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American Mathematical Society eBooks

Quantum control : mathematical and numerical challenges : CRM workshop, October 6-11, 2002, Montréal, Canada

André D. Bandrauk, Michel C. Delfour, Claude Le Bris

Published Jan 1, 2003• 1 citation
OpenAlex

Abstract

Molecular alignment and orientation: From laser-induced mechanisms to optimal control by O. Atabek and C. M. Dion Overview and software guide of evolutionary algorithms A case study in quantum control by A. Auger, A. Ben Haj-Yedder, and M. Schoenauer Laser control of molecular states--Nonperturbative examples by A. D. Bandrauk, F. Legare, and H. T. Yu Coherent control: Principles and semiclassical implementations by V. S. Batista and P. Brumer Mathematical models of contemporary elementary quantum computing devices by G. Chen, D. A. Church, B.-G. Englert, and M. S. Zubairy Addendum and remarks on doubly conservative numerical schemes for the nonlinear Schrodinger equation and its control by M. C. Delfour A note on the exact internal control of nonlinear Schrodinger equations by R. Illner, H. Lange, and H. Teismann Towards efficient numerical approaches for quantum control by C. Le Bris, Y. Maday, and G. Turinici Multichannel quantum defect study of the control in the frequency domain: Example of HI by H. Lefebvre-Brion Development of solution algorithms for quantum optimal control equations in product spaces by Y. Ohtsuki and H. Rabitz Using contracted basis functions to solve the Schrodinger equation by X.-G. Wang and T. Carrington, Jr. Remarks on the controllability of the Schrodinger equation by E. Zuazua.

Related papers

Coherent control in atoms, molecules, and semiconductors : proceedings of an international workshop held in Chicago, USA, 19-22 May 1998

W. Pötz, W. Andreas Schroeder

Preface. Part I: Atomic and Molecular systems. Scenarios in Coherent Control P. Brumer, M. Shapiro. Quantum Control in Atomic Systems D. Charalambidis, et al. Laser-phase-sensitive Coherent Control of Photoion Product States in 2-Photon vs. 2-Photon Ionization of Atomic Barium, F. Wang, et al. Coherent Phase control of Photoionization and Photodissociation R.J. Gordon, et al. Coherent Control of Atomic Photoionization and Autoionization P. Lambropoulos, T. Nakajima. Lasing Without Inversion via Interference of Double-Dark Resonances in Atomic and Quantum Well Systems M.O. Scully, et al. Part II: Coherent Control in Semiconductors. Coherence Control of Free Carriers in Bulk Semiconductors H.M. van Driel, et al. Coherent Control of 2D Excitons Probed by Time-Resolved Secondary Emission X. Marie, et al. Many-Body Effects in the Ultrafast Population Transfer in p-Doped Semiconductor Quantum Wells R. Binder, M. Lindberg. Coherent Control of Heavy-Light Hole and Phonon Quantum Beats T. Kuhn, et al. Coherent Control Schemes in Semiconductor Double Wells X. Hu, W. Poetz. Terahertz-Control of Charge-Carrier Wavepackets in Semiconductor Quantum Wells S. Hughes, D.S. Citrin. Microcavity Exciton-Polariton H. Cao. Part III: Coherent Phenomena and Related Topics. Coherent Control of Quantum Localization M. Holthaus. Tunneling of Low-Energetic Electrons in the Presence of Intense Laser Fields: The Formation of Dynamical Barrier States M. Wagner. Hopping Mechanism of Coherent Photovoltaic Effect and Photoinduced Polar Anisotropy in Glass E.M. Baskin, M.V. Entin. Heterodyning Scheme Employing Quantum Interference J.B. Khurgin, et al. Generation of Widely Tunable THz-Wave Using Nonlinear Optics K. Kawase, H.Ito. List of Participants.

