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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
Fresnel lens
Nano-lithographie
Chaos-assisted tunneling
Collisions ultrafroides
Maxwell's demon
Contrôle optimal quantique
Atom chip
Mélasse optique
Nano-lithography
Condensat Bose-Einstein
Hamiltonian
Optical lattice
Fluid
Microscopie de fluorescence
Diffraction de Bragg
Quantum gases
Chaos
Current constraint
Optical tweezers
Entropy production
Condensat de Bose-Einstein
Atomes ultrafroids dans un réseau optique
Quantum optimal control
Quantum gas
Chaos quantique
Condensation
Lattice optical
Piège magnéto-optique à miroir
Condensation de bose-Einstein
Matter waves
Contrôle optimal
Physique quantique
Mechanics
Optical lattices
Mirror-magneto-optical trap
Gaz quantiques
Bose-Einstein condensate
Phase space
Initial state
Réseau optique
Electromagnetic field time dependence
Plasmon polariton de surface
Dynamical tunneling
Condensats de Bose– Einstein
Bose-Einstein Condensates
Numerical methods
Césium
Optical molasses
Levitodynamics
Lentille de Fresnel
Quantum simulation
Optimal control theory
Approximation semi-classique et variationnelle
Atomes froids
Effet tunnel dynamique
Bose-Einstein condensates
Cold atoms
Fluorescence microscopy
Condensats de Bose Einstein
Bragg scattering
Masques matériels nanométriques
Periodic potentials
Optique atomique
Experimental results
Couches mono-moléculaire auto assemblées
Espace des phases
Effet tunnel
Engineering
Bose Einstein Condensation
Quantum chaos
Effet rochet
Condensats de Bose-Einstein
Gaz quantique
Bose-Einstein
Atom optics
Onde de matière
Quantum simulator
Field equations stochastic
Bragg Diffraction
Bose Einstein condensate
Ultracold atoms
Quantum physics
Bose–Einstein condensates
Floquet theory
Dimension 1
Bose-Einstein Condensate
Matter wave
Bose-Einstein condensates Coherent control Cold atoms and matter waves Cold gases in optical lattices
Atomic beam
Théorie de Floquet
Ouvertures métalliques sub-longueur d'onde
Quantum collisions
Réseaux optiques
Jet atomique
Effet tunnel assisté par le chaos
Quantum control
Puce atomique
Beam splitter
Atom laser
Non-adiabatic regime