Which notion of energy for bilinear quantum systems?

In this note we investigate what is the best L^p-norm in order to describe the relation between the evolution of the state of a bilinear quantum system with the L^p-norm of the external field. Although L^2 has a structure more easy to handle, the L^1 norm is more suitable for this purpose. Indeed for every p>1, it is possible to steer, with arbitrary precision, a generic bilinear quantum system from any eigenstate of the free Hamiltonian to any other with a control of arbitrary small L^p norm. Explicit optimal costs for the L^1 norm are computed on an example

Controllability of the bilinear Schr\"odinger equation with several controls and application to a 3D molecule

We show the approximate rotational controllability of a polar linear molecule by means of three nonresonant linear polarized laser fields. The result is based on a general approximate controllability result for the bilinear Schr\"odinger equation, with wavefunction varying in the unit sphere of an infinite-dimensional Hilbert space and with several control potentials, under the assumption that the internal Hamiltonian has discrete spectrum

Quantum control of molecular rotation

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 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, we 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.Comment: 52 pages, 11 figures, 607 reference

Approximate controllability of the Schrödinger equation with a polarizability term

International audienceThis paper is concerned with the controllability of quantum systems in the case where the standard dipolar approximation, involving the permanent dipole moment of the system, has to be corrected by a so-called polarizability term, involving the field induced dipole moment. Sufficient conditions for controllability between eigenstates of the free Hamiltonian are derived and control laws are explicitly given. As an illustration, the results are applied to the planar rotation of the HCN molecule

Periodic control laws for bilinear quantum systems with discrete spectrum

We provide bounds on the error between dynamics of an infinite dimensional bilinear Schr\"odinger equation and of its finite dimensional Galerkin approximations. Standard averaging methods are used on the finite dimensional approximations to obtain constructive controllability results. As an illustration, the methods are applied on a model of a 2D rotating molecule.Comment: 6 pages, submitted to ACC 201

Periodic excitations of bilinear quantum systems

A well-known method of transferring the population of a quantum system from an eigenspace of the free Hamiltonian to another is to use a periodic control law with an angular frequency equal to the difference of the eigenvalues. For finite dimensional quantum systems, the classical theory of averaging provides a rigorous explanation of this experimentally validated result. This paper extends this finite dimensional result, known as the Rotating Wave Approximation, to infinite dimensional systems and provides explicit convergence estimates.Comment: Available online http://www.sciencedirect.com/science/article/pii/S000510981200286

Quantum control of ro-vibrational dynamics and application to light-induced molecular chirality

Achiral molecules can be made temporarily chiral by excitation with electric fields, in the sense that an average over molecular orientations displays a net chiral signal [Tikhonov et al., Sci. Adv. 8, eade0311 (2022)]. Here, we go beyond the assumption of molecular orientations to remain fixed during the excitation process. Treating both rotations and vibrations quantum mechanically, we identify conditions for the creation of chiral vibrational wavepackets -- with net chiral signals -- in ensembles of achiral molecules which are initially randomly oriented. Based on the analysis of symmetry and controllability, we derive excitation schemes for the creation of chiral wavepackets using a combination of (a) microwave and IR pulses and (b) a static field and a sequence of IR pulses. These protocols leverage quantum rotational dynamics for pump-probe spectroscopy of chiral vibrational dynamics, extending the latter to regions of the electromagnetic spectrum other than the UV.Comment: 16 pages, 8 figure

Small time reachable set of bilinear quantum systems

This note presents an example of bilinear conservative system in an infinite dimensional Hilbert space for which approximate controllability in the Hilbert unit sphere holds for arbitrary small times. This situation is in contrast with the finite dimensional case and is due to the unboundedness of the drift operator

Time Minimal Trajectories for a Spin 1/2 Particle in a Magnetic Field

In this paper we consider the minimum time population transfer problem for the $z$-component of the spin of a (spin 1/2) particle driven by a magnetic field, controlled along the x axis, with bounded amplitude. On the Bloch sphere (i.e. after a suitable Hopf projection), this problem can be attacked with techniques of optimal syntheses on 2-D manifolds. Let $(-E,E)$ be the two energy levels, and $|\Omega(t)|\leq M$ the bound on the field amplitude. For each couple of values $E$ and $M$, we determine the time optimal synthesis starting from the level $-E$ and we provide the explicit expression of the time optimal trajectories steering the state one to the state two, in terms of a parameter that can be computed solving numerically a suitable equation. For $M/E<<1$, every time optimal trajectory is bang-bang and in particular the corresponding control is periodic with frequency of the order of the resonance frequency $\omega_R=2E$. On the other side, for $M/E>1$, the time optimal trajectory steering the state one to the state two is bang-bang with exactly one switching. Fixed $E$ we also prove that for $M\to\infty$ the time needed to reach the state two tends to zero. In the case $M/E>1$ there are time optimal trajectories containing a singular arc. Finally we compare these results with some known results of Khaneja, Brockett and Glaser and with those obtained by controlling the magnetic field both on the $x$ and $y$ directions (or with one external field, but in the rotating wave approximation). As byproduct we prove that the qualitative shape of the time optimal synthesis presents different patterns, that cyclically alternate as $M/E\to0$, giving a partial proof of a conjecture formulated in a previous paper.Comment: 31 pages, 10 figures, typos correcte