Application of the Pontryagin maximum principle to the time-optimal control in a chain of three spins with unequal couplings

We solve a time-optimal control problem in a linear chain of three coupled spins 1/2 with unequal couplings. We apply the Pontryagin maximum principle and show that the associated Hamiltonian system is the one of a three-dimensional rigid body. We express the optimal control fields in terms of the components of the classical angular momentum of the rigid body. The optimal trajectories and the minimum control time are given in terms of elliptic functions and elliptic integrals.

Observation of time-invariant coherence in a room temperature quantum simulator

The ability to live in coherent superpositions is a signature trait of quantum systems and constitutes an irreplaceable resource for quantum-enhanced technologies. However, decoherence effects usually destroy quantum superpositions. It has been recently predicted that, in a composite quantum system exposed to dephasing noise, quantum coherence in a transversal reference basis can stay protected for indefinite time. This can occur for a class of quantum states independently of the measure used to quantify coherence, and requires no control on the system during the dynamics. Here, such an invariant coherence phenomenon is observed experimentally in two different setups based on nuclear magnet…

Time-optimal control of spin-1/2 particles with dissipative and generalized radiation-damping effects

We analyze the time-optimal control of spin-1/2 particles with bounded field amplitudes in the presence of dissipative and radiation damping effects. Using tools of geometric optimal control theory, we determine different optimal syntheses for specific values of the system parameters. We show the nontrivial role of the effective radiation damping effect on the optimal control law.

Understanding the global structure of two-level quantum systems with relaxation: Vector fields organized through the magic plane and the steady-state ellipsoid

Nuclear magnetic resonance: The contrast imaging problem

Starting as a tool for characterization of organic molecules, the use of NMR has spread to areas as diverse as pharmacology, medical diagnostics (medical resonance imaging) and structural biology. Recent advancements on the study of spin dynamics strongly suggest the efficiency of geometric control theory to analyze the optimal synthesis. This paper focuses on a new approach to the contrast imaging problem using tools from geometric optimal control. It concerns the study of an uncoupled two-spin system and the problem is to bring one spin to the origin of the Bloch ball while maximizing the modulus of the magnetization vector of the second spin. It can be stated as a Mayer-type optimal prob…

Optimal control of the signal-to-noise ratio per unit time of a spin 1/2 particle: The crusher gradient and the radiation damping cases

We show to which extent the signal to noise ratio per unit time of a spin 1/2 particle can be maximized. We consider a cyclic repetition of experiments made of a measurement followed by a radio-frequency magnetic field excitation of the system, in the case of unbounded amplitude. In the periodic regime, the objective of the control problem is to design the initial state of the system and the pulse sequence which leads to the best signal to noise performance. We focus on two specific issues relevant in nuclear magnetic resonance, the crusher gradient and the radiation damping cases. Optimal control techniques are used to solve this non-standard control problem. We discuss the optimality of t…

Geometric optimal control of the contrast problem in Magnetic Resonance Imaging

Abstract The control of the dynamics of spin systems by magnetic fields has opened intriguing possibilities in quantum computing and in Nuclear Magnetic Resonance spectroscopy. In this framework, optimal control theory has been used to design control fields able to realize a given task while minimizing a prescribed cost such as the energy of the field or the duration of the process. However, some of the powerful tools of optimal control had not been used yet for NMR applications in medical imagery. Here, we show that the geometric control theory approach can be advantageously combined with NMR methods to crucially optimize the imaging contrast. This approach is applied to a benchmark proble…

Geometric versus numerical optimal control of a dissipative spin-12particle

We analyze the saturation of a nuclear magnetic resonance (NMR) signal using optimal magnetic fields. We consider both the problems of minimizing the duration of the control and its energy for a fixed duration. We solve the optimal control problems by using geometric methods and a purely numerical approach, the grape algorithm, the two methods being based on the application of the Pontryagin maximum principle. A very good agreement is obtained between the two results. The optimal solutions for the energy-minimization problem are finally implemented experimentally with available NMR techniques.

