Flatness based trajectory generation of quantum systems

Abstract

A two-states quantum system with one control is proved to be flat. This provides a simple procedure to design smooth open-loop controls that steer in finite time from one eigen-state to the other one. A three-states quantum system with one control is not flat in general. Following the Rabi oscillations used by physicists to control stimulated transition, we associate to this system an averaged control system where the number of controls is increased and where flatness-based motion planning techniques can be used. This allows to steer directly from one eigen-state to any other one without using an additional intermediate eigen-state. In certain energy configurations our method is a noticeable improvement of the standard Rabi transitions strategy and leads to " active tunnelling control", as illustrated by simulations. This method can be extended without major difficulties to higher dimensional cases.

Time-optimal control of a 3-level quantum system and its generalization to an n-level system

“…where A(ω 1 , ω 3 ) is given in (11). The dynamics in (26) can be derived byL i = {L i , H}, i = 1, 2, 3.…”

“…. .with the initial and final conditionsand the control constraints In the context of quantum mechanics, the model considered in this paper is a finite-dimensional low-energy approximation of a Schrödinger equation driven by rotating fields and averaged over a time interval longer than the inverse energy splittings, where each x i corresponds to the coefficient of the eigen wave function of the i-th energy level, and controls u i 's correspond to the amplitudes of lasers [11], [5].Various open-loop control problems for quantum systems have been already studied. In particular, the energy-optimal control problem for the dynamics in (4) without any magnitude constraints on control was studied at the level of Lie groups in [7], [4].…”

“…For that problem, the author in [7] combined Lie-Poisson reduction theory with the Pontryagin Maximum Principle (PMP), and the authors in [4] utilized sub-Riemannian geometry with the PMP. In [11], the trajectory generation problem for the dynamics was studied via flatness theory. In [3] the time-optimal control problem for the dynamics in (1)-(3) using sub-Riemmanian geometry with the PMP.…”

“…In the context of quantum mechanics, the model considered in this paper is a finite-dimensional low-energy approximation of a Schrödinger equation driven by rotating fields and averaged over a time interval longer than the inverse energy splittings, where each x i corresponds to the coefficient of the eigen wave function of the i-th energy level, and controls u i 's correspond to the amplitudes of lasers [11], [5].…”

“…The same problem and its generalization are studied in this article using a different approach. Our main tool, distinct from those in [3], [4], [6], [7], [11], is the detection and exploitation of both continuous and discrete symmetry in the problem. An example of this is an S 1 continuous symmetry and a Z 2 × S 3 discrete symmetry in the dynamics (1)-(3).…”