Quantum simulation is one of the most active trends in ultracold-atom physics. A quantum simulator is a system that mimics the dynamics of another system; the advantage being that the quantum simulator is simpler than the original, either from the experimental or the theoretical point of view (or both).The Anderson model is a paradigm for quantum disorder, originally introduced in the helm of solid-state physics; however, quantum solid-state disordered systems are very difficult to study experimentally, and even numerically (except in the one-dimensional case). The kicked rotor is a paradigm for classical and quantum Hamiltonian chaos, essentially because it is very easy to simulate it in a computer. Its quantum realization with ultracold atoms is also relatively simple, and is a “clean” (e.g. from decoherence) and controllable system. Amazingly, the quantum kicked rotor can be mathematically mapped onto the Anderson model. It is thus a quite subtle quantum simulator of the physics of disordered systems. In this talk I will describe a series of experiments we have performed in Lille taking advantage of the correspondence between disordered condensed matter systems and the atomic kicked rotor to investigate experimentally in great detail the Anderson metal-insulator transition. I will discuss the new possibilities opened by evolutions of this system, in particular by introducing nonlinearity via atom-atom interactions.
Quantum simulation is one of the most active trends in ultracold-atom physics. A quantum simulator is a system that mimics the dynamics of another system; the advantage being that the quantum simulator is simpler than the original, either from the experimental or the theoretical point of view (or both).The Anderson model is a paradigm for quantum disorder, originally introduced in the helm of solid-state physics; however, quantum solid-state disordered systems are very difficult to study experimentally, and even numerically (except in the one-dimensional case). The kicked rotor is a paradigm for classical and quantum Hamiltonian chaos, essentially because it is very easy to simulate it in a computer. Its quantum realization with ultracold atoms is also relatively simple, and is a “clean” (e.g. from decoherence) and controllable system. Amazingly, the quantum kicked rotor can be mathematically mapped onto the Anderson model. It is thus a quite subtle quantum simulator of the physics of disordered systems. In this talk I will describe a series of experiments we have performed in Lille taking advantage of the correspondence between disordered condensed matter systems and the atomic kicked rotor to investigate experimentally in great detail the Anderson metal-insulator transition. I will discuss the new possibilities opened by evolutions of this system, in particular by introducing nonlinearity via atom-atom interactions.
The Anderson model is a paradigm for quantum disordered physical systems. Originally conceived in a condensed matter context, it is however difficult to study experimentally in solids. The quantum kicked rotor which has also a paradigmatic status for classical and quantum Hamiltonian chaotic systems, can be easily realized using laser-cooled atoms in a standing wave. Somewhat surprisingly, there is a mathematical mapping between these two systems, and the latter can be used as a "quantum simulator" for the former. In this talk, I will describe a series of experiments using the atomic kicked rotor allowing a rather complete characterization of the Anderson physics, and in particular a precise determination of the critical exponent of the Anderson metal-insulator transition.
The kicked rotor provides a simple yet powerful model for introducing many of the central concepts of classical and quantum chaos. Despite its apparent simplicity, it exhibits rich dynamical behavior and has found applications across a wide range of fields, including atomic and optical physics, condensed matter physics, and emerging quantum technologies. This chapter begins by exploring foundational ideas using the kicked rotor as a unifying framework. We first discuss the transition from regular to chaotic motion in the classical system, and then introduce key quantum phenomena such as dynamical localization and quantum resonances. Special attention is devoted to the emergence of characteristic time scales and their role in the quantum-classical correspondence. To make these ideas more concrete, we also provide a brief overview of experimental realizations of the kicked rotor and its variants, illustrating how theoretical concepts are implemented in practice. In the second part of the chapter, we guide the reader toward more recent and advanced developments. Topics include near-resonant dynamics, topological features of kicked systems, the emergence of quantum dynamical phases inferred from classical transport properties, and extensions to non-Hermitian physics. We conclude with a discussion of open problems and future perspectives, outlining directions in which the kicked rotor continues to offer valuable insights.
