September  2015, 35(9): 4095-4114. doi: 10.3934/dcds.2015.35.4095

Integrability methods in the time minimal coherence transfer for Ising chains of three spins

1. 

Institut de mathématiques de Bourgogne, UMR 5584 CNRS Université de Bourgogne, Faculté des Sciences Mirande 9 avenue Alain Savary, France, France, France

Received  April 2014 Revised  October 2014 Published  April 2015

The objective of this article is to analyze the integrability properties of extremal solutions of Pontryagin Maximum Principle in the time minimal control of a linear spin system with Ising coupling in relation with conjugate and cut loci computations. Restricting to the case of three spins, the problem is equivalent to analyze a family of almost-Riemannian metrics on the sphere $S^{2}$, with Grushin equatorial singularity. The problem can be lifted into a SR-invariant problem on $SO(3)$, this leads to a complete understanding of the geometry of the problem and to an explicit parametrization of the extremals using an appropriate chart as well as elliptic functions. This approach is compared with the direct analysis of the Liouville metrics on the sphere where the parametrization of the extremals is obtained by computing a Liouville normal form. Finally, an algebraic approach is presented in the framework of the application of differential Galois theory to integrability.
Citation: Bernard Bonnard, Thierry Combot, Lionel Jassionnesse. Integrability methods in the time minimal coherence transfer for Ising chains of three spins. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4095-4114. doi: 10.3934/dcds.2015.35.4095
References:
[1]

A. Agrachev, U. Boscain and M. Sigalotti, A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds, Discrete Contin. Dyn. Syst., 20 (2008), 801-822. doi: 10.3934/dcds.2008.20.801.

[2]

V. I. Arnold, Mathematical Methods of Classical Mechanics, Translated from the Russian by K. Vogtmann and A. Weinstein. Second edition. Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.

[3]

M. Audin, Les Systèmes Hamiltoniens Et Leur Intégrabilité, (French) [Hamiltonian Systems and Their Integrability], Cours Spécialisés [Specialized Courses], 8. Société Mathématique de France, Paris; EDP Sciences, Les Ulis, 2001. viii+170 pp.

[4]

D. Birkhoff, Dynamical Systems, American society colloquium publications, vol. IX, 1927.

[5]

A. V. Bolsinov and A. T. Fomenko, Integrable Geodesic Flows on Two-Dimensional Surfaces, Monographs in contemporary mathematrics, Kluwer Academic, 2000. doi: 10.1007/978-1-4615-4307-7.

[6]

B. Bonnard and J.-B. Caillau, Metrics with equatorial singularities on the sphere, Ann. Mat. Pura Appl., 193 (2014), 1353-1382. doi: 10.1007/s10231-013-0333-y.

[7]

B. Bonnard, O. Cots and L. Jassionnesse, Geometric and numerical techniques to compute conjugate and cut loci on Riemannian surfaces, Geometric Control Theory and Sub-Riemannian Geometry, Springer INdAM Series, 5 (2014), 53-72. doi: 10.1007/978-3-319-02132-4_4.

[8]

B. Bonnard, O. Cots, J.-B. Pomet and N. Shcherbakova, Riemannian metrics on 2d-manifolds related to the Euler-Poinsot rigid body motion, ESAIM Control Optim. Calc. Var., 20 (2014), 864-893. doi: 10.1051/cocv/2013087.

[9]

U. Boscain, T. Chambrion and G. Charlot, Nonisotropic 3-level quantum systems: Complete solutions for minimum time and minimum energy, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 957-990. doi: 10.3934/dcdsb.2005.5.957.

[10]

J. I. Itoh and K. Kiyohara, The cut loci and the conjugate loci on ellipsoids, Manuscripta Math., 114 (2004), 247-264. doi: 10.1007/s00229-004-0455-z.

[11]

J. I. Itoh and K. Kiyohara, Cut loci and conjugate loci on Liouville surfaces, Manuscripta Math., 136 (2011), 115-141. doi: 10.1007/s00229-011-0433-1.

[12]

L. Jassionnesse, Contrôle Optimal et Métriques de Clairaut-Liouville Avec Applications, Ph.D thesis, Université de Bourgogne, 2014.

[13]

V. Jurdjevic, Geometric Control Theory, Studies in Advanced Mathematics, 52. Cambridge University Press, Cambridge, 1997. xviii+492 pp.

[14]

N. Khaneja, S. J. Glaser and R. Brockett, Sub-Riemannian geometry and time optimal control of three spin systems: quantum gates and coherence transfer, Phys. Rev. A (3), 65 (2002), part A, 032301, 11 pp. doi: 10.1103/PhysRevA.65.032301.

[15]

J. J. Kovacic, An algorithm for solving second order linear homogeneous differential equations, J. Symbolic Comput., 2 (1986), 3-43. doi: 10.1016/S0747-7171(86)80010-4.

[16]

D. F. Lawden, Elliptic Functions and Applications, Applied Mathematical Sciences, 80. Springer-Verlag, New York, 1989. xiv+334 pp. doi: 10.1007/978-1-4757-3980-0.

[17]

H. Levitt, Spin Dynamics - Basics of Nuclear Magnetic Resonance, (2001) Wiley (686 pages).

[18]

J. J. Morales-Ruiz and J. P. Ramis, Integrability of dynamical systems through differential Galois theory: A practical guide, Differential algebra, complex analysis and orthogonal polynomials, Contemp. Math., Amer. Math. Soc., Providence, RI, 509 (2010), 143-220. doi: 10.1090/conm/509/09980.

[19]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Translated from the Russian by K. N. Trirogoff; edited by L. W. Neustadt Interscience Publishers John Wiley & Sons, Inc. New York-London, 1962.

