedletzky, N. C. Nielsen, O. W. Sørensen and C. Griesinger,
Unitary control in quantum ensembles: Maximizing signal intensity in coherent spectroscopy, Science, 280 (1998), 421-424.
doi: 10.1126/science.280.5362.421. |
| [17] |
S. J. Glaser, U. Boscain, T. Calarco, C. P. Koch, W. Köckenberger, R. Kosloff, I. Kuprov, B. Luy, S. Schirmer, T. Schulte-Herbrüggen, D. Sugny and F. K. Wilhelm, Training Schrödinger's cat: Quantum optimal control, The European Physical Journal D, 69 (2015), 279.
doi: 10.1140/epjd/e2015-60464-1. |
| [18] |
M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities, Springer-Verlag, New York-Heidelberg, 1973, Graduate Texts in Mathematics, Vol. 14. |
| [19] |
M. Goresky and R. MacPherson, Stratified Morse Theory, Springer Berlin Heidelberg, Berlin, Heidelberg, 1988.
doi: 10.1007/978-3-642-71714-7. |
| [20] |
U. Helmke and M. Schönlein,
Uniform ensemble controllability for one-parameter families of time-invariant linear systems, Systems Control Lett., 71 (2014), 69-77.
doi: 10.1016/j.sysconle.2014.05.015. |
| [21] |
Z. Leghtas, A. Sarlette and P. Rouchon, Adiabatic passage and ensemble control of quantum systems, Journal of Physics B: Atomic, Molecular and Optical Physics, 44 (2011), 154017.
doi: 10.1088/0953-4075/44/15/154017. |
| [22] |
J.-S. Li and N. Khaneja,
Ensemble control of Bloch equations, IEEE Trans. Automat. Control, 54 (2009), 528-536.
doi: 10.1109/TAC.2009.2012983. |
| [23] |
J.-S. Li and J. Qi,
Ensemble control of time-invariant linear systems with linear parameter variation, IEEE Trans. Automat. Control, 61 (2016), 2808-2820.
doi: 10.1109/TAC.2015.2503698. |
| [24] |
P. Libermann and C.-M. Marle, Symplectic Geometry and Analytical Mechanics, vol. 35 of Mathematics and its Applications, D. Reidel Publishing Co., Dordrecht, 1987, Translated from the French by Bertram Eugene Schwarzbach.
doi: 10.1007/978-94-009-3807-6. |
| [25] |
M. Reed and B. Simon, Methods of Modern Mathematical Physics, Ⅳ. Analysis of Operators, Academic Press, New York, 1978. |
| [26] |
M. Schönlein and U. Helmke,
Controllability of ensembles of linear dynamical systems, Math. Comput. Simulation, 125 (2016), 3-14.
doi: 10.1016/j.matcom.2015.10.006. |
| [27] |
E. A. Shapiro, V. Milner and M. Shapiro, Complete transfer of populations from a single state to a preselected superposition of states using piecewise adiabatic passage: Theory, Phys. Rev. A, 79 (2009), 023422.
doi: 10.1103/PhysRevA.79.023422. |
| [28] |
B. W. Shore, The Theory of Coherent Atomic Excitation, , Volume 1, Simple Atoms and Fields, 1990. |
| [29] |
T. E. Skinner, T. O. Reiss, B. Luy, N. Khaneja and S. J. Glaser,
Application of optimal control theory to the design of broadband excitation pulses for high-resolution NMR, Journal of Magnetic Resonance, 163 (2003), 8-15.
doi: 10.1016/S1090-7807(03)00153-8. |
| [30] |
E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, vol. 43 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, 1993, With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, Ⅲ. |
| [31] |
S. Teufel, Adiabatic Perturbation Theory in Quantum Dynamics, vol. 1821 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2003.
doi: 10.1007/b13355. |
| [32] |
L. Van Damme, Q. Ansel, S. J. Glaser and D. Sugny, Robust optimal control of two-level quantum systems, Phys. Rev. A, 95 (2017), 063403.
doi: 10.1103/PhysRevA.95.063403. |
| [33] |
J. von Neumann and E. P. Wigner, Über das Verhalten von Eigenwerten bei adiabatischen Prozessen, The Collected Works of Eugene Paul Wigner, 1993,294–297
doi: 10.1007/978-3-662-02781-3_20. |
show all references
References:
| [1] |
A. Agrachev, D. Barilari and U. Boscain, A Comprehensive Introduction to Sub-Riemannian Geometry, vol. 181 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2020. |
| [2] |
A. Agrachev, Y. Baryshnikov and A. Sarychev,
Ensemble controllability by Lie algebraic methods, ESAIM Control Optim. Calc. Var., 22 (2016), 921-938.
