This chapter describes basic methods to realise optical potentials and optical lattices, listing in detail what can be controlled in ultracold atomic systems. It revisits the properties of non-interacting particles in periodic lattices: band structure, Bloch functions, and Wannier states. The chapter derives the Hubbard model in the tight binding approximation, and discusses Bose–Einstein condensates in optical lattices in a weak interacting limit, and in a strongly correlated regime.
This chapter describes basic methods to realise optical potentials and optical lattices, listing in detail what can be controlled in ultracold atomic systems. It revisits the properties of non-interacting particles in periodic lattices: band structure, Bloch functions, and Wannier states. The chapter derives the Hubbard model in the tight binding approximation, and discusses Bose–Einstein condensates in optical lattices in a weak interacting limit, and in a strongly correlated regime.
Originally, the Hubbard model was derived for describing the behavior of strongly correlated electrons in solids. However, for over a decade now, variations of it have also routinely been implemented with ultracold atoms in optical lattices, allowing their study in a clean, essentially defect-free environment. Here, we review some of the vast literature on this subject, with a focus on more recent non-standard forms of the Hubbard model. After giving an introduction to standard (fermionic and bosonic) Hubbard models, we discuss briefly common models for mixtures, as well as the so-called extended Bose-Hubbard models, that include interactions between neighboring sites, next-neighbor sites, and so on. The main part of the review discusses the importance of additional terms appearing when refining the tight-binding approximation for the original physical Hamiltonian. Even when restricting the models to the lowest Bloch band is justified, the standard approach neglects the density-induced tunneling (which has the same origin as the usual on-site interaction). The importance of these contributions is discussed for both contact and dipolar interactions. For sufficiently strong interactions, the effects related to higher Bloch bands also become important even for deep optical lattices. Different approaches that aim at incorporating these effects, mainly via dressing the basis, Wannier functions with interactions, leading to effective, density-dependent Hubbard-type models, are reviewed. We discuss also examples of Hubbard-like models that explicitly involve higher p orbitals, as well as models that dynamically couple spin and orbital degrees of freedom. Finally, we review mean-field nonlinear Schrödinger models of the Salerno type that share with the non-standard Hubbard models nonlinear coupling between the adjacent sites. In that part, discrete solitons are the main subject of consideration. We conclude by listing some open problems, to be addressed in the future.
Abstract This chapter discusses some features exhibited by quantum gases in the presence of periodic optical potentials (optical lattices). The behaviour of the single-particle excitation spectrum exhibiting the typical Bloch band structure is briefly summarized. The equilibrium properties of a Bose–Einstein condensate and of the ideal Fermi gas are discussed, with special focus on the behaviour of the momentum distribution. It also considers the effects of localization in one-dimensional quasiperiodic potentials and the nature of Bloch oscillations. The chapter discusses the structure of the elementary excitations in an optical potential, including the motion of the centre of mass in the presence of an additional harmonic potential. Finally, the chapter mentions the problem of quantum fluctuations, and it introduces the physics of the Bose–Hubbard and Fermi–Hubbard models, whose systematic investigation lies outside the scope of this book.
Originally, the Hubbard model has been derived for describing the behaviour of strongly-correlated electrons in solids. However, since over a decade now, variations of it are also routinely being imple-mented with ultracold atoms in optical lattices, allowing their study in a clean, essentially defect-free environment. Here, we review some of the rich literature on this subject, with a focus on more recent non-standard forms of the Hubbard model. After an introduction to standard (fermionic and bosonic) Hubbard models, we discuss briefly common models for mixtures, as well as the so called extended Bose-Hubbard models, that include interactions between neighboring sites, next-neighboring sites, and so on. The main part of the review discusses the importance of additional terms appearing when refining the tight-binding approximation on the original physical Hamiltonian. Even when restricting the models to the lowest Bloch band is justified, the standard approach neglects the density-induced tunneling (which has the same origin as the usual on-site interaction). The importance of these contri-
Ultracold quantum gases in optical lattices form almost ideal conditions to analyze the physics of strongly correlated quantum phases in periodic potentials. Such strongly correlated quantum phases are of fundamental interest in condensed matter physics, because they lie at the heart of topical quantum materials, such as high-Tc superconductors and quantum magnets, which pose a challenge to our basic understanding of interacting many-body systems. Quite generally, such strongly interacting quantum phases arise when the interaction energy between two particles dominates over the kinetic energy of the two particles. Such a regime can either be achieved by increasing the interaction strength between the atoms via Feshbach resonances or by decreasing the kinetic energy, such that eventually the interaction energy is the largest energy scale in the system. The latter can, for example, simply be achieved by increasing the optical lattice depth. This chapter tries gives an introduction into the field of optical lattices and the physics of strongly interacting quantum phases. A prominent example hereof is the superfluid-to-Mott insulator transition, which transforms a weakly interacting quantum gas into a strongly correlated many body system. Dominating interactions between the particles are in fact crucial for the Mott insulator transition and also for the realization of controlled interaction-based quantum gates, of which several have been successfully realized experimentally.
In the present thesis we study several schemes where the detection of the emitted light from an atomic ensemble gives information about the state of the atoms. In chapter 1 we review the basic theory for the light matter interaction in second quantized form. In chapter 2 we show how conservative potentials for the atoms, such as dispersive optical lattices, can be realized experimentally due to the dispersive interaction with a far detuned laser We then review some basic properties of the physics of a particle in a periodic potential. We show that the ground state and the lowest lying excitations of a Bose-Einstein condensate in a dispersive optical lattice can be described by the Bose-Hubbard Hamiltonian. In chapter 3 we study the photonic bandstructure of a chain of fixed pointlike atoms in a biperiodic configuration. We calculate the photonic spectra and the probe transmission if the atoms are in free space and inside a standing wave optical resonator. In chapter 4 we consider light scattering from ultracold atoms in an optical lattice where we calculate the photonic scattering cross section as a function of energy and direction of emission along the Mott-insulator-superfluid phase transition. We take into account the finite tunneling rate for the atoms, when evaluating the scattering cross section of the photons and the effect of photon recoil onto the atoms. In chapter 5 we consider a gas of ultracold atoms trapped at two spatially separated regions in space. Using a two Raman lasers to couple atoms out of the systems we show that it is possible to measure the mean value of the atomic field operator of the two systems by measuring the scattered light intensity of one of the Raman lasers. We show how this setup might be used to measure the temperature of a Bose-Einstein condensate and to monitor the superfluid to Mott-insulator phase transition for a gas of ultracold atoms in an optical lattice.
We review our studies on Bose and Fermi superfluids of cold atomic gases in optical lattices at zero temperature. Especially, we focus on superfluid Fermi gases along the crossover between the Bardeen-Cooper-Schrieffer (BCS) and the Bose-Einstein condensate (BEC) states, which enable us to study the Bose and the Fermi superfluids in a unified point of view. We discuss basic static and long-wavelength properties (such as the equation of state, incompressibility, and effective mass), energetic stability, and energy band structures of the superfluid Fermi gases in an optical lattice periodic along one spatial direction. The periodic potential causes pairs of atoms to be strongly bound, and this can affect the static and long-wavelength properties and the stability of the superflow. Regarding the band structure, a peculiar loop structure called “swallowtail” can appear in superfluid Fermi gases and in the Bose case, but the mechanism of emergence in the Fermi case is very different from that in bosonic case. Other quantum phases that the cold atomic gases in optical lattices can show are also briefly discussed based on their roles as quantum simulators of Hubbard models.