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Quantum optimal control is a set of methods for designing time-varying electromagnetic fields to perform operations in quantum technologies. This tutorial paper introduces the basic elements of this theory based on the Pontryagin maximum principle, in a physicist-friendly way. An analogy with classical Lagrangian and Hamiltonian mechanics is proposed to present the main results used in this field. Emphasis is placed on the different numerical algorithms to solve a quantum optimal control problem. Several examples ranging from the control of two-level quantum systems to that of Bose-Einstein Condensates (BEC) in a one-dimensional optical lattice are studied in detail, using both analytical and numerical methods. Codes based on shooting method and gradient-based algorithms are provided. The connection between optimal processes and the quantum speed limit is also discussed in two-level quantum systems. In the case of BEC, the experimental implementation of optimal control protocols is described, both for two-level and many-level cases, with the current constraints and limitations of such platforms. This presentation is illustrated by the corresponding experimental results.
Optimal control is a valuable tool for quantum simulation, allowing for the optimized preparation, manipulation, and measurement of quantum states. Through the optimization of a time-dependent control parameter, target states can be prepared to initialize or engineer specific quantum dynamics. In this work, we focus on the tailoring of a unitary evolution leading to the stroboscopic stabilization of quantum states of a Bose-Einstein condensate in an optical lattice. We show how, for states with space and time symmetries, such an evolution can be derived from the initial state-preparation controls; while for a general target state we make use of quantum optimal control to directly generate a stabilizing Floquet operator. Numerical optimizations highlight the existence of a quantum speed limit for this stabilization process, and our experimental results demonstrate the efficient stabilization of a broad range of quantum states in the lattice.
Control of stochastic systems is a challenging open problem in statistical physics, with potential applications in a wealth of systems from biology to granulates. Unlike most cases investigated so far, we aim here at controlling a genuinely out-of-equilibrium system, the two dimensional Active Brownian Particles model in a harmonic potential, a paradigm for the study of self-propelled bacteria. We search for protocols for the driving parameters (stiffness of the potential and activity of the particles) bringing the system from an initial passive-like stationary state to a final active-like one, within a chosen time interval. The exact analytical results found for this prototypical system of self-propelled particles brings control techniques to a wider class of out-of-equilibrium systems.
We discuss the emulation of non-Hermitian dynamics during a given time window using a low-dimensional quantum system coupled to a finite set of equidistant discrete states acting as an effective continuum. We first emulate the decay of an unstable state and map the quasi-continuum parameters, enabling the precise approximation of non-Hermitian dynamics. The limitations of this model, including in particular short- and long-time deviations, are extensively discussed. We then consider a driven two-level system and establish criteria for non-Hermitian dynamics emulation with a finite quasi-continuum. We quantitatively analyze the signatures of the finiteness of the effective continuum, addressing the possible emergence of non-Markovian behavior during the time interval considered. Finally, we investigate the emulation of dissipative dynamics using a finite quasi-continuum with a tailored density of states. We show through the example of a two-level system that such a continuum can reproduce non-Hermitian dynamics more efficiently than the usual equidistant quasi-continuum model.
We report on the design of a Hamiltonian ratchet exploiting periodically at rest integrable trajectories in the phase space of a modulated periodic potential, leading to the linear non-diffusive transport of particles. Using Bose-Einstein condensates in a modulated one-dimensional optical lattice, we make the first observations of this spatial ratchet, which provides way to coherently transport matter waves with possible applications in quantum technologies. In the semiclassical regime, the quantum transport strongly depends on the effective Planck constant due to Floquet state mixing. We also demonstrate the interest of quantum optimal control for efficient initial state preparation into the transporting Floquet states to enhance the transport periodicity.
Sujets
Condensat de Bose-Einstein
Quantum control
Contrôle optimal
Mélasse optique
Fresnel lens
Plasmon polariton de surface
Physique quantique
Atom optics
Lentille de Fresnel
Masques matériels nanométriques
Jet atomique
Théorie de Floquet
Bragg scattering
Entropy production
Gaz quantique
Onde de matière
Quantum simulator
Césium
Non-adiabatic regime
Experimental results
Contrôle optimal quantique
Fluorescence microscopy
Nano-lithographie
Mirror-magneto-optical trap
Dynamical tunneling
Collisions ultrafroides
Condensats de Bose Einstein
Initial state
Bose-Einstein Condensates
Atomes ultrafroids dans un réseau optique
Bose-Einstein condensate
Quantum physics
Matter waves
Atomic beam
Quantum chaos
Optical lattices
Optimal control theory
Atom laser
Mechanics
Condensation de bose-Einstein
Levitodynamics
Atomes froids
Espace des phases
Electromagnetic field time dependence
Bose-Einstein condensates Coherent control Cold atoms and matter waves Cold gases in optical lattices
Matter wave
Chaos
Effet tunnel
Engineering
Bose Einstein Condensation
Dimension 1
Piège magnéto-optique à miroir
Bose-Einstein condensates
Condensats de Bose– Einstein
Approximation semi-classique et variationnelle
Hamiltonian
Bose-Einstein
Quantum gases
Beam splitter
Quantum simulation
Current constraint
Effet tunnel assisté par le chaos
Réseau optique
Bragg Diffraction
Effet rochet
Microscopie de fluorescence
Quantum optimal control
Floquet theory
Optical lattice
Bose–Einstein condensates
Lattice optical
Condensation
Field equations stochastic
Puce atomique
Optique atomique
Condensat Bose-Einstein
Periodic potentials
Réseaux optiques
Fluid
Nano-lithography
Numerical methods
Diffraction de Bragg
Gaz quantiques
Quantum collisions
Condensats de Bose-Einstein
Phase space
Couches mono-moléculaire auto assemblées
Chaos-assisted tunneling
Bose Einstein condensate
Maxwell's demon
Ultracold atoms
Optical tweezers
Atom chip
Effet tunnel dynamique
Cold atoms
Bose-Einstein Condensate
Optical molasses
Ouvertures métalliques sub-longueur d'onde
Chaos quantique
Quantum gas