Kluwer eBooks•Jan 1, 1999•2 citations

Mathematical results in quantum mechanics : a conference on QMATH-8, Mathematical Results in Quantum Mechanics Universidad Nacional Autónoma de México, Taxco, México, December 10-14, 2001

Ricardo Weder, Pavel Exner, Benoît Grébert

Blow-up for Schrodinger equation with pointwise nonlinearity by R. Adami Quantum electrodynamics of relativistic bound states with cutoffs. II by J.-M. Barbaroux, M. Dimassi, and J.-C. Guillot Scattering by a slab: An exact calculation by F. Bentosela and M. Tater Crystalline order in one dimensional Thomas-Fermi type models by X. Blanc and C. Le Bris New results on the spectrum of lattice Schrodinger operators and applications by J. Bourgain Reconstruction of the memory from partial boundary measurements by A. L. Bukhgeim, G. V. Dyatlov, and G. Uhlmann Asymptotics for nonlinear dissipative equations in the super critical case by R. E. Cardiel and P. I. Naumkin Edge states for quantum Hall Hamiltonians by J.-M. Combes, P. D. Hislop, and E. Soccorsi A semiclassical approach in linear response theory by M. Combescure and D. Robert Wigner-von Neumann-like potentials by J. Cruz-Sampedro Equilibrium potentials in scattering theory by M. Demuth and A. Noll Straight quantum waveguides with combined boundary conditions by J. Dittrich and J. Kriz Perturbation theory for the quantum time-evolution in rotating potentials by V. Enss, V. Kostrykin, and R. Schrader Two dimensional Pauli operator via scalar potential by L. Erdos and V. Vougalter A max-min principle for the ground state of the Dirac-Fock functional by M. J. Esteban and E. Sere On the regularity of the density of electronic wave functions by S. Fournais, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, and T. O. Sorensen Enhanced binding through coupling to a photon field by C. Hainzl The large radius limit for coherent states on spheres by B. C. Hall and J. J. Mitchell Quantum scattering for potentials homogeneous of degree zero by I. Herbst and E. Skibsted Existence of the density of states for one-dimensional alloy-type potentials with small support by W. Kirsch and I. Veselic Statistical ensembles and density of states by V. Kostrykin and R. Schrader The Cauchy problem for the Toda lattice with a class of non-stabilized initial data by M. Kudryavtsev Absolutely continuous spectrum of Jacobi matrices by A. Laptev, S. Naboko, and O. Safronov Stability of matter in relativistic quantum electrodynamics by E. H. Lieb and M. Loss The algebra of observables in a magnetic field by M. Mantoiu and R. Purice Recent developments in value distribution theory for Schrodinger operators on the half line by D. B. Pearson Long time propagation results in quantum mechanics by D. Robert Comments on adiabatic quantum algorithms by M. B. Ruskai Tunneling destruction for a nonlinear Schrodinger equation by A. Sacchetti Symmetry breaking in a model of a rotating Bose gas by R. Seiringer Euler's constant in Segal-Bargmann analysis by S. B. Sontz Maxwell with a touch of Schrodinger by A. Tip Exponentially accurate semiclassical asymptotics by J. H. Toloza Some properties of the eigenvalues of a Schrodinger equation with energy-dependent potential by C. van der Mee and V. Pivovarchik The Bargmann transform for $L^2(S^3)$ and regularization of the Kepler problem by C. Villegas-Blas Schrodinger equations with superquadratic potentials by K. Yajima and G. Zhang Band spectrum of the Laplacian on a slab with the Dirichlet boundary condition on a grid by K. Yoshitomi Other talks Participants.

American Mathematical Society eBooks•Jan 1, 2002•0 citations

Limit points the monotonic schemes for quantum control

Julien Salomon

Many numerical simulations in quantum (bilinear) control use the monotonically convergent algorithms of Krotov (introduced by Tannor in [12]), Zhu & Rabitz ([11]) or the general form of Maday & Turinici ([13]). This paper presents an analysis of the limit set of controls provided by these algorithms and a proof of convergence in a particular case.

arXiv:quant-ph/0503072•IEEE Conference on Decision and Control (CDC)•Oct 4, 2006•6 citations

Developement of new techniques of Optimal Control in Quantum Dynamics (from nuclear magnetic resonance to molecular physics)

