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

* Corresponding author: Nicolas Augier

Received  August 2019 Revised  January 2020 Published  December 2020 Early access  March 2020

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.

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. AgrachevY. 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. AugierU. 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. BeauchardJ.-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. BoscainJ.-P. GauthierF. 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. BoscainF. ChittaroP. 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. ChelkowskiA. 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. GaubatzP. RudeckiS. 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. GlaserT. Schulte-HerbrüggenM. SievekingO. SchedletzkyN. C. NielsenO. 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. SkinnerT. O. ReissB. LuyN. 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. AgrachevY. 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. AugierU. 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. BeauchardJ.-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. BoscainJ.-P. GauthierF. 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. BoscainF. ChittaroP. 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. ChelkowskiA. 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. GaubatzP. RudeckiS. 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. GlaserT. Schulte-HerbrüggenM. SievekingO. SchedletzkyN. C. NielsenO. 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. SkinnerT. O. ReissB. LuyN. 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 1.  Conical intersection as a function of the controls $ (u,v)\in {\mathbb{R}}^2 $
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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