## Section: Application Domains

### Second axis: Quantum control

The goal of quantum control is to design efficient protocols for tuning the occupation probabilities of the energy levels of a system. This task is crucial in atomic and molecular physics, with applications ranging from photochemistry to nuclear magnetic resonance and quantum computing. A quantum system may be controlled by exciting it with one or several external fields, such as magnetic or electric fields. The goal of quantum control theory is to adapt the tools originally developed by control theory and to develop new specific strategies that tackle and exploit the features of quantum dynamics (probabilistic nature of wavefunctions and density operators, measure and wavefunction collapse, decoherence, ...). A rich variety of relevant models for controlled quantum dynamics exist, encompassing low-dimensional models (e.g., single-spin systems) and PDEs alike, with deterministic and stochastic components, making it a rich and exciting area of research in control theory.

The controllability of quantum system
is
a well-established topic when the state space
is finite-dimensional [102], thanks to general controllability methods for left-invariant control systems on compact Lie groups [92], [112].
When the state space
is
infinite-dimensional, it is known that in general
the bilinear SchrÃ¶dinger equation is not exactly controllable [154]. Nevertheless, weaker
controllability properties, such as approximate controllability or controllability between eigenstates of the internal Hamiltonian
(which are the most relevant physical states), may hold.
In certain cases, when the state space
is a function space on a 1D manifold, some rather precise description of the set of reachable states has
been provided [77]. A similar description for higher-dimensional manifolds seems intractable and at the moment only approximate controllability results
are available [128], [134], [113].
The most widely applicable tests for controllability of quantum systems in infinite-dimensional Hilbert spaces are based on the **Lieâ€“Galerkin technique** [96], [85], [86]. They allow, in particular, to show that the controllability property is generic among this class of systems [125].

A family of algorithms
which are specific to quantum systems are those based on adiabatic evolution [158], [157], [116].
The basic principle of adiabatic control is
that the flow of a slowly varying Hamiltonian can be approximated (up to a phase factor) by a quasi-static evolution, with a precision proportional to the velocity of variation of the Hamiltonian.
The advantage of the **adiabatic approach** is that it is constructive and produces control laws which are both smooth and
robust to parameter uncertainty. The paradigm is based on
the adiabatic perturbation theory developed in mathematical physics [83], [133], [150],
where it plays an important role for understanding molecular dynamics.
Approximation theory by adiabatic perturbation can be used to
describe the evolution of the occupation probabilities of
the energy levels of a slowly varying
Hamiltonian.
Results from the last 15 years, including those by members of our team [64], [89], have highlighted the effectiveness of control techniques based on adiabatic path following.