doi: 10.1051/cocv/2016029. |
| [3] |
A. A. Agrachev and Y. L. Sachkov, Control Theory from the Geometric Viewpoint, vol. 87 of Encyclopaedia of Mathematical Sciences, Springer-Verlag, Berlin, 2004, Control Theory and Optimization, Ⅱ.
doi: 10.1007/978-3-662-06404-7. |
| [4] |
N. Augier, U. Boscain and M. Sigalotti,
Adiabatic ensemble control of a continuum of quantum systems, SIAM J. Control Optim., 56 (2018), 4045-4068.
doi: 10.1137/17M1140327. |
| [5] |
K. Beauchard, J.-M. Coron and P. Rouchon,
Controllability issues for continuous-spectrum systems and ensemble controllability of Bloch equations, Comm. Math. Phys., 296 (2010), 525-557.
doi: 10.1007/s00220-010-1008-9. |
| [6] |
U. Boscain, J.-P. Gauthier, F. Rossi and M. Sigalotti,
Approximate controllability, exact controllability, and conical eigenvalue intersections for quantum mechanical systems, Comm. Math. Phys., 333 (2015), 1225-1239.
doi: 10.1007/s00220-014-2195-6. |
| [7] |
U. V. Boscain, F. Chittaro, P. Mason and M. Sigalotti,
Adiabatic control of the Schrödinger equation via conical intersections of the eigenvalues, IEEE Trans. Automat. Control, 57 (2012), 1970-1983.
doi: 10.1109/TAC.2012.2195862. |
| [8] |
S. Chelkowski, A. D. Bandrauk and P. B. Corkum,
Efficient molecular dissociation by a chirped ultrashort infrared laser pulse, Phys. Rev. Lett., 65 (1990), 2355-2358.
doi: 10.1103/PhysRevLett.65.2355. |
| [9] |
C. Chen, D. Dong, R. Long, I. R. Petersen and H. A. Rabitz, Sampling-based learning control of inhomogeneous quantum ensembles, Phys. Rev. A, 89 (2014), 023402.
doi: 10.1103/PhysRevA.89.023402. |
| [10] |
F. C. Chittaro and J.-P. Gauthier,
Asymptotic ensemble stabilizability of the Bloch equation, Systems Control Lett., 113 (2018), 36-44.
doi: 10.1016/j.sysconle.2018.01.008. |
| [11] |
F. C. Chittaro and P. Mason,
Approximate controllability via adiabatic techniques for the three-inputs controlled Schrödinger equation, SIAM J. Control Optim., 55 (2017), 4202-4226.
doi: 10.1137/15M1041419. |
| [12] |
Y. Colin de Verdière, The level crossing problem in semi-classical analysis. Ⅰ. The symmetric case, in Proceedings of the International Conference in Honor of Frédéric Pham (Nice, 2002), 53 (2003), 1023–1054.
doi: 10.5802/aif.1973. |
| [13] |
Y. Colin de Verdière, The level crossing problem in semi-classical analysis. Ⅱ. The Hermitian case, Ann. Inst. Fourier (Grenoble), 54 (2004), 1423–1441, xv, xx–xxi.
doi: 10.5802/aif.2054. |
| [14] |
G. Dirr,
Ensemble controllability of bilinear systems, Oberwolfach Rep., 9 (2012), 661-732.
doi: 10.4171/OWR/2012/12. |
| [15] |
U. Gaubatz, P. Rudecki, S. Schiemann and K. Bergmann,
Population transfer between molecular vibrational levels by stimulated raman scattering with partially overlapping laser fields. a new concept and experimental results, The Journal of Chemical Physics, 92 (1990), 5363-5376.
doi: 10.1063/1.458514. |
| [16] |
S. J. Glaser, T. Schulte-Herbrüggen, M. Sieveking, O. Schedletzky, N. C. Nielsen, O. W. Sørensen and C. Griesinger,
Unitary control in quantum ensembles: Maximizing signal intensity in coherent spectroscopy, Science, 280 (1998), 421-424.
