Explicit approximate controllability of the Schrödinger equation with a polarizability term

We consider a controlled Schrödinger equation with a dipolar and a polarizability term, used when the dipolar approximation is not valid. The control is the amplitude of the external electric field, it acts nonlinearly on the state. We extend in this infinite dimensional framework previous techniques used by Coron, Grigoriu, Lefter and Turinici for stabilization in finite dimension. We consider a highly oscillating control and prove the semi-global weak $$H^2$$ stabilization of the averaged system using a Lyapunov function introduced by Nersesyan. Then it is proved that the solutions of the Schrödinger equation and of the averaged equation stay close on every finite time horizon provided that the control is oscillating enough. Combining these two results, we get approximate controllability to the ground state for the polarizability system with explicit controls. Numerical simulations are presented to illustrate those theoretical results.

[1]  Karine Beauchard,et al.  Semi-global weak stabilization of bilinear Schrödinger equations , 2010 .

[2]  Compactness theorems for abstract evolution problems , 2002 .

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

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

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

[6]  Gabriel Turinici,et al.  Beyond bilinear controllability : applications to quantum control , 2008, 0808.1840.

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

[8]  Mazyar Mirrahimi,et al.  Real-time synchronization feedbacks for single-atom frequency standards: V - and Λ-structure systems , 2010, 49th IEEE Conference on Decision and Control (CDC).

[9]  Jean-Michel Coron,et al.  Quantum control design by Lyapunov trajectory tracking for dipole and polarizability coupling , 2009 .

[10]  Pierre Rouchon.  Control of a Quantum Particle in a Moving Potential Well , 2003 .

[11]  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.

[12]  Georges Bastin,et al.  A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws , 2004, CDC.

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

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

[15]  Jerrold E. Marsden,et al.  Controllability for Distributed Bilinear Systems , 1982 .

[16]  Lyapunov control of Schrödinger equation:beyond the dipole approximations , 2009 .

[17]  Mazyar Mirrahimi,et al.  Lyapunov control of bilinear Schrödinger equations , 2005, Autom..

[18]  T. Cazenave.  Semilinear Schrodinger Equations , 2003 .

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

[20]  Brigitte d'Andréa-Novel,et al.  Stabilization of a rotating body beam without damping , 1998, IEEE Trans. Autom. Control..

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

[22]  Nabile Boussaid,et al.  Weakly Coupled Systems in Quantum Control , 2011, IEEE Transactions on Automatic Control.

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

[24]  André D. Bandrauk,et al.  Two-frequency IR laser orientation of polar molecules. Numerical simulations for HCN , 1999 .

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

[26]  Nabile Boussaid,et al.  Approximate controllability of the Schrödinger equation with a polarizability term , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[27]  J. Couchouron.  Strong stabilization of controlled vibrating systems , 2011 .

[28]  André D. Bandrauk,et al.  LASER-INDUCED ALIGNMENT DYNAMICS OF HCN : ROLES OF THE PERMANENT DIPOLE MOMENT AND THE POLARIZABILITY , 1999 .

[29]  Jack K. Hale,et al.  Averaging in infinite dimensions , 1990 .

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