Optimizing MRI contrast with B1 pulses using optimal control theory

The variety of achievable contrasts by MRI makes it a highly flexible and valuable diagnostic tool. Contrast results from relaxation time differences, which are intrinsic properties of each tissue. Using optimal control theory, one can control the obtained contrast by applying excitation pulses that bring the magnetization in a user-defined target state. Simulation results are presented to illustrate the feasibility and the flexibility of using optimal contrast pulses. The robustness to experimental variable parameters such as field inhomogeneities is also studied. Finally, an in-vitro contrast experiment is performed on a small-animal MRI showing a reasonable match with the simulation resu…

Simultaneous time-optimal control of the inversion of two spin-12particles

We analyze the simultaneous time-optimal control of two-spin systems. The two noncoupled spins, which differ in the value of their chemical offsets, are controlled by the same magnetic fields. Using an appropriate rotating frame, we restrict the study to the case of opposite shifts. We then show that the optimal solution of the inversion problem in a rotating frame is composed of a pulse sequence of maximum intensity and is similar to the optimal solution for inverting only one spin by using a nonresonant control field in the laboratory frame. An example is implemented experimentally using nuclear magnetic resonance techniques.

Towards the time-optimal control of dissipative spin-1/2 particles in nuclear magnetic resonance

International audience; We consider the time-optimal control of a spin 1/2 particle whose dynamics is governed by the Bloch equations with both longitudinal and transverse relaxation terms. We solve this control problem by using geometric optimal control techniques. We show the crucial role of singular extremals in the time-optimal synthesis. This role can mainly be attributed to the presence of dissipation. We also analyze the robustness of the optimal control sequence when both the maximum amplitude of the control field and the dissipative parameters are varied. Finally, we present an experimental implementation of the different solutions using techniques of Nuclear Magnetic Resonance.

Training Schrödinger’s cat: quantum optimal control

It is control that turns scientific knowledge into useful technology: in physics and engineering itprovides a systematic way for driving a dynamical system from a given initial state into a desired targetstate with minimized expenditure of energy and resources. As one of the cornerstones for enabling quantumtechnologies, optimal quantum control keeps evolving and expanding into areas as diverse as quantumenhancedsensing, manipulation of single spins, photons, or atoms, optical spectroscopy, photochemistry,magnetic resonance (spectroscopy as well as medical imaging), quantum information processing and quantumsimulation. In this communication, state-of-the-art quantum control techniques are r…

Observation of time-invariant coherence in a nuclear magnetic resonance quantum simulator

The ability to live in coherent superpositions is a signature trait of quantum systems and constitutes an irreplaceable resource for quantum-enhanced technologies. However, decoherence effects usually destroy quantum superpositions. It was recently predicted that, in a composite quantum system exposed to dephasing noise, quantum coherence in a transversal reference basis can stay protected for an indefinite time. This can occur for a class of quantum states independently of the measure used to quantify coherence, and it requires no control on the system during the dynamics. Here, such an invariant coherence phenomenon is observed experimentally in two different setups based on nuclear magne…

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.

Singular Extremals for the Time-Optimal Control of Dissipative Spin 1/2 Particles

We consider the time-optimal control by magnetic fields of a spin 1/2 particle in a dissipative environment. This system is used as an illustrative example to show the role of singular extremals in the control of quantum systems. We analyze a simple case where the control law is explicitly determined. We experimentally implement the optimal control using techniques of nuclear magnetic resonance. To our knowledge, this is the first experimental demonstration of singular extremals in quantum systems with bounded control amplitudes.

Optimal control of the inversion of two spins in Nuclear Magnetic Resonance

International audience; We investigate the optimal control of the inversion of two spin 1/2 particles in Nuclear Magnetic Resonance. The two spins, which differ by their resonance offset, are controlled by the same radio frequency magnetic field. Using the Pontryagin Maximum Principle, we compute the optimal control sequence which allows to reach the target state in a given time, while minimizing the energy of the magnetic field. A comparison with the time-optimal solution for bounded control amplitude realizing the same control in the same time is made. An experimental illustration is done using techniques of Nuclear Magnetic Resonance.

Application of the small-tip-angle approximation in the toggling frame for the design of analytic robust pulses in quantum control

We apply the Small Tip-Angle Approximation in the Toggling Frame in order to analytically design robust pulses against resonance offsets for state to state transfer in two-level quantum systems. We show that a broadband or a local robustness up to an arbitrary order can be achieved. We provide different control parameterizations to satisfy experimental constraints and limitations on the amplitude or energy of the pulse. A comparison with numerical optimal solutions is made.

