Time-Minimal Control of Dissipative Two-level Quantum Systems: the Generic Case

The objective of this article is to complete preliminary results concerning the time-minimal control of dissipative two-level quantum systems whose dynamics is governed by Lindblad equations. The extremal system is described by a 3D-Hamiltonian depending upon three parameters. We combine geometric techniques with numerical simulations to deduce the optimal solutions.Comment: 24 pages, 16 figures. submitted to IEEE transactions on automatic contro

Geometric optimal control of the contrast imaging problem in Nuclear Magnetic Resonance

The objective of this article is to introduce the tools to analyze the contrast imaging problem in Nuclear Magnetic Resonance. Optimal trajectories can be selected among extremal solutions of the Pontryagin Maximum Principle applied to this Mayer type optimal problem. Such trajectories are associated to the question of extremizing the transfer time. Hence the optimal problem is reduced to the analysis of the Hamiltonian dynamics related to singular extremals and their optimality status. This is illustrated by using the examples of cerebrospinal fluid / water and grey / white matter of cerebrum.Comment: 30 pages, 13 figur

Energy minimization problem in two-level dissipative quantum control: meridian case

International audienceWe analyze the energy-minimizing problem for a two-level dissipative quantum system described by the Kossakowsky-Lindblad equation. According to the Pontryagin Maximum Principle (PMP), minimizers can be selected among normal and abnormal extremals whose dynamics are classified according to the values of the dissipation parameters. Our aim is to improve our previous analysis concerning 2D solutions in the case where the Hamiltonian dynamics are integrable

Time-optimal Unitary Operations in Ising Chains II: Unequal Couplings and Fixed Fidelity

We analytically determine the minimal time and the optimal control laws required for the realization, up to an assigned fidelity and with a fixed energy available, of entangling quantum gates ($\mathrm{CNOT}$) between indirectly coupled qubits of a trilinear Ising chain. The control is coherent and open loop, and it is represented by a local and continuous magnetic field acting on the intermediate qubit. The time cost of this local quantum operation is not restricted to be zero. When the matching with the target gate is perfect (fidelity equal to one) we provide exact solutions for the case of equal Ising coupling. For the more general case when some error is tolerated (fidelity smaller than one) we give perturbative solutions for unequal couplings. Comparison with previous numerical solutions for the minimal time to generate the same gates with the same Ising Hamiltonian but with instantaneous local controls shows that the latter are not time-optimal.Comment: 11 pages, no figure

Monotonically convergent optimal control theory of quantum systems with spectral constraints on the control field

We propose a new monotonically convergent algorithm which can enforce spectral constraints on the control field (and extends to arbitrary filters). The procedure differs from standard algorithms in that at each iteration the control field is taken as a linear combination of the control field (computed by the standard algorithm) and the filtered field. The parameter of the linear combination is chosen to respect the monotonic behavior of the algorithm and to be as close to the filtered field as possible. We test the efficiency of this method on molecular alignment. Using band-pass filters, we show how to select particular rotational transitions to reach high alignment efficiency. We also consider spectral constraints corresponding to experimental conditions using pulse shaping techniques. We determine an optimal solution that could be implemented experimentally with this technique.Comment: 16 pages, 4 figures. To appear in Physical Review

Geometric Approach to Pontryagin's Maximum Principle

Since the second half of the 20th century, Pontryagin's Maximum Principle has been widely discussed and used as a method to solve optimal control problems in medicine, robotics, finance, engineering, astronomy. Here, we focus on the proof and on the understanding of this Principle, using as much geometric ideas and geometric tools as possible. This approach provides a better and clearer understanding of the Principle and, in particular, of the role of the abnormal extremals. These extremals are interesting because they do not depend on the cost function, but only on the control system. Moreover, they were discarded as solutions until the nineties, when examples of strict abnormal optimal curves were found. In order to give a detailed exposition of the proof, the paper is mostly self\textendash{}contained, which forces us to consider different areas in mathematics such as algebra, analysis, geometry.Comment: Final version. Minors changes have been made. 56 page

Hamiltonian dynamics and constrained variational calculus: continuous and discrete settings

The aim of this paper is to study the relationship between Hamiltonian dynamics and constrained variational calculus. We describe both using the notion of Lagrangian submanifolds of convenient symplectic manifolds and using the so-called Tulczyjew's triples. The results are also extended to the case of discrete dynamics and nonholonomic mechanics. Interesting applications to geometrical integration of Hamiltonian systems are obtained.Comment: 33 page

Saturation of a spin 1/2 particle by generalized Local control

We show how to apply a generalization of Local control design to the problem of saturation of a spin 1/2 particle by magnetic fields in Nuclear Magnetic Resonance. The generalization of local or Lyapunov control arises from the fact that the derivative of the Lyapunov function does not depend explicitly on the control field. The second derivative is used to determine the local control field. We compare the efficiency of this approach with respect to the time-optimal solution which has been recently derived using geometric methods.Comment: 12 pages, 4 figures, submitted to new journal of physics (2011

Outcome of alimentary tract duplications operated on by minimally invasive surgery: a retrospective multicenter study by the GECI (Groupe d'Etude en Coeliochirurgie Infantile).

BACKGROUND: Alimentary tract duplications (ATD) are a rare cause of intestinal obstruction in childhood. There are many case reports but few series about laparoscopy or thoracoscopy for ATD. The aim of our study was to report the outcome of minimally invasive surgery (MIS) for ATD. METHODS: This was a retrospective multicenter study from the GECI (Groupe d\u27Etude en Coeliochirurgie Infantile). We reviewed the charts of 114 patients operated on by MIS for ATD from 1994 to 2009. RESULTS: Sixty-two patients (54 %) had a prenatal diagnosis. Forty-nine patients (43 %) were symptomatic before surgery: 33 of those patients (63 %) with postnatal diagnosis compared to 16 (25 %) with prenatal diagnosis (P < 0.01). In this last group, the median age at onset of symptoms was 16 days (range = 0-972). One hundred and two patients had laparoscopy (esophageal to rectal duplications) and 12 patients had thoracoscopy for esophageal duplications. The mean operative time was 90 min (range = 82-98). There were 32 (28 %) resection anastomoses, 55 (48 %) enucleations, and 27 (24 %) unroofings. The conversion rate was 32 %, and in a multivariate analysis, it was significantly higher, up to 41 % for patients weighing <10 kg (P < 0.01). Ten patients (8 %) had unintentional perioperative opening of the digestive tract during the dissection. Eight patients had nine postoperative complications, including six small bowel obstructions. The median length of hospital stay was 4 days (range = 1-21) without conversion and 6 days (range = 1-27) with conversion (P = 0.01). The median follow-up was 3 months (range = 1-120). Eighteen of the 27 patients who underwent partial surgery had an ultrasound examination during follow-up. Five (18 %) of them had macroscopic residue. CONCLUSION: This study showed that MIS for ATD is feasible with a low rate of complications. Patients with prenatal diagnosis should have prompt surgery to prevent symptoms, despite a high rate of conversion in small infants