Quantum systems and control 1

This paper describes several methods used by physicists for manipulations of quantum states. For each method, we explain the model, the various time-scales, the performed approximations and we propose an interpretation in terms of control theory. These various interpretations underlie open questions on controllability, feedback and estimations. For 2-level systems we consider: the Rabi oscillations in connection with averaging; the Bloch-Siegert corrections associated to the second order terms; controllability versus parametric robustness of open-loop control and an interesting controllability problem in infinite dimension with continuous spectra. For 3-level systems we consider: Raman pulses and the second order terms. For spin/spring systems we consider: composite systems made of 2-level sub-systems coupled to quantized harmonic oscillators; multi-frequency averaging in infinite dimension; controllability of 1D partial differential equation of Shrodinger type and affine versus the control; motion planning for quantum gates. For open quantum systems subject to decoherence with continuous measures we consider: quantum trajectories and jump processes for a 2-level system; Lindblad-Kossakovsky equation and their controllability.

[1]  K. Lendi,et al.  Quantum Dynamical Semigroups and Applications , 1987 .

[2]  Pierre Rouchon,et al.  Flatness-Based Control of a Single Qubit Gate , 2008, IEEE Transactions on Automatic Control.

[3]  P. Cheinet,et al.  Conception et réalisation d'un gravimètre à atomes froids , 2006 .

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

[5]  R. Murray,et al.  Flat systems, equivalence and trajectory generation , 2003 .

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

[7]  M. Fliess,et al.  Linéarisation par bouclage dynamique et transformations de Lie-Bäcklund , 1993 .

[8]  L. P. Yatsenko,et al.  Topology of adiabatic passage , 2002 .

[9]  J. Lévine,et al.  Sufficient conditions for dynamic state feedback linearization , 1991 .

[10]  Francesco Petruccione,et al.  The Theory of Open Quantum Systems , 2002 .

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

[12]  M. Fliess,et al.  Flatness and defect of non-linear systems: introductory theory and examples , 1995 .

[13]  J. Gauthier,et al.  Optimal control in laser-induced population transfer for two- and three-level quantum systems , 2002 .

[14]  Mazyar Mirrahimi,et al.  Stabilizing Feedback Controls for Quantum Systems , 2005, SIAM J. Control. Optim..

[15]  M. Fliess,et al.  Flatness, motion planning and trailer systems , 1993, Proceedings of 32nd IEEE Conference on Decision and Control.

[16]  F. Verhulst,et al.  Averaging Methods in Nonlinear Dynamical Systems , 1985 .

[17]  P. Rouchon,et al.  On the controllability of some quantum electro-dynamical systems , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[18]  Ramon van Handel,et al.  Feedback control of quantum state reduction , 2005, IEEE Transactions on Automatic Control.

[19]  J. Coron.  Control and Nonlinearity , 2007 .

[20]  Stefan Teufel,et al.  Adiabatic perturbation theory in quantum dynamics , 2003 .

[21]  E. Jaynes,et al.  Comparison of quantum and semiclassical radiation theories with application to the beam maser , 1962 .

[22]  Domenico D'Alessandro,et al.  Notions of controllability for bilinear multilevel quantum systems , 2003, IEEE Trans. Autom. Control..

[23]  H. Rabitz,et al.  Wavefunction controllability in quantum systems , 2003 .

[24]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[25]  F. Wilczek,et al.  Geometric Phases in Physics , 1989 .

[26]  Mazyar Mirrahimi,et al.  Real-Time Synchronization Feedbacks for Single-Atom Frequency Standards , 2009, SIAM J. Control. Optim..

[27]  N. Khaneja,et al.  Control of inhomogeneous quantum ensembles , 2006 .

[28]  C. Lobry.  Contr^olabilite des systemes non lineaires , 1970 .

[29]  I. Chuang,et al.  Quantum Computation and Quantum Information: Introduction to the Tenth Anniversary Edition , 2010 .