M. Lapert, Dominique Sugny

L objectif de cette thèse est d appliquer la théorie du contrôle optimal à la dynamique de systèmes quantiques. Le premier point consiste à introduire dans le domaine du contrôle quantique des outils de contrôle optimal initialement développés en mathématique. Cette approche a ensuite été appliquée sur différent types de systèmes quantiques décrit par une grande ou une petite dimension. La première partie du manuscrit introduit les différents outils de contrôles utilisés avec une approche adaptée à un public de physiciens. Dans la seconde partie, ces techniques sont utilisées pour contrôler la dynamique des spins en RMN et IRM. La troisième partie s intéresse au développement de nouveaux algorithmes itératifs de contrôle optimal appliqués au contrôle par champ laser de la dynamique rotationnelle des molécules linéaires en phases gazeuse ainsi qu au développement d une stratégie de contrôle simple permettant de délocaliser une molécule dans un plan. La quatrième partie traite le contrôle en temps minimum d un condensat de Bose-Einstein à deux composantes. La dernière partie permet de comparer qualitativement et quantitativement les différentes méthodes de contrôle optimal utilisées. Les seconde et troisième parties ont également bénéficier de l implémentation expérimentale des solutions de contrôle optimal obtenues.

•Jan 1, 2011•0 citations

Krotov method for optimal control of closed quantum systems

Oleg Vasilevich Morzhin, Alexander Nikolaevich Pechen

Abstract The mathematics of optimal control of quantum systems is of great interest in connection with fundamental problems of physics as well as with existing and prospective applications to quantum technologies. One important problem is the development of methods for constructing controls for quantum systems. One of the commonly used methods is the Krotov method, which was initially proposed outside of quantum control theory in articles by Krotov and Feldman (1978, 1983). This method was used to develop a novel approach to finding optimal controls for quantum systems in [64] (Tannor, Kazakov, and Orlov, 1992), [65] (Somlói, Kazakov, and Tannor, 1993), and in many other works by various scientists. Our survey discusses mathematical aspects of this method for optimal control of closed quantum systems. It outlines various modifications with different forms of the improvement function (for example, linear or linear-quadratic), different constraints on the control spectrum and on the admissible states of the quantum system, different regularisers, and so on. The survey describes applications of the Krotov method to controlling molecular dynamics and Bose–Einstein condensates, and to quantum gate generation. This method is compared with the GRAPE (GRadient Ascent Pulse Engineering) method, the CRAB (Chopped Random-Basis) method, and the Zhu–Rabitz and Maday–Turinici methods. Bibliography: 158 titles.

arXiv:1809.09562•Russian Mathematical Surveys•Oct 1, 2019•52 citations

Convergence of the time-discretized monotonic schemes

Julien Salomon

Many numerical simulations in (bilinear) quantum control use the monotonically convergent Krotov algorithms (introduced by Tannor et al. [Time Dependent Quantum Molecular Dynamics (1992) 347–360]), Zhu and Rabitz [J. Chem. Phys. (1998) 385–391] or their unified form described in Maday and Turinici [J. Chem. Phys. (2003) 8191–8196]. In Maday et al. [Num. Math. (2006) 323–338], a time discretization which preserves the property of monotonicity has been presented. This paper introduces a proof of the convergence of these schemes and some results regarding their rate of convergence.

