The angular momentum of molecules, or, equivalently, their rotation in three-dimensional space, is ideally suited for quantum control. Molecular angular momentum is naturally quantized, time evolution is governed by a well-known Hamiltonian with only a few accurately known parameters, and transitions between rotational levels can be driven by external fields from various parts of the electromagnetic spectrum. Control over the rotational motion can be exerted in one-, two-, and many-body scenarios, thereby allowing one to probe Anderson localization, target stereoselectivity of bimolecular reactions, or encode quantum information to name just a few examples. The corresponding approaches to quantum control are pursued within separate, and typically disjoint, subfields of physics, including ultrafast science, cold collisions, ultracold gases, quantum information science, and condensed-matter physics. It is the purpose of this review to present the various control phenomena, which all rely on the same underlying physics, within a unified framework. To this end, recall the Hamiltonian for free rotations, assuming the rigid rotor approximation to be valid, and summarize the different ways for a rotor to interact with external electromagnetic fields. These interactions can be exploited for control—from achieving alignment, orientation, or laser cooling in a one-body framework, steering bimolecular collisions, or realizing a quantum computer or quantum simulator in the many-body setting.

Optimal control theory is a powerful mathematical tool, which has known a rapid development since the 1950s, mainly for engineering applications. More recently, it has become a widely used method to improve process performance in quantum technologies by means of highly efficient control of quantum dynamics. This tutorial aims at providing an introduction to key concepts of optimal control theory that is accessible to physicists and engineers working in quantum control or in related fields. The different mathematical results are introduced intuitively, before being rigorously stated. This tutorial describes modern aspects of optimal control theory, with a particular focus on the Pontryagin maximum principle, which is the main tool for determining open-loop control laws without experimental feedback. The different steps to solve an optimal control problem are discussed, before moving on to more advanced topics such as the existence of optimal solutions or the definition of the different types of extremals, namely normal, abnormal, and singular. The tutorial covers various quantum control issues and describes their mathematical formulation suitable for optimal control. The connection between the Pontryagin maximum principle and gradient-based optimization algorithms used for high-dimensional quantum systems is described. The optimal solution of different low-dimensional quantum systems is presented in detail, illustrating how the mathematical tools are applied in a practical way.

We consider the time-optimal control by magnetic fields of a spin 12 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.

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 for 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.

We investigate the time- and 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 than or equal to $2N$ for a field robust to the $N\mathrm{th}$ order, which leads to an estimate of its complexity.

We propose an analysis of the time-optimal control of a dissipative two-level quantum system whose dynamics is governed by the Lindblad equation. This simple system allows one to use tools of geometric control theory and to construct its optimal synthesis, i.e., to determine the set of all the optimal trajectories starting from a given initial point. We study different processes such as conversion of a pure state into a mixed state and purification of a mixed state. In particular cases, we show that dissipation is not undesirable and can help accelerating the control.