The electric field intensity of a standing-wave laser field is periodic in space. Due to the ac Stark effect, this gives rise to a spatially periodic potential acting on an atom, as explained in Chapter 4 (see, e.g., Eq. (4.31)). This is the physical principle behind the generation of optical lattices. By superimposing a number of different laser beams it is possible to generate potentials which are periodic in one, two or three dimensions. The suggestion that standing light waves may be used to confine the motion of atoms dates back to 1968 and is due to Letokhov. The first experimental realization of an optical lattice was achieved in 1987 for a classical gas of cesium atoms. The study of atoms in such potentials has many different facets. At the simplest level, it is possible to study the energy band structure of atoms moving in these potentials and to explore experimentally a number of effects that are difficult to observe for electrons in the periodic lattice of a solid. Interactions between atoms introduce qualitatively new effects. Within mean-field theory, which applies when the number of atoms in the vicinity of a single minimum of the potential is sufficiently large, one finds that interatomic interactions give rise to novel features in the band structure. These include multivaluedness of the energy for a given band and states possessing a periodicity different from that of the optical lattice.
The electric field intensity of a standing-wave laser field is periodic in space. Due to the ac Stark effect, this gives rise to a spatially periodic potential acting on an atom, as explained in Chapter 4 (see, e.g., Eq. (4.31)). This is the physical principle behind the generation of optical lattices. By superimposing a number of different laser beams it is possible to generate potentials which are periodic in one, two or three dimensions. The suggestion that standing light waves may be used to confine the motion of atoms dates back to 1968 and is due to Letokhov. The first experimental realization of an optical lattice was achieved in 1987 for a classical gas of cesium atoms. The study of atoms in such potentials has many different facets. At the simplest level, it is possible to study the energy band structure of atoms moving in these potentials and to explore experimentally a number of effects that are difficult to observe for electrons in the periodic lattice of a solid. Interactions between atoms introduce qualitatively new effects. Within mean-field theory, which applies when the number of atoms in the vicinity of a single minimum of the potential is sufficiently large, one finds that interatomic interactions give rise to novel features in the band structure. These include multivaluedness of the energy for a given band and states possessing a periodicity different from that of the optical lattice.
An optical lattice is a periodic potential for atoms, created using a standing wave pattern of light. Due to the interaction between the light and the atoms, the atoms are attracted to either the nodes or the anti-nodes of the standing wave, depending on the exact wave lenght of the light. This means that if such a lattice is loaded with a sufficiently high number of ultracold atoms, a periodic array of atoms is obtained, we an interatomic distance of a few tenths of a micrometer. In order to obtain such a high number of cold atoms, one first has to create a so-called Bose-Einstein condensate. When an optical lattice is loaded from a Bose-Einstein condensate, it is possible to create a system in which every atom is in the lowest band of the lattice and there is on average one atom in each lattice site. Because the lattice potential is created with laser light, the depth of the lattice can easily be tuned by changing the intensity of the laser. When the intensity of the laser light is low, the atoms can tunnel from one site to the next. Due to this tunneling, the gas of atoms in the lattice will remain superfluid. However, if the intensity of the laser light is increased to above a certain critical value, a quantum phase transition occurs to a so-called Mott insulator. In this state, the atoms can no longer tunnel due to the fact that the on-site interaction between atoms becomes more important then the tunneling probability. In this PhD thesis, a description is given of the experimental setup that is being constructed in our group to create these systems in our lab. Also, a theoretical description is given of these systems and several important quantities our derived, such as the gap of the Mott-insulating state. Furthermore, an experiment is proposed that can be used to accurately measure this gap.