ESAIM Mathematical Modelling and Numerical Analysis•Jan 1, 2007•20 citations
Molecular alignment and orientation: From laser-induced mechanisms to optimal control by O. Atabek and C. M. Dion Overview and software guide of evolutionary algorithms A case study in quantum control by A. Auger, A. Ben Haj-Yedder, and M. Schoenauer Laser control of molecular states--Nonperturbative examples by A. D. Bandrauk, F. Legare, and H. T. Yu Coherent control: Principles and semiclassical implementations by V. S. Batista and P. Brumer Mathematical models of contemporary elementary quantum computing devices by G. Chen, D. A. Church, B.-G. Englert, and M. S. Zubairy Addendum and remarks on doubly conservative numerical schemes for the nonlinear Schrodinger equation and its control by M. C. Delfour A note on the exact internal control of nonlinear Schrodinger equations by R. Illner, H. Lange, and H. Teismann Towards efficient numerical approaches for quantum control by C. Le Bris, Y. Maday, and G. Turinici Multichannel quantum defect study of the control in the frequency domain: Example of HI by H. Lefebvre-Brion Development of solution algorithms for quantum optimal control equations in product spaces by Y. Ohtsuki and H. Rabitz Using contracted basis functions to solve the Schrodinger equation by X.-G. Wang and T. Carrington, Jr. Remarks on the controllability of the Schrodinger equation by E. Zuazua.
Preface. Part I: Atomic and Molecular systems. Scenarios in Coherent Control P. Brumer, M. Shapiro. Quantum Control in Atomic Systems D. Charalambidis, et al. Laser-phase-sensitive Coherent Control of Photoion Product States in 2-Photon vs. 2-Photon Ionization of Atomic Barium, F. Wang, et al. Coherent Phase control of Photoionization and Photodissociation R.J. Gordon, et al. Coherent Control of Atomic Photoionization and Autoionization P. Lambropoulos, T. Nakajima. Lasing Without Inversion via Interference of Double-Dark Resonances in Atomic and Quantum Well Systems M.O. Scully, et al. Part II: Coherent Control in Semiconductors. Coherence Control of Free Carriers in Bulk Semiconductors H.M. van Driel, et al. Coherent Control of 2D Excitons Probed by Time-Resolved Secondary Emission X. Marie, et al. Many-Body Effects in the Ultrafast Population Transfer in p-Doped Semiconductor Quantum Wells R. Binder, M. Lindberg. Coherent Control of Heavy-Light Hole and Phonon Quantum Beats T. Kuhn, et al. Coherent Control Schemes in Semiconductor Double Wells X. Hu, W. Poetz. Terahertz-Control of Charge-Carrier Wavepackets in Semiconductor Quantum Wells S. Hughes, D.S. Citrin. Microcavity Exciton-Polariton H. Cao. Part III: Coherent Phenomena and Related Topics. Coherent Control of Quantum Localization M. Holthaus. Tunneling of Low-Energetic Electrons in the Presence of Intense Laser Fields: The Formation of Dynamical Barrier States M. Wagner. Hopping Mechanism of Coherent Photovoltaic Effect and Photoinduced Polar Anisotropy in Glass E.M. Baskin, M.V. Entin. Heterodyning Scheme Employing Quantum Interference J.B. Khurgin, et al. Generation of Widely Tunable THz-Wave Using Nonlinear Optics K. Kawase, H.Ito. List of Participants.
Blow-up for Schrodinger equation with pointwise nonlinearity by R. Adami Quantum electrodynamics of relativistic bound states with cutoffs. II by J.-M. Barbaroux, M. Dimassi, and J.-C. Guillot Scattering by a slab: An exact calculation by F. Bentosela and M. Tater Crystalline order in one dimensional Thomas-Fermi type models by X. Blanc and C. Le Bris New results on the spectrum of lattice Schrodinger operators and applications by J. Bourgain Reconstruction of the memory from partial boundary measurements by A. L. Bukhgeim, G. V. Dyatlov, and G. Uhlmann Asymptotics for nonlinear dissipative equations in the super critical case by R. E. Cardiel and P. I. Naumkin Edge states for quantum Hall Hamiltonians by J.-M. Combes, P. D. Hislop, and E. Soccorsi A semiclassical approach in linear response theory by M. Combescure and D. Robert Wigner-von Neumann-like potentials by J. Cruz-Sampedro Equilibrium potentials in scattering theory by M. Demuth and A. Noll Straight quantum waveguides with combined boundary conditions by J. Dittrich and J. Kriz Perturbation theory for the quantum time-evolution in rotating potentials by V. Enss, V. Kostrykin, and R. Schrader Two dimensional Pauli operator via scalar potential by L. Erdos and V. Vougalter A max-min principle for the ground state of the Dirac-Fock functional by M. J. Esteban and E. Sere On the regularity of the density of electronic wave functions by S. Fournais, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, and T. O. Sorensen Enhanced binding through coupling to a photon field by C. Hainzl The large radius limit for coherent states on spheres by B. C. Hall and J. J. Mitchell Quantum scattering for potentials homogeneous of degree zero by I. Herbst and E. Skibsted