This paper reviews the physics of quantum disorder in relation with a series of experiments using laser-cooled atoms exposed to “kicks” of a standing wave, realizing a paradigmatic model of quantum chaos, the kicked rotor. This dynamical system can be mapped onto a tight-binding Hamiltonian with pseudo-disorder, formally equivalent to the Anderson model of quantum disorder, with quantum chaos playing the role of disorder. This provides a very good quantum simulator for the Anderson physics.
This paper reviews the physics of quantum disorder in relation with a series of experiments using laser-cooled atoms exposed to “kicks” of a standing wave, realizing a paradigmatic model of quantum chaos, the kicked rotor. This dynamical system can be mapped onto a tight-binding Hamiltonian with pseudo-disorder, formally equivalent to the Anderson model of quantum disorder, with quantum chaos playing the role of disorder. This provides a very good quantum simulator for the Anderson physics.
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.
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 systems with Hofstadter's butterfly spectrum are of fundamental interest to many research areas. Based upon slight modifications of existing cold-atom experiments, a cold-atom realization of quantum maps with Hofstadter's butterfly spectrum is proposed. Connections and differences between our realization and the kicked Harper model are identified. This work also exposes, for the first time, a simple connection between the kicked Harper model and the kicked rotor model, the two paradigms of classical and quantum chaos.
We discuss recent experimental and theoretical investigations of quantum dynamics in periodically kicked chaotic systems. These investigations are centred on a modified version of the classic δ-kicked rotor, as realized using laser-cooled caesium atoms. Dramatic phenomena, principally quantum accelerator modes, have been observed and these have inspired novel theoretical approaches to quantum chaos. The latter work has, in turn, prompted further experiments and the combined experimental and theoretical effort continues to lead to exciting advances in this field.
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.
This thesis presents an account of experimental and numerical investigations of two quantum systems whose respective classical analogues are chaotic. These are the delta-kicked rotor, a paradigm in classical chaos theory, and the novel delta-kicked accelerator, created by application of a constant external acceleration or torque to the rotor. The experimental realisation of these systems has been achieved by the exposure of laser-cooled caesium atoms to approximate delta-kicks from a pulsed, high-intensity, vertical standing wave of laser light. Gravity’s effect on the atoms can be controlled by appropriate shifting of the profile of the standing wave. Numerical simulations of the systems are based on a diffractive model of the potential’s effect. Each system’s dynamics are characterised by the final form of the momentum distribution and the dependence of the atoms’ mean kinetic energy on the number and time period of the delta-kicks. The phenomena of dynamical localisation and quantum resonances in the delta-kicked rotor, which have no counterparts in the system’s classical analogue, are observed and investigated. Similar experiments on the delta-kicked accelerator reveal the striking phenomenon of the quantum accelerator mode, in which a large momentum is transferred to a substantial fraction of the atomic ensemble. This feature, absent in the system’s classical analogue, is characterised and an analytic explanation is presented. The effect on each quantum system of decoherence, introduced through spontaneous emission in the atoms, is examined and comparison is made with the results of classical simulations. While having little effect on the classical systems, the level of decoherence used is found to degrade quantum signatures of behaviour. Classical-like behaviour is, to some extent, restored, although significant quantum features remain. Possible applications of the quantum accelerator mode are discussed. These include use as a tool in atom optics and interferometry, a technique for measuring gravity, and a method of preparing atoms in a particular region of phase space. This may allow measurement of quantum phase space stability, and hence investigation of quantum chaos and quantum-classical correspondence.