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  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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

G. Dirr, Ensemble controllability of bilinear systems, Oberwolfach Rep., 9 (2012), 661-732.  doi: 10.4171/OWR/2012/12.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities, Springer-Verlag, New York-Heidelberg, 1973, Graduate Texts in Mathematics, Vol. 14.  Google Scholar

[19]

M. Goresky and R. MacPherson, Stratified Morse Theory, Springer Berlin Heidelberg, Berlin, Heidelberg, 1988. doi: 10.1007/978-3-642-71714-7.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

M. Reed and B. Simon, Methods of Modern Mathematical Physics, Ⅳ. Analysis of Operators, Academic Press, New York, 1978.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

B. W. Shore, The Theory of Coherent Atomic Excitation, , Volume 1, Simple Atoms and Fields, 1990.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

S. Teufel, Adiabatic Perturbation Theory in Quantum Dynamics, vol. 1821 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2003. doi: 10.1007/b13355.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

G. Dirr, Ensemble controllability of bilinear systems, Oberwolfach Rep., 9 (2012), 661-732.  doi: 10.4171/OWR/2012/12.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities, Springer-Verlag, New York-Heidelberg, 1973, Graduate Texts in Mathematics, Vol. 14.  Google Scholar

[19]

M. Goresky and R. MacPherson, Stratified Morse Theory, Springer Berlin Heidelberg, Berlin, Heidelberg, 1988. doi: 10.1007/978-3-642-71714-7.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

M. Reed and B. Simon, Methods of Modern Mathematical Physics, Ⅳ. Analysis of Operators, Academic Press, New York, 1978.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

B. W. Shore, The Theory of Coherent Atomic Excitation, , Volume 1, Simple Atoms and Fields, 1990.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

S. Teufel, Adiabatic Perturbation Theory in Quantum Dynamics, vol. 1821 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2003. doi: 10.1007/b13355.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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