Abstract
In this paper we analyze the Lyapunov trajectory tracking of the Schrödinger equation for a coupling control operator containing both a linear (dipole) and a quadratic (polarizability) term. We show numerically that the contribution of the quadratic part cannot be exploited by standard trajectory tracking tools and propose two improvements: discontinuous feedback and periodic (time-dependent) feedback. For both cases we present theoretical results and support them by numerical illustrations.
Highlights
We consider in this work the evolution of a quantum system with wavefunction Ψ(t) under the external influence of a laser field; the system satisfies the Time Dependent
This positive result for degenerate system shows that the theoretical results are sufficient but not necessary; the approach may fail in some particular degenerate cases.This is consistent with the literature on quantum control that shows that degenerate cases have special structure
We focus in this paper on designing trajectory tracking procedures for a control system with polarizability terms u2(t)H2 present
Summary
We consider in this work the evolution of a quantum system with wavefunction Ψ(t) under the external influence of a laser field; the system satisfies the Time Dependent. Our method is valid to track any eigenstate trajectory of a Schrodinger equation (2) when the Hamiltonian includes a second order coupling operator. Both sections present theoretical results on the convergence illustrated by numerical simulations.
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