[20]

M. Singer and F. Ulmer, Galois groups of second and third order linear differential equations, J. Symbolic Comput., 16 (1993), 9-36. doi: 10.1006/jsco.1993.1032.

[21]

M. van der Put and M. Singer, Galois Theory of Linear Differential Equations, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-642-55750-7.

[22]

H. Yuan, Geometry, Optimal Control and Quantum Computing, Ph.D thesis, Harvard, 2006.

[23]

H. Yuan, R. Zeier and N. Khaneja, Elliptic functions and efficient control of Ising spin chains with unequal couplings, Phys. Rev. A, 77, (2008), 032340. doi: 10.1103/PhysRevA.77.032340.

show all references

References:
[1]

A. Agrachev, U. Boscain and M. Sigalotti, A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds, Discrete Contin. Dyn. Syst., 20 (2008), 801-822. doi: 10.3934/dcds.2008.20.801.

[2]

V. I. Arnold, Mathematical Methods of Classical Mechanics, Translated from the Russian by K. Vogtmann and A. Weinstein. Second edition. Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.

[3]

M. Audin, Les Systèmes Hamiltoniens Et Leur Intégrabilité, (French) [Hamiltonian Systems and Their Integrability], Cours Spécialisés [Specialized Courses], 8. Société Mathématique de France, Paris; EDP Sciences, Les Ulis, 2001. viii+170 pp.

[4]

D. Birkhoff, Dynamical Systems, American society colloquium publications, vol. IX, 1927.

[5]

A. V. Bolsinov and A. T. Fomenko, Integrable Geodesic Flows on Two-Dimensional Surfaces, Monographs in contemporary mathematrics, Kluwer Academic, 2000. doi: 10.1007/978-1-4615-4307-7.

[6]

B. Bonnard and J.-B. Caillau, Metrics with equatorial singularities on the sphere, Ann. Mat. Pura Appl., 193 (2014), 1353-1382. doi: 10.1007/s10231-013-0333-y.

[7]

B. Bonnard, O. Cots and L. Jassionnesse, Geometric and numerical techniques to compute conjugate and cut loci on Riemannian surfaces, Geometric Control Theory and Sub-Riemannian Geometry, Springer INdAM Series, 5 (2014), 53-72. doi: 10.1007/978-3-319-02132-4_4.

[8]

B. Bonnard, O. Cots, J.-B. Pomet and N. Shcherbakova, Riemannian metrics on 2d-manifolds related to the Euler-Poinsot rigid body motion, ESAIM Control Optim. Calc. Var., 20 (2014), 864-893. doi: 10.1051/cocv/2013087.

[9]

U. Boscain, T. Chambrion and G. Charlot, Nonisotropic 3-level quantum systems: Complete solutions for minimum time and minimum energy, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 957-990. doi: 10.3934/dcdsb.2005.5.957.

[10]

J. I. Itoh and K. Kiyohara, The cut loci and the conjugate loci on ellipsoids, Manuscripta Math., 114 (2004), 247-264. doi: 10.1007/s00229-004-0455-z.

[11]

J. I. Itoh and K. Kiyohara, Cut loci and conjugate loci on Liouville surfaces, Manuscripta Math., 136 (2011), 115-141. doi: 10.1007/s00229-011-0433-1.

[12]

L. Jassionnesse, Contrôle Optimal et Métriques de Clairaut-Liouville Avec Applications, Ph.D thesis, Université de Bourgogne, 2014.

[13]

V. Jurdjevic, Geometric Control Theory, Studies in Advanced Mathematics, 52. Cambridge University Press, Cambridge, 1997. xviii+492 pp.

[14]

N. Khaneja, S. J. Glaser and R. Brockett, Sub-Riemannian geometry and time optimal control of three spin systems: quantum gates and coherence transfer, Phys. Rev. A (3), 65 (2002), part A, 032301, 11 pp. doi: 10.1103/PhysRevA.65.032301.

[15]

J. J. Kovacic, An algorithm for solving second order linear homogeneous differential equations, J. Symbolic Comput., 2 (1986), 3-43. doi: 10.1016/S0747-7171(86)80010-4.

[16]

D. F. Lawden, Elliptic Functions and Applications, Applied Mathematical Sciences, 80. Springer-Verlag, New York, 1989. xiv+334 pp. doi: 10.1007/978-1-4757-3980-0.

[17]

H. Levitt, Spin Dynamics - Basics of Nuclear Magnetic Resonance, (2001) Wiley (686 pages).

[18]

J. J. Morales-Ruiz and J. P. Ramis, Integrability of dynamical systems through differential Galois theory: A practical guide, Differential algebra, complex analysis and orthogonal polynomials, Contemp. Math., Amer. Math. Soc., Providence, RI, 509 (2010), 143-220. doi: 10.1090/conm/509/09980.

[19]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Translated from the Russian by K. N. Trirogoff; edited by L. W. Neustadt Interscience Publishers John Wiley & Sons, Inc. New York-London, 1962.

[20]

M. Singer and F. Ulmer, Galois groups of second and third order linear differential equations, J. Symbolic Comput., 16 (1993), 9-36. doi: 10.1006/jsco.1993.1032.

[21]

M. van der Put and M. Singer, Galois Theory of Linear Differential Equations, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-642-55750-7.

[22]

H. Yuan, Geometry, Optimal Control and Quantum Computing, Ph.D thesis, Harvard, 2006.

[23]

H. Yuan, R. Zeier and N. Khaneja, Elliptic functions and efficient control of Ising spin chains with unequal couplings, Phys. Rev. A, 77, (2008), 032340. doi: 10.1103/PhysRevA.77.032340.

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