Weakly Coupled Systems in Quantum Control

Weakly coupled systems are a class of infinite dimensional conservative bilinear control systems with discrete spectrum. An important feature of these systems is that they can be precisely approached by finite dimensional Galerkin approximations. This property is of particular interest for the approximation of quantum system dynamics and the control of the bilinear Schrödinger equation. The present study provides rigorous definitions and analysis of the dynamics of weakly coupled systems and gives sufficient conditions for an infinite dimensional quantum control system to be weakly coupled. As an illustration we provide examples chosen among common physical systems.

[1]  Claude Le Bris,et al.  Mathematical models and methods for ab initio quantum chemistry , 2000 .

[2]  Sylvain Ervedoza,et al.  Approximate Controllability for a System of Schrödinger Equations Modeling a Single Trapped Ion , 2009 .

[3]  Riccardo Adami,et al.  Controllability of the Schrödinger Equation via Intersection of Eigenvalues , 2005 .

[4]  M. Reed,et al.  Methods of Modern Mathematical Physics. 2. Fourier Analysis, Self-adjointness , 1975 .

[5]  M. Reed.  Methods of Modern Mathematical Physics. I: Functional Analysis , 1972 .

[6]  Mazyar Mirrahimi,et al.  Practical Stabilization of a Quantum Particle in a One-Dimensional Infinite Square Potential Well , 2009, SIAM J. Control. Optim..

[7]  Mark S. C. Reed,et al.  Method of Modern Mathematical Physics , 1972 .

[8]  Reinhard Illner,et al.  Limitations on the control of Schrödinger equations , 2006 .

[9]  Nabile Boussaid,et al.  Periodic control laws for bilinear quantum systems with discrete spectrum , 2012, 2012 American Control Conference (ACC).

[10]  Thomas Chambrion,et al.  Periodic excitations of bilinear quantum systems , 2011, Autom..

[11]  Mazyar Mirrahimi,et al.  Controllability of quantum harmonic oscillators , 2004, IEEE Transactions on Automatic Control.

[12]  Mazyar Mirrahimi,et al.  Lyapunov control of a quantum particle in a decaying potential , 2008, 0805.0910.

[13]  Karine Beauchard,et al.  Local controllability of a 1-D Schrödinger equation , 2005 .

[14]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[15]  P. Mason,et al.  Generic Controllability Properties for the Bilinear Schrödinger Equation , 2009 .

[16]  Mario Sigalotti,et al.  A Weak Spectral Condition for the Controllability of the Bilinear Schrödinger Equation with Application to the Control of a Rotating Planar Molecule , 2011, ArXiv.

[17]  P. Buser,et al.  Geometry and Spectra of Compact Riemann Surfaces , 1992 .

[18]  Navin Khaneja,et al.  Sub-Riemannian geometry and time optimal control of three spin systems: Quantum gates and coherence transfer , 2002 .

[19]  A. I. Solomon,et al.  Complete controllability of quantum systems , 2000, quant-ph/0010031.

[20]  Law,et al.  Arbitrary control of a quantum electromagnetic field. , 1996, Physical review letters.

[21]  E. Davies,et al.  Spectral Theory and Differential Operators: Index , 1995 .

[22]  D. D’Alessandro.  Introduction to Quantum Control and Dynamics , 2007 .

[23]  M. Slemrod,et al.  Controllability of distributed bilinear systems , 1981, 1981 20th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes.

[24]  Claudio Altafini,et al.  Controllability of quantum mechanical systems by root space decomposition of su(N) , 2002 .

[25]  U. Boscain,et al.  Controllability of the Schrödinger Equation via Intersection of Eigenvalues , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[26]  Gabriel Turinici,et al.  On the controllability of bilinear quantum systems , 2000 .

[27]  Karine Beauchard,et al.  Local controllability of 1D linear and nonlinear Schr , 2010, 1001.3288.

[28]  Nabile Boussaid,et al.  Small time reachable set of bilinear quantum systems , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[29]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[30]  C. cohen-tannoudji,et al.  Quantum Mechanics: , 2020, Fundamentals of Physics II.

[31]  V. Nersesyan.  Growth of Sobolev Norms and Controllability of the Schrödinger Equation , 2008, 0804.3982.

[32]  Roger W. Brockett,et al.  Finite Controllability of Infinite-Dimensional Quantum Systems , 2010, IEEE Transactions on Automatic Control.

[33]  Mario Sigalotti,et al.  Generic controllability properties for the bilinear Schrödinger equation , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[34]  Vahagn Nersesyan,et al.  Global approximate controllability for Schr\"odinger equation in higher Sobolev norms and applications , 2009, 0905.2438.

[35]  David E. Edmunds,et al.  Spectral Theory and Differential Operators , 1987, Oxford Scholarship Online.

[36]  A. Agrachev,et al.  Control Theory from the Geometric Viewpoint , 2004 .

[37]  Ian R. Petersen,et al.  Quantum control theory and applications: A survey , 2009, IET Control Theory & Applications.

[38]  Nabile Boussaid,et al.  Implementation of logical gates on infinite dimensional quantum oscillators , 2011, 2012 American Control Conference (ACC).

[39]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[40]  Mario Sigalotti,et al.  Controllability of the discrete-spectrum Schrödinger equation driven by an external field , 2008, 0801.4893.