doi: 10.1126/science.280.5362.421. |
| [17] |
S. J. Glaser, U. Boscain, T. Calarco, C. P. Koch, W. Köckenberger, R. Kosloff, I. Kuprov, B. Luy, S. Schirmer, T. Schulte-Herbrüggen, D. Sugny and F. K. Wilhelm, Training Schrödinger's cat: Quantum optimal control, The European Physical Journal D, 69 (2015), 279.
doi: 10.1140/epjd/e2015-60464-1. |
| [18] |
M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities, Springer-Verlag, New York-Heidelberg, 1973, Graduate Texts in Mathematics, Vol. 14. |
| [19] |
M. Goresky and R. MacPherson, Stratified Morse Theory, Springer Berlin Heidelberg, Berlin, Heidelberg, 1988.
doi: 10.1007/978-3-642-71714-7. |
| [20] |
U. Helmke and M. Schönlein,
Uniform ensemble controllability for one-parameter families of time-invariant linear systems, Systems Control Lett., 71 (2014), 69-77.
doi: 10.1016/j.sysconle.2014.05.015. |
| [21] |
Z. Leghtas, A. Sarlette and P. Rouchon, Adiabatic passage and ensemble control of quantum systems, Journal of Physics B: Atomic, Molecular and Optical Physics, 44 (2011), 154017.
doi: 10.1088/0953-4075/44/15/154017. |
| [22] |
J.-S. Li and N. Khaneja,
Ensemble control of Bloch equations, IEEE Trans. Automat. Control, 54 (2009), 528-536.
doi: 10.1109/TAC.2009.2012983. |
| [23] |
J.-S. Li and J. Qi,
Ensemble control of time-invariant linear systems with linear parameter variation, IEEE Trans. Automat. Control, 61 (2016), 2808-2820.
doi: 10.1109/TAC.2015.2503698. |
| [24] |
P. Libermann and C.-M. Marle, Symplectic Geometry and Analytical Mechanics, vol. 35 of Mathematics and its Applications, D. Reidel Publishing Co., Dordrecht, 1987, Translated from the French by Bertram Eugene Schwarzbach.
doi: 10.1007/978-94-009-3807-6. |
| [25] |
M. Reed and B. Simon, Methods of Modern Mathematical Physics, Ⅳ. Analysis of Operators, Academic Press, New York, 1978. |
| [26] |
M. Schönlein and U. Helmke,
Controllability of ensembles of linear dynamical systems, Math. Comput. Simulation, 125 (2016), 3-14.
doi: 10.1016/j.matcom.2015.10.006. |
| [27] |
E. A. Shapiro, V. Milner and M. Shapiro, Complete transfer of populations from a single state to a preselected superposition of states using piecewise adiabatic passage: Theory, Phys. Rev. A, 79 (2009), 023422.
doi: 10.1103/PhysRevA.79.023422. |
| [28] |
B. W. Shore, The Theory of Coherent Atomic Excitation, , Volume 1, Simple Atoms and Fields, 1990. |
| [29] |
T. E. Skinner, T. O. Reiss, B. Luy, N. Khaneja and S. J. Glaser,
Application of optimal control theory to the design of broadband excitation pulses for high-resolution NMR, Journal of Magnetic Resonance, 163 (2003), 8-15.
doi: 10.1016/S1090-7807(03)00153-8. |
| [30] |
E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, vol. 43 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, 1993, With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, Ⅲ. |
| [31] |
S. Teufel, Adiabatic Perturbation Theory in Quantum Dynamics, vol. 1821 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2003.
doi: 10.1007/b13355. |
| [32] |
L. Van Damme, Q. Ansel, S. J. Glaser and D. Sugny, Robust optimal control of two-level quantum systems, Phys. Rev. A, 95 (2017), 063403.
doi: 10.1103/PhysRevA.95.063403. |
| [33] |
J. von Neumann and E. P. Wigner, Über das Verhalten von Eigenwerten bei adiabatischen Prozessen, The Collected Works of Eugene Paul Wigner, 1993,294–297
doi: 10.1007/978-3-662-02781-3_20. |
Figure 2.
Semi-conical intersection of eigenvalues as a function of the controls $ (u,v)\in {\mathbb{R}}^2 $
Figure 3.
Semi-conical intersection for the STIRAP as a function of the controls $ (u,v)\in {\mathbb{R}}^2 $
Figure 4.
A curve $ (u,v) $ as in the statement of Theorem 1.3
Figure 5.
A control path passing at a semi-conical intersection in the non-conical direction as a function of the controls $ (u,v)\in {\mathbb{R}}^2 $
Figure 6.
A graphical representation of condition (C)
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