Existence of the density of states for one-dimensional alloy-type potentials with small support by W. Kirsch and I. Veselic Statistical ensembles and density of states by V. Kostrykin and R. Schrader The Cauchy problem for the Toda lattice with a class of non-stabilized initial data by M. Kudryavtsev Absolutely continuous spectrum of Jacobi matrices by A. Laptev, S. Naboko, and O. Safronov Stability of matter in relativistic quantum electrodynamics by E. H. Lieb and M. Loss The algebra of observables in a magnetic field by M. Mantoiu and R. Purice Recent developments in value distribution theory for Schrodinger operators on the half line by D. B. Pearson Long time propagation results in quantum mechanics by D. Robert Comments on adiabatic quantum algorithms by M. B. Ruskai Tunneling destruction for a nonlinear Schrodinger equation by A. Sacchetti Symmetry breaking in a model of a rotating Bose gas by R. Seiringer Euler's constant in Segal-Bargmann analysis by S. B. Sontz Maxwell with a touch of Schrodinger by A. Tip Exponentially accurate semiclassical asymptotics by J. H. Toloza Some properties of the eigenvalues of a Schrodinger equation with energy-dependent potential by C. van der Mee and V. Pivovarchik The Bargmann transform for L2(S3)L^2(S^3)L2(S3) and regularization of the Kepler problem by C. Villegas-Blas Schrodinger equations with superquadratic potentials by K. Yajima and G. Zhang Band spectrum of the Laplacian on a slab with the Dirichlet boundary condition on a grid by K. Yoshitomi Other talks Participants.
L objectif de cette thèse est d appliquer la théorie du contrôle optimal à la dynamique de systèmes quantiques. Le premier point consiste à introduire dans le domaine du contrôle quantique des outils de contrôle optimal initialement développés en mathématique. Cette approche a ensuite été appliquée sur différent types de systèmes quantiques décrit par une grande ou une petite dimension. La première partie du manuscrit introduit les différents outils de contrôles utilisés avec une approche adaptée à un public de physiciens. Dans la seconde partie, ces techniques sont utilisées pour contrôler la dynamique des spins en RMN et IRM. La troisième partie s intéresse au développement de nouveaux algorithmes itératifs de contrôle optimal appliqués au contrôle par champ laser de la dynamique rotationnelle des molécules linéaires en phases gazeuse ainsi qu au développement d une stratégie de contrôle simple permettant de délocaliser une molécule dans un plan. La quatrième partie traite le contrôle en temps minimum d un condensat de Bose-Einstein à deux composantes. La dernière partie permet de comparer qualitativement et quantitativement les différentes méthodes de contrôle optimal utilisées. Les seconde et troisième parties ont également bénéficier de l implémentation expérimentale des solutions de contrôle optimal obtenues.
Abstract The mathematics of optimal control of quantum systems is of great interest in connection with fundamental problems of physics as well as with existing and prospective applications to quantum technologies. One important problem is the development of methods for constructing controls for quantum systems. One of the commonly used methods is the Krotov method, which was initially proposed outside of quantum control theory in articles by Krotov and Feldman (1978, 1983). This method was used to develop a novel approach to finding optimal controls for quantum systems in [64] (Tannor, Kazakov, and Orlov, 1992), [65] (Somlói, Kazakov, and Tannor, 1993), and in many other works by various scientists. Our survey discusses mathematical aspects of this method for optimal control of closed quantum systems. It outlines various modifications with different forms of the improvement function (for example, linear or linear-quadratic), different constraints on the control spectrum and on the admissible states of the quantum system, different regularisers, and so on. The survey describes applications of the Krotov method to controlling molecular dynamics and Bose–Einstein condensates, and to quantum gate generation. This method is compared with the GRAPE (GRadient Ascent Pulse Engineering) method, the CRAB (Chopped Random-Basis) method, and the Zhu–Rabitz and Maday–Turinici methods. Bibliography: 158 titles.
Many numerical simulations in (bilinear) quantum control use the monotonically convergent Krotov algorithms (introduced by Tannor et al. [Time Dependent Quantum Molecular Dynamics (1992) 347–360]), Zhu and Rabitz [J. Chem. Phys. (1998) 385–391] or their unified form described in Maday and Turinici [J. Chem. Phys. (2003) 8191–8196]. In Maday et al. [Num. Math. (2006) 323–338], a time discretization which preserves the property of monotonicity has been presented. This paper introduces a proof of the convergence of these schemes and some results regarding their rate of convergence.