On the control of quantum oscillators

where ψ is the complex probability amplitude vector (belongs to an Hilbet space of finite or infinite dimension), H0 is the free Hamiltonian and H1 is the Hamiltonian associated to the scalar control u (corresponding to a classical electro-magnetic field). We discuss here the controllability and/or the following motion planning problem: for two pure states, ψa and ψb of free energy Eb and Eb (H0ψa = Eaψa and H0ψb = Ebψb), find an open-loop control [0, T ] t → u(t) steering the state ψ form ψa at t = 0 to the state ψb at t = T > 0. It seems that according to [10], such motion planning problem is meaning full. In this report, we consider several Hamiltonian H0 and H1: • The 1D harmonic oscillator where ψ(q, t) is an L complex function of q ∈ R the space position, H0 = p/2 + q/2 and H1 = −q. p corresponds to the operator ı/ ∂ ∂q and q to the multiplication by q. • A 1D particle with H0 = p/2 + V (q) and H1 = −q in the quasi-classic approximation ( ≈ 0). • A two states system ψ ∈ C where we exploit the fictitious spin description.