Exploring the Physical Limits of Saturation Contrast in Magnetic Resonance Imaging

International audience; Magnetic Resonance Imaging has become nowadays an indispensable tool with applications ranging from medicine to material science. However, so far the physical limits of the maximum achievable experimental contrast were unknown. We introduce an approach based on principles of optimal control theory to explore these physical limits, providing a benchmark for numerically optimized robust pulse sequences which can take into account experimental imperfections. This approach is demonstrated experimentally using a model system of two spatially separated liquids corresponding to blood in its oxygenated and deoxygenated forms.

Optimal control design of preparation pulses for contrast optimization in MRI

Abstract This work investigates the use of MRI radio-frequency (RF) pulses designed within the framework of optimal control theory for image contrast optimization. The magnetization evolution is modeled with Bloch equations, which defines a dynamic system that can be controlled via the application of the Pontryagin Maximum Principle (PMP). This framework allows the computation of optimal RF pulses that bring the magnetization to a given state to obtain the desired contrast after acquisition. Creating contrast through the optimal manipulation of Bloch equations is a new way of handling contrast in MRI, which can explore the theoretical limits of the system. Simulation experiments carried out…

Robust optimal control of two-level quantum systems

We investigate the time and the energy minimum optimal solutions for the robust control of two-level quantum systems against offset or control field uncertainties. Using the Pontryagin Maximum Principle, we derive the global optimal pulses for the first robustness orders. We show that the dimension of the control landscape is lower or equal to 2N for a field robust to the N th order, which leads to an estimate of its complexity.

Optimal control of an inhomogeneous spin ensemble coupled to a cavity

We apply optimal control techniques to an inhomogeneous spin ensemble coupled to a cavity. A general procedure is proposed for designing the control strategies. We numerically show the extent to which optimal control fields robust against system uncertainties help enhancing the sensitivity of the detection process. The parameters of the numerical simulations are taken from recent Electron Spin Resonance experiments. The low and high cooperativity regimes are explored.

Time-optimal selective pulses of two uncoupled spin-1/2 particles

We investigate the time-optimal solution of the selective control of two uncoupled spin 1/2 particles. Using the Pontryagin Maximum Principle, we derive the global time-optimal pulses for two spins with different offsets. We show that the Pontryagin Hamiltonian can be written as a one-dimensional effective Hamiltonian. The optimal fields can be expressed analytically in terms of elliptic integrals. The time-optimal control problem is solved for the selective inversion and excitation processes. A bifurcation in the structure of the control fields occurs for a specific offset threshold. In particular, we show that for small offsets, the optimal solution is the concatenation of regular and sin…

Time-optimal control of SU(2) quantum operations

We propose an analysis of the time-optimal control of SU(2) quantum operations. By using the Pontryagin Maximum Principle, we show how to determine the optimal trajectory reaching a given target state. Explicit analytical solutions are given for two specific examples. We discuss the role of the detuning in the construction of the optimal synthesis.

A simplified framework to optimize MRI contrast preparation

PURPOSE This article proposes a rigorous optimal control framework for the design of preparation schemes that optimize MRI contrast based on relaxation time differences. METHODS Compared to previous optimal contrast preparation schemes, a drastic reduction of the optimization parameter number is performed. The preparation scheme is defined as a combination of several block pulses whose flip angles, phase terms and inter-pulse delays are optimized to control the magnetization evolution. RESULTS The proposed approach reduces the computation time of B 0 -robust preparation schemes to around a minute (whereas several hours were required with previous schemes), with negligible performance loss. …

Discrete-valued-pulse optimal control algorithms: Application to spin systems

International audience; This article is aimed at extending the framework of optimal control techniques to the situation where the control field values are restricted to a finite set. We propose generalizations of the standard GRAPE algorithm suited to this constraint. We test the validity and the efficiency of this approach for the inversion of an inhomogeneous ensemble of spin systems with different offset frequencies. It is shown that a remarkable efficiency can be achieved even for a very limited number of discrete values. Some applications in nuclear magnetic resonance are discussed.