Many interesting physical systems, e.g. condensed matter systems, are hard to study. From the experimental point of view, in such systems most parameters are ''frozen" for a given sample. From the theoretical point of view, they often cannot be simply modelled from first principles. In contrast, laser-matter interactions are very well known theoretically, and a very high degree of control of laser beams is available for the experimentalist. In the last years, one learned to ''engineer" quantum hamiltonians using ultracold atoms interacting with electromagnetic fields. For example, laser beams may form an interference pattern that is ''seen" by the atoms as a periodic potential, thus creating an analog of a crystal. However, the analog has great advantages over the original system, as one has a larger control of its properties: One can vary the parameters of the potential by changing the properties of the laser beams, one can make a ''mesoscopic" quantum system by using a Bose-Einstein condensate, one can change it to a degenerate Fermi gas by using a fermionic isotope, one can control atom-atom interactions by using the so-called . Building on seminal ideas of Feynman, the notion of ''quantum simulator" -- a ''simpler" system able to mimic the physics of a complicate one -- recently became a reality. In a variety of situations of experimental interest, weakly interacting degenerate boson gases (Bose-Einstein condensates) are very well described by the Nonlinear Schödinger Equation with a cubic nonlinearity (called, in the present context, the {\it Gross-Pitaevskii equation}), which is much simpler than the underlying many-body problem. This opened a new field for both theoretical and experimental studies: The {\it nonlinear} quantum mechanics, where the familiar techniques like the spectral analysis in terms of eigenvalues and eigenvectors, which derive from the linearity of the usual Schödinger equation, cannot be used. The mathematical framework of this new quantum mechanics is still largely to be invented. Meanwhile, one can resort to numerical simulations -- and to experiments. In this talk, I will discuss two simple yet important examples illustrating these ideas. First, I will show how one can ''quantum simulate" with ultracold atoms the celebrated Anderson model, which describes quantum disordered systems. One can even enrich this system by adding particle-particle interactions in a controlled way. Second, I will show a simple model displaying ''quasiclassical" chaos, that is, chaotic behavior related to sensitivity to the initial conditions, which cannot exist in {\it linear} quantum mechanics. I will show that the observed behavior, although intrinsically quantum, can be described by the KAM theorem.
Disorder plays a fundamental role in many fields of physics such as condensed-matter physics, optics, acoustics, or atomic physics. It is responsible for a number of surprizing effects, such as Anderson localization, metal-insulator transitions and intriguing glass phases. The inherent complexity of disordered systems poses outstanding challenges to the full understanding of these phenomena. In the last years, disorder has emerged as a major line of research in the field of ultracold quantum gases. The latter offer fascinating perspectives to better understand the effects of disorder in quantum systems, thanks not only to an unprecedented control of parameters, but also to their original properties. This Habilitation thesis reviews our recent contributions to the theory of disordered quantum gases along three main lines: - Anderson localization in disordered quantum gases; - Disorder in interacting Bose gases; - Simulating extended Hubbard and spin models with ultracold gases. On the one hand, we propose and analyze experiments aiming at realizing quantum simulators to address open questions of fundamental importance for the field of disordered systems. In this respect, we show that quantum gases offer promising perspectives. On the other hand, we lead prospective works, which in particular show that quantum gases have original properties. They hence shed new light on issues of broad interest in the field of disordered systems.
This chapter extends the investigation of ultracold atoms in optical lattices to the emerging field of ‘quantum simulation’, in which atoms are used to experimentally realize basic condensed-matter models to precisely investigate their properties and their quantum phase transitions in an ultimately clean setting, where decoherence or unwanted interactions with the environment can be avoided. The chapter discusses the superfluid/metal-to-insulator transition exhibited by strongly-interacting atoms realizing Hubbard models, as well as the Anderson localization determining the suppression of transport for atoms moving in a disordered potential. It concludes with an illustration of the most recent developments in the field, discussing novel experimental techniques and important frontiers in this research.