An optical lattice is the periodic potential that atoms experience via the ac-Stark shift when they are illuminated by counter-propagating laser beams that form a standing-wave pattern. Optical lattices have been widely used as a versatile platform in cooling, trapping, controlling atoms, and also for the study of a variety of problems in physics. The many-body states of ultracold atoms in optical lattices can be characterized by the quantum-correlations encoded in time-of-flight images. In this thesis, we mainly discuss the use of correlations function as a natural framework for characterizing quantum states in optical lattices.\nThe outline of the thesis is as follows. Chapter 1 gives a brief introduction to optical lattice potentials and ways for describing particles moving in a periodic potential. Chapter 2 explains the importance of correlations and discusses common methods to detect them in cold atom experiments.\nChapter 3 presents work done to study the many-body Schrödinger equation in a quasiperiodic potential and discusses its connection with the Kolmogorov-Arnold-Moser (KAM) problem of classical mechanics. We posed a possible visualization of such a connection in experimentally accessible many-body observables. These observables are useful probes for the three characteristic phases of the problem: the metallic, Anderson, and band-insulator phases. In addition, they exhibit fingerprints of nonlinear phenomena such as bifurcations and devil's staircases. Our numerical treatment is complemented with a perturbative analysis that provides insight on the underlying physics. The perturbative-theory approach is particularly useful in illuminating the distinction between the Anderson-insulator and the band-insulator phases in terms of paired sets of dimerized states.\nChapter 4 discusses several theoretical procedures developed to understand a recent experiment on macroscopic quantum self-trapping (ST) performed in a 2D optical lattice. Mean field and truncated-Wigner-approximation (TWA) calculations are performed trying to reproduce the experimental observations. The discrepancy between the theory and the experiment lead to the hypothesis of a new type of ST caused by strong correlations. We analyze toy models to support it.
We report the first observation of atomic Wannier-Stark ladders in a potential formed by an accelerating standing wave of light.1 This is a fundamental effect of quantum transport in optical lattices. Wannier-Stark ladder resonances were predicted to occur in the band structure of an electron moving in a periodic lattice with a dc electric field, and have been recently observed in superlattices. Our work is the first spectroscopic study of Wannier-Stark ladders free from complications such as the impurities, lattice vibrations, and multi-particle interactions present in solids.
The subject of this chapter is perhaps the most spectacular application of atom trapping: optical lattices. Optical lattices are light fields whose intensity exhibits periodicity in one, two, and three dimensions and that resemble the structure of the crystal lattices of solid bodies (explaining why they are termed optical lattices). Their significance, as will be presented in this chapter and the next, extends beyond a similarity to crystal lattices in terms of form and geometry. Optical lattices actually open up new possibilities in atomic and condensed matter physics.
Cold atoms in optical lattices are an ideal test bed to explore ratchet physics. This is largely due to the extreme tunability of optical lattice systems, which allows one to precisely control the shape of the potential and of any applied force, as well as to vary at will the level of dissipation. Since the very first demonstration of directed motion in a cold atom system by Mennerat-Robilliard et al. (1999), a number of experimental investigations with cold atoms in optical lattices have explored different aspects of the physics of ratchets. Here, the focus is first on the experiments in the classical regime which explored the relationship between symmetry and transport in a.c. driven ratchets. Experiments in the full quantum Hamiltonian regime are then described, and the unique features of such a regime highlighted. Ratchets in dissipative optical lattices Dissipative optical lattices Optical lattices are periodic potentials for atoms created by the interference of two or more laser fields.The detuning between the laser fields and the nearest atomic transition will emerge as a key parameter to describe the light–atom interaction. It is thus important to distinguish two very different situations: the case of fardetuned laser fields, and the case of near-resonant laser fields. For far-detuned laser fields, a purely conservative potential is produced. As will be discussed in the following sections, far-detuned optical lattices are ideal to model Hamiltonian systems. In the case of near-resonant laser fields, the interaction between the laser and the atoms may lead, under appropriate conditions, to dissipative dynamics. That is, for an appropriately arranged set of near-resonant optical lattices, a dissipative optical lattice, is produced, where the set of laser fields produce at once the periodic potential acting on the atoms and the cooling mechanism, which decreases their kinetic energy and lead to the trapping of the atoms at the bottom of potential wells. The principles behind the cooling mechanism, termed Sisyphus cooling for reasons which will become apparent in the following discussion, can be captured by considering the minimal configuration of a one-dimensional optical lattice and a J g = 1 / 2 → J e = 3 / 2 atomic transition. This is the simplest configuration in which Sisyphus cooling takes place.
Matter waves inside periodic potentials are well known from solid-state physics, where electrons interacting with a crystal lattice are considered. Atomic Bose-Einstein condensates inside light-induced periodic potentials (optical lattices) share many features with electrons in solids, but also with light waves in nonlinear materials and other nonlinear systems. Generally, atom-atom interactions in Bose-Einstein condensates lead to rich and interesting nonlinear effects. Furthermore, the experimental control over the parameters of the periodic potential and the condensate make it possible to enter regimes inaccessible in other systems. In this review, an introduction to the physics of ultracold bosonic atoms in optical lattices is given and an overview of the theoretical and experimental advances to date.